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0.133.2
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First, observe that if $S$ is contained in a threefold $T\subset
\mathbb P^5$ of dimension $3$ and minimal degree $3$, then $T$ is
necessarily a {\it smooth} rational normal scroll \cite[p. 76]{DGF}.
Moreover, observe that we may apply the same argument as in \cite[p.
75-76]{DGF} in order to exclude the case $S$ is contained in a
threefold of degree $4$. In fact the argument works for $d>24$
\cite[p. 76, first line after formula (13)]{DGF}.
In conclusion, assuming $r=5$, $S$ is a scroll, and (\ref{bound}),
it remains to exclude that $S$ is not contained in a threefold of
degree $<5$, when $26\leq d \leq 30$.
Assume $S$ is not contained in a threefold of degree $<5$. Denote by
$\Gamma\subset \mathbb P^3$ the general hyperplane section of $H$.
Recall that $26\leq d \leq 30$.
$\bullet$ Case I: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))\geq 2$.
It is impossible. In fact, if $d>4$, by monodromy \cite[Proposition
2.1]{CCD}, $\Gamma$ should be contained in a reduced and irreducible
space curve of degree $\leq 4$, and so, for $d>20$, $S$ should be
contained in a threefold of degree $\leq 4$ \cite[Theorem
(0.2)]{CC}.
$\bullet$ Case II: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))=1$ and $h^0(\mathbb P^3,\mathcal I_{\Gamma}(3))>4$.
As before, if $d>6$, by monodromy, $\Gamma$ is contained in a
reduced and irreducible space curve $X$ of degree $\deg(X)\leq 6$.
Again as before, if $\deg(X)\leq 4$, then $S$ is contained in a
threefold of degree $\leq 4$. So we may assume $5\leq \deg(X)\leq
6$.
Since $d\geq 26$, by Bezout's Theorem we have $h_{\Gamma}(i)=h_X(i)$
for all $i\leq 4$. Let $X'$ be the general plane section of $X$.
Since $h_X(i)\geq \sum_{j=0}^{i}h_{X'}(j)$, we have $h_X(3)\geq 14$
and $h_X(4)\geq 19$ \cite[pp. 81-87]{EH}. Therefore, when $d\geq
26$, taking into account \cite[Corollary (3.5)]{EH}, we get:
$$
h_{\Gamma}(1)=4,\, h_{\Gamma}(2)=9,\, h_{\Gamma}(3)\geq 14,\,
h_{\Gamma}(4)\geq 19,
$$
$$
h_{\Gamma}(5)\geq 22, \,
h_{\Gamma}(6)\geq \min\{d,\, 27\},\, h_{\Gamma}(7)=d.
$$
It follows that:
$$
p_a(C)\leq \sum_{i=1}^{+\infty}d-h_{\Gamma}(i)\leq
(d-4)+(d-9)+(d-14)+(d-19)+(d-22)+3=5d-65,
$$
which is $<\frac{1}{8}d(d-6)+1$ for $d \geq 26$. This is in contrast
with (\ref{bound}).
$\bullet$ Case III: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))=1$ and $h^0(\mathbb P^3,\mathcal I_{\Gamma}(3))=4$.
We have:
$$
h_{\Gamma}(1)=4,\, h_{\Gamma}(2)=9,\, h_{\Gamma}(3)=16, \,
h_{\Gamma}(4)\geq 19,
h_{\Gamma}(5)\geq 24, \, h_{\Gamma}(6)=d.
$$
It follows that:
$$
p_a(C)\leq \sum_{i=1}^{+\infty}d-h_{\Gamma}(i)\leq
(d-4)+(d-9)+(d-16)+(d-19)+(d-24)=5d-72,
$$
which is $< \frac{1}{8}d(d-6)+1$ for $d \geq 26$. This is in
contrast with (\ref{bound}).
$\bullet$ Case IV: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))=0$.
We have:
$$
h_{\Gamma}(1)=4,\, h_{\Gamma}(2)=10,\, h_{\Gamma}(3)\geq 13, \,
h_{\Gamma}(4)\geq 19,
$$
$$
h_{\Gamma}(5)\geq 22,\, h_{\Gamma}(6)\geq \min\{d,\, 28\}, \,
h_{\Gamma}(7)=d.
$$
It follows that:
$$
p_a(C)\leq \sum_{i=1}^{+\infty}d-h_{\Gamma}(i)\leq
(d-4)+(d-10)+(d-13)+(d-19)+(d-22)+2=5d-66,
$$
which is $< \frac{1}{8}d(d-6)+1$ for $d \geq 26$. This is in
contrast with (\ref{bound}).
This concludes the proof of Theorem \ref{lbound}.
\begin{remark}\label{altro}
$(i)$ Let $Q\subseteq \mathbb P^3$ be a smooth quadric, and
$H\in|\mathcal O_Q(1,d-1)|$ be a smooth rational curve of degree $d$
\cite[p. 231, Exercise 5.6]{Hartshorne}. Let $S\subseteq\mathbb P^4$
be the projective cone over $H$. A computation, which we omit,
proves that
$$
\chi (\mathcal O_S)=1-\binom{d-1}{3}.
$$
Therefore, if $S$ is singular, it may happen that $\chi (\mathcal
O_S)<-\frac{1}{8}d(d-6)$. One may ask whether $1-\binom{d-1}{3}$ is
a lower bound for $\chi(\mathcal O_S)$ for every {\it integral}
surface.
$(ii)$ Let $(S,\mathcal L)$ be a smooth surface, polarized by a very
ample line bundle $\mathcal L$ of degree $d$. By Harris' bound for
the geometric genus $p_g(S)$ of $S$ \cite{H}, we see that
$p_g(S)\leq \binom{d-1}{3}$. Taking into account that for a smooth
surface one has $\chi(\mathcal O_S)=h^0(S,\mathcal
O_S)-h^1(S,\mathcal O_S)+h^2(S,\mathcal O_S) \leq 1+h^2(S,\mathcal
O_S)=1+p_g(S)$, from Theorem \ref{lbound} we deduce (the first
inequality only when $d>25$):
$$
-\binom{\frac{d}{2}-1}{2}\leq \chi (\mathcal O_S)\leq
1+\binom{d-1}{3}.
$$
\end{remark}
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\section{Proof of Corollary \ref{Veronese}}
$\bullet$ First, assume that $\chi(\mathcal
O_S)=-\frac{1}{8}d(d-6)$.
By Theorem \ref{lbound}, we know that $r=5$. Moreover, $S$ is
contained in a nonsingular threefold $T\subseteq \mathbb P^5$ of
minimal degree $3$. Therefore, the general hyperplane section
$H=S\cap L$ of $S$ ($L\cong \mathbb P^4$ denotes the general
hyperplane of $\mathbb P^5$) is contained in a smooth surface
$\Sigma=T\cap L$ of $L\cong \mathbb P^4$, of minimal degree $3$.
This surface $\Sigma$ is isomorphic to the blowing-up of $\mathbb
P^2$ at a point, and, for a suitable point $x\in V\backslash L$, the
projection of $\mathbb P^5\backslash\{x\}$ on $L\cong \mathbb P^4$
from $x$ restricts to an isomorphism
$$p_x:V\backslash\{x\}\to \Sigma\backslash E,$$ where $E$ denotes
the exceptional line of $\Sigma$ \cite[p. 58]{BV}.
Since $S$ is linearly equivalent on $T$ to $\frac{d}{2}(H_T- W_T)$
($H_T$ denotes the hyperplane section of $T$, and $W_T$ the ruling),
it follows that $H$ is linearly equivalent on $\Sigma$ to
$\frac{d}{2}(H_{\Sigma}- W_{\Sigma})$ (now $H_{\Sigma}$ denotes the
hyperplane section of $\Sigma$, and $W_{\Sigma}$ the ruling of
$\Sigma$). Therefore, $H$ does not meet the exceptional line
$E=H_{\Sigma}- 2W_{\Sigma}$. In fact, since $H_{\Sigma}^2=3$,
$H_{\Sigma}\cdot W_{\Sigma}=1$, and $W_{\Sigma}^2=0$, one has:
$$(H_{\Sigma}- W_{\Sigma})\cdot (H_{\Sigma}-
2W_{\Sigma})= H_{\Sigma}^2-3H_{\Sigma}\cdot
W_{\Sigma}+2W_{\Sigma}^2=0.$$
This implies that $H$ is contained in $\Sigma\backslash E$, and the
assertion of Corollary \ref{Veronese} follows.
$\bullet$ Conversely, assume there exists a curve $C$ on the
Veronese surface $V\subseteq \mathbb P^5$, and a point $x\in
V\backslash C$, such that $H$ is the projection $p_x(C)$ of $C$ from
the point $x$.
In particular, $d$ is an even number, and $H$ is contained in a
smooth surface $\Sigma\subseteq L\cong \mathbb P^4$ of minimal
degree, and is disjoint from the exceptional line $E\subseteq
\Sigma$. By \cite[Theorem (0.2)]{CC}, $S$ is contained in a
threefold $T\subseteq \mathbb P^5$ of minimal degree. $T$ is
nonsingular. In fact, otherwise, $H$ should be a Castelnuovo's curve
in $\mathbb P^4$ \cite[p. 76]{DGF}. On the other hand, by our
assumption, $H$ is isomorphic to a plane curve of degree
$\frac{d}{2}$. Hence, we should have:
$$
g=\frac{d^2}{6}-\frac{2}{3}d+1=\frac{d^2}{8}-\frac{3}{4}d+1
$$
(the first equality because $H$ is Castelnuovo's, the latter because
$H$ is isomorphic to a plane curve of degree $\frac{d}{2}$). This is
impossible when $d>0$.
Therefore, $S$ is contained in a smooth threefold $T$ of minimal
degree in $\mathbb P^5$.
Now observe that in $\Sigma$ there are only two families of curves
of degree even $d$ and genus $g=\frac{d^2}{8}-\frac{3}{4}d+1$. These
are the curves linearly equivalent on $\Sigma$ to
$\frac{d}{2}(H_{\Sigma}- W_{\Sigma})$, and the curves equivalent to
$\frac{d+2}{6}H_{\Sigma}+ \frac{d-2}{2}W_{\Sigma}$. But only in the
first family the curves do not meet $E$. Hence, $H$ is linearly
equivalent on $\Sigma$ to $\frac{d}{2}(H_{\Sigma}- W_{\Sigma})$.
Since the restriction ${\text{Pic}}(T)\to {\text{Pic}}(\Sigma)$ is
bijective, it follows that $S$ is linearly equivalent on $T$ to
$\frac{d}{2}(H_{T}- W_{T})$. By Theorem \ref{lbound}, $S$ is a
fortiori linearly normal, and of minimal Euler characteristic
$\chi(\mathcal O_S)=-\frac{1}{8}d(d-6)$.
\end{document}
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\begin{document}
\title{\large E\MakeLowercase{xistence of flips and minimal models for 3-folds in char $p$}
\begin{abstract}
We will prove the following results for $3$-fold pairs $(X,B)$
over an algebraically closed field $k$ of characteristic $p>5$:
log flips exist for $\mathbb Q$-factorial dlt pairs $(X,B)$;
log minimal models exist for projective klt pairs $(X,B)$ with pseudo-effective $K_X+B$;
the log canonical ring $R(K_X+B)$ is finitely generated for projective klt pairs $(X,B)$ when $K_X+B$ is a
big $\mathbb Q$-divisor; semi-ampleness holds for a nef and big $\mathbb Q$-divisor $D$ if $D-(K_X+B)$ is nef and big
and $(X,B)$ is projective klt;
$\mathbb Q$-factorial dlt models exist for lc pairs $(X,B)$; terminal models exist for klt pairs $(X,B)$;
ACC holds for lc thresholds; etc.
\end{abstract}
\tableofcontents
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\section{Introduction}
We work over an algebraically closed field $k$ of characteristic (char) $p>0$.
The pairs $(X,B)$ we consider in this paper always have $\mathbb R$-boundaries $B$ unless otherwise
stated.
Higher dimensional birational geometry in char $p$ is still largely conjectural. Even the most basic problems
such as base point freeness are not solved in general.
Ironically though Mori's work on existence of rational curves which plays an
important role in characteristic $0$ uses reduction mod $p$ techniques.
There are two reasons among others which have held back progress in char $p$:
resolution of singularities is not known and Kawamata-Viehweg vanishing fails.
However, it was expected that one can work
out most components of the minimal model program in dimension $3$. This is because
resolution of singularities is known in dimension $3$ and many problems can be
reduced to dimension $2$ hence one can use special features of surface geometry.
On the positive side there has been some good progress toward understanding birational
geometry in char $p$. People have tried to replace the characteristic $0$ tools that fail
in char $p$. For example,
Keel [\ref{Keel}] developed techniques for dealing with the base point free problem and
semi-ampleness questions in general without relying on Kawamata-Viehweg
vanishing type theorems. On the other hand, motivated by questions in commutative
algebra, people have introduced Frobenius-singularities whose definition do not
require resolution of singularities and they are very similar to singularities in
characteristic $0$ (cf. [\ref{Schwede}]).
More recently Hacon-Xu [\ref{HX}] proved the existence
of flips in dimension $3$ for pairs $(X,B)$ with $B$ having standard coefficients, that is, coefficients in
$\mathfrak{S}=\{1-\frac{1}{n}\mid n\in \mathbb N\cup \{\infty\}\}$, and char $p>5$. From this they could derive
existence of minimal models for $3$-folds with canonical singularities.
In this paper, we rely on their results and ideas.
The requirement $p>5$ has to do with the behavior of singularities on surfaces, eg a klt surface singularity
over $k$ of char $p>5$ is strongly $F$-regular.\\
{\textbf{\sffamily{Log flips.}}} Our first result is on the existence of flips.
\begin{thm}\label{t-flip-1}
Let $(X,B)$ be a $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of char $p>5$.
Let $X\to Z$ be a $K_X+B$-negative extremal flipping projective contraction. Then its flip exists.
\end{thm}
The conclusion also holds if $(X,B)$ is klt but not necessarily $\mathbb Q$-factorial. This
follows from the finite generation below (\ref{t-fg}).
The theorem is proved in Section \ref{s-flips} when $X$ is projective. The quasi-projective case
is proved in Section \ref{s-mmodels}.
We reduce the theorem to the case when $X$ is projective, $B$ has standard coefficients,
and some component of $\rddown{B}$ is negative on the extremal ray:
this case is [\ref{HX}, Theorem 4.12] which is one of the main results of that paper.
A different approach is taken in [\ref{CGS}] to prove \ref{t-flip-1}
when $B$ has hyperstandard coefficients and $p\gg 0$ (these coefficients are of
the form $\frac{n-1}{n}+\sum \frac{l_ib_i}{n}$ where $n \in\mathbb N\cup \{\infty\}$, $l_i\in \mathbb Z^{\ge 0}$
and $b_i$ are in some fixed DCC set).
To prove Theorem \ref{t-flip-1} we actually first prove the existence of \emph{generalized
flips} [\ref{HX}, after Theorem 5.6]. See Section \ref{s-flips} for more details.\\
{\textbf{\sffamily{Log minimal models.}}}
In [\ref{HX}, after Theorem 5.6], using generalized flips, a \emph{generalized LMMP} is defined which is
used to show the existence of minimal models for varieties with canonical singularities
(or for pairs with canonical singularities and "good" boundaries). Using weak Zariski decompositions
as in [\ref{B-WZD}], we construct log minimal models for klt pairs in general.
\begin{thm}\label{t-mmodel}
Let $(X,B)$ be a klt pair of dimension $3$ over $k$ of char $p>5$ and let
$X\to Z$ be a projective contraction. If $K_X+B$ is pseudo-effective$/Z$, then $(X,B)$ has a log minimal model over $Z$.
\end{thm}
The theorem is proved in Section \ref{s-mmodels}.
Alternatively, one can apply the methods of [\ref{B-mmodel}] to construct log minimal models
for lc pairs $(X,B)$ such that $K_X+B\equiv M/Z$ for some $M\ge 0$.
Note that when $X\to Z$ is a semi-stable fibration over a curve and $B=0$, the theorem was
proved much earlier by Kawamata [\ref{Kawamata}].\\
{\textbf{\sffamily{Remark on Mori fibre spaces.}}}
Let $(X,B)$ be a projective klt pair of dimension $3$ over $k$ of char $p>5$ such that $K_X+B$ is not
pseudo-effective. An important question is whether $(X,B)$ has a Mori fibre space.
There is an ample $\mathbb R$-divisor $A\ge 0$ such that
$K_X+B+A$ is pseudo-effective but $K_X+B+(1-\epsilon)A$
is not pseudo-effective for any $\epsilon>0$. Moreover,
we may assume that $(X,B+A)$ is klt as well (\ref{l-ample-dlt}). By Theorem \ref{t-mmodel},
$(X,B+A)$ has a log minimal model $(Y,B_Y+A_Y)$. Since
$K_Y+B_Y+A_Y$ is not big, $K_Y+B_Y+A_Y$ is numerically trivial on some covering
family of curves by [\ref{CTX}](see also \ref{t-aug-b-non-big} below).
Again by [\ref{CTX}], there is a nef reduction map $Y\dashrightarrow T$ for $K_Y+B_Y+A_Y$
which is projective over the generic point of $T$. Although $Y\dashrightarrow T$ is not necessarily a
Mori fibre space but in some sense it is similar.\\
{\textbf{\sffamily{Finite generation, base point freeness, and contractions.}}}
We will prove finite generation in the big case from which we can derive base point freeness
and contractions of extremal rays in many cases. These are proved in Section \ref{s-fg}.
\begin{thm}\label{t-fg}
Let $(X,B)$ be a klt pair of dimension $3$ over $k$ of char $p>5$ and $X\to Z$ a projective contraction.
Assume that $K_X+B$ is a $\mathbb Q$-divisor which is big$/Z$.
Then the relative log canonical algebra $\mathcal{R}(K_X+B/Z)$ is finitely generated over $\mathcal{O}_Z$.
\end{thm}
Assume that $Z$ is a point.
If $K_X+B$ is not big, then ${R}(K_X+B/Z)$ is still finitely generated if
$\kappa(K_X+B)\le 1$.
It remains to show the finite generation when $\kappa(K_X+B)=2$: this can probably be
reduced to dimension $2$ using an appropriate canonical bundle formula, for example as
in [\ref{CTX}].
A more or less immediate consequence of the above finite generation is the following base point
freeness.
\begin{thm}\label{t-bpf}
Let $(X,B)$ be a projective klt pair of dimension $3$ over $k$ of char $p>5$ and $X\to Z$ a projective contraction
where $B$ is a $\mathbb Q$-divisor. Assume that $D$ is a $\mathbb Q$-divisor such that
$D$ and $D-(K_X+B)$ are both nef and big$/Z$.
Then $D$ is semi-ample$/Z$.
\end{thm}
Assume that $Z$ is a point. When $D-(K_X+B)$ is nef and big but $D$ is nef with numerical
dimension $\nu(D)$ one or two,
semi-ampleness of $D$ is proved in [\ref{CTX}] under some restrictions on the coefficients.
\begin{thm}\label{t-contraction}
Let $(X,B)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of char $p>5$, and
$X\to Z$ a projective contraction. Let $R$ be a $K_X+B$-negative extremal ray$/Z$.
Assume that there is a nef and big$/Z$ $\mathbb Q$-divisor $N$ such that $N\cdot R=0$.
Then $R$ can be contracted by a projective morphism.
\end{thm}
Note that if $K_X+B$ is pseudo-effective$/Z$, then for every $K_X+B$-negative extremal
ray $R/Z$ there exists $N$ as in the theorem (see \ref{ss-ext-rays-II}). Therefore such extremal rays
can be contracted by projective morphisms.
Theorems \ref{t-bpf} and \ref{t-contraction} have been proved by Xu [\ref{Xu}]
independently and more or less at the same time but using a different approach. His proof also relies on our
results on flips and minimal models.\\
{\textbf{\sffamily{Dlt and terminal models.}}}
The next two results are standard consequences of the LMMP (more precisely, of special termination).
They are proved in Section \ref{s-crepant-models}.
\begin{thm}\label{cor-dlt-model}
Let $(X,B)$ be an lc pair of dimension $3$ over $k$ of char $p>5$.
Then $(X,B)$ has a (crepant) $\mathbb Q$-factorial dlt model. In particular, if $(X,B)$
is klt, then $X$ has a $\mathbb Q$-factorialization by a small morphism.
\end{thm}
The theorem was proved in [\ref{HX}, Theorem 6.1] for pairs with standard coefficients.
\begin{thm}\label{cor-terminal-model}
Let $(X,B)$ be a klt pair of dimension $3$ over $k$ of char $p>5$.
Then $(X,B)$ has a (crepant) $\mathbb Q$-factorial terminal model.
\end{thm}
The theorem was proved in [\ref{HX}, Theorem 6.1] for pairs with standard coefficients and
canonical singularities.\\
{\textbf{\sffamily{The connectedness principle with applications to semi-ampleness.}}}
The next result concerns the Koll\'ar-Shokurov connectedness principle.
In characteristic $0$, the surface case was proved by Shokurov by taking a resolution and
then calculating intersection numbers [\ref{Shokurov}, Lemma 5.7]
but the higher dimensional case was proved by Koll\'ar by deriving it from the Kawamata-Viehweg vanishing
theorem [\ref{Kollar+}, Theorem 17.4].
\begin{thm}\label{t-connectedness-d-3}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ over $k$ of char $p>5$.
Let $f\colon X\to Z$ be a birational contraction such that
$-(K_X+B)$ is ample$/Z$. Then for any closed point $z\in Z$, the non-klt locus of
$(X,B)$ is connected in any neighborhood of the fibre $X_z$.
\end{thm}
The theorem is proved in Section \ref{s-connectedness}.
To prove it we use the LMMP rather than vanishing theorems.
When $\dim X=2$, the theorem holds in a stronger form (see \ref{t-connectedness-d-2}).
We will use the connectedness principle on surfaces to prove some semi-ampleness results
on surfaces and $3$-folds.
Here is one of them:
\begin{thm}\label{t-sa-reduced-boundary}
Let $(X,B+A)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of
char $p>5$. Assume that $A,B\ge 0$ are $\mathbb Q$-divisors such that $A$ is ample and
$(K_X+B+A)|_{\rddown{B}}$ is nef. Then $(K_X+B+A)|_{\rddown{B}}$ is semi-ample.
\end{thm}
Note that if one could show that $\rddown{B}$ is semi-lc, then the result would follow from Tanaka
[\ref{Tanaka-2}]. In order to show
that $\rddown{B}$ is semi-lc one needs to check that it satisfies the Serre condition $S_2$.
In characteristic $0$ this is a consequence of Kawamata-Viehweg vanishing (see Koll\'ar [\ref{Kollar+}, Corollary 17.5]).
The $S_2$ condition can be used to glue sections on the various irreducible components of $\rddown{B}$.
To prove the above semi-ampleness we instead use a result of Keel [\ref{Keel}, Corollary 2.9] to glue sections.\\
{\textbf{\sffamily{Log canonical thresholds.}}}
As in characteristic $0$, we will derive the following result from existence of
$\mathbb Q$-factorial dlt models and boundedness results on Fano surfaces.
\begin{thm}\label{t-ACC}
Suppose that $\Lambda\subseteq [0,1]$ and $\Gamma\subseteq
\mathbb{R}$ are DCC sets. Then the set
$$\{\lct(M,X,B)|\mbox{ $(X,B)$ is lc of dimension $\le 3$}\}$$
{\flushleft satisfies} the ACC where $X$ is over $k$ with char $p>5$,
the coefficients of $B$ belong to $\Lambda$, $M\ge 0$ is an
$\mathbb R$-Cartier divisor with coefficients in $\Gamma$, and $\lct(M,X,B)$ is the lc
threshold of $M$ with respect to $(X,B)$.
\end{thm}
With some work it seems that using the above ACC one can actually prove termination for those lc pairs
$(X,B)$ of dimension $3$ such that $K_X+B\equiv M$ for some $M\ge 0$ following the ideas in [\ref{B-acc}].
But we will not pursue this here.\\
{\textbf{\sffamily{Numerically trivial family of curves in the non-big case.}}}
We will also give a somewhat different proof of the following result which was proved
by Cascini-Tanaka-Xu [\ref{CTX}] in char $p$. This was also proved independently by M$^{\rm c}$Kernan much earlier but unpublished.
He informed us that his proof was inspired by [\ref{KMM}].
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Theorems \ref{t-bpf} and \ref{t-contraction} have been proved by Xu [\ref{Xu}]
independently and more or less at the same time but using a different approach. His proof also relies on our
results on flips and minimal models.\\
{\textbf{\sffamily{Dlt and terminal models.}}}
The next two results are standard consequences of the LMMP (more precisely, of special termination).
They are proved in Section \ref{s-crepant-models}.
\begin{thm}\label{cor-dlt-model}
Let $(X,B)$ be an lc pair of dimension $3$ over $k$ of char $p>5$.
Then $(X,B)$ has a (crepant) $\mathbb Q$-factorial dlt model. In particular, if $(X,B)$
is klt, then $X$ has a $\mathbb Q$-factorialization by a small morphism.
\end{thm}
The theorem was proved in [\ref{HX}, Theorem 6.1] for pairs with standard coefficients.
\begin{thm}\label{cor-terminal-model}
Let $(X,B)$ be a klt pair of dimension $3$ over $k$ of char $p>5$.
Then $(X,B)$ has a (crepant) $\mathbb Q$-factorial terminal model.
\end{thm}
The theorem was proved in [\ref{HX}, Theorem 6.1] for pairs with standard coefficients and
canonical singularities.\\
{\textbf{\sffamily{The connectedness principle with applications to semi-ampleness.}}}
The next result concerns the Koll\'ar-Shokurov connectedness principle.
In characteristic $0$, the surface case was proved by Shokurov by taking a resolution and
then calculating intersection numbers [\ref{Shokurov}, Lemma 5.7]
but the higher dimensional case was proved by Koll\'ar by deriving it from the Kawamata-Viehweg vanishing
theorem [\ref{Kollar+}, Theorem 17.4].
\begin{thm}\label{t-connectedness-d-3}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ over $k$ of char $p>5$.
Let $f\colon X\to Z$ be a birational contraction such that
$-(K_X+B)$ is ample$/Z$. Then for any closed point $z\in Z$, the non-klt locus of
$(X,B)$ is connected in any neighborhood of the fibre $X_z$.
\end{thm}
The theorem is proved in Section \ref{s-connectedness}.
To prove it we use the LMMP rather than vanishing theorems.
When $\dim X=2$, the theorem holds in a stronger form (see \ref{t-connectedness-d-2}).
We will use the connectedness principle on surfaces to prove some semi-ampleness results
on surfaces and $3$-folds.
Here is one of them:
\begin{thm}\label{t-sa-reduced-boundary}
Let $(X,B+A)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of
char $p>5$. Assume that $A,B\ge 0$ are $\mathbb Q$-divisors such that $A$ is ample and
$(K_X+B+A)|_{\rddown{B}}$ is nef. Then $(K_X+B+A)|_{\rddown{B}}$ is semi-ample.
\end{thm}
Note that if one could show that $\rddown{B}$ is semi-lc, then the result would follow from Tanaka
[\ref{Tanaka-2}]. In order to show
that $\rddown{B}$ is semi-lc one needs to check that it satisfies the Serre condition $S_2$.
In characteristic $0$ this is a consequence of Kawamata-Viehweg vanishing (see Koll\'ar [\ref{Kollar+}, Corollary 17.5]).
The $S_2$ condition can be used to glue sections on the various irreducible components of $\rddown{B}$.
To prove the above semi-ampleness we instead use a result of Keel [\ref{Keel}, Corollary 2.9] to glue sections.\\
{\textbf{\sffamily{Log canonical thresholds.}}}
As in characteristic $0$, we will derive the following result from existence of
$\mathbb Q$-factorial dlt models and boundedness results on Fano surfaces.
\begin{thm}\label{t-ACC}
Suppose that $\Lambda\subseteq [0,1]$ and $\Gamma\subseteq
\mathbb{R}$ are DCC sets. Then the set
$$\{\lct(M,X,B)|\mbox{ $(X,B)$ is lc of dimension $\le 3$}\}$$
{\flushleft satisfies} the ACC where $X$ is over $k$ with char $p>5$,
the coefficients of $B$ belong to $\Lambda$, $M\ge 0$ is an
$\mathbb R$-Cartier divisor with coefficients in $\Gamma$, and $\lct(M,X,B)$ is the lc
threshold of $M$ with respect to $(X,B)$.
\end{thm}
With some work it seems that using the above ACC one can actually prove termination for those lc pairs
$(X,B)$ of dimension $3$ such that $K_X+B\equiv M$ for some $M\ge 0$ following the ideas in [\ref{B-acc}].
But we will not pursue this here.\\
{\textbf{\sffamily{Numerically trivial family of curves in the non-big case.}}}
We will also give a somewhat different proof of the following result which was proved
by Cascini-Tanaka-Xu [\ref{CTX}] in char $p$. This was also proved independently by M$^{\rm c}$Kernan much earlier but unpublished.
He informed us that his proof was inspired by [\ref{KMM}].
\begin{thm}\label{t-aug-b-non-big}
Assume that $X$ is a normal projective variety of dimension $d$ over an algebraically
closed field (of any characteristic),
and that $B,A\ge 0$ are $\mathbb R$-divisors.
Moreover, suppose $A$ is nef and big and $D=K_X+B+A$ is nef. If $D^d=0$, then
for each general closed point $x\in X$ there is a rational curve $L_x$ passing through
$x$ with $D\cdot L_x=0$.
\end{thm}
The theorem is independent of the rest of this paper. Its proof is an application of the
bend and break theorem.\\
{\textbf{\sffamily{Some remarks about this paper.}}}
In writing this paper we have tried to give as much details as possible even
if the arguments are very similar to the characteristic $0$ case. This is for convenience,
future reference, and to avoid any unpleasant surprise having to do with positive characteristic.
The main results are proved in the following order: \ref{t-flip-1} in the projective case,
\ref{cor-dlt-model}, \ref{cor-terminal-model}, \ref{t-mmodel}, \ref{t-flip-1} in general,
\ref{t-connectedness-d-3},
\ref{t-sa-reduced-boundary}, \ref{t-fg}, \ref{t-bpf}, \ref{t-contraction}, \ref{t-ACC}, and
\ref{t-aug-b-non-big}.\\
{\textbf{\sffamily{Acknowledgements.}}}
Part of this work was done when I visited National Taiwan University in September 2013 with the support of the
Mathematics Division (Taipei Office) of the National Center for Theoretical Sciences. The visit was arranged by Jungkai A. Chen.
I would like to thank them for their hospitality.
This work was partially supported by a Leverhulme grant.
I would also like to thank Paolo Cascini, Christopher Hacon, Janos Koll\'ar, James M$^{\rm{c}}$Kernan,
Burt Totaro and Chenyang Xu for their
comments and suggestions. Finally I would like to thank the referee for valuable corrections
and suggestions.
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\section{Preliminaries}
We work over an algebraically closed field $k$ of characteristic $p>0$ fixed throughout the paper
unless stated otherwise.
\subsection{Contractions}\label{ss-contractions}
A \emph{contraction} $f\colon X\to Z$ of algebraic spaces over $k$ is a proper morphism such that
$f_*\mathcal{O}_X=\mathcal{O}_Z$.
When $X,Z$ are quasi-projective varieties over $k$ and $f$ is
projective, we refer to $f$ as a \emph{projective contraction} to avoid confusion.
Let $f\colon X\to Z$ be a projective contraction of normal varieties.
We say $f$ is \emph{extremal} if the relative Kleiman-Mori cone of curves $\overline{NE}(X/Z)$ is one-dimensional.
Such a contraction is a \emph{divisorial contraction} if it is birational and it contracts
some divisor. It is called a \emph{small contraction} if it is birational and it
contracts some subvariety of codimension $\ge 2$ but no divisors.
Let $f\colon X\to Z$ be a small contraction and
$D$ an $\mathbb R$-Cartier divisor such that $-D$ is ample$/Z$. We refer to $f$ as a
\emph{$D$-flipping contraction} or just a flipping contraction for short.
We say the \emph{$D$-flip} of $f$
exists if there is a small contraction $X^+\to Z$ such that the birational transform
$D^+$ is ample$/Z$.
\subsection{Some notions related to divisors}\label{ss-divisors}
Let $X$ be a normal projective variety over $k$ and $L$ a nef $\mathbb R$-Cartier
divisor. We define $L^\perp:=\{\alpha\in \overline{NE}(X) \mid L\cdot \alpha=0\}$.
This is an extremal face of $\overline{NE}(X)$ cut out by $L$.
Let $f\colon X\to Z$ be a projective morphism of normal varieties over $k$, and
let $D$ be an $\mathbb R$-divisor on $X$. We define the algebra of $D$ over $Z$
as $\mathcal{R}(D/Z)=\bigoplus_{m\in\mathbb Z^{\ge 0}} f_*\mathcal{O}_X(\rddown{mD})$.
When $Z$ is a point we denote the algebra by $R(D)$.
When $D=K_X+B$ for a pair $(X,B)$ we call the algebra the \emph{log canonical algebra}
of $(X,B)$ over $Z$.
Now let $\phi\colon X\dashrightarrow Y$ be a birational map of normal projective varieties over $k$ whose inverse does
not contract divisors. Let $D$ be an $\mathbb R$-Cartier divisor on $X$ such that $D_Y:=\phi_*D$ is
$\mathbb R$-Cartier too. We say that $\phi$ is
\emph{$D$-negative} if there is a common resolution $f\colon W\to X$ and $g\colon W\to Y$
such that $f^*D-g^*D_Y$ is effective and exceptional$/Y$, and its support contains the birational transform of
all the prime divisors on $X$ which are contracted$/Y$.
\subsection{The negativity lemma}\label{ss-negativity}
The negativity lemma states that if $f\colon Y\to X$ is a projective birational
contraction of normal quasi-projective varieties over $k$ and $D$ is an $\mathbb R$-Cartier
divisor on $Y$ such that $-D$ is nef$/X$ and $f_*D\ge 0$, then $D\ge 0$
(since this is a local statement over $X$, it also holds if we assume
$X$ is an algebraic space and $f$ is proper). See
[\ref{Shokurov}, Lemma 1.1] for the characteristic $0$ case. The proof there also works
in char $p>0$ and we reproduce it for convenience.
Assume that the lemma does not hold. We reduce the problem to the surface case.
Let $P$ be the image of the negative components of $D$. If $\dim P>0$,
we take a general hypersurface section $H$ on $X$, let $G$ be the normalization
of the birational transform of $H$ on $Y$ and reduce the problem to the contraction
$G\to H$ and the divisor $D|_G$. But if $\dim X>2$ and $\dim P=0$, we take a general hypersurface
section $G$ on $Y$, let $H$ be the normalization of $f(G)$, and reduce
the problem to the induced contraction $G\to H$ and divisor $D|_G$. So we can reduce the
problem to the case when $X,Y$ are surfaces, $P$ is just one point, and $f$ is an isomorphism
over $X\setminus \{P\}$.
Taking a resolution enables us to assume $Y$ is smooth. Now let $E\ge 0$ be a divisor whose support is
equal to the exceptional locus of $f$ and such that $-E$ is nef$/X$: pick a Cartier divisor $L\ge 0$
passing through $P$ and write $f^*L=L^\sim+E$ where $L^\sim$ is the birational
transform of $L$; then $E$ satisfies the requirements. Let $e$ be the smallest
number such that $D+eE\ge 0$. Now there is a component $C$ of $E$ whose coefficient
in $D+eE$ is zero and that $C$ intersects $\Supp (D+eE)$. But then
$(D+eE)\cdot C>0$, a contradiction.
\subsection{Resolution of singularities}\label{ss-resolution}
Let $X$ be a quasi-projective variety of dimension $\le 3$ over $k$ and $P\subset X$
a closed subset. Assume that
there is an open set $U\subset X$ such that $P\cap U$ is a divisor
with simple normal crossing (snc) singularities. Then there is a \emph{log resolution}
of $X,P$ which is an isomorphism over $U$, that is, there is a projective
birational morphism $f\colon Y\to X$ such that the union of the exceptional locus of $f$
and the birational transform of $P$ is an snc divisor, and
$f$ is an isomorphism over $U$. This follows from Cutkosky [\ref{Cut-sing}, Theorems 1.1, 1.2, 1.3]
when $k$ has char $p>5$, and from Cossart-Piltant [\ref{CP-sing}, Theorems 4.1, 4.2][\ref{CP-sing-2}, Theorem]
in general (see also [\ref{HX}, Theorem 2.1]).
\subsection{Pairs}\label{ss-pairs}
A \emph{pair} $(X,B)$ consists of a normal quasi-projective variety $X$ over $k$
and an \emph{$\mathbb R$-boundary} $B$, that is an $\mathbb R$-divisor $B$ on $X$ with coefficients in $[0,1]$,
such that $K_X+B$ is $\mathbb{R}$-Cartier. When $B$ has rational coefficients we
say $B$ is a \emph{$\mathbb Q$-boundary} or say $B$ is \emph{rational}.
We say that $(X,B)$ is \emph{log smooth} if $X$ is smooth
and $\Supp B$ has simple normal crossing singularities.
Let $(X,B)$ be a pair. For a prime divisor $D$ on some birational model of $X$ with a
nonempty centre on $X$, $a(D,X,B)$ denotes the \emph{log discrepancy} which is defined
by taking a projective birational morphism $f\colon Y\to X$ from a normal variety
containing $D$ as a prime divisor and putting $a(D,X,B)=1-b$ where $b$ is the
coefficient of $D$ in $B_Y$ and $K_Y+B_Y=f^*(K_X+B)$.
As in characteristic $0$, we can define various types of singularities using
log discrepancies. Let $(X,B)$ be a pair.
We say that the pair is \emph{log canonical} or lc for short (resp. \emph{Kawamata log terminal} or {klt}
for short) if
$a(D,X,B)\ge 0$ (resp. $a(D,X,B)>0$) for any prime divisor $D$ on birational models of $X$.
An \emph{lc centre} of $(X,B)$ is the image in $X$ of a $D$ with $a(D,X,B)=0$.
The pair $(X,B)$ is \emph{terminal} if $a(D,X,B)>1$ for any prime divisor $D$ on birational models of $X$ which is
exceptional$/X$ (such pairs are sometime called terminal in codimension $\ge 2$).
On the other hand, we say that $(X,B)$ is \emph{dlt} if there is a closed subset $P\subset X$
such that $(X,B)$ is log smooth
outside $P$ and no lc centre of $(X,B)$ is inside $P$. In particular,
the lc centres of $(X,B)$ are exactly the components of $S_1\cap \cdots \cap S_r$ where $S_i$
are among the components of $\rddown{B}$. Moreover, there is a log resolution
$f\colon Y\to X$ of $(X,B)$ such that $a(D,X,B)>0$ for any prime divisor $D$ on $Y$
which is exceptional$/X$, eg take a log resolution $f$
which is an isomorphism over $X\setminus P$.
Finally, we say that $(X,B)$ is \emph{plt} if it is dlt and each connected component of
$\rddown{B}$ is irreducible. In particular, the only lc centres of $(X,B)$ are
the components of $\rddown{B}$.
\subsection{Ample divisors on log smooth pairs}\label{ss-log-smooth}
Let $(X,B)$ be a projective log smooth pair over $k$ and let $A$ be an ample $\mathbb Q$-divisor.
We will argue that there is $A'\sim_\mathbb Q A$ such that $A'\ge 0$ and that
$(X,B+A')$ is log smooth. The argument was suggested to us by several people
independently. We may assume that $B$ is reduced. Let $S_1,\dots, S_r$
be the components of $B$ and let $\mathcal{S}$ be the set of the components
of $S_{i_1}\cap \cdots \cap S_{i_n}$ for all the choices
$\{i_1,\dots,i_n\}\subseteq \{1,\cdots,r\}$. By Bertini's theorem,
there is a sufficiently divisible integer $l>0$ such that for any $T\in\mathcal{S}$, a general element of
$|lA|_T|$ is smooth. Since $lA$ is sufficiently ample, such general elements
are restrictions of general elements of $|lA|$. Therefore, we can choose a
general $G\sim lA$ such that $G$ is smooth and $G|_T$ is smooth for any $T\in \mathcal{S}$.
This means that $(X,B+G)$ is log smooth. Now let $A'=\frac{1}{l}G$.
\subsection{Models of pairs}\label{ss-mmodels}
Let $(X,B)$ be a pair and $X\to Z$ a projective contraction over $k$.
A pair $(Y,B_Y)$ with a projective contraction $Y\to Z$ and a birational map
$\phi\colon X\dashrightarrow Y/Z$ is a \emph{log birational model} of $(X,B)$
if $B_Y$ is the sum of the birational transform of $B$
and the reduced exceptional divisor of $\phi^{-1}$.
We say that $(Y,B_Y)$ is a \emph{weak lc model} of $(X,B)$ over $Z$ if in addition\\\\
(1) $K_Y+B_Y$ is nef/$Z$.\\
(2) for any prime divisor $D$ on $X$ which is exceptional/$Y$, we have
$$
a(D,X,B)\le a(D,Y,B_Y)
$$
And we call $(Y,B_Y)$ a \emph{log minimal model} of $(X,B)$ over $Z$ if in addition\\\\
(3) $(Y,B_Y)$ is $\mathbb Q$-factorial dlt,\\
(4) the inequality in (2) is strict.\\
When $K_X+B$ is big$/Z$, the \emph{lc model} of $(X,B)$ over $Z$ is a weak lc model $(Y,B_Y)$
over $Z$ with $K_Y+B_Y$ ample$/Z$.
On the other hand, a log birational model $(Y,B_Y)$ of $(X,B)$ is called a
\emph{Mori fibre space} of $(X,B)$ over $Z$ if there is a $K_Y+B_Y$-negative extremal projective
contraction $Y\to T/Z$, and if for any prime divisor $D$ on birational models of $X$ we have
$$
a(D,X,B)\le a(D,Y,B_Y)
$$
with strict inequality if $D\subset X$ and if it is exceptional/$Y$,
Note that the above definitions are slightly different from the traditional definitions. However,
if $(X,B)$ is plt (hence also klt) the definitions coincide.
Let $(X,B)$ be an lc pair over $k$. A $\mathbb Q$-factorial dlt pair $(Y,B_Y)$ is
a \emph{$\mathbb Q$-factorial dlt model} of $(X,B)$ if there is a projective birational
morphism $f\colon Y\to X$ such that $K_Y+B_Y=f^*(K_X+B)$ and such that every exceptional prime
divisor of $f$ has coefficient $1$ in $B_Y$. On the other hand,
when $(X,B)$ is klt, a pair $(Y,B_Y)$ with terminal singularities
is a \emph{terminal model} of $(X,B)$ if there is a projective birational
morphism $f\colon Y\to X$ such that $K_Y+B_Y=f^*(K_X+B)$.
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\subsection{Ample divisors on log smooth pairs}\label{ss-log-smooth}
Let $(X,B)$ be a projective log smooth pair over $k$ and let $A$ be an ample $\mathbb Q$-divisor.
We will argue that there is $A'\sim_\mathbb Q A$ such that $A'\ge 0$ and that
$(X,B+A')$ is log smooth. The argument was suggested to us by several people
independently. We may assume that $B$ is reduced. Let $S_1,\dots, S_r$
be the components of $B$ and let $\mathcal{S}$ be the set of the components
of $S_{i_1}\cap \cdots \cap S_{i_n}$ for all the choices
$\{i_1,\dots,i_n\}\subseteq \{1,\cdots,r\}$. By Bertini's theorem,
there is a sufficiently divisible integer $l>0$ such that for any $T\in\mathcal{S}$, a general element of
$|lA|_T|$ is smooth. Since $lA$ is sufficiently ample, such general elements
are restrictions of general elements of $|lA|$. Therefore, we can choose a
general $G\sim lA$ such that $G$ is smooth and $G|_T$ is smooth for any $T\in \mathcal{S}$.
This means that $(X,B+G)$ is log smooth. Now let $A'=\frac{1}{l}G$.
\subsection{Models of pairs}\label{ss-mmodels}
Let $(X,B)$ be a pair and $X\to Z$ a projective contraction over $k$.
A pair $(Y,B_Y)$ with a projective contraction $Y\to Z$ and a birational map
$\phi\colon X\dashrightarrow Y/Z$ is a \emph{log birational model} of $(X,B)$
if $B_Y$ is the sum of the birational transform of $B$
and the reduced exceptional divisor of $\phi^{-1}$.
We say that $(Y,B_Y)$ is a \emph{weak lc model} of $(X,B)$ over $Z$ if in addition\\\\
(1) $K_Y+B_Y$ is nef/$Z$.\\
(2) for any prime divisor $D$ on $X$ which is exceptional/$Y$, we have
$$
a(D,X,B)\le a(D,Y,B_Y)
$$
And we call $(Y,B_Y)$ a \emph{log minimal model} of $(X,B)$ over $Z$ if in addition\\\\
(3) $(Y,B_Y)$ is $\mathbb Q$-factorial dlt,\\
(4) the inequality in (2) is strict.\\
When $K_X+B$ is big$/Z$, the \emph{lc model} of $(X,B)$ over $Z$ is a weak lc model $(Y,B_Y)$
over $Z$ with $K_Y+B_Y$ ample$/Z$.
On the other hand, a log birational model $(Y,B_Y)$ of $(X,B)$ is called a
\emph{Mori fibre space} of $(X,B)$ over $Z$ if there is a $K_Y+B_Y$-negative extremal projective
contraction $Y\to T/Z$, and if for any prime divisor $D$ on birational models of $X$ we have
$$
a(D,X,B)\le a(D,Y,B_Y)
$$
with strict inequality if $D\subset X$ and if it is exceptional/$Y$,
Note that the above definitions are slightly different from the traditional definitions. However,
if $(X,B)$ is plt (hence also klt) the definitions coincide.
Let $(X,B)$ be an lc pair over $k$. A $\mathbb Q$-factorial dlt pair $(Y,B_Y)$ is
a \emph{$\mathbb Q$-factorial dlt model} of $(X,B)$ if there is a projective birational
morphism $f\colon Y\to X$ such that $K_Y+B_Y=f^*(K_X+B)$ and such that every exceptional prime
divisor of $f$ has coefficient $1$ in $B_Y$. On the other hand,
when $(X,B)$ is klt, a pair $(Y,B_Y)$ with terminal singularities
is a \emph{terminal model} of $(X,B)$ if there is a projective birational
morphism $f\colon Y\to X$ such that $K_Y+B_Y=f^*(K_X+B)$.
\subsection{Keel's results}\label{ss-Keel}
We recall some of the results of Keel which will be used in this paper.
For a nef $\mathbb Q$-Cartier divisor $L$ on a projective scheme $X$ over $k$, the
\emph{exceptional locus} $\mathbb{E}(L)$ is the union of those
positive-dimensional integral subschemes $Y\subseteq X$ such that $L|_Y$ is not big, i.e. $(L|_Y)^{\dim Y}=0$.
By [\ref{CMM}], $\mathbb{E}(L)$ coincides with the augmented base locus ${\bf{B_+}}(L)$.
We say $L$ is \emph{endowed with a map} $f\colon X\to V$, where $V$ is an algebraic space over $k$,
if: an integral subscheme $Y$ is contracted by $f$ (i.e. $\dim Y>\dim f(Y)$)
if and only if $L|_Y$ is not big.
\begin{thm}[{[\ref{Keel}, 1.9]}]\label{t-Keel-1}
Let $X$ be a projective scheme over $k$ and $L$ a nef $\mathbb Q$-Cartier divisor on $X$.
Then
$\bullet$ $L$ is semi-ample if and only if $L|_{\mathbb{E}(L)}$ is semi-ample;
$\bullet$ $L$ is endowed with a map if and only if $L|_{\mathbb{E}(L)}$ is endowed with a map.
\end{thm}
The theorem does not hold if $k$ is of characteristic $0$. When $L|_{\mathbb{E}(L)}\equiv 0$, then
$L|_{\mathbb{E}(L)}$ is automatically endowed with the constant map $\mathbb{E}(L)\to \rm{pt}$ hence
$L$ is endowed with a map. This is particularly useful for studying $3$-folds because it is often
not difficult to show that $L|_{\mathbb{E}(L)}$ is endowed with a map, eg when $\dim \mathbb{E}(L)=1$.
\begin{thm}[{[\ref{Keel}, 0.5]}]\label{t-Keel-2}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ over $k$ with $B$ a $\mathbb Q$-divisor.
Assume that $A$ is an ample $\mathbb Q$-divisor such that
$L=K_X+B+A$ is nef and big. Then $L$ is endowed with a map.
\end{thm}
In particular, when $L^\perp$ is an extremal ray,
then we can contract $R$ to an algebraic space by the map associated to $L$.
Thus such an extremal ray is generated by the class of some curve.
We also recall the following cone theorems which we will use repeatedly in Section \ref{s-ext-rays}.
Note that these theorems (as well as \ref{t-Keel-2}) do not assume singularities to be lc.
\begin{thm}[{[\ref{Keel}, 0.6]}]\label{t-Keel-3}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ over $k$ with $B$ a $\mathbb Q$-divisor.
Assume that $K_X+B\sim_\mathbb Q M$ for some $M\ge 0$. Then there is a countable
number of curves $\Gamma_i$ such that
$\bullet$ $\overline{NE}(X)=\overline{NE}(X)_{K_X+B\ge 0}+\sum_i \mathbb R [\Gamma_i]$,
$\bullet$ all but finitely many of the $\Gamma_i$ are rational curves satisfying
$-3\le (K_X+B)\cdot \Gamma_i<0$, and
$\bullet$ the rays $\mathbb R [\Gamma_i]$ do not accumulate inside $\overline{NE}(X)_{K_X+B<0}$.\\
\end{thm}
\begin{thm}[{[\ref{Keel}, 5.5.2]}]\label{t-Keel-4}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ over $k$.
Assume that
$$
L=K_X+B+H\sim_\mathbb R A+M
$$
is nef where $H,A$ are ample $\mathbb R$-divisors, and $M\ge 0$.
Then any extremal ray of $L^\perp$ is generated by some curve $\Gamma$
such that either
$\bullet$ $\Gamma$ is a component of the singular locus of $B+M$ union
with the singular locus of $X$, or
$\bullet$ $\Gamma$ is a rational curve satisfying $-3\le (K_X+B)\cdot \Gamma<0$.\\
\end{thm}
\begin{rem}\label{ss-good-exc-locus}
Let $(X,B)$ a projective lc pair of dimension $3$ over $k$ with $B$ a $\mathbb Q$-boundary,
and $H$ an ample $\mathbb Q$-divisor. Assume that $L=K_X+B+H$ is
nef and big. Moreover, suppose that each connected component of
$\mathbb{E}(L)$ is inside some normal irreducible component $S$ of $\rddown{B}$.
Then $L|_S$ is semi-ample for such components (cf. [\ref{Tanaka}]) hence $L|_{\mathbb{E}(L)}$ is semi-ample
and this in turn implies that $L$ is semi-ample by Theorem \ref{t-Keel-1}.
\end{rem}
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\section{Extremal rays and special kinds of LMMP}\label{s-ext-rays}
As usual the varieties and algebraic spaces in this section are defined over $k$ of char $p>0$.
\subsection{Extremal curve of a ray} \label{ss-ext-curve}
Let $X$ be a projective variety and $H$ a fixed ample Cartier divisor.
Let $R$ be a ray of $\overline{NE}(X)$ which is generated by some curve $\Gamma$.
Assume that
$$
H\cdot \Gamma=\min\{H\cdot C\mid \mbox{$C$ generates $R$} \}
$$
In this case, we say $\Gamma$
is an \emph{extremal curve} of $R$ (in practice we do not mention $H$ and assume that
it is already fixed). Let $C$ be any other curve
generating $R$.
Assume that $D\cdot R<0$ for some $\mathbb R$-Cartier divisor $D$. Since $\Gamma$ and $C$ both generate $R$,
$$
\frac{D\cdot C}{H\cdot C}=\frac{D\cdot \Gamma}{H\cdot \Gamma}
$$
hence
$$
D\cdot \Gamma={D\cdot C}(\frac{H\cdot \Gamma}{H\cdot C})\ge D\cdot C
$$
which implies that
$$
D\cdot \Gamma=\max\{D\cdot C\mid \mbox{$C$ generates $R$} \}
$$
\subsection{Negative extremal rays}\label{ss-ext-rays}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$.
Let $R$ be a $K_X+B$-negative extremal ray. Assume that there is a
boundary $\Delta$ such that $K_X+\Delta$ is pseudo-effective and
$(K_X+\Delta)\cdot R<0$. By adding a small ample divisor and perturbing the
coefficients we can assume that $\Delta$ is rational and that $K_X+\Delta$ is big.
Then by Theorem \ref{t-Keel-3},
$R$ is generated by some extremal curve and $R$ is an isolated extremal ray of
$\overline{NE}(X)$.
Now assume that $K_X+B$ is pseudo-effective and let $A$ be an ample $\mathbb R$-divisor.
Then for any $\epsilon>0$, there are only finitely many $K_X+B+\epsilon A$-negative
extremal rays: assume that this is not the case; then we can find a $\mathbb Q$-boundary $\Delta$
such that $K_X+\Delta$ is big and
$$
K_X+B+\epsilon A\sim_\mathbb R K_X+\Delta+G
$$ where
$G$ is ample; so there are also infinitely many $K_X+\Delta$-negative
extremal rays; but $K_X+\Delta$ is
big hence by Theorem \ref{t-Keel-3} all but finitely many of the
$K_X+\Delta$-negative extremal
rays are generated by extremal curves $\Gamma$ with
$-3\le (K_X+\Delta)\cdot \Gamma<0$; if $(K_X+B+\epsilon A)\cdot \Gamma<0$,
then $G\cdot \Gamma\le 3$; since $G$ is ample, there can be only
finitely many such $\Gamma$ up to numerical equivalence.
Let $R$ be a $K_X+B$-negative extremal ray where $K_X+B$ is not necessarily pseudo-effective.
But assume that there is a pseudo-effective $K_X+\Delta$ with
$(K_X+\Delta)\cdot R<0$. By the remarks above we may assume $\Delta$ is rational, $K_X+\Delta$ big,
and that there are only finitely many $K_X+\Delta$-negative extremal rays.
Therefore, we can find an ample $\mathbb Q$-divisor
$H$ such that $L=K_X+\Delta+H$ is nef and big and $L^\perp=R$. That is, $L$ is a \emph{supporting divisor}
of $R$. Moreover,
$R$ can be contracted to an algebraic space, by Theorem \ref{t-Keel-2}.
More precisely, there is a
contraction $X\to V$ to an algebraic space such that
it contracts a curve $C$ if and only if $L\cdot C=0$ if and only if the class $[C]\in R$.
\subsection{More on negative extremal rays}\label{ss-ext-rays-II}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$.
Let $\mathcal{C}\subset \overline{NE}(X)$ be one of the following:
$(1)$ $\mathcal{C}=\overline{NE}(X/Z)$ for a given projective contraction $X\to Z$ such that
$K_X+B\equiv P+M/Z$ where $P$ is nef$/Z$ and $M\ge 0$ (this is a \emph{weak Zariski decomposition};
see \ref{ss-WZD}); or
$(2)$ $\mathcal{C}=N^\perp$ for some nef and big $\mathbb Q$-divisor $N$;
We will show that in both cases, each $K_X+B$-negative extremal ray
$R$ of $\mathcal{C}$ is generated by an extremal curve $\Gamma$, and for all but finitely many of
those rays we have $-3\le (K_X+B)\cdot \Gamma<0$.
We first deal with case (1). Fix a
$K_X+B$-negative extremal ray $R$ of $\mathcal{C}$. By replacing $P$ we can
assume that $K_X+B=P+M$. Let $A$ be an ample $\mathbb R$-divisor and
$T$ be the pullback of a sufficiently ample divisor on $Z$
so that $K_X+B+A+T$ is big and $(K_X+B+A+T)\cdot R<0$.
By \ref{ss-ext-rays}, there is a nef and big $\mathbb Q$-divisor $L$ with $L^\perp=R$. Moreover,
we may assume that if
$l\gg 0$, then
$$
Q_1:=K_X+B+T+lL+A
$$
is nef and big and $Q_1^\perp=R$.
By construction, $T+lL+A$ is ample, $P+T+lL+A$ is also ample, and
$$
K_X+B+T+lL+A=P+T+lL+A+M
$$
Therefore, by Theorem \ref{t-Keel-4}, $R$ is generated by some curve
$\Gamma$ satisfying $-3\le (K_X+B)\cdot \Gamma<0$ or $R$ is generated by some curve
in the singular locus of $B+M$ or $X$. There are only finitely many possibilities in the latter
case. The claim then follows.
Now we deal with case (2). Fix a $K_X+B$-negative extremal ray $R$ of $\mathcal{C}$.
Since $N$ is nef and big, for some $n>0$,
$$
K_X+B+nN\sim_\mathbb R G+S
$$
where $G$ is ample and $S\ge 0$.
By \ref{ss-ext-rays}, there is a nef and big $\mathbb Q$-divisor $L$ with $L^\perp=R$.
Moreover, for some
$l\gg 0$ and some ample $\mathbb R$-divisor $A$,
$$
Q_2:=K_X+B+nN+lL+A
$$
is nef and big with $Q_2^\perp=R$.
Now, $nN+lL+A$ is ample, $G+lL+A$ is ample, and
$$
K_X+B+nN+lL+A\sim_\mathbb R G+lL+A+S
$$
Therefore, by Theorem \ref{t-Keel-4}, $R$ is generated by some curve
$\Gamma$ satisfying $-3\le (K_X+B)\cdot \Gamma<0$ or $R$ is generated by some curve
in the singular locus of $B+S$ or $X$. There are only finitely many possibilities in the latter
case. The claim then follows.
Assume that $R$ is a $K_X+B$-negative extremal ray of $\mathcal{C}$, in either case.
Then the above arguments show that there is a $\mathbb Q$-boundary $\Delta$ and an ample $\mathbb Q$-divisor
$H$ such that $K_X+\Delta$ is big, $(K_X+\Delta)\cdot R<0$, and $L=K_X+\Delta+H$ is nef and
big with $L^\perp=R$. Therefore, as in \ref{ss-ext-rays}, $R$ can be contracted via a contraction $X\to V$ to an
algebraic space. Moreover, if $B$ is rational, then we can find an ample $\mathbb Q$-divisor
$H'$ such that $L'=K_X+B+H'$ is nef and big and again $L'^\perp=R$.
\subsection{Extremal rays given by scaling.}\label{ss-ext-rays-scaling}
Let $(X,B)$ be a projective $\mathbb Q$-factorial
pair of dimension $3$. Assume that either $\mathcal{C}=\overline{NE}(X/Z)$ for some
projective contraction $X\to Z$ such that $K_X+B\equiv M/Z$ for some $M\ge 0$, or $\mathcal{C}=N^\perp$ for some
nef and big $\mathbb Q$-divisor $N$.
In addition assume that $(X,B+C)$ is a pair for some $C\ge 0$ and that $K_X+B+C$ is nef on
$\mathcal{C}$, that is, $(K_X+B+C)\cdot R\ge 0$ for every extremal ray $R$ of $\mathcal{C}$.
Let
$$
\lambda=\inf\{t\ge 0\mid K_X+B+tC \mbox{~is nef on $\mathcal{C}$}\}
$$
Then we will see that either $\lambda=0$ or there is an extremal ray $R$ of $\mathcal{C}$
such that $(K_X+B+\lambda C)\cdot R=0$ and $(K_X+B)\cdot R<0$. Assume $\lambda>0$.
If the claim is not true, then there exist a sequence of numbers $t_1<t_2<\cdots$ approaching
$\lambda$ and extremal rays $R_i$ of $\mathcal{C}$ such that
$(K_X+B+t_i C)\cdot R_i=0$ and $(K_X+B)\cdot R_i<0$.
First assume that $\mathcal{C}=N^\perp$ for some nef and big $\mathbb Q$-divisor $N$.
We can write a finite sum $K_X+B=\sum_j r_j(K_X+B_j)$ where $r_j\in (0,1]$, $\sum r_j=1$, and
$(X,B_j)$ are pairs with $B_j$ being rational. By \ref{ss-ext-rays-II}, we may assume
that each $R_i$ is generated by some extremal curve $\Gamma_i$ with $-3\le (K_X+B_j)\cdot \Gamma_i$
for each $j$. This implies that there are only finitely many possibilities
for the numbers $(K_X+B)\cdot \Gamma_i$. A similar reasoning shows that there are only finitely many
possibilities for the numbers $(K_X+B+\frac{\lambda}{2}C)\cdot \Gamma_i$ hence there are also only finitely many
possibilities for the numbers $C\cdot \Gamma_i$. But then this implies that there are finitely
many $t_i$, a contradiction.
Now assume that $\mathcal{C}=\overline{NE}(X/Z)$ for some projective contraction $X\to Z$
such that $K_X+B\equiv M/Z$ for some $M\ge 0$.
Then we can write $K_X+B=\sum_j r_j(K_X+B_j)$ and $M=\sum_j r_jM_j$
where $r_j\in (0,1]$, $\sum r_j=1$, $(X,B_j)$ are pairs with $B_j$ being rational,
$K_X+B_j\equiv M_j/Z$, and $M_j\ge 0$.
To find such a decomposition we argue as in [\ref{BP}, pages 96-97].
Let $V$ and $W$ be the $\mathbb R$-vector spaces generated by the
components of $B$ and $M$ respectively. For a vector $v\in V$ (resp. $w\in W$) we denote the
corresponding $\mathbb R$-divisor by
$B_v$ (resp. $M_w$). Let $F$ be the set of those $(v,w)\in V\times W$ such that
$(X,B_v)$ is a pair, $M_w\ge 0$, and $K_X+B_v\equiv M_w/Z$.
Then $F$ is defined by a finite
number of linear equalities and
inequalities with rational coefficients. If $B=B_{v_0}$ and $M=M_{w_0}$ are the given divisors, then $(v_0,w_0)\in F$
hence it belongs to some polytope in $F$ with rational vertices. The vertices of the polytope give the $B_j,M_j$.
The rest of the proof is as in the last paragraph.
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\subsection{Extremal rays given by scaling.}\label{ss-ext-rays-scaling}
Let $(X,B)$ be a projective $\mathbb Q$-factorial
pair of dimension $3$. Assume that either $\mathcal{C}=\overline{NE}(X/Z)$ for some
projective contraction $X\to Z$ such that $K_X+B\equiv M/Z$ for some $M\ge 0$, or $\mathcal{C}=N^\perp$ for some
nef and big $\mathbb Q$-divisor $N$.
In addition assume that $(X,B+C)$ is a pair for some $C\ge 0$ and that $K_X+B+C$ is nef on
$\mathcal{C}$, that is, $(K_X+B+C)\cdot R\ge 0$ for every extremal ray $R$ of $\mathcal{C}$.
Let
$$
\lambda=\inf\{t\ge 0\mid K_X+B+tC \mbox{~is nef on $\mathcal{C}$}\}
$$
Then we will see that either $\lambda=0$ or there is an extremal ray $R$ of $\mathcal{C}$
such that $(K_X+B+\lambda C)\cdot R=0$ and $(K_X+B)\cdot R<0$. Assume $\lambda>0$.
If the claim is not true, then there exist a sequence of numbers $t_1<t_2<\cdots$ approaching
$\lambda$ and extremal rays $R_i$ of $\mathcal{C}$ such that
$(K_X+B+t_i C)\cdot R_i=0$ and $(K_X+B)\cdot R_i<0$.
First assume that $\mathcal{C}=N^\perp$ for some nef and big $\mathbb Q$-divisor $N$.
We can write a finite sum $K_X+B=\sum_j r_j(K_X+B_j)$ where $r_j\in (0,1]$, $\sum r_j=1$, and
$(X,B_j)$ are pairs with $B_j$ being rational. By \ref{ss-ext-rays-II}, we may assume
that each $R_i$ is generated by some extremal curve $\Gamma_i$ with $-3\le (K_X+B_j)\cdot \Gamma_i$
for each $j$. This implies that there are only finitely many possibilities
for the numbers $(K_X+B)\cdot \Gamma_i$. A similar reasoning shows that there are only finitely many
possibilities for the numbers $(K_X+B+\frac{\lambda}{2}C)\cdot \Gamma_i$ hence there are also only finitely many
possibilities for the numbers $C\cdot \Gamma_i$. But then this implies that there are finitely
many $t_i$, a contradiction.
Now assume that $\mathcal{C}=\overline{NE}(X/Z)$ for some projective contraction $X\to Z$
such that $K_X+B\equiv M/Z$ for some $M\ge 0$.
Then we can write $K_X+B=\sum_j r_j(K_X+B_j)$ and $M=\sum_j r_jM_j$
where $r_j\in (0,1]$, $\sum r_j=1$, $(X,B_j)$ are pairs with $B_j$ being rational,
$K_X+B_j\equiv M_j/Z$, and $M_j\ge 0$.
To find such a decomposition we argue as in [\ref{BP}, pages 96-97].
Let $V$ and $W$ be the $\mathbb R$-vector spaces generated by the
components of $B$ and $M$ respectively. For a vector $v\in V$ (resp. $w\in W$) we denote the
corresponding $\mathbb R$-divisor by
$B_v$ (resp. $M_w$). Let $F$ be the set of those $(v,w)\in V\times W$ such that
$(X,B_v)$ is a pair, $M_w\ge 0$, and $K_X+B_v\equiv M_w/Z$.
Then $F$ is defined by a finite
number of linear equalities and
inequalities with rational coefficients. If $B=B_{v_0}$ and $M=M_{w_0}$ are the given divisors, then $(v_0,w_0)\in F$
hence it belongs to some polytope in $F$ with rational vertices. The vertices of the polytope give the $B_j,M_j$.
The rest of the proof is as in the last paragraph.
\subsection{LMMP with scaling}\label{ss-g-LMMP-scaling}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$.
Assume that either $\mathcal{C}=\overline{NE}(X/Z)$ for some
projective contraction $X\to Z$ such that $K_X+B\equiv M/Z$ for some $M\ge 0$, or $\mathcal{C}=N^\perp$ for some
nef and big $\mathbb Q$-divisor $N$.
In addition assume that $(X,B+C)$ is a pair for some $C\ge 0$ and that $K_X+B+C$ is nef on
$\mathcal{C}$.
If $K_X+B$ is not nef on $\mathcal{C}$, by \ref{ss-ext-rays-scaling}, there is an extremal ray $R$
of $\mathcal{C}$ such that $(K_X+B+\lambda C)\cdot R=0$ and $(K_X+B)\cdot R<0$
where $\lambda$ is the smallest number such that $K_X+B+\lambda C$ is nef on
$\mathcal{C}$. Assume that $R$ can be contracted by a projective morphism. The contraction is birational
because $L\cdot R=0$ for some nef and big $\mathbb Q$-Cartier divisor $L$ (see \ref{ss-ext-rays-II}).
Assume that $X\dashrightarrow X'$ is the corresponding divisorial contraction or flip, and assume that
$X'$ is $\mathbb Q$-factorial.
Let $\mathcal{C}'$ be the cone given by $\mathcal{C}'=\overline{NE}(X'/Z)$ or $\mathcal{C}'=(N')^\perp$
corresponding to the above cases.
Let $\lambda'$ be the smallest nonnegative number such that $K_{X'}+B'+\lambda' C'$ is nef on
$\mathcal{C}'$. If $\lambda'>0$, then there is an extremal ray $R'$
of $\mathcal{C}'$ such that $(K_{X'}+B'+\lambda' C')\cdot R'=0$ and $(K_{X'}+B')\cdot R'<0$.
Assume that $R'$ can be contracted and so on. Assuming that all the necessary ingredients
exist, the process gives a special kind of
LMMP which we may refer to as \emph{LMMP$/\mathcal{C}$ on $K_X+B$ with scaling of $C$}.
Note that $\lambda\ge \lambda'\ge \cdots$
If $\mathcal{C}=\overline{NE}(X/Z)$, we also refer to the above LMMP as the LMMP$/Z$
on $K_X+B$ with scaling of $C$. If $\mathcal{C}=N^\perp$, and if
$N$ is endowed with a map $X\to V$ to an algebraic space, we refer to the
above LMMP as the LMMP$/V$ on $K_X+B$ with scaling of $C$.
In practice, when we run an LMMP with scaling, $(X,B)$ is $\mathbb Q$-factorial dlt
and each extremal ray in the process intersects some component of $\rddown{B}$
negatively. In particular, such rays can be contracted by projective morphisms and
the $\mathbb Q$-factorial property is preserved by the LMMP (see \ref{ss-pl-ext-rays}).
If the required
flips exist then the LMMP terminates by special termination (see \ref{ss-special-termination}).
\subsection{Extremal rays given by a weak Zariski decomposition}\label{ss-ext-ray-WZD}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ and $X\to Z$ a projective contraction such that
\begin{enumerate}
\item $K_X+B\equiv P+M/Z$, $P$ is nef$/Z$, $M\ge 0$, and
\item $\Supp M\subseteq \rddown{B}$.
\end{enumerate}
Let
$$
\mu=\sup \{t\in [0,1] \mid P+tM ~~~\mbox{is nef$/Z$}\}.
$$
Assume that $\mu<1$. We will show that there is an extremal ray $R/Z$
such that $(K_X+B)\cdot R<0$ and $(P+\mu M)\cdot R=0$.
Replacing $P$ with $P+\mu M$ we may assume that $\mu=0$.
Then by definition of $\mu$, $P+\epsilon'M$ is not nef$/Z$ for any $\epsilon'>0$. In particular, for any
$\epsilon'>0$ there is a $K_X+B$-negative extremal ray $R/Z$
such that $(P+\epsilon'M)\cdot R<0$ but $(P+\epsilon M)\cdot R=0$ for some $\epsilon\in [0,\epsilon')$.
If there is no $K_X+B$-negative extremal ray $R/Z$ such that $P\cdot R=0$, then there is an infinite
strictly decreasing
sequence of sufficiently small positive real numbers $\epsilon_i$ and $K_X+B$-negative extremal rays $R_i/Z$
such that $\lim_{i\to \infty} \epsilon_i=0$ and
$(P+\epsilon_i M)\cdot R_i=0$.
We may assume that for each $i$, there is an extremal curve
$\Gamma_i$ generating $R_i$ such that $-3\le (K_X+B)\cdot \Gamma_i<0$ (see \ref{ss-ext-rays-II}).
Since $\Supp M\subseteq \rddown{B}$,
there is a small $\delta>0$ such that $(K_X+B-\delta M)\cdot \Gamma_i<0$
for each $i$, $B-\delta M\ge 0$,
and $\Supp (B-\delta M)=\Supp B$.
We have
$$
K_X+B-\delta M\equiv P+(1-\delta)M/Z
$$
By replacing the sequence of extremal rays with a subsequence,
we can assume that each component $S$ of $M$ satisfies: either $S\cdot R_i\ge 0$ for every $i$, or
$S\cdot R_i<0$ for every $i$. Pick a component $S$. If $S\cdot R_i\ge 0$ for each $i$,
then by \ref{ss-ext-rays-II}, we may assume that
$$
-3\le (K_X+B-\delta M)\cdot \Gamma_i<0
$$
and
$$
-3\le (K_X+B-\delta M-\tau S)\cdot \Gamma_i<0
$$
for every $i$ where $\tau>0$ is a small number.
In particular, this means that $S\cdot \Gamma_i$
is bounded from below and above. On the other hand, if $S\cdot R_i< 0$ for each $i$, then by considering
$K_X+B-\delta M+\tau S$ and arguing similarly we can show that again $S\cdot \Gamma_i$
is bounded from below and above. In particular, there are only finitely many
possibilities for the numbers $M\cdot \Gamma_i$.
Therefore,
$$
\lim_{i\to \infty} P\cdot \Gamma_i=\lim_{i\to \infty} -\epsilon_iM\cdot \Gamma_i=0
$$
Write $K_X+B=\sum_j r_j(K_X+B_j)$ where $r_j\in (0,1]$, $\sum r_j=1$, and
$(X,B_j)$ are pairs with $B_j$ being rational.
We can assume that
each component of $B-B_j$ has irrational coefficient in $B$ hence
$B-B_j$ and $M$ have no common components because $\Supp M\subseteq \rddown{B}$.
Assume $(K_X+B_j)\cdot \Gamma_i<0$ for some $i,j$. Let $S$ be a component of $M$ such that
$S\cdot \Gamma_i<0$, and let $S^\nu$ be its normalization.
Let $K_{S^\nu}+B_{j,S^\nu}=(K_X+B_j)|_{S^\nu}$ (see Section \ref{s-adjunction} for adjunction
formulas of this type). On the other hand, by \ref{ss-ext-rays-II},
there is an ample $\mathbb Q$-divisor $H$ such that $Q=K_X+B_j+H$ is nef and big and $R_i=Q^\perp$.
Now the face $(Q|_{S^\nu})^\perp$ of $\overline{NE}(S^\nu/Z)$ is generated by finitely many curves
$\Lambda_1^\nu, \dots, \Lambda_r^\nu$ such that $\alpha_j\le (K_{S^\nu}+B_{j,S^\nu})\cdot \Lambda_l^\nu<0$
where $\alpha_j$ depends on $({S^\nu},B_{j,S^\nu})$ but does not depend on $i$,
by Tanaka [\ref{Tanaka}, Theorem 4.4, Remark 4.5]. Let $\Lambda_l$ be the image of $\Lambda_l^\nu$
under the map $S^\nu\to X$.
Since $R_i=Q^\perp$ and $Q\cdot \Lambda_l=0$, each $\Lambda_l$ also generates $R_i$. But as $\Gamma_i$ is
extremal, perhaps after replacing the $\alpha_j$, we get
$$
\alpha_j\le (K_X+B_j)\cdot \Lambda_l\le (K_X+B_j)\cdot \Gamma_i<0
$$
by \ref{ss-ext-curve}.
On the other hand,
since
$$
-3\le (K_X+B)\cdot \Gamma_i=\sum_j r_j(K_X+B_j)\cdot \Gamma_i<0
$$
for each $i$, we deduce that $(K_X+B_j)\cdot \Gamma_i$ is
bounded from below and above for each $i,j$ which in turn implies that there are only finitely many
possibilities for $(K_X+B)\cdot \Gamma_i$. Recalling that there are also finitely many
possibilities for $M\cdot \Gamma_i$, we get a contradiction as
$$
0<P\cdot \Gamma_i=(K_X+B)\cdot \Gamma_i-M\cdot \Gamma_i
$$
but $\lim_{i\to \infty} P\cdot \Gamma_i=0$.
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\subsection{LMMP using a weak Zariski decomposition}\label{ss-LMMP-WZD}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ and $X\to Z$ a projective contraction such that
$K_X+B\equiv P+M/Z$ where $P$ is nef$/Z$, $M\ge 0$, and
$\Supp M\subseteq \rddown{B}$. Let $\mu$ be the largest number such that $P+\mu M$ is nef$/Z$.
Assume $\mu<1$. Then, by \ref{ss-ext-ray-WZD}, there is an extremal ray $R/Z$ such that
$(K_X+B)\cdot R<0$ and $(P+\mu M)\cdot R=0$. By replacing $P$ with $P+\mu M$ we may assume that
$P\cdot R=0$. Assume that $R$ can be contracted by a projective morphism and that
it gives a divisorial contraction or a log flip
$X\dashrightarrow X'/Z$ with $X'$ being $\mathbb Q$-factorial. Obviously,
$K_{X'}+B'\equiv P'+M'/Z$
where $P'$ is nef$/Z$, $M'\ge 0$, and $\Supp M'\subseteq \rddown{B'}$.
Continuing this process we obtain a particular kind of LMMP
which we will refer to as the \emph{LMMP using a weak Zariski decomposition} or more specifically
the \emph{LMMP$/Z$ on $K_X+B$ using $P+M$}. When we need this LMMP below we will make sure that
all the necessary ingredients exist.\\
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\section{Adjunction}\label{s-adjunction}
The varieties in this section are over $k$ of arbitrary characteristic.
We will use some of the
results of Koll\'ar [\ref{Kollar-sing}] to prove an adjunction formula.
Let $\Lambda$ be a DCC set of numbers in $[0,1]$.
Then the hyperstandard set
$$
\mathfrak{S}_\Lambda=\{\frac{m-1}{m}+\sum \frac{l_ib_i}{m}\le 1 \mid m\in \mathbb N\cup\{\infty\}, l_i\in \mathbb Z^{\ge 0}, b_i\in \Lambda\}
$$
also satisfies DCC.
Now let $(X,B)$ be a pair and $S$ a component of
$\rddown{B}$. Let $S^\nu \to S$ be the normalization.
Following a suggestion of Koll\'ar,
we will show that the pullback of $K_X+B$ to $S^{\nu}$ can be canonically written
as $K_{S^\nu}+B_{S^\nu}$ for some $B_{S^\nu}\ge 0$ which is called the \emph{different}.
Moreover, if $(X,B)$ is lc outside a codimension $3$ closed subset and if
the coefficients of $B$ belong to $\Lambda$, then we show $B_{S^\nu}$ is a boundary with coefficients in
$\mathfrak{S}_\Lambda$. When there is a
log resolution $f\colon W\to X$, it is easy to define $B_{S^\nu}$:
let $K_W+B_W=f^*(K_X+B)$ and let
$K_T+B_T=(K_W+B_W)|_T$ where $T$ is the birational transform of $S$.
Next, let $B_{S^\nu}$ be the pushdown of $B_T$ via $T\to S^\nu$.
However, since existence of log resolutions is not known in general, we follow a different path, that is,
that of [\ref{Kollar-sing}, Section 4.1].
Actually, in this paper we will need this construction
only when $\dim X\le 3$ in which case log resolutions exist.
The characteristic $0$ case of the results mentioned is due to Shokurov [\ref{Shokurov}, Corollary 3.10].
His idea is to cut by appropriate hyperplane sections and reduce the problem to the case when $X$ is a surface.
If the index of $K_X+S$ is $1$ one proves the claim by direct calculations on a
resolution. If the index is more than $1$ one then uses the index $1$ cover. Unfortunately this
does not work in positive characteristic.
\begin{prop}\label{p-adjunction-existence}
Let $(X,B)$ be a pair, $S$ be a component of $\rddown{B}$, and $S^\nu\to S$ be the normalization.
Then there is a canonically determined $\mathbb R$-divisor $B_{S^\nu}\ge 0$
such that
$$
K_{S^\nu}+B_{S^\nu}\sim_\mathbb R (K_X+B)|_{S^\nu}
$$
where $|_{S^\nu}$ means pullback to ${S^\nu}$ by the induced morphism $S^\nu\to X$.
\end{prop}
\begin{proof}
If $K_X+B$ is $\mathbb Q$-Cartier, then the statement is proved in
[\ref{Kollar-sing}, 4.2 and 4.5]. In fact, [\ref{Kollar-sing}, 4.2] defines $\Delta_{S^\nu}$
in general when $\Delta$ is a $\mathbb Q$-divisor with arbitrary rational coefficients,
$S$ is a component of $\Delta$ with coefficient $1$, and $K_X+\Delta$ is $\mathbb Q$-Cartier (but $\Delta_{S^\nu}$ is not effective in general).
Let $U$ be the
$\mathbb R$-vector space generated by the components of $B$. There is a
rational affine subspace $V$ of $U$ containing $B$ and with minimal dimension.
Since $V$ has minimal dimension, $\Delta-B$ is supported in the irrational part of
$B$ for every $\Delta\in V$. Thus the coefficient of $S$ in $\Delta$ is $1$
for every $\Delta\in V$.
Let $V_\mathbb Q$ be the underlying $\mathbb Q$-affine space of $V$. Let
$$
W_\mathbb Q=\{\Delta_{S^\nu} \mid \Delta\in V_\mathbb Q\}
$$
If $\Delta=\sum r_j\Delta^j$ where $r_j> 0$ is rational, $\sum r_j=1$, and $\Delta^j\in V_\mathbb Q$, then
the construction of [\ref{Kollar-sing}, 4.2] shows that $\Delta_{S^\nu}=\sum r_j\Delta^j_{S^\nu}$.
Therefore, $W_\mathbb Q$ is a $\mathbb Q$-affine space and the map $\alpha \colon V_\mathbb Q\to W_\mathbb Q$
sending $\Delta$ to $\Delta_{S^\nu}$ is an affine map. Letting $W$ be the $\mathbb R$-affine space generated by $W_\mathbb Q$,
we get an induced affine map $V\to W$ which sends $B$ to some element $B_{S^\nu}$.
Writing $B=\sum r_j\Delta^j$ where $r_j> 0$, $\sum r_j=1$, and $0\le \Delta^j\in V_\mathbb Q$, we see that
$B_{S^\nu}=\sum r_j\Delta^j_{S^\nu}\ge 0$. Moreover, by construction
$$
K_{S^\nu}+B_{S^\nu}=\sum r_j(K_{S^\nu}+\Delta^j_{S^\nu})\sim_\mathbb R \sum r_j(K_X+\Delta^j)|_{S^\nu} =(K_X+B)|_{S^\nu}
$$\\
\end{proof}
Note that in general $B_{S^\nu}$ is not a boundary, i.e. its coefficients may not
be in $[0,1]$.
\begin{prop}\label{p-adjunction-DCC}
Let $\Lambda\subseteq [0,1]$ be a DCC set of real numbers.
Let $(X,B)$ be a pair, $S$ be a component of $\rddown{B}$, $S^\nu\to S$ be the normalization,
and $B_{S^\nu}$ be the divisor given by Proposition \ref{p-adjunction-existence}.
Assume that
$\bullet$ $(X,B)$ is lc outside a codimension $3$ closed subset, and
$\bullet$ the coefficients of $B$ are in $\Lambda$.\\
Then $B_{S^\nu}$ is a boundary with coefficients in $\mathfrak{S}_\Lambda$.
More precisely: write $B=S+\sum_{i\ge 2} b_iB_i$,
let $V^\nu$ be a prime divisor on $S^\nu$ and let $V$ be its image on $S$;
then there exists $m\in\mathbb N\cup \{\infty\}$ depending only on $X,S$ and $V$, and there exist
nonnegative integers $l_i$ depending only on $X,S,B_i$ and $V$, such that
the coefficient of $V^\nu$ in $B_{S^\nu}$ is equal to
$$
\frac{m-1}{m}+\sum_{i\ge 2} \frac{l_ib_i}{m}
$$
\end{prop}
\begin{lem}\label{l-lc-codim2}
Let $(X,B)$ be a pair which is lc outside a codimension $3$ closed subset.
Then we can write $B=\sum r_jB^j$ where $r_j> 0$, $\sum r_j=1$, $B^j$ are $\mathbb Q$-boundaries,
and $(X,B^j)$ are lc outside a codimension $3$ closed subset.
\end{lem}
\begin{proof}
As in the proof of Proposition \ref{p-adjunction-existence}, there is a
rational affine space $V$ of divisors, containing $B$, such that
$K_X+\Delta$ is $\mathbb R$-Cartier for every $\Delta\in V$. The set of those $\Delta\in V$
with coefficients in $[0,1]$ is a rational polytope $\mathcal{P}$ containing $B$.
We want to show that there is a rational polytope $\mathcal{L}\subseteq \mathcal{P}$,
containing $B$, such that $(X,\Delta)$ is lc outside a fixed codimension $3$ closed subset,
for every $\Delta\in \mathcal{L}$.
If $(X,B)$ has a log resolution, then existence of $\mathcal{L}$ can be proved using
the same arguments as in [\ref{Shokurov}, 1.3.2].
The pair $(X,B)$ is log smooth outside some codimension $2$ closed subset $Y$.
In particular, $(X,\Delta)$ is lc outside $Y$, for every $\Delta\in \mathcal{P}$.
Shrinking $X$ we can assume $Y$ is of pure codimension $2$ and that $(X,B)$ is lc everywhere.
Assume that
for each component $R$ of $Y$, there is a rational polytope $\mathcal{L}_R\subseteq \mathcal{P}$,
containing $B$, such that $(X,\Delta)$ is lc near the generic point of $R$,
for every $\Delta\in \mathcal{L}_R$. Then we can take $\mathcal{L}$ to be any rational
polytope, containing $B$, inside the intersection of the $\mathcal{L}_R$.
Existence of $\mathcal{L}_R$ is a local problem near the generic point of $R$.
By replacing $X$ with $\Spec \mathcal{O}_{X,R}$ we are reduced to the situation in which
$X$ is a normal excellent scheme of dimension $2$ (see
[\ref{Kollar-sing}, 3.3] for notion of lc pairs in this setting).
Now $(X,B)$ has a log resolution (cf. see [\ref{Shafarevich}, page 28 and following remarks, and page 72]).
So existence of $\mathcal{L}_R$ can be proved again as in [\ref{Shokurov}, 1.3.2].\\
\end{proof}
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Note that in general $B_{S^\nu}$ is not a boundary, i.e. its coefficients may not
be in $[0,1]$.
\begin{prop}\label{p-adjunction-DCC}
Let $\Lambda\subseteq [0,1]$ be a DCC set of real numbers.
Let $(X,B)$ be a pair, $S$ be a component of $\rddown{B}$, $S^\nu\to S$ be the normalization,
and $B_{S^\nu}$ be the divisor given by Proposition \ref{p-adjunction-existence}.
Assume that
$\bullet$ $(X,B)$ is lc outside a codimension $3$ closed subset, and
$\bullet$ the coefficients of $B$ are in $\Lambda$.\\
Then $B_{S^\nu}$ is a boundary with coefficients in $\mathfrak{S}_\Lambda$.
More precisely: write $B=S+\sum_{i\ge 2} b_iB_i$,
let $V^\nu$ be a prime divisor on $S^\nu$ and let $V$ be its image on $S$;
then there exists $m\in\mathbb N\cup \{\infty\}$ depending only on $X,S$ and $V$, and there exist
nonnegative integers $l_i$ depending only on $X,S,B_i$ and $V$, such that
the coefficient of $V^\nu$ in $B_{S^\nu}$ is equal to
$$
\frac{m-1}{m}+\sum_{i\ge 2} \frac{l_ib_i}{m}
$$
\end{prop}
\begin{lem}\label{l-lc-codim2}
Let $(X,B)$ be a pair which is lc outside a codimension $3$ closed subset.
Then we can write $B=\sum r_jB^j$ where $r_j> 0$, $\sum r_j=1$, $B^j$ are $\mathbb Q$-boundaries,
and $(X,B^j)$ are lc outside a codimension $3$ closed subset.
\end{lem}
\begin{proof}
As in the proof of Proposition \ref{p-adjunction-existence}, there is a
rational affine space $V$ of divisors, containing $B$, such that
$K_X+\Delta$ is $\mathbb R$-Cartier for every $\Delta\in V$. The set of those $\Delta\in V$
with coefficients in $[0,1]$ is a rational polytope $\mathcal{P}$ containing $B$.
We want to show that there is a rational polytope $\mathcal{L}\subseteq \mathcal{P}$,
containing $B$, such that $(X,\Delta)$ is lc outside a fixed codimension $3$ closed subset,
for every $\Delta\in \mathcal{L}$.
If $(X,B)$ has a log resolution, then existence of $\mathcal{L}$ can be proved using
the same arguments as in [\ref{Shokurov}, 1.3.2].
The pair $(X,B)$ is log smooth outside some codimension $2$ closed subset $Y$.
In particular, $(X,\Delta)$ is lc outside $Y$, for every $\Delta\in \mathcal{P}$.
Shrinking $X$ we can assume $Y$ is of pure codimension $2$ and that $(X,B)$ is lc everywhere.
Assume that
for each component $R$ of $Y$, there is a rational polytope $\mathcal{L}_R\subseteq \mathcal{P}$,
containing $B$, such that $(X,\Delta)$ is lc near the generic point of $R$,
for every $\Delta\in \mathcal{L}_R$. Then we can take $\mathcal{L}$ to be any rational
polytope, containing $B$, inside the intersection of the $\mathcal{L}_R$.
Existence of $\mathcal{L}_R$ is a local problem near the generic point of $R$.
By replacing $X$ with $\Spec \mathcal{O}_{X,R}$ we are reduced to the situation in which
$X$ is a normal excellent scheme of dimension $2$ (see
[\ref{Kollar-sing}, 3.3] for notion of lc pairs in this setting).
Now $(X,B)$ has a log resolution (cf. see [\ref{Shafarevich}, page 28 and following remarks, and page 72]).
So existence of $\mathcal{L}_R$ can be proved again as in [\ref{Shokurov}, 1.3.2].\\
\end{proof}
\begin{proof}(of Proposition \ref{p-adjunction-DCC})
Assume that the proposition holds whenever $K_X+B$ is $\mathbb Q$-Cartier.
In the general case, that is, when $K_X+B$ is only $\mathbb R$-Cartier,
we can use Lemma \ref{l-lc-codim2} to write
$B=\sum r_jB^j$ where $r_j> 0$, $\sum r_j=1$, $B^j$ are $\mathbb Q$-boundaries, and $(X,B^j)$ are lc
outside a codimension $3$ closed subset. Moreover, we can assume $S$ is a component of
$\rddown{B^j}$ for each $j$ since we can choose the $B^j$ so that $B-B^j$ are supported on the
irrational part of $B$.
Then $B_{S^\nu}=\sum r_jB^j_{S^\nu}$ (see the proof of Proposition \ref{p-adjunction-existence}).
Write $B^j=S+\sum_{i\ge 2} b^j_iB_i$.
By assumption, there exists $m\in\mathbb N\cup \{\infty\}$ depending only on $X,S$ and $V$, and
there exist nonnegative integers $l_i$ depending only on $X,S,B_i$ and $V$, such that
the coefficient of $V^\nu$ in $B^j_{S^\nu}$ is equal to
$$
\frac{m-1}{m}+\sum \frac{l_ib^j_i}{m}
$$
Therefore, the coefficient of $V^\nu$ in $B_{S^\nu}$ is equal to
$$
\frac{m-1}{m}+\sum_j r_j (\sum_i \frac{l_ib^j_i}{m})=\frac{m-1}{m}+\sum_i l_i (\sum_j \frac{r_jb^j_i}{m})=\frac{m-1}{m}+\sum_i \frac{l_ib_i}{m}
$$
So from now on we can assume that $K_X+B$ is $\mathbb Q$-Cartier. Determining the coefficient of
$V^\nu$ in $B_{S^\nu}$ is a local problem near the generic point of $V$.
As in the proof of Lemma \ref{l-lc-codim2},
we can replace $X$ with $\Spec \mathcal{O}_{X,V}$ hence
assume that $X$ is a normal excellent scheme of
dimension $2$, $S$ is one-dimensional, and $V$ is a closed point. Now $(X,B)$ is lc and
the fact that $B_{S^\nu}$ is a boundary is proved in [\ref{Kollar-sing}, 4.5].
Assume that $X$ is regular at $V$. If $S$ is not regular at $V$, then $B=S$
and the coefficient of $V^\nu$ in $B_{S^\nu}$ is equal to $1$ (by [\ref{Kollar-sing}, 3.45] or
by blowing up $V$ and working on the blow up).
But if $S$ is regular at $V$, then $S^\nu \to S$ is an isomorphism, $(K_X+S)|_{S^\nu}=K_{S^\nu}$,
$m=1$, and $B_{S^\nu}=B|_{S^\nu}$. From these we can get the formula for the coefficient of $V^\nu$
as claimed.
Thus we can assume $X$ is not regular at $V$.
Since $(X,B)$ is lc, $(X,S)$ is numerically lc (see [\ref{Kollar-sing}, 3.3] for definition of numerical lc
which is the same as lc except that $K_X+S$ may not be $\mathbb Q$-Cartier). If
$(X,S)$ is not numerically plt, i.e. if there is an exceptional divisor over $V$ whose
log discrepancy with respect to $(X,S)$ is $0$, then in fact $B=S$, and
the coefficient of $V^\nu$ in $B_{S^\nu}$ is equal to $1$
by [\ref{Kollar-sing}, 3.45]. Thus we can assume $(X,S)$ is numerically plt
which in particular implies that $S$ is regular and that $S^\nu \to S$ is an isomorphism,
by [\ref{Kollar-sing}, 3.35].
Let $f\colon Y\to X$ be a log minimal resolution of $(X,S)$ as in [\ref{Kollar-sing}, 2.25]
and let $S^\sim$ be the birational transform of $S$. Then $S^\sim\to S$ is an isomorphism
and the extended dual
graph of the resolution is of the form
$$
\mathscrymatrix{
\bullet \ar@{-}[r] & c_1 \ar@{-}[r] & c_2 \ar@{-}[r] &\cdots \ar@{-}[r] & c_r
}
$$
where $\bullet$ corresponds to $S^\sim$, $c_i=-E_i^2$, and $E_1,\dots, E_r$ are the exceptional
curves of $f$.
Let $M=[-E_i\cdot E_j]$ be the minus of the intersection matrix of the resolution, and let
$m=\det M$. Then by [\ref{Kollar-sing}, 3.35.1] we have
$$
K_Y+S^\sim+\sum e_jE_j\equiv 0/X
$$
for certain $e_j>0$ and $e_1=\frac{m-1}{m}$.
Let $D\neq 0$ be an effective Weil divisor on $X$ with coefficients in $\mathbb N$. Let $d_i$ be the numbers so that
$D^\sim+\sum d_iE_i\equiv 0/X$ where $D^\sim$ is the birational transform of $D$.
The $d_i$ satisfy the equations
$$
(\sum d_jE_j)\cdot E_t =-D^\sim\cdot E_t
$$
Since $M$ has integer entries and the numbers $-D^\sim\cdot E_t$ are integers,
by Cramer's rule, we can write $d_j=\frac{n_j}{m}$ for certain
$n_j\in \mathbb N$. Applying this to $D=B_i$, we have $B_i^\sim+\sum d_{i,j}E_j\equiv 0/X$ for certain
$d_{i,j}=\frac{n_{i,j}}{m}$ with $n_{i,j}\in \mathbb N$. But then
$$
K_Y+B^\sim+\sum e_j'E_j\equiv 0/X
$$
where $B^\sim$ is the birational transform of $B$ and $e_j'=e_j+\sum_{i\ge 2} \frac{n_{i,j}b_i}{m}$.
In particular, $e_1'=\frac{m-1}{m}+\sum_{i\ge 2} \frac{l_ib_i}{m}$ where we put $l_i:=n_{i,1}$.
Now the coefficient of $V^\nu$ in $B_{S^\nu}$ is
simply the coefficient of the divisor $e_1'E_1|_{S^\sim}$ which is nothing but $e_1'$.\\
\end{proof}
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\section{Special termination}\label{s-st}
All the varieties and algebraic spaces in this section are
over $k$ of char $p>0$ unless stated otherwise.
\subsection{Reduced components of boundaries of dlt pairs}
\begin{lem}\label{l-plt-normal}
Let $(X,B)$ be an lc pair of dimension $3$ and $S$ a component of
$\rddown{B}$. Then we have:
$(1)$ if the coefficients of $B$ are standard, then the coefficients of $B_{S^\nu}$ are also standard;
$(2)$ if $k$ has char $p>5$ and $(X,B)$ is $\mathbb Q$-factorial dlt, then $S$ is normal.
\end{lem}
\begin{proof}
(1) This follows from Koll\'ar [\ref{Kollar-sing}, Corollary 3.45] (see also [\ref{Kollar-sing}, 4.4]).
(2) We may assume $B=S$ by discarding all the other components, in particular, $(X,B)$ is plt
hence $({S^\nu},B_{S^\nu})$ is klt.
By (1), $B_{S^\nu}$ has standard coefficients. By [\ref{HX}, Theorem 3.1], $({S^\nu},B_{S^\nu})$
is actually strongly $F$-regular. Therefore, $S$ is normal by [\ref{HX}, Theorem 4.1].\\
\end{proof}
\subsection{Pl-extremal rays}\label{ss-pl-ext-rays}
Let $(X,B)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$.
A $K_X+B$-negative extremal ray $R$ is called a \emph{pl-extremal ray} if
$S\cdot R<0$ for some component $S$ of $\rddown{B}$. This is named after
Shokurov's pl-flips.
Assume that $k$ has char $p>5$.
Now as in \ref{ss-ext-rays-II}, assume that $\mathcal{C}=\overline{NE}(X/Z)$ for
some projective contraction $X\to Z$ such that
$K_X+B\equiv P+M/Z$ where $P$ is nef$/Z$ and $M\ge 0$, or $\mathcal{C}=N^\perp$
for some nef and big $\mathbb Q$-divisor $N$. Let $R$ be a $K_X+B$-negative pl-extremal ray
of $\mathcal{C}$. By \ref{ss-ext-rays-II}, we can find a $\mathbb Q$-boundary $\Delta$ and
an ample $\mathbb Q$-divisor
$H$ such that $\rddown{\Delta}=S$, $(K_X+\Delta)\cdot R<0$ and $L=K_X+\Delta+H$ is nef and
big with $L^\perp=R$. Let $X\to V$ be the contraction associated to $L$ which contracts
$R$ to an algebraic space. Every curve contracted by $X\to V$ is inside $S$.
So the exceptional locus $\mathbb{E}(L)$ of $L$ is inside $S$. Thus
$L$ is semi-ample by \ref{ss-good-exc-locus}.
Therefore $X\to V$ is a projective contraction.
In other words, pl-extremal rays can be contracted by projective morphisms.
This was proved in
[\ref{HX}, Theorem 5.4] when $K_X+B$ is pseudo-effective. The
extremal rays that appear below are often pl-extremal rays.
If $X\to V$ is a divisorial contraction put $X'=V$ but if it is a flipping contraction
assume $X\dashrightarrow X'/V$ is its flip. Then it is not hard to see that in any case $X'$ is $\mathbb Q$-factorial,
by the following argument [\ref{Xu}]: we treat the divisorial case;
the flipping case can be proved similarly. We can assume that $B$ is a $\mathbb Q$-boundary and $\Delta=B$.
Let $D'$ be a prime divisor on $X'$ and
$D$ its birational transform on $X$.
There are rational numbers $\epsilon>0$ and $\delta$ such that
$M:=K_X+B+H+\epsilon D+\delta S$ is nef and big, $M\equiv 0/V$, $H+\epsilon D+\delta S$ is ample,
and $\mathbb{E}(M)=\mathbb{E}(L)=S$. Since $M|_{S}$
is semi-ample, $M$ is semi-ample by Theorem \ref{t-Keel-1}. That is, $M$
is the pullback of some ample divisor $M'$ on $X'$. But then $\epsilon D'=M'-L'$ is
$\mathbb Q$-Cartier hence $D'$ is $\mathbb Q$-Cartier.
\subsection{Special termination}\label{ss-special-termination}
The following important result is proved just like in characteristic $0$.
We include the proof for convenience.
\begin{prop}\label{p-st}
Let $(X,B)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of char $p>5$.
Assume that we are given an LMMP on $K_X+B$, say $X_i\dashrightarrow X_{i+1}/Z_i$ where $X_1=X$
and each $X_i\dashrightarrow X_{i+1}/Z_i$ is a flip, or a divisorial contraction with $X_{i+1}=Z_i$.
Then after finitely many steps, each remaining step of the LMMP is an isomorphism near
the lc centres of $(X,B)$.
\end{prop}
\begin{proof}
There are only finitely many lc centres and no new one
can be created in the process, so we may assume that the LMMP does not contract any lc centre.
In particular, we can assume that the LMMP is an isomorphism near each lc centre of
dimension zero.
Now let $C$ be an lc centre of dimension one. Since $(X,B)$ is dlt, $C$ is a component
of the intersection of two components $S,S'$ of $\rddown{B}$. Let $C_i,S_i\subset X_i$ be the birational
transforms of $C,S$. Applying Lemma \ref{l-plt-normal}, we can see that $C_i,S_i$ are normal.
By adjunction,
we can write $(K_{X_i}+B_i)|_{S_i}=K_{S_i}+B_{S_i}$ where
the coefficient of $C_i$ in $B_{S_i}$ is one. Applying adjunction once more,
we can write the pullback of $K_{S_i}+B_{S_i}$ to $C_i$ as $K_{C_i}+B_{C_i}$
for some boundary $B_{C_i}$. Since $C_i\simeq {C_{i+1}}$, we will use the notation
$(C,B_{i,C})$ instead of $({C_i},B_{C_i})$.
Since each step of the LMMP makes the divisor $K_X+B$ "smaller",
$$
K_{C}+B_{i,C}\ge K_{C}+B_{i+1,C}
$$
hence $B_{i,C}\ge B_{i+1,C}$ for every $i$. By Propositions
\ref{p-adjunction-DCC}, the
coefficients of $B_{S_i}$ and $B_{i,C}$ belong to some fixed DCC set.
Therefore $B_{i,C}=B_{i+1,C}$ for every $i\gg 0$ which implies that
after finitely many steps, each remaining step of the LMMP
is an isomorphism near $C_i$.
From now on we may assume that all the steps of the LMMP are flips.
Let $S$ be any lc centre of dimension $2$, i.e. a component of $B$ with coefficient one.
If $S_i$ intersects the exceptional locus $E_i$ of $X_i\to Z_i$, then no other component of $\rddown{B_i}$
can intersect the exceptional locus: assume that another component $T_i$ intersects the exceptional
locus; if either $S_i$ or $T_i$ contains $E_i$, then $S_i\cap T_i$ intersects $E_i$;
but $S_i\cap T_i$ is a union of lc centres of dimension one and this contradicts the
last paragraph; so none of $S_i,T_i$ contains $E_i$. But then both contain the exceptional
locus of $X_{i+1}\to Z_i$ and similar arguments give a contradiction.
Assume $D_i\subset S_i$ is a component of the exceptional locus of
$X_i\to Z_{i-1}$ where $i>1$. Then the log discrepancy of $D_i$ with
respect to $(S_1,B_{S_1})$ is less than one. Moreover, we can assume that the generic point of the centre of $D_i$
on $S_1$ is inside the klt locus of $(S_1,B_{S_1})$ by the last paragraph. But there can be at most finitely
many such $D_i$ (as prime divisors on birational models of $S_1$).
Since the coefficients of $D_i$ in $B_{S_i}$ belongs to a DCC set,
the coefficient of $D_i$ stabilizes. Therefore after finitely many steps, $S_i$
cannot contain any component of the exceptional locus of $X_i\to Z_{i-1}$.
So we get a sequence $S_i\dashrightarrow S_{i+1}$ of birational morphisms which are isomorphisms
if $i\gg 0$. In particular, $S_i$ is disjoint from $E_i$ for $i\gg 0$.\\
\end{proof}
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\section{Existence of log flips}\label{s-flips}
In this section, we first prove that generalized flips exist (\ref{t-flip-2}).
Next we prove Theorem \ref{t-flip-1}
in the projective case, that is, when $X$ is projective. The general case of
Theorem \ref{t-flip-1} is proved in Section \ref{s-mmodels} where $X$ is quasi-projective.
\subsection{Divisorial and flipping extremal rays}
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ over $k$ of char $p>0$, and
let $R$ be a $K_X+B$-negative extremal ray. Assume that there is a nef and big $\mathbb Q$-divisor
$L$ such that $R=L^\perp$. We say $R$ is a
\emph{divisorial extremal ray} if $\dim \mathbb{E}(L)=2$. But we say
$R$ is a \emph{flipping extremal ray} if $\dim \mathbb{E}(L)=1$. By \ref{ss-ext-rays-II},
such rays can be contracted to algebraic spaces. By \ref{ss-ext-rays}, when $K_X+B$ is pseudo-effective,
each $K_X+B$-negative extremal ray is either a divisorial extremal ray or a
flipping extremal ray.
We will show below (\ref{t-contraction})
that any divisorial or flipping extremal ray can actually be contracted by a projective
morphism if $(X,B)$ is dlt and $p>5$. However, we still need contractions to algebraic spaces as an auxiliary tool.
\subsection{Existence of generalized flips}
We recall the definition of \emph{generalized flips} which was introduced in [\ref{HX}].
Let $(X,B)$ be a projective $\mathbb Q$-factorial pair of dimension $3$ over $k$ char $p>0$, and
let $R$ be a $K_X+B$-negative flipping extremal ray. We say that the {generalized
flip} of $R$ exists (see [\ref{HX}, after Theorem 5.6])
if there is a birational map $X\dashrightarrow X^+/V$ which is an isomorphism in
codimension one, $X^+$ is $\mathbb Q$-factorial projective, and $K_{X^+}+B^+$ is numerically
positive on any curve contracted by $X^+\to V$.
\begin{thm}\label{t-flip-2}
Let $(X,B)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of char $p>5$.
Let $R$ be a $K_X+B$-negative flipping extremal ray. Then
the generalized flip of $R$ exists.
\end{thm}
The theorem was proved in [\ref{HX}, Theorem 5.6] when $B$ has standard coefficients and
$K_X+B$ is pseudo-effective.
\begin{proof}
This proof (as well as the proof of [\ref{HX}, Theorem 5.6]) is modeled on the proof of Shokurov's
reduction theorem [\ref{Shokurov-pl}, Theorem 1.2].
Since $R$ is a flipping extremal ray, by definition, there is a nef and big $\mathbb Q$-divisor
$L$ such that $R=L^\perp$. Moreover, $L$ is endowed with a map $X\to V$ to an algebraic space
which contracts the curves generating $R$.
Note that if $B'$ is another boundary such that $(K_X+B')\cdot R<0$, then the generalized flip
exists for $(X,B)$ if and only if it exists for $(X,B')$. This follows from the fact that
$K_X+B\equiv t(K_X+B')/V$ for some number $t>0$ where the numerical equivalence means that
$K_X+B-t(K_X+B')$ is numerically trivial on any curve contracted by $X\to V$.
Let $\mathfrak{S}$ be the set of standard coefficients as defined in the introduction.
Define
$$
\zeta(X,B)=\#\{S \mid \mbox{$S$ is a component of $B$ and its coefficient is not in~} \mathfrak{S}\}
$$
Assume that the generalized flip of $R$ does not exist. We will derive a contradiction.
We can assume that $\zeta(X,B)$
is minimal, that is, we may assume that generalized flips always exist for
pairs with smaller $\zeta$. We can decrease the coefficients of $\rddown{B}$ slightly so that
$(X,B)$ becomes klt and $\zeta(X,B)$ is unchanged. In addition, each
component $S$ of $B$ whose coefficient is not in $\mathfrak{S}$ satisfies $S\cdot R<0$
otherwise we can discard $S$ and decrease $\zeta(X,B)$ which is not possible by the
minimality assumption.
First assume that $\zeta(X,B)>0$.
Choose a component $S$ of $B$ whose coefficient $b$ is not
in $\mathfrak{S}$. There is a positive number $a$ such that $K_X+B\equiv aS/V$.
Let $g\colon W\to X$ be a log resolution, and let $B_W=B^\sim+E$
and $\Delta_W=B_W+(1-b)S^\sim$
where $E$ is the reduced exceptional divisor of $g$ and $B^\sim,S^\sim$ are
birational transforms. Note that $\rddown{B_W}=E$ and $\rddown{\Delta_W}=S^\sim+E$.
Since $(X,B)$ is klt,
$$
K_W+\Delta_W=K_W+B_W+(1-b)S^\sim=g^*(K_X+B)+G+(1-b)S^\sim
$$
where $G$ is effective and its support is equal to the support of $E$. Thus
$$
K_W+\Delta_W\equiv g^*(aS)+G+(1-b)S^\sim=(a+1-b)S^\sim+F/V
$$
where $F$ is effective and $\Supp F=\Supp E$. By construction, we have
$$
\mbox{$\Supp (S^\sim+F)=\rddown{\Delta_W}~~$ and $~~\zeta(W,\Delta_W)<\zeta(X,B)$}
$$
Run an LMMP$/V$ on $K_W+\Delta_W$ with scaling of some ample divisor, as in \ref{ss-g-LMMP-scaling}.
Recall that this is an LMMP$/\mathcal{C}$
on $K_W+\Delta_W$ where $\mathcal{C}=N^\perp$ and $N$ is the pullback of the nef and big $\mathbb Q$-divisor $L$.
In each step some component of $\rddown{\Delta_W}$
is negative on the corresponding extremal ray. So such extremal rays are pl-extremal rays,
they can be contracted by projective
morphisms, and the $\mathbb Q$-factorial property is preserved (see \ref{ss-pl-ext-rays}).
Moreover, if we encounter a flipping contraction, then
its generalized flip exists because
$\zeta(W,\Delta_W)<\zeta(X,B)$ and because we chose $\zeta(X,B)$ to be minimal;
the flip is a usual one since its extremal ray is contracted projectively.
By special termination (\ref{p-st}),
the LMMP terminates on some model $Y/V$.
Now run an LMMP$/V$ on $K_Y+B_Y$ with scaling of $(1-b)S_Y$
where $B_Y$ is the pushdown of
$B_W$ and $S_Y$ is the pushdown of $S^\sim$. Since we have the numerical equivalence
$K_Y+B_Y\equiv a S_Y+F_Y/V$ and $\Supp F_Y=\rddown{B_Y}$,
in each step of the LMMP the corresponding extremal ray intersects
some component of $\rddown{B_Y}$ negatively hence they are pl-extremal rays and
they can be contracted by projective morphisms (\ref{ss-pl-ext-rays}). Moreover,
if one of these rays gives a
flipping contraction, then its generalized flip exists because $K_Y+B_Y-bS_Y$
is negative on that ray and $\zeta(Y,B_Y-bS_Y)<\zeta(X,B)$.
Note that again such flips are usual flips.
The LMMP terminates on a model $X^+$ by special termination.
Let $h\colon W'\to X$ and $e\colon W'\to X^+$ be a common resolution.
Now the negativity lemma (\ref{ss-negativity})
applied to the divisor $h^*(K_X+B)-{e^*}(K_{X^+}+B^+)$ over $X$ implies
that
$$
h^*(K_X+B)-{e^*}(K_{X^+}+B^+)\ge 0
$$
Thus
every component $D$ of $E$ is contracted over $X^+$ because
$$
0<a(D,X,B)\le a(D,X^+,B^+)
$$
Therefore $X\dashrightarrow X^+$ is an isomorphism in codimension one. It is enough to show that
$K_{X^+}+B^+$ is numerically positive$/V$. Let $H^+$ be an ample
divisor on $X^+$ and $H$ its birational transform on $X$. There is a positive number
$c$ such that $K_X+B\equiv cH/V$ hence $K_{X^+}+B^+\equiv cH^+/V$ which implies that
$K_{X^+}+B^+$ is numerically positive$/V$.
So we have constructed the generalized flip and this contradicts our assumptions above.
Now assume that $\zeta(X,B)=0$. If $K_X+B$ is pseudo-effective, then we can simply
apply [\ref{HX}, Theorem 5.6] to get a contradiction. Unfortunately, $K_X+B$ may not be pseudo-effective (note that
even if we originally start with a pseudo-effective log divisor we may end up with a
non-pseudo-effective $K_X+B$ since we decreased some coefficients). However, this is not
a problem because the proof of [\ref{HX}, Theorem 5.6] still works. Since there is a
nef and big $\mathbb Q$-divisor $L$ with $L\cdot R=0$, there is a prime divisor $S$ with $S\cdot R<0$.
There is a number $a>0$ such that $K_X+B\equiv aS/V$. Now take a
log resolution $g\colon W\to X$ and define $B_W$ and $\Delta_W$ as above (if $S$ is not
a component of $B$ simply let $b=0$). Run an LMMP$/V$
on $K_W+\Delta_W$. The extremal rays in the process are all pl-extremal rays hence
they can be contracted by projective morphisms. Moreover, if we encounter a
flipping contraction, then its flip exists by [\ref{HX}, Theorem 4.12] because
all the coefficients of $\Delta_W$ are standard. The LMMP terminates on some model $Y$ by the
special termination. Next, run the LMMP$/V$ on $K_Y+B_Y$ with scaling of $(1-b)S_Y$.
Again, the extremal rays in the process are all pl-extremal rays hence
they can be contracted by projective morphisms. Moreover, if we encounter a
flipping contraction, then its flip exists by [\ref{HX}, Theorem 4.12] because
all the coefficients of $B_Y$ are standard.
The LMMP terminates on some model $X^+$ by the
special termination. The rest of the argument goes as before.\\
\end{proof}
\subsection{Proof of \ref{t-flip-1} in the projective case}
\begin{proof}(of Theorem \ref{t-flip-1} in the projective case)
Assume that $X$ is projective. Then by Theorem \ref{t-flip-2}, the generalized
flip of the extremal ray of $X\to Z$ exists. But since $X\to Z$ is a projective contraction,
the generalized flip is a usual flip.
If $X$ is only quasi-projective, we postpone the proof to Section \ref{s-mmodels}. Until then we need
flips only in the projective case.\\
\end{proof}
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\section{Crepant models}\label{s-crepant-models}
\subsection{Divisorial extremal rays}
The next lemma is essentially {[\ref{HX}, Theorem 5.6(2)]}.
\begin{lem}\label{l-div-ray}
Let $(X,B)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of char $p>5$.
Let $R$ be a $K_X+B$-negative divisorial extremal ray. Then $R$ can be
contracted by a projective morphism $X\to Z$ where $Z$ is $\mathbb Q$-factorial.
\end{lem}
\begin{proof}
We may assume that $(X,B)$ is klt.
Since $R$ is a divisorial extremal ray, by definition, there is a nef and big $\mathbb Q$-divisor
$L$ such that $R=L^\perp$ and $\dim \mathbb{E}(L)=2$.
Moreover, $R$ can be contracted by a map $X\to V$ to an algebraic space.
There is a prime divisor $S$ with $S\cdot R<0$. In particular, $\mathbb{E}(L)\subseteq S$ and
$S$ is the only prime divisor contracted by $X\to V$.
There is a number $a>0$ such that $K_X+B\equiv aS/V$.
Let $g\colon W\to X$ be a log resolution and define $\Delta_W$ as in the
proof of Theorem \ref{t-flip-2}. Run an LMMP$/V$ on $K_W+\Delta_W$. As in \ref{t-flip-2},
the extremal rays in the process are pl-extremal rays hence they are contracted projectively
and the LMMP terminates with a model $Z$.
We are done if we show that $Z\to V$ is an isomorphism (the $\mathbb Q$-factoriality claim follows from
\ref{ss-pl-ext-rays}). Assume this is not the case.
Recall that
$$
K_W+\Delta_W\equiv (a+1-b)S^\sim+F/V
$$
and now $(a+1-b)S^\sim+F$ is exceptional$/V$. In particular, $(a+1-b)S_Z+F_Z$ is
effective, exceptional and nef$/V$.
Let $H_Z$ be a general ample divisor on $Z$
and $H$ its birational transform on $X$. There is a number $t\ge 0$ such that
$H+tS\equiv 0/V$. Therefore there is an effective and exceptional$/V$ divisor
$P_Z$ such that $H_Z+P_Z\equiv 0/V$. Note that $\Supp P_Z$ contains all the exceptional
divisors of $Z\to V$ hence $\Supp P_Z=\Supp F_Z$. Moreover, $P_Z\neq 0$ otherwise $H_Z\equiv 0/V$
hence $Z\to V$ is an isomorphism which is not the case by assumption. This also shows that
$F_Z\neq 0$.
Let $s$ be the smallest number such that
$$
Q_Z:=(a+1-b)S_Z+F_Z-sP_Z\le 0
$$
Then $Q_Z$ is numerically positive over $V$ and there is some prime exceptional$/V$
divisor $D$ which is not a component of $Q_Z$. This is not possible since
$Q_Z$ cannot be numerically positive on the general curves of $D$ contracted$/V$.\\
\end{proof}
\subsection{Projectivization and dlt models}
\begin{lem}\label{l-reduced-Cartier}
Let $X$ be a normal projective variety over $k$ and $D\neq X$ a closed subset. Then
there is a reduced effective Cartier divisor $H$ whose support contains $D$.
\end{lem}
\begin{proof}
We may assume that each irreducible component of $D$ is a prime divisor hence we
can think of $D$ as a reduced Weil divisor. Let $A$ be a sufficiently ample divisor.
Let $U$ be the smooth locus of $X$. Since $(A-D)|_U$ is sufficiently ample,
we can choose a reduced effective divisor $H'$ with no common components with $D$
such that $H'|_U\sim (A-D)|_U$. This
extends to $X$ and gives $H'\sim A-D$. Now $H:=H'+D\sim A$ is Cartier and satisfies the
requirements.\\
\end{proof}
The next few results are standard consequences of special termination (cf.
[\ref{B-mmodel}, Lemma 3.3][\ref{HX}, Theorem 6.1]).
\begin{lem}\label{l-dlt-model-proj}
Let $(X,B)$ be an lc pair of dimension $3$ over $k$ of char $p>5$, and let $\overline{X}$ be a projectivization of
$X$. Then there is a projective $\mathbb Q$-factorial dlt pair $(\overline{Y},B_{\overline{Y}})$
with a birational morphism $\overline{Y}\to \overline{X}$
satisfying the following:
$\bullet$ $K_{\overline{Y}}+B_{\overline{Y}}$ is nef$/\overline{X}$,
$\bullet$ let $Y$ be the inverse image of $X$ and $B_Y=B_{\overline{Y}}|_Y$;
then $(Y,B_Y)$ is a $\mathbb Q$-factorial dlt model of $(X,B)$.
\end{lem}
\begin{proof}
We may assume that $\overline{X}$ is normal.
By Lemma \ref{l-reduced-Cartier}, there is a reduced effective Cartier divisor $H$ containing the complement of $X$ in
$\overline{X}$. We may assume that $H$ has no common components with $B$.
Let $f\colon \overline{W}\to \overline{X}$ be a log resolution.
Now let $B_{\overline{W}}$ be the sum of the reduced exceptional divisor of $f$ and the birational
transform of $B$, and let $\Delta_{\overline{W}}$ be the sum of $B_{\overline{W}}$ and the birational
transform of $H$.
Run the LMMP$/\overline{X}$ on $K_{\overline{W}}+\Delta_{\overline{W}}$ inductively as follows.
Assume that we have arrived at a model $\overline{Y}$. Let $R$ be a
$K_{\overline{Y}}+\Delta_{\overline{Y}}$-negative extremal ray$/\overline{X}$.
Let $\overline{Y}\to \overline{Z}$ be the contraction of $R$
to an algebraic space, and let $L$ be a nef and big $\mathbb Q$-divisor with $L^\perp=R$. Any curve contracted
by $\overline{Y}\to \overline{Z}$ is also contracted over $\overline{X}$.
If $\dim \mathbb{E}(L)=2$, then $R$ is a
divisorial extremal ray hence $\overline{Y}\to \overline{Z}$ is a projective contraction
by Lemma \ref{l-div-ray}. In this case, we continue the program with $\overline{Z}$.
Now assume that $\dim \mathbb{E}(L)=1$.
Let $C$ be a connected component of
$\mathbb{E}(L)$ and $P$ its image in $\overline{X}$ which is just a point.
If $P\in \Supp H$, then $C$ is contained in some
component of the pullback of $H$ hence it is contained in some component of
$\rddown{\Delta_{\overline{Y}}}$. In this case, $\overline{Y}\to \overline{Z}$
is again a projective contraction by \ref{ss-good-exc-locus}. Now assume that $P$ does not belong to the
support of $H$. Since $(X,B)$ is lc, over $X\setminus H$ the divisor
$$
K_{\overline{W}}+\Delta_{\overline{W}}-f^*(K_X+B)
$$
is effective and exceptional$/\overline{X}$ hence some component of $\Delta_{\overline{Y}}$ intersects
$R$ negatively which implies again that the contraction $\overline{Y}\to \overline{Z}$ is projective.
Therefore in any case $R$ can be contracted by a projective morphism and we can continue the LMMP as usual.
The required flips exist by the results of Section \ref{s-flips}.
By special termination (\ref{p-st}), the LMMP terminates say on ${{\overline{Y}}}$.
Next, we run the LMMP$/\overline{X}$ on $K_{\overline{Y}}+B_{\overline{Y}}$ with scaling of
$\Delta_{\overline{Y}}-B_{\overline{Y}}$ as in \ref{ss-g-LMMP-scaling}. Note that $\Delta_{\overline{Y}}-B_{\overline{Y}}$ is
nothing but the
birational transform of $H$. Since the pullback of $H$ is numerically trivial over $\overline{X}$,
each extremal ray in the process intersects some exceptional divisor negatively hence
such extremal rays can be contracted by projective morphisms. Moreover, the required
flips exist and by special termination the LMMP
terminates on a model which we may again denote by ${\overline{Y}}$. Now let $Y$
be the inverse image of $X$ under $g\colon \overline{Y}\to \overline{X}$ and let $B_Y$
be the restriction of $B_{\overline{Y}}$ to $Y$. Then $(Y,B_Y)$ is a $\mathbb Q$-factorial
dlt model of $(X,B)$ because $K_Y+B_Y-g^*(K_X+B)$ is effective and exceptional hence zero
as it is nef$/X$.\\
\end{proof}
\begin{proof}(of Theorem \ref{cor-dlt-model})
This is already proved in Lemma \ref{l-dlt-model-proj}.\\
\end{proof}
\subsection{Extraction of divisors and terminal models}
\begin{lem}\label{l-extraction}
Let $(X,B)$ be an lc pair of dimension $3$ over $k$ of char $p>5$ and let $\{D_i\}_{i\in I}$ be a finite set
of exceptional$/X$ prime divisors (on birational
models of $X$) such that $a(D_i,X,B)\le 1$. Then there is a $\mathbb Q$-factorial dlt pair $(Y,B_Y)$
with a projective birational morphism $Y\to X$ such that\\
$\bullet$ $K_Y+B_Y$ is the crepant pullback of $K_X+B$,\\
$\bullet$ every exceptional/$X$ prime divisor $E$ of $Y$ is one of the $D_i$ or $a(E,X,B)=0$,\\
$\bullet$ the set of exceptional/$X$ prime divisors of $Y$ includes $\{D_i\}_{i\in I}$.
\end{lem}
\begin{proof}
By Lemma \ref{l-dlt-model-proj}, we can assume that $(X,B)$ is projective
$\mathbb Q$-factorial dlt. Let $f\colon W\to X$ be a log resolution and let $\{E_j\}_{j\in J}$
be the set of prime exceptional divisors of $f$. We can assume that
for some $J'\subseteq J$, $\{E_j\}_{j\in J'}=\{D_i\}_{i\in I}$.
Now define
$$
K_W+B_W:=f^*(K_{X}+B)+\sum_{j\notin J'} a(E_j,X,B)E_j
$$
which ensures that if $j\notin J'$, then $E_j$ is a component of $\rddown{B_W}$.
Run an LMMP$/X$ on $K_W+B_W$ which would be an LMMP on $\sum_{j\notin J'} a(E_j,X,B)E_j$.
So each extremal ray in the process intersects some component of $\rddown{B_W}$ negatively
hence such rays can be contracted by projective morphisms (\ref{ss-pl-ext-rays}),
the required flips exists (Section \ref{s-flips}),
and the LMMP terminates by special termination (\ref{p-st}), say on a model $Y$.
Now $(Y,B_Y)$ satisfies all the requirements.\\
\end{proof}
\begin{proof}(of Corollary \ref{cor-terminal-model})
Apply Lemma \ref{l-extraction} by taking $\{D_i\}_{i\in I}$ to be the set of all
prime divisors with log discrepancy $a(D_i,X,B)\le 1$.\\
\end{proof}
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\section{Existence of log minimal models}\label{s-mmodels}
\subsection{Weak Zariski decompositions}\label{ss-WZD}
Let $D$ be an $\mathbb R$-Cartier divisor on a normal variety $X$ and $X\to Z$ a projective contraction
over $k$.
A \emph{weak Zariski decomposition$/Z$} for $D$
consists of a projective birational morphism $f\colon W\to X$ from a normal variety, and a numerical equivalence
$f^*D\equiv P+M/Z$ such that
\begin{enumerate}
\item $P$ and $M$ are $\mathbb R$-Cartier divisors,
\item $P$ is nef$/Z$, and $M\ge 0$.
\end{enumerate}
We then define ${\theta}(X,B,M)$ to be the number of those components of $f_*M$ which are not components of
$\rddown{B}$.
\subsection{From weak Zariski decompositions to minimal models}
We use the methods of [\ref{B-WZD}], which is somewhat similar to [\ref{BCHM}, \S 5],
to prove the following result.
\begin{prop}\label{p-WZD}
Let $(X,B)$ be a projective lc pair of dimension $3$ over $k$ of char $p>5$,
and $X\to Z$ a projective contraction.
Assume that $K_X+B$ has a weak Zariski decomposition$/Z$. Then $(X,B)$ has a
log minimal model over $Z$.
\end{prop}
\begin{proof}
Assume that $\mathfrak{W}$ is the set of pairs $(X,B)$ and projective contractions $X\to Z$
such that\\
\begin{description}
\item[L] $(X,B)$ is projective, lc of dimension $3$ over $k$,
\item[Z] $K_X+B$ has a weak Zariski decomposition$/Z$, and
\item[N] $(X,B)$ has no log minimal model over $Z$.\\
\end{description}
Clearly, it is enough to show that $\mathfrak{W}$ is empty. Assume otherwise and let $(X,B)$ and
$X\to Z$ be in $\mathfrak{W}$.
Let $f\colon W\to X$, $P$ and $M$ be the data given by
a weak Zariski decomposition$/Z$ for $K_X+B$ as in \ref{ss-WZD}. Assume in addition that
${\theta}(X,B,M)$ is minimal. Perhaps after replacing $f$ we can assume that $f$ gives a
log resolution of $(X, \Supp (B+f_*M))$. Let $B_W=B^\sim+E$ where $B^\sim$ is the birational
transform of $B$ and $E$ is the reduced exceptional divisor of $f$. Then
$$
K_W+B_W=f^*(K_X+B)+F\equiv P+M+F/Z
$$
is a weak Zariski decomposition where $F\ge 0$ is exceptional$/X$. Moreover,
$$
{\theta}(W,B_W,M+F)={\theta}(X,B,M)
$$
and any log minimal model of
$(W,B_W)$ is also a log minimal model of $(X,B)$ [\ref{B-WZD}, Remark 2.4].
So by replacing $(X,B)$ with $(W,B_W)$ and $M$ with $M+F$ we may assume
that $W=X$, $(X,\Supp (B+M))$ is log smooth, and that $K_X+B\equiv P+M/Z$.
First assume that ${\theta}(X,B,M)=0$, that is, $\Supp M\subseteq \rddown{B}$.
Run the LMMP$/Z$ on $K_X+B$ using $P+M$
as in \ref{ss-LMMP-WZD}. Obviously, $M$ negatively intersects each
extremal ray in the process, and since $\Supp M\subseteq \rddown{B}$,
the rays are pl-extremal rays. Therefore those rays can be contracted by projective morphisms
(\ref{ss-pl-ext-rays}), the required flips exist (Section \ref{s-flips}), and the LMMP terminates by special
termination (\ref{ss-special-termination}). Thus we get a log minimal model of $(X,B)$ over $Z$ which
contradicts the assumption that $(X,B)$ and $X\to Z$ belong to $\mathfrak{W}$.
For the rest of the proof we do not use LMMP.
From now on we assume that ${\theta}(X,B,M)>0$. Define
$$
\alpha:=\min\{t>0~|~~\rddown{(B+tM)^{\le 1}}\neq \rddown{B}~\}
$$
where for a divisor $D=\sum d_iD_i$ we define $D^{\le 1}=\sum d_i'D_i$ with $d_i'=\min\{d_i,1\}$.
In particular, $(B+\alpha M)^{\le 1}=B+C$ for some $C\ge 0$
supported in $\Supp M$, and $\alpha M=C+A$
where $A\ge 0$ is supported in $\rddown{B}$ and $C$ has no common components with $\rddown{B}$. Note that
${\theta}(X,B,M)$ is equal to the number of components of $C$.
The pair $(X,B+C)$ is lc and the expression
$$
K_X+B+C\equiv P+M+C/Z
$$
is a weak Zariski decomposition$/Z$. By construction
$$
{\theta}(X,B+C,M+C)<{\theta}(X,B,M)
$$
so $(X,B+C)$ has a log minimal model over $Z$ by minimality of ${\theta}(X,B,M)$
and the definition of $\mathfrak W$. Let $(Y,(B+C)_Y)$ be the minimal model.
Let $g\colon V\to X$ and $h\colon V\to Y$ be a common resolution.
By definition, $K_Y+(B+C)_Y$ is nef/$Z$.
In particular, the expression
$$
g^*(K_X+B+C)= P'+M'
$$
is a weak Zariski decomposition$/Z$ of $K_X+B+C$ where $P'=h^*(K_Y+(B+C)_Y)$
and $M'\ge 0$ is exceptional$/Y$ (cf. [\ref{B-WZD}, Remark 2.4 (2)]).
Moreover,
$$
g^*(K_X+B+C)=P'+M'\equiv g^*P+g^*(M+C)/Z
$$
Since $M'$ is exceptional$/Y$,
$$
h_*(g^*(M+C)-M')\ge 0
$$
On the other hand,
$$
g^*(M+C)-M'\equiv P'-g^*P/Z
$$
is anti-nef$/Y$ hence by the negativity lemma,
$g^*(M+C)-M'\ge 0$.
Therefore $\Supp M'\subseteq \Supp g^*(M+C)=\Supp g^*M$.
Now,
\begin{equation*}
\begin{split}
(1+\alpha)g^*(K_X+B) & \equiv g^*(K_X+B)+\alpha g^*P+ \alpha g^*M\\
& \equiv g^*(K_X+B)+\alpha g^*P+g^* C+g^*A\\
& \equiv P'+\alpha g^*P+M'+g^*A/Z
\end{split}
\end{equation*}
hence we get a weak Zariski decomposition$/Z$ as
$$
g^*(K_X+B)\equiv P''+M''/Z
$$
where
$$
P''=\frac{1}{1+\alpha}(P'+ \alpha g^*P) \mbox{\hspace{0.5cm} and \hspace{0.5cm}} M''=\frac{1}{1+\alpha}(M'+g^*A)
$$
and $\Supp M''\subseteq \Supp g^* M$ hence $\Supp g_*M''\subseteq \Supp M$. Since ${\theta}(X,B,M)$ is minimal,
$$
{\theta}(X,B,M)={\theta}(X,B,M'')
$$
So every component of $C$ is also a component
of $g_*M''$ which in turn implies that every component of $C$ is also a
component of $g_*M'$. But $M'$ is exceptional$/Y$
hence so is $C$ which means that $(B+C)_Y=B^\sim+C^\sim+E=B^\sim+E=B_Y$ where $\sim$ stands for birational transform
and $E$ is the reduced exceptional divisor of $Y\dashrightarrow X$. Thus we have $P'=h^*(K_Y+B_Y)$. Although $K_Y+B_Y$
is nef$/Z$, $(Y,B_Y)$ is not necessarily a
log minimal model of $(X,B)$ over $Z$ because condition (4) of definition
of log minimal models may not be satisfied (see \ref{ss-mmodels}).
Let $G$ be the largest $\mathbb R$-divisor such that $G\le g^*C$ and $G\le {M}'$. By letting $\tilde{C}=g^*C-G$ and
$\tilde{M}'= M'-G$ we get the expression
$$
g^*(K_X+B)+\tilde{C}= P'+\tilde{M}'
$$
where $\tilde{C}$ and $\tilde{M}'$ are effective with no common components.
Assume that $\tilde{C}$ is exceptional$/X$. Then $g^*(K_X+B)-P'=\tilde{M}'-\tilde{C}$ is antinef$/X$
so by the negativity lemma $\tilde{M}'-\tilde{C}\ge 0$ which implies that $\tilde{C}=0$ since
$\tilde{C}$ and $\tilde{M}'$
have no common components. Thus
$$
g^*(K_X+B)-h^*(K_Y+B_Y)=\sum_D a(D,Y,B_Y)D-a(D,X,B)D=\tilde{M}'
$$
where $D$ runs over the prime divisors on $V$.
If $\Supp g_*\tilde{M}'=\Supp g_*M'$, then $\Supp \tilde{M}'$ contains the birational transform of
all the prime exceptional$/Y$ divisors on $X$
hence $(Y,B_Y)$ is a log minimal model of $(X,B)$ over $Z$, a contradiction. Thus
$$
\Supp (g_*M'-g_*G)=\Supp g_*\tilde{M}'\subsetneq \Supp g_* M'\subseteq \Supp M
$$
so some component of $C$ is not a component of $g_*\tilde{M}'$ because $\Supp g_*G\subseteq \Supp C$. Therefore
$$
{\theta}(X,B,M)>{\theta}(X,B,\tilde{M}')
$$
which gives a contradiction again by minimality of ${\theta}(X,B,M)$ and the assumption
that $(X,B)$ has no log minimal model over $Z$.
So we may assume that $\tilde{C}$ is not exceptional$/X$. Let
$\beta>0$ be the smallest number such that $\tilde{A}:=\beta g^* M-\tilde{C}$ satisfies $g_*\tilde{A}\ge 0$. Then
there is a component of $g_*\tilde{C}$ which is not a component of $g_*\tilde{A}$.
Now
\begin{equation*}
\begin{split}
(1+\beta)g^*(K_X+B) & \equiv g^*(K_X+B)+\beta g^*M+\beta g^*P\\
& \equiv g^*(K_X+B)+\tilde{C}+\tilde{A}+\beta g^*P\\
& \equiv P'+\beta g^*P+ \tilde{M}'+\tilde{A}/Z
\end{split}
\end{equation*}
where $\tilde{M}'+\tilde{A}\ge 0$ by the negativity lemma.
Thus we get a weak Zariski decomposition$/Z$ as $g^*(K_X+B)\equiv P'''+M'''/Z$
where
$$
P'''=\frac{1}{1+\beta}(P'+\beta g^*P) \mbox{\hspace{0.5cm} and \hspace{0.5cm}} M'''=\frac{1}{1+\beta}(\tilde{M}'+\tilde{A})
$$
and $\Supp g_*M'''\subseteq \Supp M$.
Moreover, by construction, there is a component $D$ of $g_*\tilde{C}$ which is not a component of $g_*\tilde{A}$.
Since $g_*\tilde{C}\le C$, $D$ is a component of $C$ hence of $M$, and since $\tilde{C}$ and $\tilde{M}'$
have no common components, $D$ is not a component of $g_*\tilde{M}'$. Therefore $D$ is not a component of
$g_*M'''=\frac{1}{1+\beta}(g_*\tilde{M}'+g_*\tilde{A})$ which implies that
$$
{\theta}(X,B,M)>{\theta}(X,B,M''')
$$
giving a contradiction again.\\
\end{proof}
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\subsection{Proofs of \ref{t-mmodel} and \ref{t-flip-1}}
\begin{proof}(of Theorem \ref{t-mmodel})
By applying Lemma \ref{l-dlt-model-proj}, we can reduce the problem to the
case when $X,Z$ are projective. We can find a log resolution $f\colon W\to X$
and a $\mathbb Q$-boundary $B_W$ such that
$$
K_W+B_W=f^*(K_X+B)+E
$$
where $E\ge 0$ and its support is equal to the union of the
exceptional divisors of $f$, and $(W,B_W)$ has terminal singularities. It is enough to
construct a log minimal model for $(W,B_W)$ over $Z$. So by replacing $(X,B)$
with $(W,B_W)$ we can assume $(X,B)$ has terminal singularities and that $X$ is $\mathbb Q$-factorial.
Let
$$
\mathcal{E}=\{B' \mid K_X+B' ~~\mbox{is pseudo-effective$/Z$ and $0\le B'\le B$}\}
$$
which is a compact subset of the $\mathbb R$-vector space $V$ generated by the components of $B$.
Let $B'$ be an element in $\mathcal{E}$ which has minimal
distance from $0$ with respect to the standard metric on $V$. So either $B'=0$, or
$K_X+B''$ is not pseudo-effective$/Z$ for any $0\le B''\lneq B'$.
Run the generalized LMMP$/Z$ on $K_X+B'$ as follows [\ref{HX}, proof of Theorem 5.6]:
let $R$ be a $K_{X}+B'$-negative extremal ray$/Z$. By \ref{ss-ext-rays-II}, $R$ is either
a divisorial extremal ray or a flipping extremal ray (see the beginning of Section \ref{s-flips}
for definitions), and $R$ can be contracted to an algebraic space.
If $R$ is a divisorial extremal ray, then it can actually be contracted by a projective
morphism, by Lemma \ref{l-div-ray}, and we continue the process. But if $R$ is a
flipping extremal ray, then we use the generalized flip, which exists by Theorem \ref{t-flip-2},
and then continue the process.
No component of $B'$ is contracted by the LMMP: otherwise let $X_i\dashrightarrow X_{i+1}$
be the sequence of log flips and divisorial contractions of this LMMP where $X=X_1$.
Pick $j$ so that $\phi_j\colon X_j\dashrightarrow X_{j+1}$ is a divisorial contraction
which contracts a component $D_j$ of $B_j'$, the birational transform of $B'$. Now there is $a>0$
such that
$$
K_{X_j}+B_j'=\phi_j^*(K_{X_{j+1}}+B_{j+1}')+aD_j
$$
Since $K_{X_{j+1}}+B_{j+1}'$ is pseudo-effective$/Z$,
$K_{X_{j}}+B_{j}'-aD_j$ is pseudo-effective$/Z$ which implies that $K_X+B'-bD$ is pseudo-effective$/Z$
for some $b>0$ where
$D$ is the birational transform of $D_j$, a contradiction.
Therefore every $(X_j,B'_j)$ has terminal singularities.
The LMMP terminates for reasons similar to the characteristic $0$ case
[\ref{Shokurov-nv}, Corollary 2.17][\ref{Kollar-Mori}, Theorem 6.17] (see also [\ref{HX}, proof of Theorem 1.2]).
So we get a log minimal model of $(X,B')$ over $Z$, say $(Y,B'_{Y})$.
Let $g\colon V\to X$ and $h\colon V\to Y$ be a common resolution. By letting
$P=h^*(K_{Y}+B'_{Y})$ and
$$
M=g^*(K_X+B)-h^*(K_{Y}+B'_{Y})
$$
we get a weak Zariski decomposition$/Z$ as
$
g^*(K_X+B)=P+M/Z.
$
Note that $M\ge 0$ because $g^*(K_X+B')-h^*(K_{Y}+B'_{Y})\ge 0$. Therefore $(X,B)$ has a log minimal model
over $Z$ by Proposition \ref{p-WZD}.\\
\end{proof}
\begin{proof}(of Theorem \ref{t-flip-1} in general case)
Recall that we proved the theorem when $X$ is projective, in Section \ref{s-flips}.
By perturbing the coefficients, we can assume that $(X,B)$ is klt.
By Theorem \ref{t-mmodel}, $(X,B)$ has a log minimal model over $Z$, say $(X^+,B^+)$.
Since $(X,B)$ is klt, $X\dashrightarrow X^+$ is an isomorphism in codimension one.
Let $H^+$ be an ample$/Z$ divisor on $X^+$ and let $H$ be its birational transform
on $X$. Since $X\to Z$ is a $K_X+B$-negative extremal contraction, $K_X+B\equiv hH/Z$
for some $h>0$. Thus $K_{X^+}+B^+\equiv hH^+/Z$ which means that
$K_{X^+}+B^+$ is ample$/Z$ so we are done.\\
\end{proof}
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\section{The connectedness principle with applications to semi-ampleness}\label{s-connectedness}
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\subsection{Connectedness}
In this subsection, we prove the connectedness principle in dimension $\le 3$.
The proof is based on LMMP rather than vanishing theorems.
The following lemma is essentially [\ref{Xu}, Proposition 2.3]. We recall its proof for convenience.
\begin{lem}\label{l-ample-dlt}
Let $(X,B)$ be a projective pair of dimension $\le 3$ over $k$.
Assume that $(X,B)$ is klt (resp. dlt) and that $A$ is a nef and big (resp. ample) $\mathbb R$-divisor.
Then there is $0\le A'\sim_\mathbb R A$ such that $(X,B+A')$ is klt (resp. dlt).
\end{lem}
\begin{proof}
First we deal with the dlt case.
Let $f\colon W\to X$ be a log resolution of $(X,B)$ which extracts only prime divisors with positive
log discrepancy with respect to $(X,B)$.
This exists by the definition of dlt pairs. The resolution is
obtained by a sequence of blow ups with smooth centers, hence there is an $\mathbb R$-divisor
$E'$ exceptional$/X$ with sufficiently small coefficients such that $-E'$ is ample$/X$ and
$\Supp E'$ is the union of all the prime exceptional$/X$ divisors on $W$. Note that by the
negativity lemma (\ref{ss-negativity}), $E'\ge 0$. Moreover, $f^*A-E'$ is ample$/X$.
Let $B_W$ be given by
$$
K_W+B_W= f^*(K_X+B)
$$
By assumption, $B_W$ has coefficients at most $1$ and the coefficient of any
prime exceptional$/X$ divisor is less than $1$.
Let $A_W'\sim_\mathbb R f^*A-E'$ be general and let $A':=f_*A_W'$. Then $A'\sim_\mathbb R A$ and we can write
$$
K_W+B_W+A_W'+E'= f^*(K_X+B+A')
$$
where we can make sure that the coefficients of $B_W+A_W'+E'$ are at most $1$ and that the coefficient of
any prime exceptional$/X$ divisor is less than $1$ because the
coefficients of $E'$ are sufficiently small. This implies that $(X,B+A')$ is dlt.
Now we deal with the klt case. Since $A$ is nef and big, by definition,
$A\sim_\mathbb R G+D$ with $G\ge 0$ ample and $D\ge 0$. So by
replacing $A$ with $(1-\epsilon)A+\epsilon G$ and replacing $B$ with $B+\epsilon D$
we can assume that $A$ is ample. Now apply the dlt case.\\
\end{proof}
\begin{proof}(of Theorem \ref{t-connectedness-d-3})
Assume that the statement does not hold for some $z$.
By Lemma \ref{l-dlt-model-proj}, there is a $\mathbb Q$-factorial dlt pair $(Y,B_Y)$ and a
birational morphism $g\colon Y\to X$ with $K_Y+B_Y$ nef$/X$, every exceptional divisor of
$g$ is a component of $\rddown{B_Y}$, and $g_*B_Y=B$.
Moreover, $K_Y+B_Y+E_Y=f^*(K_X+B)$ for some $E_Y\ge 0$ with $\Supp E_Y\subseteq \rddown{B_Y}$.
Also the non-klt locus of $(Y,B_Y)$, that is $\rddown{B_Y}$, maps surjectively onto the non-klt locus of $(X,B)$
hence $\rddown{B_Y}$ is not connected in some neighborhood of $Y_z$.
Now by assumptions, $K_Y+B_Y+E_Y+L_Y\sim_\mathbb R 0/Z$ for some globally nef and big $\mathbb R$-divisor $L_Y$.
Since $X$ is $\mathbb Q$-factorial, we can write $L_Y\sim_\mathbb R A_Y+D_Y$ where $A_Y$ is ample and $D_Y\ge 0$
is exceptional$/X$. In particular, $\Supp D_Y\subset \rddown{B_Y}$. By picking
a general
$$
G_Y\sim_\mathbb R \epsilon A_Y+(1-\epsilon)L_Y-\delta \rddown{B_Y}
$$
for some small $\delta>0$
and applying Lemma \ref{l-ample-dlt} we can assume that $(Y,B_Y+G_Y)$
is dlt. By construction,
$$
K_Y+B_Y+G_Y\sim_\mathbb R P_Y:=-\epsilon D_Y-E_Y-\delta\rddown{B_Y}/Z
$$
and $\Supp P_Y=\rddown{B_Y}$.
Run a generalized LMMP$/Z$ on $K_Y+B_Y+G_Y$ as in the proof of Theorem \ref{t-mmodel}.
We show that this is actually a usual LMMP hence it terminates by special termination (\ref{p-st}).
Assume that we have arrived at a model $Y'$
and let $R$ be a $K_{Y'}+B_{Y'}+G_{Y'}$-negative extremal ray$/Z$. Since $Y'\to Z$ is birational,
$R$ is either a divisorial extremal ray or a flipping extremal ray.
In the former case $R$ can be contracted by a projective morphism by Lemma \ref{l-div-ray}.
So assume $R$ is a flipping extremal ray. Then the generalized flip $Y'\dashrightarrow Y''/V$ exists by
Theorem \ref{t-flip-2} where $Y'\to V$ is the contraction of $R$ to the algebraic space $V$.
Since $P_{Y'}\cdot R<0$, some component $S_{Y'}$ of $\rddown{B_{Y'}}$ intersects $R$ positively.
Now there is a boundary $\Delta_{Y'}$ such that $(Y',\Delta_{Y'})$ is plt, $S_{Y'}=\rddown{\Delta_{Y'}}$,
and $(K_{Y'}+\Delta_{Y'})\cdot R=0$. But then we can find $N_{Y''}\ge 0$ such that
$(Y'',\Delta_{Y''}+N_{Y''})$ is plt and $(K_{Y''}+\Delta_{Y''}+N_{Y''})\cdot R<0$.
Therefore by \ref{ss-pl-ext-rays} and \ref{ss-good-exc-locus}, $Y''\to V$ is a projective morphism which implies that
$Y'\to V$ is also a projective morphism and that the flip is a usual flip.
We claim that the connected components of $\rddown{B_Y}$ over $z$ remain disjoint over $z$
in the course of the LMMP: assume not and let $Y'$
be the first model in the process such that there are irreducible components
$S_Y,T_Y$ of $\rddown{B_Y}$ belonging to disjoint connected components over $z$
such that $S_{Y'},T_{Y'}$ intersect over $z$.
Let $\Delta_Y=B_Y-\tau (\rddown{B_Y}-S_Y-T_Y)$ for some small $\tau>0$. Then $(Y,\Delta_Y+G_Y)$
is plt in some neighborhood of $Y_z$ because $\rddown{\Delta_Y+G_Y}=S_Y+T_Y$ and $S_Y,T_Y$
are disjoint over $z$. Moreover, $Y\dashrightarrow Y'$ is a partial LMMP
on $K_Y+\Delta_Y+G_Y$ hence $(Y',\Delta_{Y'}+G_{Y'})$ is also plt over $z$. But since $S_{Y'},T_{Y'}$
intersect over $z$, $(Y',\Delta_{Y'}+G_{Y'})$ cannot be plt over $z$,
a contradiction.
Next we claim that no connected component of $\rddown{B_Y}$ over $z$
can be contracted by the LMMP (although some of their
irreducible components might be contracted). By construction $-P_Y\ge 0$ and $\Supp -P_Y=\rddown{B_Y}$, and
$-P_Y$ is positive on each extremal ray in the LMMP. Write $-P_Y=\sum -P_Y^i$ where $-P_Y^i$
are the connected components of $-P_Y$ over $z$. By the previous paragraph, $-P_Y^i$ and $-P_Y^j$
remain disjoint during the LMMP if $i\neq j$. Moreover, if we arrive a model $Y'$ in the LMMP
on which we contract an extremal ray $R$, then $-P_{Y'}^j\cdot R>0$ for some $j$ and
$-P_{Y'}^i\cdot R=0$ for $i\neq j$. Therefore the contraction
of $R$ cannot contract any of the $-P_{Y'}^i$.
The LMMP ends up with a log minimal model $(Y',B_{Y'}+G_{Y'})$ over $Z$. Then
$P_{Y'}$ is nef$/Z$. Assume that $Y_z'\nsubseteq \Supp P_{Y'}$ set-theoretically. Since $Y_z'$ intersects
$\Supp P_{Y'}$, there is some curve $C\subset Y_z'$ not contained in $\Supp P_{Y'}$
but intersects it. Then as $-P_{Y'}\ge 0$ we have $-P_{Y'}\cdot C>0$ hence $P_{Y'}\cdot C<0$,
a contradiction.
Now since $Y_z'$ is connected, it is contained in exactly one connected
component of $\rddown{B_{Y'}}$ over $z$. This is a contradiction because by assumptions
at least two connected components of $\rddown{B_{Y'}}$ over $z$ intersect the fibre $Y_z'$.\\
\end{proof}
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We now show that a strong form of the connectedness principle holds on surfaces.
\begin{thm}\label{t-connectedness-d-2}
Let $(X,B)$ be a $\mathbb Q$-factorial projective pair of dimension $2$ over $k$.
Let $f\colon X\to Z$ be a projective contraction (not necessarily birational) such that
$-(K_X+B)$ is ample$/Z$. Then for any closed point $z\in Z$, the non-klt locus $N$ of
$(X,B)$ is connected in any neighborhood of the fibre $X_z$ over $z$. More strongly, $N\cap X_z$
is connected.
\end{thm}
\begin{proof}
It is enough to prove the last claim.
Assume that $N\cap X_z$ is not connected for some $z$.
We use the notation and the arguments of the proof of Theorem \ref{t-connectedness-d-3}.
Let $(Y,B_Y)$ be the pair constructed over $X$ and $Y\dashrightarrow Y'$ the LMMP$/Z$
on $K_Y+B_Y+G_Y\sim_\mathbb R P_Y$ and $h\colon Y'\to Z$ the corresponding map. The same arguments
of the proof of Theorem \ref{t-connectedness-d-3} show that the connected components of
$P_Y$ over $z$ remain disjoint in the course of the LMMP and none of them will be contracted.
By assumptions,
$\rddown{B_Y}\cap Y_z$ is not connected.
We claim that the same holds in the course of the LMMP.
If not, then at some step of the LMMP we arrive at a model $W$ with a $K_W+B_W+G_W$-negative
extremal birational contraction $\phi\colon W\to V$ such that $\rddown{B_W}\cap W_z$ is not connected but
$\rddown{B_V}\cap V_z$ is connected. Let $C$ be the exceptional curve of $W\to V$.
Now $\phi(\rddown{B_W})=\rddown{B_V}$: the inclusion $\supseteq$ is clear;
the inclusion $\subseteq$ follows from the fact that if $C$ is a component of $\rddown{B_W}$,
then at least one other irreducible component of $\rddown{B_W}$ intersects $C$ because $P_W\cdot C<0$.
Therefore $\phi(\rddown{B_W}\cap W_z)=\rddown{B_V}\cap V_z$. Since $\rddown{B_V}\cap V_z$ is connected
but $\rddown{B_W}\cap W_z$ is not connected, there exist two connected components of
$\rddown{B_W}\cap W_z$ whose images under $\phi$ intersect. So there are closed points
$w,w'$ belonging to different connected components of $\rddown{B_W}\cap W_z$ such that $\phi(w)=\phi(w')$.
In particular, $w,w'\in C$. Note that $C$ is not a component of $\rddown{B_W}$ otherwise
$C\subset \rddown{B_W}\cap W_z$ connects $w,w'$ which contradicts the assumptions.
Therefore $\rddown{B_W}\cap C$ is a finite set of closed points with more than one element.
Now perturbing the coefficients of
$B_W$ we can find a $\Gamma_W\le \rddown{B_W}$ such that
$(W,\Gamma_W)$ is plt in a neighborhood of $C$, $(K_W+\Gamma_W)\cdot C<0$ and such
that $\rddown{\Gamma_W}\cap C$ is a finite set of closed points with more than one element.
Then in a formal neighborhood of $\phi(w)$, $\rddown{\Gamma_V}$ has at least two branches which implies that
$\rddown{\Gamma_V}$ is not normal which in turn contradicts the plt property of
$(V,\Gamma_V)$.
Since $\rddown{B_{Y'}}\cap Y_z'$ is not connected, there is a component $D$ of $Y_z'$
not contained in $\Supp P_{Y'}=\rddown{B_{Y'}}$ but intersects it. Thus $P_{Y'}$ cannot be nef$/Z$
as $-P_{Y'}\ge 0$. Therefore
the LMMP terminates with a Mori fibre space $Y'\to Z'/Z$.
If $Z'$ is a point, then $\rddown{B_{Y'}}$ has at least two disjoint
irreducible components which contradicts the fact that the Picard number $\rho(Y')=1$ in this case.
So we can assume that $Z'$ is a curve.
Assume that $Z$ is also a curve in which case $Z'=Z$.
Let $F$ be the reduced variety associated to a general fibre of $Y'\to Z'$.
Then by the adjunction formula we get $F\simeq \mathbb P^1$, $K_{Y'}\cdot F=-2$, and
$(B_{Y'}+G_{Y'})\cdot F<2$. On the other hand,
since $\rddown{B_{Y'}}\cap Y'_z$ has at least two points, $\rddown{B_{Y'}}\cap F$ also
has at least two points hence
$$
(B_{Y'}+G_{Y'})\cdot F\ge (\rddown{B_{Y'}}+G_{Y'})\cdot F>2
$$
which is a contradiction.
Now assume that $Z$ is a point. Since $\rddown{B_{Y'}}\cap Y_z'$ is not connected,
$\rddown{B_{Y'}}$ has at least two disjoint
connected components, say $M_{Y'},N_{Y'}$. On the other hand, since $P_{Y'}\cdot F<0$,
we may assume that $M_{Y'}$
intersects $F$ (hence $M_{Y'}$ intersects every fibre of $Y'\to Z'$).
If some component of $N_{Y'}$ is vertical$/Z'$,
then $M_{Y'},N_{Y'}$ intersect a contradiction. Thus each component of $N_{Y'}$ is horizontal$/Z'$
hence they intersect each
fibre of $Y'\to Z'$. But then we can get a contradiction as in the $Z'=Z$ case.\\
\end{proof}
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\subsection{Semi-ampleness}
We use the connectedness principle on surfaces to prove some semi-ampleness results in dimension $2$ and $3$.
These are not only interesting on their own but also useful for the proof of the finite generation (\ref{t-fg}).
\begin{proof}(of Theorem \ref{t-sa-reduced-boundary})
Let $S\le \rddown{B}$ be a reduced divisor. Assume that
$(K_X+B+A)|_S$ is not semi-ample. We will derive a contradiction.
We can assume that if $S'\lneq S$ is any other reduced divisor, then
$(K_X+B+A)|_{S'}$ is semi-ample. Note that $S$ cannot be irreducible by abundance for surfaces (cf.
[\ref{Tanaka}]). Using the ample divisor $A$ and applying Lemma \ref{l-ample-dlt}, we can perturb the coefficients
of $B$ so that we can assume $S=\rddown{B}$.
Let $T$ be an irreducible component of $S$ and let $S'=S-T$.
By assumptions, $(K_X+B+A)|_{T}$ and $(K_X+B+A)|_{S'}$ are both semi-ample.
Let $g\colon T\to Z$ be the projective contraction associated to $(K_X+B+A)|_{T}$.
By adjunction define $K_T+B_T:=(K_X+B)|_T$ and $A_T=A|_T$. Since $K_T+B_T+A_T\sim_\mathbb Q 0/Z$ and
since $A_T$ is ample, $-(K_T+B_T)$ is ample$/Z$. Moreover, $S'\cap T=\rddown{B_T}$ as topological spaces.
By the connectedness principle for surfaces (\ref{t-connectedness-d-2}), $\rddown{B_T}\to Z$ has
connected fibres hence $S'\cap T\to Z$ also has connected fibres.
Now apply Keel [\ref{Keel}, Corollary 2.9].\\
\end{proof}
\begin{thm}\label{t-sa-reduced-boundary-2}
Let $(X,B+A)$ be a projective $\mathbb Q$-factorial dlt pair of dimension $3$ over $k$ of
char $p>5$. Assume that
$\bullet$ $A,B\ge 0$ are $\mathbb Q$-divisors with $A$ ample,
$\bullet$ $(Y,B_Y+A_Y)$ is a $\mathbb Q$-factorial weak lc model of $(X,B+A)$,
$\bullet$ $Y\dashrightarrow X$ does not contract any divisor,
$\bullet$ $\Supp A_{Y}$ does not contain any lc centre of $(Y,B_{Y}+A_{Y})$,
$\bullet$ if $\Sigma$ is a connected component of $\mathbb{E}(K_{Y}+B_{Y}+A_{Y})$ and
$\Sigma \nsubseteq \rddown{B_{Y}}$, then
$(K_{Y}+B_{Y}+A_{Y})|_{\Sigma}$ is semi-ample.\\
Then $K_{Y}+B_{Y}+A_{Y}$ is semi-ample.
\end{thm}
\begin{proof}
Note that if $K_X+B+A$ is not big, then $\mathbb{E}(K_{Y}+B_{Y}+A_{Y})=Y$
hence the statement is trivial. So we can assume that $K_X+B+A$ is big.
Let $\phi$ denote the map $X\dashrightarrow Y$ and let $U$ be the largest open set
over which $\phi$ is an isomorphism. Then since $A$ is ample and $X$ is $\mathbb Q$-factorial, $\Supp A_Y$
contains $Y\setminus \phi(U)$: indeed let $y\in Y\setminus \phi(U)$ be a closed point and let $W$ be the normalization
of the graph of $\phi$, and $\alpha\colon W\to X$ and $\beta\colon W\to Y$ be the
corresponding morphisms; first assume that $\dim \beta^{-1}\{y\}>0$; then
$\alpha^*A$ intersects $\beta^{-1}\{y\}$ because $A$ is ample hence $\Supp A_Y$ contains
$y$; now assume that $\dim \beta^{-1}\{y\}=0$; then $\beta$ is an isomorphism over
$y$; on the other hand, $\alpha$ cannot be an isomorphism near $\beta^{-1}\{y\}$ otherwise
$\phi$ would be an isomorphism near $\alpha(\beta^{-1}\{y\})$ hence $y\in \phi(U)$,
a contradiction; thus as $X$ is $\mathbb Q$-factorial, $\alpha$ contracts some prime divisor
$E$ containing $\beta^{-1}\{y\}$; but then $Y\dashrightarrow X$ contracts a divisor, a contradiction.
Let $C\ge 0$ be any $\mathbb Q$-divisor
such that $(X,B+A+C)$ is dlt. Then $(Y,B_{Y}+A_{Y}+\epsilon C_Y)$ is dlt
for any sufficiently small $\epsilon>0$ because $(Y,B_{Y}+A_{Y})$ has no lc
centre inside $Y\setminus \phi(U)\subset \Supp A_Y$.
Now let $G_{Y}\ge 0$ be a general small ample $\mathbb Q$-divisor
on $Y$ and $G$ its birational transform on $X$. Since $G$ is small, $A-G$ is ample.
Let $C\sim_\mathbb Q A-G$ be a general $\mathbb Q$-divisor.
Let
$$
\Gamma_{Y}:=B_{Y}+(1-\epsilon)A_{Y}+\epsilon C_{Y}+\epsilon G_{Y}
$$
Then
$$
K_{Y}+\Gamma_{Y}\sim_\mathbb Q K_{Y}+B_{Y}+A_{Y}
$$
and $\rddown{B_{Y}}=\rddown{\Gamma_{Y}}$.
Moreover, by the above remarks and by Lemma \ref{l-ample-dlt} we can assume that
$(Y,\Gamma_{Y})$ is dlt.
Now by Theorem \ref{t-sa-reduced-boundary}, $(K_{Y}+\Gamma_{Y})|_{\rddown{\Gamma_Y}}$ is semi-ample
hence $(K_{Y}+B_{Y}+A_{Y})|_{\rddown{B_Y}}$ is semi-ample. Therefore $(K_{Y}+B_{Y}+A_{Y})|_{\Sigma}$
is semi-ample for any connected component of $\mathbb{E}(K_{Y}+B_{Y}+A_{Y})$ hence we can
apply Theorem \ref{t-Keel-1}.\\
\end{proof}
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\section{Finite generation and base point freeness}\label{s-fg}
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\subsection{Finite generation}
In this subsection we prove Theorem \ref{t-fg}.
\begin{lem}\label{l-fg-decrease}
Let $(X,B)$ be a pair
and $M$ a $\mathbb Q$-divisor satisfying the following properties:
$(1)$ $(X,\Supp(B+M))$ is projective log smooth of dimension $3$ over $k$ of char $p>5$,
$(2)$ $K_X+B$ is a big $\mathbb Q$-divisor,
$(3)$ $K_X+B\sim_\mathbb Q M\ge 0$ and $\rddown{B}\subset \Supp M\subseteq \Supp B$,
$(4)$ $M=A+D$ where $A$ is an ample $\mathbb Q$-divisor and $D\ge 0$,
$(5)$ $\alpha M=N+C$ for some rational number $\alpha>0$ such that $N,C\ge 0$ are $\mathbb Q$-divisors,
$\Supp N=\rddown{B}$, and $(X,B+C)$ is dlt,
$(6)$ there is an ample $\mathbb Q$-divisor $ A'\ge 0$ such that $A'\le A$ and $A'\le C$.\\
If $(X,B+tC)$ has an lc model for some real number $t\in (0,1]$, then
$(X,B+(t-\epsilon)C)$ also has an lc model for any
sufficiently small $\epsilon>0$.
\end{lem}
\begin{proof}
We can assume that $C\neq 0$.
If we let $\Delta=B-\delta(N+C)$ for some small rational number $\delta>0$,
then $(X,\Delta)$ is klt and $K_X+B$ is a positive multiple of $K_X+\Delta$ up to
$\mathbb Q$-linear equivalence.
Similarly, for any $s\in (0,1]$, there is $s'\in (0,s)$ such that $(X,\Delta+s'C)$ is klt
and $K_X+B+sC$ is a positive multiple of $K_X+\Delta+s'C$ up to
$\mathbb Q$-linear equivalence. So if $(Y,\Delta_Y+s'C_Y)$ is a log minimal model of
$(X,\Delta+s'C)$, which exists by Theorem \ref{t-mmodel}, then $(Y,B_Y+sC_Y)$ is a $\mathbb Q$-factorial
weak lc model of $(X,B+sC)$ such that $Y\dashrightarrow X$
does not contract divisors and $X\dashrightarrow Y$ is $K_X+B+sC$-negative (see \ref{ss-divisors} for
this notion).
We will make use of this observation below.
Let $T$ be the lc model of $(X,B+tC)$ and let $(Y,B_Y+tC_Y)$ be a $\mathbb Q$-factorial weak lc model of $(X,B+tC)$
such that $X\dashrightarrow Y$ is $K_X+B+tC$-negative and its inverse does not contract divisors.
Then the induced map $Y\dashrightarrow T$ is a morphism and $K_T+B_T+tC_T$ pulls back to
$K_Y+B_Y+tC_Y$.
First assume that $t$ is irrational. Then $C_Y\equiv 0/T$. Moreover,
$C_T$ is $\mathbb Q$-Cartier because the set of those $s\in\mathbb R$ such that $K_T+B_T+sC_T$ is $\mathbb R$-Cartier
forms a rational affine subspace of $\mathbb R$ (this can be proved using simple
linear algebra similar to \ref{ss-ext-rays-scaling}). Since $t$ belongs to this affine subspace and $t$ is
not rational, the affine subspace is equal to $\mathbb R$ hence $K_T+B_T+sC_T$ is $\mathbb R$-Cartier for
every $s$ which implies that $C_T$ is $\mathbb Q$-Cartier. Thus $C_Y\sim_\mathbb Q 0/T$ hence
$K_T+B_T+(t-\epsilon) C_T$ pulls back to $K_Y+B_Y+(t-\epsilon)C_Y$ and the former is ample
for every sufficiently small $\epsilon>0$. This means that $T$ is also the lc model
of $(X,B+(t-\epsilon)C)$.
From now on we assume that $t$ is rational. Replace $Y$ with a $\mathbb Q$-factorial weak lc model
of $(Y,B_Y+(t-\epsilon)C_Y)$ over $T$ so that $X\dashrightarrow Y$ is still $K_X+B+(t-\epsilon)C$-negative.
Since $K_T+B_T+tC_T$ is ample,
by choosing $\epsilon$ to be small enough, we can assume that
$K_Y+B_Y+(t-\epsilon)C_Y$ is nef globally, by \ref{ss-ext-rays-II}. Then $(Y,B_Y+(t-\epsilon)C_Y)$
is a weak lc model of $(X,B+(t-\epsilon)C)$ hence it is enough to show that
$K_Y+B_Y+(t-\epsilon)C_Y$ is semi-ample.
Perhaps after replacing $\epsilon$ with a smaller number we can assume that
$K_Y+B_Y+(t-\epsilon')C_Y$ also nef globally for some $\epsilon'>\epsilon$ and that $t-\epsilon$ is rational.
Let $Y\to V$ be the
contraction to an algebraic space associated to $K_Y+B_Y+(t-\epsilon)C_Y$.
Any curve contracted by $Y\to V$ is also contracted by $Y\to T$ because $K_Y+B_Y+tC_Y$
and $K_Y+B_Y+(t-\epsilon')C_Y$ are both nef and $\epsilon'>\epsilon$. Thus we get an induced map $V\to T$.
Moreover, there is a small contraction $Y'\to V$ from a $\mathbb Q$-factorial normal projective
variety $Y'$: recall that $(Y,\Lambda_Y:=\Delta_Y+t'C_Y)$ is klt where $\Delta$ and $t'$ are
as in the first paragraph; now $Y'$ can be obtained by taking a log resolution $W\to Y$, defining
$\Lambda_W$ to be the birational transform of $\Lambda_V$ plus the reduced exceptional
divisor of $W\to V$, running an LMMP$/V$ on $K_W+\Lambda_W$, using special termination
and the fact that $K_W+\Lambda_W\equiv E/V$ for some $E\ge 0$ whose support is
equal to the reduced exceptional divisor of $W\to V$, and applying the negativity lemma (\ref{ss-negativity}).
Since $K_Y+B_Y+(t-\epsilon)C_Y\equiv 0/V$, $K_{Y'}+B_{Y'}+(t-\epsilon)C_{Y'})$ is also nef and
the former is semi-ample if and only if the latter is. So by replacing $Y$ with
$Y'$, we can in addition assume that $Y\to V$ is a small contraction.
Let $\Sigma$ be a connected component of the exceptional set of $Y\to V$. Since
$Y\to V$ is a small morphism, $\Sigma$ is one-dimensional. On the other hand,
since
$$
K_{Y}+B_Y+(t-\epsilon)C_Y\equiv 0/V
$$
and
$$
K_{Y}+B_Y+tC_Y\equiv 0/V
$$
we get $C_Y\equiv 0/V$ hence $N_Y\equiv 0/V$.
Therefore either $\Sigma\subset \Supp N_Y$ or $\Sigma\cap \Supp N_Y=\emptyset$.
Moreover, if $\Sigma\cap \Supp N_Y=\emptyset$, then $(K_Y+B_Y+(t-\epsilon)C_Y)|_\Sigma$
is semi-ample because near $\Sigma$ the divisor
$K_Y+B_Y+(t-\epsilon)C_Y$ is a multiple of $K_Y+B_Y+tC_Y$
and the latter is semi-ample.
We can assume that $ A'$ in (6) has small coefficients. Let $B'=B+(t-\epsilon)C-A'$. Since $(Y,B_Y'+A_Y'+\epsilon C_Y)$
is lc, $\Supp C_Y$ (hence also $\Supp A'_Y$) does not contain any lc centre of $(Y,B_Y'+A_Y')$.
Now applying Theorem \ref{t-sa-reduced-boundary-2} to
$(X,B'+A')$ shows that
$K_Y+B_Y+(t-\epsilon)C_Y$ is semi-ample (note that the exceptional locus of $Y\to V$ is
equal to $\mathbb{E}(K_Y+B_Y'+A_Y')$). Therefore, $K_Y+B_Y+sC_Y$ is semi-ample for every $s\in[t-\epsilon,t]$.\\
\end{proof}
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\begin{prop}\label{p-fg-1-4}
Let $(X,B)$ be a pair and $M$ a $\mathbb Q$-divisor satisfying properties $(1)$ to $(4)$
of Lemma \ref{l-fg-decrease}. Then the lc ring $R(K_X+B)$ is finitely generated.
\end{prop}
\begin{proof}
\emph{Step 1.}
We follow the proof of [\ref{B-mmodel}, Proposition 3.4], which is similar to
[\ref{BCHM}, \S 5], but with some twists.
Assume that $R(K_X+B)$ is not finitely generated.
We will derive a contradiction.
By replacing $A$ with $\frac{1}{m}S$ where $m$ is sufficiently divisible and $S$ is a
general member of $|mA|$, and changing $M,B$ accordingly, we can assume that
$(7)$ $S:=\Supp A$ is irreducible and $K_X+S+\Delta$ is ample for any boundary $\Delta$
supported on $\Supp(B)-S$.\\
Let ${\theta}(X,B,M)$ be the number of those components of $M$ which are not components
of $\rddown{B}$ (such $\theta$ functions were defined in \ref{ss-WZD} in a more general setting).
By (7), $S$ is not a component of $\rddown{B}$, hence ${\theta}(X,B,M)>0$
otherwise $K_X+B$ is ample and $R(K_X+B)$ is finitely generated,
a contradiction.
Define
$$
\alpha:=\min\{t>0~|~~\rddown{(B+tM)^{\le 1}}\neq \rddown{B}~\}
$$
where for a divisor $R=\sum r_iR_i$ we define $R^{\le 1}=\sum r_i'R_i$ with $r_i'=\min\{r_i,1\}$.
In particular, $(B+\alpha M)^{\le 1}=B+C$ for some $C\ge 0$
supported in $\Supp M$, and $\alpha M=C+N$
where $N\ge 0$ is supported in $\rddown{B}$ and $C$ has no common components with $\rddown{B}$.
Property (3) ensures that $\Supp N=\rddown{B}$, and by property (7) we have $\alpha A\le C$.
So $(X,B)$ and $M$ also satisfy properties (5) and (6) of \ref{l-fg-decrease} with $A'=\alpha' A$
for some $\alpha'>0$.
\emph{Step 2.}
Let $B':=B+C$ and let $M':=M+C$.
Then the pair $(X,B')$ is log smooth dlt and
$$
{\theta}(X,B',M')<{\theta}(X,B,M)
$$
Assume that $R(K_X+B')$ is not finitely generated. By (7), $S$ is not a
component of $\rddown{B'}$ and ${\theta}(X,B',M')>0$.
Now replace $(X,B)$ with $(X,B')$, replace $D$ with $D':=D+C$,
and replace $M$ with $M'$. By construction, all the properties (1) to (4) of \ref{l-fg-decrease}
and property (7) above are still satisfied.
Repeating the above process we get to the situation in which either
$R(K_X+B')$ is finitely generated, or ${\theta}(X,B',M')=0$ and $K_X+B'$ is ample.
Thus in any case we can assume $R(K_X+B')$ is finitely generated.
\emph{Step 3.} Let
$$
\mathcal T=\{t\in [0,1]~|~ (X,B+tC)~~\mbox{has an lc model}\}
$$
Since $R(K_X+B'=K_X+B+C)$ is finitely generated,
$1\in\mathcal T$ hence $\mathcal T\neq \emptyset$. Moreover, if $t\in\mathcal T\cap (0,1]$,
then by Lemma \ref{l-fg-decrease}, $[t-\epsilon,t]\subset \mathcal{T}$ for some $\epsilon>0$.
Now let $\tau=\inf \mathcal{T}$. If $\tau\in \mathcal{T}$, then $\tau=0$ which implies that
$R(K_X+B)$ is finitely generated, a
contradiction. So we may assume $\tau\notin \mathcal{T}$. There is a sequence
$t_1>t_2>\cdots$ of rational numbers in $\mathcal{T}$ approaching $\tau$.
For each $i$, there is a $\mathbb Q$-factorial weak lc model $(Y_i,B_{Y_i}+t_iC_{Y_i})$
of $(X,B+t_iC)$ such that $Y_i\dashrightarrow X$ does not
contract divisors (see the beginning of the proof of Lemma \ref{l-fg-decrease}).
By taking a subsequence, we can assume that
all the $Y_i$ are isomorphic in codimension one. In particular,
$N_\sigma(K_{Y_1}+B_{Y_1}+\tau C_{Y_1})=0$.
Arguing as in the proof of Theorem \ref{t-sa-reduced-boundary-2}, we can show that
$({Y_1},B_{Y_1}+\tau C_{Y_1})$ is dlt because $\alpha A\le C$ is ample and $\Supp A_{Y_1}$
does not contain any lc centre of $({Y_1},B_{Y_1}+\tau C_{Y_1})$.
Run the LMMP on $K_{Y_1}+B_{Y_1}+\tau C_{Y_1}$ with scaling of $(t_1-\tau)C_{Y_1}$ as in \ref{ss-g-LMMP-scaling}.
Since $\alpha M_{Y_1}=N_{Y_1}+C_{Y_1}$, the LMMP is also an LMMP on $N_{Y_1}$.
Thus each extremal ray in the process is a pl-extremal ray hence they can
be contracted by projective morphisms (\ref{ss-pl-ext-rays}). Moreover,
the required flips exist by Theorem \ref{t-flip-1}, and the LMMP terminates
with a model $Y$ on which $K_{Y}+B_{Y}+\tau C_{Y}$ is nef, by special termination (\ref{p-st}).
Note that the LMMP does not contract any divisor by the $N_\sigma=0$ property.
Moreover, $K_{Y}+B_{Y}+(\tau+\delta) C_{Y}$ is nef for some $\delta>0$. Now, by replacing the sequence
we can assume that $K_{Y}+B_{Y}+t_i C_{Y}$ is nef for every $i$ and by replacing each $Y_i$ with $Y$
we can assume that $Y_i=Y$ for every $i$.
A simple comparison of discrepancies (cf. [\ref{B-mmodel}, Claim 3.5]) shows
that $(Y,B_{Y}+\tau C_{Y})$ is a $\mathbb Q$-factorial
weak lc model of $(X,B+\tau C)$.
\emph{Step 4.} Let $T_i$ be the lc model of
$(X,B+t_iC)$. Then the map $Y\dashrightarrow T_i$ is a morphism and $K_{Y}+B_{Y}+t_i C_{Y}$ is the
pullback of an ample divisor on $T_i$.
Moreover, for each $i$, the map $T_{i+1}\dashrightarrow T_i$ is a morphism because any curve
contracted by $Y\to T_{i+1}$ is also contracted by $Y\to T_i$. So perhaps after replacing the sequence,
we can assume that $T_i$ is independent of $i$ so we can drop the subscript and simply
use $T$. Since $C\sim_\mathbb Q 0/T$, we can replace $Y$ with a $\mathbb Q$-factorialization of $T$
so that we can assume that $Y\to T$ is a small morphism (such a $\mathbb Q$-factorialization exists
by the observations in the first paragraph of the proof of Lemma \ref{l-fg-decrease}).
Assume that $\tau$ is irrational.
If $K_Y+B_Y+(\tau-\epsilon)C_Y$ is nef for
some $\epsilon>0$, then $K_Y+B_Y+\tau C_Y$ is semi-ample because in this case
$K_T+B_T+(\tau-\epsilon)C_T$ is nef and $K_T+B_T+t_i C_T$ is ample hence
$K_T+B_T+\tau C_T$ is ample. If there is no $\epsilon$ as above, then
by \ref{ss-ext-rays-scaling} and \ref{ss-ext-rays-II}, there is a curve
$\Gamma$ generating some extremal ray such that $(K_Y+B_Y+\tau C_Y)\cdot \Gamma=0$
and $C_Y\cdot \Gamma>0$.
This is not possible since $\tau$ is assumed to be irrarional.
So from now on we assume that
$\tau$ is rational.
\emph{Step 5.} Let $Y\to V$ be the contraction to an algebraic space associated to $K_Y+B_Y+\tau C_Y$.
This map factors through $Y\to T$ so we get an induced map $T\to V$.
We can write
$$
K_T+B_T+\tau C_T=a(K_T+B_T+t_iC_T)+bN_T
$$
for some $i$ and some rational numbers $a,b>0$. Since $K_T+B_T+t_iC_T$ is ample, we get
$$
\mathbb{E}(K_T+B_T+\tau C_T)\subset \Supp N_T=\rddown{B_T}
$$
Thus since $N_Y\sim_\mathbb Q 0\sim_\mathbb Q C_Y/T$, the locus
$\mathbb{E}(K_Y+B_Y+\tau C_Y)$ is a subset of the union of $\Supp N_Y=\rddown{B_Y}$ and the exceptional
set of $Y\to T$.
Let $\Lambda$ be a connected component of the exceptional
set of $Y\to T$. Then, since $N_Y\sim_\mathbb Q 0/T$ and since $\Lambda$ is one-dimensional,
either $\Lambda\subset \Supp N_{Y}$ or $\Lambda\cap \Supp N_{Y}=\emptyset$.
Therefore if $\Sigma$ is a connected component of $\mathbb{E}(K_Y+B_Y+\tau C_Y)$, then
either $\Sigma\subset \Supp N_{Y}$ or $\Sigma\cap \Supp N_{Y}=\emptyset$.
In the latter case, $(K_Y+B_Y+\tau C_Y)|_\Sigma$ is semi-ample because near $\Sigma$ the divisor
$K_Y+B_Y+\tau C_Y$ is a multiple of $K_Y+B_Y+t_i C_Y$ and the latter is semi-ample.
Finally as in the end of the proof of Lemma \ref{l-fg-decrease} we can apply Theorem
\ref{t-sa-reduced-boundary-2} to show that $K_Y+B_Y+\tau C_Y$ is semi-ample.
This is a contradiction because we assumed $\tau\notin\mathcal{T}$.\\
\end{proof}
\begin{proof}(of Theorem \ref{t-fg})
First assume that $Z$ is a point.
Pick $M\ge 0$ such that $K_X+B\sim_\mathbb Q M$. We can choose $M$ so that
$M=A+D$ where $A\ge 0$ is ample and $D\ge 0$.
Let $f\colon W\to X$ be a log resolution
of $(X,\Supp (B+M))$. Since $(X,B)$ is klt, we can write
$$
K_W+B_W=f^*(K_X+B)+E
$$
where $(W,B_W)$ is klt, $K_W+B_W$ is a $\mathbb Q$-divisor, and $E\ge 0$ is exceptional$/X$.
Moreover, there is $E'\ge 0$ exceptional$/X$
such that $-E'$ is ample$/X$ (cf. proof of Lemma \ref{l-ample-dlt}). Let $A_W\sim_\mathbb Q f^*A-E'$
be general and let $D_W=f^*D+E+E'$. Then
$$
K_W+B_W\sim_\mathbb Q M_W:=A_W+D_W
$$
Now replace $(X,B)$ with $(W,B_W)$,
replace $M$ with $M_W$, and replace $A$ and $D$ with $A_W$ and $D_W$.
Moreover, by adding a small multiple of $M$ to $B$ we can also assume that
$\Supp M\subseteq \Supp B$.
Then $(X,B)$ and $M$ satisfy the properties (1) to (4) of Lemma \ref{l-fg-decrease}.
Therefore, by Proposition \ref{p-fg-1-4}, $R(K_X+B)$ is finitely generated.
Now we treat the general case, that is, when $Z$ is not necessarily a point.
By taking projectivizations of $X,Z$ and taking a log resolution, we
may assume that $X,Z$ are projective and that $(X,B)$ is log smooth. We can also assume that
$K_X+B\sim_\mathbb Q M=A+D/Z$ where $A$ is an ample $\mathbb Q$-divisor and $D\ge 0$. By adding
some multiple of $M$ to $B$ we may assume $\Supp M\subseteq \Supp B$. Let $(Y,B_Y)$ be a log minimal model
of $(X,B)$ over $Z$. Let $H$ be the pullback of an
ample divisor on $Z$. Since $A\le B$, for each integer $m\ge 0$, there is $\Delta$ such that
$K_X+B+mH\sim_\mathbb Q K_X+\Delta$ is big globally and that $(X,\Delta)$ is klt. Moreover, $(Y,\Delta_Y)$ is a
log minimal model of $(X,\Delta)$ over $Z$. Now
by \ref{ss-ext-rays-II}, if $m\gg 0$, then $K_Y+\Delta_Y$
is big and globally nef. On the other hand, $R(K_Y+\Delta_Y)$ is finitely generated over $k$
which means that $K_Y+\Delta_Y$ is semi-ample. Therefore $K_Y+B_Y$ is semi-ample$/Z$
hence $\mathcal{R}(K_X+B/Z)$ is a finitely generated $\mathcal{O}_Z$-algebra.
\end{proof}
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\subsection{Base point freeness}
\begin{proof}(of Theorem \ref{t-bpf})
It is enough to show that $\mathcal{R}(D/Z)$ is a finitely generated $\mathcal{O}_Z$-algebra.
By taking a $\mathbb Q$-factorialization using Theorem \ref{cor-dlt-model}, we may assume that
$X$ is $\mathbb Q$-factorial.
Let $A=D-(K_X+B)$ which is nef and big$/Z$ by assumptions. By replacing $A$, and replacing $B$ accordingly,
we may assume that $A$ is ample globally.
By Lemma \ref{l-ample-dlt}, we can change $A$ up to $\mathbb Q$-linear equivalence so that
$(X,B+A)$ is klt. But then $\mathcal{R}(K_X+B+A/Z)$ is finitely generated
by Theorem \ref{t-fg} hence $\mathcal{R}(D/Z)$ is also finitely generated.
\end{proof}
\subsection{Contractions}
\begin{proof}(of Theorem \ref{t-contraction})
We may assume that $B$ is a $\mathbb Q$-divisor and that $(X,B)$ is klt. We can assume $N=H+D$ where $H$ is ample$/Z$
and $D\ge 0$. Let $G$ be the pullback of an ample divior on $Z$, and let
$N'=mG+nN+\epsilon H+\epsilon D$ where $\epsilon>0$ is sufficiently small
and $m\gg n\gg 0$. Then we can
find $A\sim_\mathbb Q N'$ such that $(X,B+A)$ is klt, $K_X+B+A$ is globally big, and $(K_X+B+A)\cdot R<0$.
By \ref{ss-ext-rays-II}, we can find an ample divisor $E$ such that $L:=(K_X+B+A+E)$
is nef and big globally and $L^\perp=R$. We can also assume that $(X,B+A+E)$ is klt
hence by Theorem \ref{t-bpf}, $L$ is semi-ample which implies that $R$ can be contracted by a
projective morphism.\\
\end{proof}
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\section{ACC for lc thresholds}\label{s-ACC}
In this section, we prove Theorem \ref{t-ACC} by a method similar to the characteristic $0$ case
(see [\ref{Kollar+}, Chapter 18] and [\ref{MP}]).
Let us recall the definition of \emph{lc threshold}. Let $(X,B)$ be an lc pair over $k$ and
$M\ge 0$ an $\mathbb R$-Cartier divisor. The lc threshold of $M$ with respect to $(X,B)$ is defined as
$$
\lct(M,X,B)=\sup \{t\mid (X,B+tM)~~\mbox{is lc}\}
$$
We first prove some results, including ACC for lc thresholds, for surfaces
before we move on to 3-folds.
\subsection{ACC for lc thresholds on surfaces}
\begin{prop}\label{p-acc-surfaces}
ACC for lc thresholds holds in dimension $2$ (formulated similar to \ref{t-ACC}).
\end{prop}
\begin{proof}
If this is not the case, then there is a sequence $(X_i,B_i)$ of lc pairs of
dimension $2$ over $k$ and $\mathbb R$-Cartier divisors $M_i\ge 0$ such that the coefficients of
$B_i$ are in $\Lambda$, the coefficients of $M_i$ are in $\Gamma$ but such that
the $t_i:=\lct(M_i,X_i,B_i)$ form a strictly increasing sequence of numbers.
If for infinitely many $i$, $(X_i,\Delta_i:=B_i+t_iM_i)$ has an lc centre of dimension one
contained in $\Supp M_i$,
then it is quite easy to get a contradiction.
We may then assume that each $(X_i,\Delta_i)$ has an lc centre $P_i$ of dimension zero
contained in $\Supp M_i$. We may also assume that $(X_i,\Delta_i)$ is plt outside $P_i$.
Let $(Y_i,\Delta_{Y_i})$ be a $\mathbb Q$-factorial dlt model of
$(X_i,\Delta_i)$ such that there are some exceptional divisors on $Y_i$
mapping to $P_i$. Such $Y_i$ exist by a version of Lemma \ref{l-extraction} in dimension $2$.
There is a prime exceptional divisor $E_i$ of $Y_i\to X_i$ such that it intersects the
birational transform of $M_i$. Note that $E_i$ is normal and actually isomorphic to $\mathbb P^1_k$
since $E_i$ is a component of $\rddown{\Delta_{Y_i}}$ and $(K_{Y_i}+\Delta_{Y_i})\cdot E_i=0$. Now by adjunction define
$K_{E_i}+\Delta_{E_i}=(K_{Y_i}+\Delta_{Y_i})|_{E_i}$. Then by Proposition \ref{p-adjunction-DCC} and
its proof, the set of all the coefficients of the
$\Delta_{E_i}$ is a subset of a fixed DCC set but they do not satisfy ACC.
This is a contradiction since $\deg \Delta_{E_i}=2$.
\end{proof}
We apply the ACC of \ref{p-acc-surfaces} to negativity of contractions.
\begin{lem}\label{l-lim-nefness}
Let $\Lambda\subset [0,1]$ be a DCC set of real numbers. Then there is $\epsilon>0$ satisfying
the following: assume we have
$\bullet$ a klt pair $(X,B)$ of dimension $2$,
$\bullet$ the coefficients of $B$ belong to $\Lambda\cup [1-\epsilon,1]$,
$\bullet$ $f\colon X\to Y$ is an extremal birational projective contraction with exceptional divisor $E$,
$\bullet$ the coefficient of $E$ in $B$ belongs to $[1-\epsilon,1]$, and
$\bullet$ $-(K_X+B)$ is nef$/Y$.\\
If $\Delta$ is obtained from $B$ by replacing each coefficient in $[1-\epsilon, 1]$ with $1$,
then $-(K_X+\Delta)$ is also nef$/Y$.
\end{lem}
\begin{proof}
Note that klt pairs of dimension $2$ are $\mathbb Q$-factorial so $K_X+\Delta$ is $\mathbb R$-Cartier.
By \ref{p-acc-surfaces},
we can pick $\epsilon>0$ so that: if $(T,C)$ is lc of dimension $2$ and $M\ge 0$ such that
the coefficients of $C$ belong to $\Lambda$ and the coefficients of $M$ belong to $\{1\}$,
then the lc threshold $\lct(M,T,C)$ does not belong to $[1-\epsilon,1)$.
Now since $(X,B)$ is klt and $-(K_X+B)$ is nef$/Y$,
$(Y,B_Y)$ is also klt. Thus $(Y,\Delta_Y-\epsilon\rddown{\Delta_Y})$ is klt because
$B_Y\ge \Delta_Y-\epsilon\rddown{\Delta_Y}$.
In particular, the lc threshold of $\rddown{\Delta_Y}$ with respect to $(Y,\Delta_Y-\rddown{\Delta_Y})$ is
at least $1-\epsilon$. Note that the coefficients of $\Delta$ belong $\Lambda\cup \{1\}$
and the coefficients of $\Delta_Y-\rddown{\Delta_Y}$ belong to $\Lambda$.
Thus by our choice of $\epsilon$, the pair $(Y,\Delta_Y)$ is lc. Therefore we can write
$$
K_X+\Delta=f^*(K_Y+\Delta_Y)+eE
$$
for some $e\ge 0$ because the coefficient of $E$ in $\Delta$ is $1$.
This implies that $-(K_X+\Delta)$ is indeed nef$/Y$.\\
\end{proof}
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\subsection{Global ACC for surfaces}
In this subsection we prove a global type of ACC for surfaces (\ref{p-ACC-global}) which
will be used in the proof of Theorem \ref{t-ACC}.
\begin{constr}\label{rem-Fano-twist}
Let $\epsilon\in (0,1)$ and let $X'$ be a klt Fano surface with $\rho(X')=1$.
Assume that $X'$ is not $\epsilon$-lc.
Pick a prime divisor $E$ (on birational models of $X'$) with log discrepancy $a(E,X',0)<\epsilon$.
By a version of Lemma \ref{l-extraction}
in dimension two, there is a
birational contraction $Y'\to X'$ which is extremal and has $E$ as the only exceptional
divisor. Under our assumptions
it is easy to find a boundary $D_{Y'}$ such that $(Y',D_{Y'})$ is klt and $K_{Y'}+D_{Y'}\sim_\mathbb R -eE$
for some $e>0$. In particular, we can run an LMMP on $-E$ which ends with a Mori fibre space $X''\to T''$
so that $E''$ positively intersects the extremal ray defining $X''\to T''$ where $E''$ is the
birational transform of $E$.
As $\rho(X')=1$, we get $\rho(Y')=2$. One of the extremal rays of $Y'$
gives the contraction $Y'\to X'$. The other one either gives $X''\to T''$ with $Y'=X''$
or it gives a birational contraction $Y'\to X''$.
If $\dim T''=0$, then $X''$ is also a klt Fano with
$\rho(X'')=1$.
\end{constr}
\begin{lem}\label{l-bnd-comps-surfaces}
Let $b\in (0,1)$ be a real number. Then there is a natural number $m$ depending only
on $b$ such that: let $(X,B)$ be a klt pair of dimension $2$ and $x\in X$ a closed point;
then the number of those components of
$B$ containing $x$ and with coefficient $\ge b$ is at most $m$.
\end{lem}
\begin{proof}
Since $(X,B)$ is klt and $\dim X=2$, $X$ is $\mathbb Q$-factorial.
We can assume that each coefficient of $B$ is equal to $b$ by discarding any component
with coefficient less than $b$ and by decreasing each coefficient which is more than $b$.
Moreover, we can assume every component of $B$ contains $x$.
Pick a nonzero $\mathbb R$-Cartier divisor $G\ge 0$ such that $(X,C:=B+G)$ is lc
near $x$ and such that $x$ is a lc centre of $(X,B+G)$: for example we can take a
log resolution $W\to X$ and let $G$ be the pushdown of an appropriate
ample $\mathbb R$-divisor on $W$. Shrinking $X$ we can assume $(X,C)$ is lc.
Since $(X,B)$ is klt, there is an extremal contraction $f\colon Y\to X$
which extracts a prime divisor $S$ with log discrepancy $a(S,X,C)=0$.
Let $B_Y$ be the sum of $S$ and the birational transform of $B$.
Then $-(K_Y+B_Y)$ is ample$/X$. Apply adjunction (\ref{p-adjunction-DCC}) and write
$K_{S^\nu}+B_{S^\nu}$ for the pullback of $K_Y+B_Y$ to the normalization of $S$.
As $-(K_{S^\nu}+B_{S^\nu})$ is ample, ${S^\nu}\simeq \mathbb P^1$ and
$\deg B_{S^\nu}<2$.
By \ref{p-adjunction-DCC}, the coefficient of each $s\in \Supp B_{S^\nu}$
is of the form $\frac{n-1}{n}+\frac{rb}{n}$ for some integer $r\ge 0$ and some $n\in \mathbb N\cup \{\infty\}$.
In particular,
the number of the components of $B_{S^\nu}$ is bounded and the number $r$ in the
formula is also bounded. This bounds the number of the components of $B$ because $r$ is
more than or equal to the number of those
components of $B_Y-S$ which pass through the image of $s$.\\
\end{proof}
\begin{prop}\label{p-ACC-global}
Let $\Lambda\subset [0,1]$ be a DCC set of real numbers. Then there is a
finite subset $\Gamma\subset \Lambda$ with the following property:
let $(X,B)$ be a pair and $X\to Z$ a projective morphism such that
$\bullet$ $(X,B)$ is lc of dimension $2$ over $k$,
$\bullet$ the coefficients of $B$ are in $\Lambda$,
$\bullet$ $K_X+B\equiv 0/Z$,
$\bullet$ $\dim X>\dim Z$.
Then the coefficient of each horizontal$/Z$ component of $B$ is in $\Gamma$.
\end{prop}
\begin{proof}
\emph{Step 1.}
We can assume that $1\in \Lambda$.
If the proposition is not true, then there is a sequence $(X_i,B_i), X_i\to Z_i$
of pairs and morphisms as in the proposition such that the set of the coefficients
of the horizontal$/Z_i$ components of all the $B_i$ put together does not
satisfy ACC. By taking $\mathbb Q$-factorial dlt
models we can assume that $(X_i,B_i)$ are $\mathbb Q$-factorial dlt.
Write $B_i=\sum b_{i,j}B_{i,j}$. We may assume that $B_{i,1}$ is horizontal$/Z_i$
and that $b_{1,1}<b_{2,1}<\cdots$.
\emph{Step 2.}
First assume that $\dim Z_i=1$ for every $i$. Run the LMMP$/Z_i$ on
$K_{X_i}+B_i-b_{i,1}B_{i,1}$ with scaling of $b_{i,1}B_{i,1}$.
This terminates with a model $X_i'$ having an extremal contraction
$X_i'\to Z_i'/Z_i$ such that $K_{X_i'}+B_i'-b_{i,1}B_{i,1}'$ is numerically negative over $Z_i'$.
Let $F_i'$ be the reduced variety associated to a general fibre of $X_i'\to Z_i'$.
Since $K_{X_i'}+B_i'\equiv 0/Z'$ and ${F'_i}^2=0$, we get $(K_{X_i'}+B_i'+F_i')\cdot F_i'=0$
hence the arithmetic genus $p_a(F_i')<0$ which implies that $F_i'\simeq \mathbb P^1_k$.
We can write
$$
\deg (K_{X_i'}+B_i'+F_i')|_{F_i'}=-2+\sum n_{i,j}b_{i,j}= 0
$$
for certain integers $n_{i,j}\ge 0$ such that $n_{i,1}>0$. Since the $b_{i,j}$
belong to the DCC set $\Lambda$, $n_{i,1}$ is bounded from above and below.
Moreover, we can assume that the sums $\sum_{j\ge 2} n_{i,j}b_{i,j}$ satisfy the DCC hence
$n_{i,1}b_{i,1}=2-\sum_{j\ge 2} n_{i,j}b_{i,j}$ satisfies the ACC, a contradiction.
\emph{Step 3.}
From now on we may assume that $\dim Z_i=0$ for every $i$. Run the
LMMP$/Z_i$ on $K_{X_i}+B_i-b_{i,1}B_{i,1}$ with scaling of $b_{i,1}B_{i,1}$.
This terminates with a model $X_i'$ having an extremal contraction
$X_i'\to Z_i'$ such that $K_{X_i'}+B_i'-b_{i,1}B_{i,1}'$ is numerically negative over $Z_i'$.
If $\dim Z_i'=1$ for infinitely many $i$, then we get a contradiction by Step 2.
So we assume that $Z_i'$ are all points hence each $X_i'$ is a Fano with Picard number one.
Assume that $({X_i}',B_i')$ is lc but not klt for every $i$. Assume that each $({X_i}',B_i')$
has an lc centre $S_i'$ of dimension one. Let $K_{S_i'}+B_{S_i'}=(K_{X_i'}+B_i')|_{S_i'}$
by adjunction. Note that $S_i'$ is normal since $({X_i'},B_i'-b_{i,1}B_{i,1}')$
is $\mathbb Q$-factorial dlt. Since $K_{S_i'}+B_{S_i'}\equiv 0$, $S_i'\simeq \mathbb P^1_k$.
If $\Supp B_{i,1}'$ contains an lc centre for infinitely many $i$, then we get a contradiction
by ACC for lc thresholds in dimension $2$. So we can assume that $\Supp B_{i,1}'$ does not
contain any lc centre, in particular, none of the points of $S_i'\cap B_{i,1}'$ is
an lc centre. Now, since $\{b_{i,j}\}$ does not satisfy ACC, by Proposition
\ref{p-adjunction-DCC}, the set of the coefficients of all the $B_{S_i'}$
satisfies DCC but not ACC which gives a contradiction as above (by considering the coefficients of
the points in $S_i'\cap B_{i,1}'$). So we can assume that each $({X_i}',B_i')$
has an lc centre of dimension zero.
By a version of Lemma \ref{l-extraction} in dimension $2$, there is a projective birational contraction $Y_i'\to X_i'$
which extracts only one prime divisor $E_i'$ and it satisfies $a(E_i',X_i',B_i')=0$.
Let $K_{Y_i'}+B_{Y_i'}$ be the pullback of $K_{X_i'}+B_i'$. By running the
LMMP on $K_{Y_i'}+B_{Y_i'}-E_i'$, we arrive on a model on which either
the birational transform of $E_i'$ intersects the birational transform of $B_{i,1}'$ for infinitely many $i$,
or we get a Mori fibre space over a curve whose general fibre intersects the birational
transform of $B_{i,1}'$ for infinitely many $i$.
In any case, we can apply the arguments above to get a contradiction.
So from now on we may assume that $({X_i}',B_i')$ are all klt.
\emph{Step 4.}
If there is $\epsilon>0$ such that $X_i'$ is $\epsilon$-lc for every $i$, then
we are done since such $X_i'$ are bounded by Alexeev [\ref{Alexeev}]. So we can assume
that the minimal log discrepancies of the $X_i'$ form a strictly decreasing
sequence of positive numbers. Since $({X_i}',B_i')$ are klt, we can assume that
the minimal log discrepancies of the $(X_i',B_i')$ also form a strictly decreasing
sequence of positive numbers. As in Construction \ref{rem-Fano-twist}, we find a
contraction ${Y}_i'\to X_i'$ extracting a prime divisor $E_i$ with minimal log discrepancy
$a(E_i,X_i',B_i')<\epsilon$ and run a $-E_i$-LMMP to get a Mori fibre structure $X_i''\to Z_i''$.
If $\dim Z_i''=1$ for each $i$, we use Step 2 to get a contradiction. So we may assume
that $\dim Z_i''=0$ for each $i$. Note that the exceptional divisor of $X_i''\dashrightarrow X_i'$
is a component of $B_i''$ with coefficient $\ge 1-\epsilon$ where $K_{X_i''}+B_i''$
is the pullback of $K_{X_i'}+B_i'$.
Write $K_{Y_i'}+{B}_{Y_i'}$ for the pullback of $K_{X_i'}+B_i'$.
By construction, the coefficients of ${B}_{Y_i'}$ belong to some DCC subset of
$\Lambda\cup [1-\epsilon,1]$. We show that if $\epsilon$ is sufficiently small, then
${Y}_i'\to X_i''$ cannot contract a component of ${B}_{Y_i'}$ with coefficient $\ge 1-\epsilon$.
Indeed let $\Delta_{Y_i'}$ be obtained from ${B}_{Y_i'}$ by replacing each
coefficient $\ge 1-\epsilon$ with $1$. Then by Lemma \ref{l-lim-nefness},
$-(K_{Y_i'}+{\Delta}_{Y_i'})$ is nef over both $X_i'$ and $X_i''$. As $\rho(Y_i')=2$,
$-(K_{Y_i'}+{\Delta}_{Y_i'})$ is nef globally. This is a contradiction because
the pushdown of $K_{Y_i'}+{\Delta}_{Y_i'}$ to $X_i''$ is ample.
\emph{Step 5.}
Now replace $({X_i'},B_i')$ with $({X_i''},B_i'')$ and repeat the process of Step 4, $m$ times.
By the last paragraph the new components of $B_i'$ that
appear in the process are not contracted again. So
we may assume that we have at least $m$ components of $B_i'$ with coefficients $\ge 1-\epsilon$.
Let $x_i'$ be the image of the exceptional divisor of $Y_i'\to X_i'$ and
let $x_i''$ be the image of the exceptional divisor of $Y_i'\to X_i''$.
Also let $m_i'$ be the number of those components of $B_i'$ with coefficient
$\ge 1-\epsilon$ and passing through $x_i'$. Define $m_i''$ similarly. Since
$\rho(Y_i')=2$, each component of $B_{Y_i'}$ intersects the exceptional
divisor of $Y_i'\to X_i'$ or the exceptional divisor of $Y_i'\to X_i''$. Therefore,
$m_i'+m_i''\ge m$.
Finally by Lemma \ref{l-bnd-comps-surfaces} both $m_i'$ and $m_i''$
are bounded hence $m$ is also bounded. This means that after finitely many times applying the
process of Step 4, we can assume there is $\epsilon>0$ such that $X_i'$ is $\epsilon$-lc for every $i$,
and then apply boundedness of such $X_i'$ [\ref{Alexeev}].
\end{proof}
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\subsection{$3$-folds}
\begin{proof}(of Theorem \ref{t-ACC})
If the theorem does not hold, then there is a sequence $(X_i,B_i)$ of lc pairs of
dimension $3$ over $k$ and $\mathbb R$-Cartier divisors $M_i\ge 0$ such that the coefficients of
$B_i$ are in $\Lambda$, the coefficients of $M_i$ are in $\Gamma$ but such that
the $t_i:=\lct(M_i,X_i,B_i)$ form a strictly increasing sequence of numbers.
We may assume that each $(X_i,\Delta_i:=B_i+t_iM_i)$ has an lc centre of dimension $\le 1$
contained in $\Supp M_i$.
Let $(Y_i,\Delta_{Y_i})$ be a $\mathbb Q$-factorial dlt model of
$(X_i,\Delta_i)$ such that there is an exceptional divisor on $Y_i$
mapping onto an lc centre inside $\Supp M_i$. Such $Y_i$ exist by Lemma \ref{l-extraction}.
There is a prime exceptional divisor $E_i$ of $Y_i\to X_i$ such that it intersects the
birational transform of $M_i$ and that it maps into $\Supp M_i$.
Note that $E_i$ is normal by Lemma \ref{l-plt-normal}.
Let $E_i\to Z_i$ be the contraction induced by $E_i\to X_i$. Now by adjunction define
$K_{E_i}+\Delta_{E_i}=(K_{Y_i}+\Delta_{Y_i})|_{E_i}$. Then the set of all the coefficients of the
horizontal$/Z_i$ components of the $\Delta_{E_i}$ satisfies DCC
but not ACC, by Proposition \ref{p-adjunction-DCC}.
This contradicts Proposition \ref{p-ACC-global}.
\end{proof}
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\section{Non-big log divisors: proof of \ref{t-aug-b-non-big}}\label{s-numerical}
\begin{lem}\label{l-movable-curve}
Let $X$ be a normal projective variety of dimension $d$ over an algebraically closed field
(of any characteristic). Let $A$ an ample $\mathbb R$-divisor and
$P$ a nef $\mathbb R$-divisor with $P^d=0$. Then for any $\epsilon>0$, there exist
$\delta\in [0,\epsilon]$ and a very ample divisor $H$ such that $(P-\delta A)\cdot H^{d-1}=0$.
\end{lem}
\begin{proof}
First we show that there is an ample divisor $H$ such that $(P-\epsilon A)\cdot H^{d-1}<0$.
Put $r(\tau):=(P-\epsilon A)(P+\tau A)^{d-1}$.
Then
$$
r(\tau)=(P-\epsilon A)(P^{d-1}+a_{d-2}\tau P^{d-2}A+\dots+a_1\tau^{d-2}PA^{d-2}+\tau^{d-1}A^{d-1})
$$
where the $a_i>0$ depend only on $d$. Put $a_{d-1}=a_0=1$, $a_{-1}=0$, and let $n$ be the smallest
integer such that $P^{d-n}A^{n}\neq 0$. Then we can write
$$
r(\tau)=\sum_{i=0}^{d-1} (a_{i-1}\tau^{d-i}-\epsilon a_i\tau^{d-i-1}) P^iA^{d-i}
$$
from which we get
$$
r(\tau)=\sum_{i=0}^{d-n} (a_{i-1}\tau^{d-i}-\epsilon a_i\tau^{d-i-1}) P^iA^{d-i}
$$
hence
$$
\frac{r(\tau)}{\tau^{n-1}}=(a_{d-n-1}\tau-\epsilon a_{d-n}) P^{d-n}A^{n}+\tau s(\tau)
$$
for some polynomial function $s(\tau)$. Now if $\tau>0$ is sufficiently small it is clear
that the right hand side is negative hence $r(\tau)<0$.
Choose $\tau>0$ so that $r(\tau)<0$. Since $P+\tau A$ is ample and ampleness
is an open condition, there is an ample $\mathbb Q$-divisor $H$ close to $P+\tau A$
such that $(P-\epsilon A)\cdot H^{d-1}<0$. By replacing $H$ with a multiple we
can assume that $H$ is very ample. Since $P\cdot H^{d-1}\ge 0$ by the nefness of $P$,
it is then obvious that there is some
$\delta\in [0,\epsilon]$ such that $(P-\delta A)\cdot H^{d-1}=0$.\\
\end{proof}
\begin{proof}(of Theorem \ref{t-aug-b-non-big})
Assume that $D^d=0$. By replacing $A$ we may assume that it is ample.
Fix $\alpha>0$. By Lemma \ref{l-movable-curve}, there exist a number $t$ sufficiently close to $1$
(possibly equal to $1$) and a very ample divisor $H$ such that
$$
(K_X+B+t(A+\alpha D))\cdot H^{d-1}=0
$$
Now we can view $H^{d-1}$ as a $1$-cycle on $X$. For each point $x\in X$, there is an effective
$1$-cycle $C_x$ whose class is the same as $H^{d-1}$ and such that $x\in C_x$. Since
$H$ is very ample, we may assume that $C_x$ is irreducible and that it is
inside the smooth locus of $X$ for general $x$.
In particular, we have
$$
(K_X+B+t(A+\alpha D))\cdot C_x= 0
$$
Pick a general $x\in X$ and let $C_x$ be the curve mentioned above.
Since $B$ is effective and $A+\alpha D$ is ample, we get
$K_X\cdot C_x<0$.
Thus by Koll\'ar [\ref{kollar}, Chapter II, Theorem 5.8],
there is a rational curve $L_x$ passing through $x$ such that
$$
0<A\cdot L_x\le (A+\alpha D)\cdot L_x\le (2d) \frac{(A+\alpha D)\cdot C_x}{-K_X\cdot C_x}
$$
$$
=\frac{2d}{t} (1+\frac{B\cdot C_x}{K_X\cdot C_x})\le \frac{2d}{t}<3d
$$
because $K_X\cdot C_x<0$, $B\cdot C_x\ge 0$, and $t$ is sufficiently close to $1$.
Note that although $K_X$ and $B$ need not be $\mathbb R$-Cartier, the intersection
numbers still make sense since $C_x$ is inside the smooth locus of $X$.
As $A$ is ample and $A\cdot L_x\le 3d$, we can
assume that such $L_x$ (for general $x$) belong to a bounded family $\mathcal{L}$ of curves
on $X$ (independent of the choice of $t,\alpha$).
Therefore there are only
finitely many possibilities for the intersection numbers $D\cdot L_x$.
If we choose $\alpha$ sufficiently large, then the inequality $(A+\alpha D)\cdot L_x\le 3d$
implies $D\cdot L_x=0$
and so we get the desired family.\\
\end{proof}
\flushleft{DPMMS}, Centre for Mathematical Sciences,\\
Cambridge University,\\
Wilberforce Road,\\
Cambridge, CB3 0WB,\\
UK\\
email: [email protected]\\
\end{document}
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\color{blue}egin{document}
\title{{\huge Open-loop and Closed-loop Local and Remote Stochastic Nonzero-sum Game with Inconsistent Information Structure }}
\author{Xin Li, Qingyuan Qi$^{*}$ and Xinbei Lv
\thanks{This work was supported by National Natural Science Foundation of China under grants 61903210, Natural Science Foundation of Shandong Province under grant ZR2019BF002, China Postdoctoral Science Foundation under grant 2019M652324, 2021T140354, Qingdao Postdoctoral Application Research Project, Major Basic Research of Natural Science Foundation of Shandong Province (ZR2021ZD14).(Corresponding author: Qingyuan Qi.)
X. Li ([email protected]) is with Institute of Complexity Science, College of Automation, Qingdao University, Qingdao, China 266071. Q. Qi ([email protected]) and X. Lv ([email protected]) are with Qingdao Innovation and Development Center of Harbin Engineering University, Qingdao, China, 266000.
}}
\maketitle
\IEEEpeerreviewmaketitle
\color{blue}egin{abstract}
In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimize its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example.
\end{abstract}
\color{blue}egin{IEEEkeywords}
Discrete-time stochastic nonzero-sum game, inconsistent information structure, open-loop and closed-loop Nash equilibrium, maximum principle.
\end{IEEEkeywords}
\def\small{\smallmall}
\def\normalsize{\normalsizeormalsize}
\def\!+\!{\!+\!}
\def\!-\!{\!-\!}
\def\!=\!{\!=\!}
\def\mathbb{E}{\mathbb{E}}
\def\mathbb{P}{\mathbb{P}}
\def\mathrm{Tr}{\mathrm{Tr}}
\def\color{blue}{\color{blue}}
\def\color{red}{\color{red}}
\smallection{Introduction}
As is well known, due to the wide applications in industry, economics, management \cite{djl2000,cl2002}, the differential games have received many scholars' interest since 1950s, and abundant research results have been obtained, see \cite{i1965,ek1972,i1999,zp2022}. As a special case of differential games, the linear quadratic (LQ) non-cooperative stochastic game is a hot research topic, in view of its rigorous solution and the potential application background. For the LQ non-cooperative stochastic game problem, the system dynamics is described with a linear stochastic differential/difference equation, and the quadratic utility functions are to be minimized. It is worth mentioning that the open-loop and closed-loop Nash equilibrium were proposed in \cite{sy2019} for the continuous-time stochastic LQ two-person nonzero-sum differential game, it was shown that the existence of the open-loop Nash equilibrium is characterized by the forward-backward stochastic differential equations, and the closed-loop Nash equilibrium is characterized by the Riccati equations. Besides, references \cite{sjz2012a,sjz2012b} studied the discrete-time LQ non-cooperative stochastic game problem, and the Nash equilibrium was derived. Please refer to \cite{r2007,h1999,sy2019,slz2011,sjz2012a,sjz2012b,z2018,szl2021,cyl2022} and the cited references therein for the recent study on the LQ non-cooperative stochastic game.
It is noted that the previous works \cite{h1999,sy2019,slz2011,sjz2012a,sjz2012b,z2018,szl2021,cyl2022} on LQ stochastic game mainly focused on the case of the information structure being consistent, i.e., the two players share the same information sets. While, the information inconsistent case remains less investigated, i.e., the information sets obtained by the two players are different. As pointed out in \cite{nglb2014}, the asymmetry of information among the controllers makes it difficult to compute or characterize Nash equilibria. In fact, due to the inconsistent of the information structure, the two players are coupled with each other and finding the Nash equilibrium strategy becomes hard. Therefore, the study of the LQ non-zero stochastic game is challenging from the theoretical aspect.
In this paper, a special case of LQ stochastic two-person nonzero-sum differential games with inconsistent information shall be studied, which is called the local and remote stochastic nonzero-sum game (LRSNG) problem. The detailed description of the networked system under consideration is shown in Figure \color{red}ef{Figure1}. It can be seen that the two players (the local player and the remote player) are included, and the uplink channel from the local player to the remote player is unreliable, while the downlink channel from the remote player to the local player is perfect. In other words, the precise state information can only be perfectly observed by the local player, and the remote player can just receive the disturbed state information delivered from the local player due to the unreliable uplink channel. Hence, the information sets accessed by the local player and the remote player are not the same, which is called inconsistent information structure. The objective of the two players is that each player should minimize its own quadratic cost function. Specifically, the two players are non-cooperative, therefore, the above problem is actually a LQ two-person nonzero-sum differential game with inconsistent information (LRSNG problem), and the goal of this paper is to find the Nash equilibrium associated with the LRSNG problem.
Actually, the corresponding local and remote control problem was originally studied in \cite{ott2016}, in which two controllers (the local controller and the remote controller) were cooperative, and the goal was to derive the jointly control law to optimize one common cost function. The optimal local and remote control problem is a decentralized control problem with asymmetric information structure, since the local controller and the remote controller can access different information sets, and the recent research results on the local and remote control can be found in \cite{qxz2020,lx2018,lqz2021,aon2019,tyw2022}. For instance, \cite{lx2018} investigated the optimal local and remote control problem, and a necessary and sufficient condition for the finite horizon optimal control problem was given by the use of maximum principle. In \cite{aon2019}, the local and remote control problem with multiple subsystems was studied, by using the common information approach, the optimal control strategy of the controller was obtained.
\color{blue}egin{figure}[htbp]
\centering
\includegraphics[width=0.38\textwidth]{Figure1.pdf}
\caption{ Description of the LRSNG problem with inconsistent information structure.}
\label{Figure1}
\end{figure}
While, the LRSNG problem studied in this paper is essentially different with the previous works \cite{qxz2020,lx2018,lqz2021,aon2019,tyw2022} on the optimal local and remote control in the following aspects: Firstly, the two players for the LRSNG problem need to be designed respectively to optimize their own cost function, i.e., they are non-cooperative. While, the two controllers for the local and remote control problem are cooperative, which is derived to minimize a common cost function. Secondly, although the local and remote control problem has been well studied, the LRSNG problem has not been solved before.
It is stressed that the study of the LRSNG problem is significant both from the theoretical and application aspects. From the theoretical point of view, as pointed out earlier, the LRSNG problem has not been solved before in view of the challenges from the inconsistent information structure. Furthermore, the LRSNG can be potentially applied in unmanned systems, manufacturing systems and autonomous vehicles, smart grid, remote surgery, etc., see \cite{lx2018,aon2019,hv2000,gc2010,hnx2007,lczsm2014}, and the cited references therein.
As analyzed above, the inconsistent information structure makes that the two players are coupled with each other, hence finding the Nash equilibrium for LRSNG problem becomes difficult. To overcome this challenge, the maximum principle and orthogonal decomposition approach are adopted in this paper to solve the LRSNG problem, and the open-loop and closed-loop Nash equilibrium are derived. Firstly, by the use of the convex variational method, the Pontryagin maximum principle is derived. Moreover, the necessary and sufficient conditions for the existence of the open-loop optimal Nash equilibrium for the LRSNG problem are derived, which are based on the solution of FBSDEs from the maximum principle. Consequently, in order to find a feedback explicit Nash equilibrium strategy, the orthogonal decomposition method and the completing square approach are applied, then the closed-loop optimal Nash equilibrium strategy for the LRSNG problem is derived for the first time, which is based on the solution to modified coupled Riccati equations. Finally, a numerical example is given to illustrate the main results of this paper.
In this paper, a special class of LQ non-cooperative stochastic game with inconsistent information (i.e., LRSNG) is firstly solved. The contributions of this paper are twofold. On the one hand, the open-loop Nash equilibrium for LRSNG problem is solved, the necessary and sufficient solvability conditions are derived. On the other hand, the closed-loop Nash equilibrium for LRSNG is developed, and the optimal feedback Nash equilibrium strategy is shown to rely on the solution to coupled Riccati equations.
The remainder of the paper is organized as follows. The LRSNG Problem is formulated in Section II. Section III solves the open-loop Nash equilibrium, and the solvability conditions are investigated. Section IV is devoted to solving the closed-loop Nash equilibrium for LRSNG problem. In Section V, the effectiveness of the main results is illustrated by numerical examples. The paper is concluded in Section VI.
For convenience, we will use the following notations throughout the paper. $\mathbb{R}^n$ signifies the $n$-dimensional Euclidean space. $I_n$ presents the unit matrix of $n\times n$ dimension. $A^\mathrm{T}$ denotes the transpose of the matrix $A$. $\mathcal F(X)$ denotes the $\smalligma$-algebra generated by the random variable $X$. $A$ $\geq 0$ $(>0)$ means that $A$ is a positive semi-definite (positive definite) matrix. $Tr(A)$ represents the trace of matrix A. $\mathbb{E}[X]$ is the mathematical expectation, and $\mathbb{E}[X|\mathcal F_k]$ means the conditional expectation with respect to $\mathcal F_k$. The superscripts $L$, $R$ denote the local player and the remote player, respectively. $\inf$ means the infimum or the greatest lower bound of a set.
\smallection{Problem Formulation}
Throughout this paper, the following system dynamics shall be considered:
\color{blue}egin{align}\label{ss1}
x_{k + 1} = Ax_k + {B^L}u_k^L + B^Ru_k^R + w_k,
\end{align}
where $x_k \in \mathbb{R}^n$ is the state, $u_k^L\in \mathbb{R}^{m_1}$ and $u_k^R\in \mathbb{R}^{m_2}$ are the local player and the remote player, respectively. $A$, $B^L$, $B^R$ are the constant system matrices with appropriate dimensions. $w_k$ is system noise with zero mean
and covariance $\Sigma_{w}$, taking values in $\mathbb{R}^n$. The initial value of state is $x_0$ with mean $\mu$ and covariance $\Sigma_{x_0}$, taking values in $\mathbb{R}^n$. $x_0$ and $w_k$, independent of each other, are Gaussian random variables.
As illustrated in Figure \color{red}ef{Figure1}, the uplink channel from the local player to the remote player is unreliable, while the downlink channel from the remote player to the local player is perfect. Thus, the information sets accessed by the local player and the remote player are given as follows, respectively:
\color{blue}egin{align}\label{is1}
\mathcal{F}_k^R&=\smalligma \left\{\gamma_0x_0, \cdots, \gamma_kx_k \color{red}ight\},\normalsizeotag\\
\mathcal{F}_k^L&=\smalligma\left\{x_0, w_0, \cdots, w_{k-1},\gamma_0x_0, \cdots, \gamma_kx_k \color{red}ight\},
\end{align}
in which $\gamma _k$ is an independent identically distributed Bernoulli random variable describing the state information transmitted through unreliable communication channel, i.e.,
\color{blue}egin{equation}\label{uk3}
\gamma_k=\left\{ \color{blue}egin{array}{ll}
0, ~~\text{with probability}~~1-p,\\
1, ~~\text{with probability}~~p.
\end{array} \color{red}ight.
\end{equation}
In the above, $\gamma _k=1$ denotes that the state can be successfully accessed by the remote player, while $\gamma _k=0$
means the dropout of the state information from the local player to the remote player.
For the purpose of simplicity, the following notations are given:
\color{blue}egin{align}\label{cs1}
\mathcal{U}^L_N =\{u_0^L,\cdots,&u_N^L|u_k^L \in \mathbb{R}^{m_1}, u_k^L~is~\mathcal{F}_{k}^L-adapted,\normalsizeotag\\
&and~\smallum\limits_{k = 0}^N \mathbb{E}[(u_k^L)^Tu_k^L]<+\infty \},\normalsizeotag\\
\mathcal{U}^R_N =\{u_0^R,\cdots,&u_N^R|u_k^R \in \mathbb{R}^{m_2}, u_k^R~is~\mathcal{F}_{k}^R-adapted,\normalsizeotag\\
&and~\smallum\limits_{k = 0}^N \mathbb{E}[(u_k^R)^Tu_k^R]<+\infty\}.
\end{align}
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As analyzed above, the inconsistent information structure makes that the two players are coupled with each other, hence finding the Nash equilibrium for LRSNG problem becomes difficult. To overcome this challenge, the maximum principle and orthogonal decomposition approach are adopted in this paper to solve the LRSNG problem, and the open-loop and closed-loop Nash equilibrium are derived. Firstly, by the use of the convex variational method, the Pontryagin maximum principle is derived. Moreover, the necessary and sufficient conditions for the existence of the open-loop optimal Nash equilibrium for the LRSNG problem are derived, which are based on the solution of FBSDEs from the maximum principle. Consequently, in order to find a feedback explicit Nash equilibrium strategy, the orthogonal decomposition method and the completing square approach are applied, then the closed-loop optimal Nash equilibrium strategy for the LRSNG problem is derived for the first time, which is based on the solution to modified coupled Riccati equations. Finally, a numerical example is given to illustrate the main results of this paper.
In this paper, a special class of LQ non-cooperative stochastic game with inconsistent information (i.e., LRSNG) is firstly solved. The contributions of this paper are twofold. On the one hand, the open-loop Nash equilibrium for LRSNG problem is solved, the necessary and sufficient solvability conditions are derived. On the other hand, the closed-loop Nash equilibrium for LRSNG is developed, and the optimal feedback Nash equilibrium strategy is shown to rely on the solution to coupled Riccati equations.
The remainder of the paper is organized as follows. The LRSNG Problem is formulated in Section II. Section III solves the open-loop Nash equilibrium, and the solvability conditions are investigated. Section IV is devoted to solving the closed-loop Nash equilibrium for LRSNG problem. In Section V, the effectiveness of the main results is illustrated by numerical examples. The paper is concluded in Section VI.
For convenience, we will use the following notations throughout the paper. $\mathbb{R}^n$ signifies the $n$-dimensional Euclidean space. $I_n$ presents the unit matrix of $n\times n$ dimension. $A^\mathrm{T}$ denotes the transpose of the matrix $A$. $\mathcal F(X)$ denotes the $\smalligma$-algebra generated by the random variable $X$. $A$ $\geq 0$ $(>0)$ means that $A$ is a positive semi-definite (positive definite) matrix. $Tr(A)$ represents the trace of matrix A. $\mathbb{E}[X]$ is the mathematical expectation, and $\mathbb{E}[X|\mathcal F_k]$ means the conditional expectation with respect to $\mathcal F_k$. The superscripts $L$, $R$ denote the local player and the remote player, respectively. $\inf$ means the infimum or the greatest lower bound of a set.
\smallection{Problem Formulation}
Throughout this paper, the following system dynamics shall be considered:
\color{blue}egin{align}\label{ss1}
x_{k + 1} = Ax_k + {B^L}u_k^L + B^Ru_k^R + w_k,
\end{align}
where $x_k \in \mathbb{R}^n$ is the state, $u_k^L\in \mathbb{R}^{m_1}$ and $u_k^R\in \mathbb{R}^{m_2}$ are the local player and the remote player, respectively. $A$, $B^L$, $B^R$ are the constant system matrices with appropriate dimensions. $w_k$ is system noise with zero mean
and covariance $\Sigma_{w}$, taking values in $\mathbb{R}^n$. The initial value of state is $x_0$ with mean $\mu$ and covariance $\Sigma_{x_0}$, taking values in $\mathbb{R}^n$. $x_0$ and $w_k$, independent of each other, are Gaussian random variables.
As illustrated in Figure \color{red}ef{Figure1}, the uplink channel from the local player to the remote player is unreliable, while the downlink channel from the remote player to the local player is perfect. Thus, the information sets accessed by the local player and the remote player are given as follows, respectively:
\color{blue}egin{align}\label{is1}
\mathcal{F}_k^R&=\smalligma \left\{\gamma_0x_0, \cdots, \gamma_kx_k \color{red}ight\},\normalsizeotag\\
\mathcal{F}_k^L&=\smalligma\left\{x_0, w_0, \cdots, w_{k-1},\gamma_0x_0, \cdots, \gamma_kx_k \color{red}ight\},
\end{align}
in which $\gamma _k$ is an independent identically distributed Bernoulli random variable describing the state information transmitted through unreliable communication channel, i.e.,
\color{blue}egin{equation}\label{uk3}
\gamma_k=\left\{ \color{blue}egin{array}{ll}
0, ~~\text{with probability}~~1-p,\\
1, ~~\text{with probability}~~p.
\end{array} \color{red}ight.
\end{equation}
In the above, $\gamma _k=1$ denotes that the state can be successfully accessed by the remote player, while $\gamma _k=0$
means the dropout of the state information from the local player to the remote player.
For the purpose of simplicity, the following notations are given:
\color{blue}egin{align}\label{cs1}
\mathcal{U}^L_N =\{u_0^L,\cdots,&u_N^L|u_k^L \in \mathbb{R}^{m_1}, u_k^L~is~\mathcal{F}_{k}^L-adapted,\normalsizeotag\\
&and~\smallum\limits_{k = 0}^N \mathbb{E}[(u_k^L)^Tu_k^L]<+\infty \},\normalsizeotag\\
\mathcal{U}^R_N =\{u_0^R,\cdots,&u_N^R|u_k^R \in \mathbb{R}^{m_2}, u_k^R~is~\mathcal{F}_{k}^R-adapted,\normalsizeotag\\
&and~\smallum\limits_{k = 0}^N \mathbb{E}[(u_k^R)^Tu_k^R]<+\infty\}.
\end{align}
The quadratic cost functions associated with system \eqref{ss1} are given by
\color{blue}egin{small}
\color{blue}egin{align}
J_N^L({u_k^L},{u_k^R})&= \smallum\limits_{k = 0}^N \mathbb{E}[x_k^T{Q^L}{x_k}+(u_k^L)^T{S^L}u_k^L+(u_k^R)^T{M^L}{u_k^R}]\normalsizeotag\\
&+\mathbb{E}[ x_{N + 1}^T{P^L_{N + 1}}{x_{N + 1}}]\label{cf1},\\
J_N^R({u_k^L},{u_k^R})&= \smallum\limits_{k = 0}^N \mathbb{E}[x_k^T{Q^R}{x_k}+(u_k^L)^T{S^R}{u_k^L}+(u_k^R)^T{M^R}{u_k^R}]\normalsizeotag\\
&+\mathbb{E}[ x_{N + 1}^T{P^R_{N + 1}}{x_{N + 1}}]\label{cf2},
\end{align}
\end{small}
where $Q^L$, $Q^R$, $S^L$, $M^L$, $S^R$, $M^R$, $P_{N+1}^L$, $P_{N+1}^R$ are given symmetric weighting matrices with compatible dimensions.
\color{blue}egin{remark}\label{rm1}
It is stressed that the information sets $\mathcal{F}_{k}^L$ and $\mathcal{F}_k^R$ available to $u_k^L$ and $u_k^R$ are different, which is different from the consistent information structure case studied in previous works on LQ stochastic games \cite{slz2011,sjz2012a, sjz2012b,sy2019}. In fact, for $k=0,\cdots, N$, we have $\mathcal{F}_k^R \smallubseteq \mathcal{F}_{k}^L$, and the inconsistent information structure property would bring essential difficulties in solving the LQ stochastic two-person nonzero-sum game.
\end{remark}
Then, the open-loop and closed-loop LRSNG problems are stated as follows:
\textbf{Problem LRSNG.} For system \eqref{ss1} and cost functions \eqref{cf1}-\eqref{cf2}, find $u_k^L\in\mathcal{U}_N^L$ and $u_k^R\in\mathcal{U}_N^R$ to minimize $J_N^L$ and $J_N^R$, respectively.
\smallection{Open-loop Nash equilibrium}
In this section, we shall discuss the open-loop Nash equilibrium for Problem LRSNG in terms of FBSDEs, and the methods used are the convex variational principle and the maximum principle.
In the first place, the definition of the open-loop Nash equilibrium will be introduced.
\color{blue}egin{definition}\label{def1}
A pair $(u_k^{L,*},u_k^{R,*})$ $\in$ $\mathcal{U}_N^L \times \mathcal{U}_N^R$ is called an open-loop Nash equilibrium of Problem LRSNG if
\color{blue}egin{align}
J_N^L\left(u_k^{L,*},u_k^{R,*} \color{red}ight) \le J_N^L\left(u_k^{L},u_k^{R,*} \color{red}ight),\forall u_k^L\in \mathcal{U}_N^L,\normalsizeotag\\
J_N^R\left( u_k^{L,*},u_k^{R,*} \color{red}ight) \le J_N^R\left(u_k^{L,*},u_k^{R} \color{red}ight), \forall u_k^R\in \mathcal{U}_N^R.\label{pi1}
\end{align}
\end{definition}
Before stating the main results of this section, the following two lemmas will be given, which serve as preliminaries.
\color{blue}egin{lemma}\label{lem1}
If we denote $(u_{k}^{L,*},u_{k}^{R,*})$ as the open-loop Nash equilibrium, then set $u_{k}^{L,\varepsilon}=u_{k}^{L,*}+\varepsilon \delta u_k^{L}$, $\delta u_k^{L}\in \mathcal{U}_N^L$, $\varepsilon\in\mathbb{R}$, ${z_{k}} = \frac{{x_{k}^\varepsilon - {x_{k}}}}{\varepsilon }$,
and denote ${x_{k}^\varepsilon}$ and ${J_N^L}(u_k^{L,\varepsilon},u_k^{R,*})$ as the corresponding state and cost function associated with $u_{k}^{L,\varepsilon}$, $k =0, \cdots N$, then there holds
\color{blue}egin{align}\label{lbf1}
&J_N^{L}(u_k^{L,\varepsilon},u_{k}^{R,*}) - {J_N^{L}(u_k^{L,*},u_{k}^{R,*})}\\
=&{\varepsilon ^2}\delta J_N^{L}(\delta u_k^{L})+ 2\varepsilon \smallum\limits_{k = 0}^N \mathbb{E}[[(B^L)^T\theta _k^L + {S^L}u_k^{L,*}]^T\delta u_k^{L}],\normalsizeotag
\end{align}
where $\delta J_N^{L}(\delta u_k^{L})$ is given by
\color{blue}egin{align}\label{lbf2}
&\delta J_N^{L}(\delta u_k^{L})=\smallum\limits_{k = 0}^N \mathbb{E}[z_k^T{Q^L}{z_k} + (\delta
u_k^{L})^T{S^L}\delta {u_k^{L}}]\normalsizeotag\\
&+ \mathbb{E}[z_{N + 1}^TP_{N + 1}^L{z_{N + 1}}].
\end{align}
In the above, the costate ${\theta}_k^L (k = 0, \cdots, N)$ satisfies the following backward stochastic difference equation
\color{blue}egin{equation}\label{lbf4}
\theta _{k-1}^{L}=Q^Lx_k+ \mathbb{E}\left[ A^T\theta _{k}^{L}|\mathcal{F}_k^L \color{red}ight],\theta_N^L=P_{N+1}^{L}x_{N+1}.
\end{equation}
\end{lemma}
\color{blue}egin{proof}
Using the notations introduced above, it can be derived that $z_k$ satisfies
\color{blue}egin{align}\label{lbf3}
z_{k+1}=Az_k+B^L\delta u_k^{L},
\end{align}
with initial condition $z_0=0$.
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\color{blue}egin{remark}\label{rm1}
It is stressed that the information sets $\mathcal{F}_{k}^L$ and $\mathcal{F}_k^R$ available to $u_k^L$ and $u_k^R$ are different, which is different from the consistent information structure case studied in previous works on LQ stochastic games \cite{slz2011,sjz2012a, sjz2012b,sy2019}. In fact, for $k=0,\cdots, N$, we have $\mathcal{F}_k^R \smallubseteq \mathcal{F}_{k}^L$, and the inconsistent information structure property would bring essential difficulties in solving the LQ stochastic two-person nonzero-sum game.
\end{remark}
Then, the open-loop and closed-loop LRSNG problems are stated as follows:
\textbf{Problem LRSNG.} For system \eqref{ss1} and cost functions \eqref{cf1}-\eqref{cf2}, find $u_k^L\in\mathcal{U}_N^L$ and $u_k^R\in\mathcal{U}_N^R$ to minimize $J_N^L$ and $J_N^R$, respectively.
\smallection{Open-loop Nash equilibrium}
In this section, we shall discuss the open-loop Nash equilibrium for Problem LRSNG in terms of FBSDEs, and the methods used are the convex variational principle and the maximum principle.
In the first place, the definition of the open-loop Nash equilibrium will be introduced.
\color{blue}egin{definition}\label{def1}
A pair $(u_k^{L,*},u_k^{R,*})$ $\in$ $\mathcal{U}_N^L \times \mathcal{U}_N^R$ is called an open-loop Nash equilibrium of Problem LRSNG if
\color{blue}egin{align}
J_N^L\left(u_k^{L,*},u_k^{R,*} \color{red}ight) \le J_N^L\left(u_k^{L},u_k^{R,*} \color{red}ight),\forall u_k^L\in \mathcal{U}_N^L,\normalsizeotag\\
J_N^R\left( u_k^{L,*},u_k^{R,*} \color{red}ight) \le J_N^R\left(u_k^{L,*},u_k^{R} \color{red}ight), \forall u_k^R\in \mathcal{U}_N^R.\label{pi1}
\end{align}
\end{definition}
Before stating the main results of this section, the following two lemmas will be given, which serve as preliminaries.
\color{blue}egin{lemma}\label{lem1}
If we denote $(u_{k}^{L,*},u_{k}^{R,*})$ as the open-loop Nash equilibrium, then set $u_{k}^{L,\varepsilon}=u_{k}^{L,*}+\varepsilon \delta u_k^{L}$, $\delta u_k^{L}\in \mathcal{U}_N^L$, $\varepsilon\in\mathbb{R}$, ${z_{k}} = \frac{{x_{k}^\varepsilon - {x_{k}}}}{\varepsilon }$,
and denote ${x_{k}^\varepsilon}$ and ${J_N^L}(u_k^{L,\varepsilon},u_k^{R,*})$ as the corresponding state and cost function associated with $u_{k}^{L,\varepsilon}$, $k =0, \cdots N$, then there holds
\color{blue}egin{align}\label{lbf1}
&J_N^{L}(u_k^{L,\varepsilon},u_{k}^{R,*}) - {J_N^{L}(u_k^{L,*},u_{k}^{R,*})}\\
=&{\varepsilon ^2}\delta J_N^{L}(\delta u_k^{L})+ 2\varepsilon \smallum\limits_{k = 0}^N \mathbb{E}[[(B^L)^T\theta _k^L + {S^L}u_k^{L,*}]^T\delta u_k^{L}],\normalsizeotag
\end{align}
where $\delta J_N^{L}(\delta u_k^{L})$ is given by
\color{blue}egin{align}\label{lbf2}
&\delta J_N^{L}(\delta u_k^{L})=\smallum\limits_{k = 0}^N \mathbb{E}[z_k^T{Q^L}{z_k} + (\delta
u_k^{L})^T{S^L}\delta {u_k^{L}}]\normalsizeotag\\
&+ \mathbb{E}[z_{N + 1}^TP_{N + 1}^L{z_{N + 1}}].
\end{align}
In the above, the costate ${\theta}_k^L (k = 0, \cdots, N)$ satisfies the following backward stochastic difference equation
\color{blue}egin{equation}\label{lbf4}
\theta _{k-1}^{L}=Q^Lx_k+ \mathbb{E}\left[ A^T\theta _{k}^{L}|\mathcal{F}_k^L \color{red}ight],\theta_N^L=P_{N+1}^{L}x_{N+1}.
\end{equation}
\end{lemma}
\color{blue}egin{proof}
Using the notations introduced above, it can be derived that $z_k$ satisfies
\color{blue}egin{align}\label{lbf3}
z_{k+1}=Az_k+B^L\delta u_k^{L},
\end{align}
with initial condition $z_0=0$.
Consequently, the variation of the cost function can be calculated as follows.
\color{blue}egin{align*}
&{J_N^L}(u_k^{L,\varepsilon},u_k^{R,*}) - {J_N^L}(u_k^{L,*},u_k^{R,*})\\
&= \smallum\limits_{k = 0}^N \mathbb{E}[{({x_k} + \varepsilon {z_k})}^T{Q^L}({x_k} + \varepsilon {z_k})\\
&+ {({u_k^{L,*}} + \varepsilon \delta {u_k^{L}})^T}{S^L}(u_k^{L,*} + \varepsilon \delta u_k^{L})\\
&+ (u_k^{R,*})^{T}{M^L}{u_k^{R,*}}]+ \mathbb{E}[({x_{N + 1}} + \varepsilon{z_{N + 1}})^T{P^L_{N + 1}}\\
&\times({x_{N + 1}}+ \varepsilon{z_{N + 1}})]- \mathbb{E}[x_{N + 1}^T{P^L_{N + 1}}{x_{N + 1}}]\\
&- \smallum\limits_{k = 0}^N \mathbb{E}[x_k^T{Q^L}{x_k} + (u_k^{L,*})^TS^Lu_k^{L,*}+(u_k^{R,*})^{T}{M^L}{u_k^{R,*}}]\\
&= 2\varepsilon \mathbb{E}[\smallum\limits_{k = 0}^N[x_k^TQ^Lz_k + (u_k^{L,*})^TS^L\delta u_k^{L}]\\
&+x_{N + 1}^TP^L_{N +1}z_{N + 1}] \\
&+ {\varepsilon ^2}\mathbb{E}[\smallum\limits_{k = 0}^N[z_k^T{Q^L}{z_k} + (\delta u_k^{L})^T{S^L}\delta
u_k^{L}]\\
&+z_{N + 1}^TP_{N + 1}^L{z_{N + 1}}].
\end{align*}
Furthermore, from \eqref{lbf4} and \eqref{lbf3}, we have
\color{blue}egin{align*}
&\mathbb{E}[\smallum\limits_{k = 0}^N[x_k^TQ^Lz_k + (u_k^{L,*})^TS^L\delta u_k^{L}]\\
&+x_{N + 1}^TP^L_{N +1}z_{N + 1}] \\
&=\mathbb{E}[\smallum\limits_{k = 0}^N[\theta _{k - 1}^L- E[A^T\theta
_k^L|\mathcal{F}_k^L]]^Tz_k \\
&+\smallum\limits_{k = 0}^N(u_k^{L,*})^TS^L\delta u_k^{L}+(\theta _N^L)^Tz_{N + 1}] \\
&=\mathbb{E}[\smallum\limits_{k = 0}^N[(B^L)^T\theta _k^L+S^Lu_k^{L,*}]^T\delta u_k^{L}].
\end{align*}
The proof is complete.
\end{proof}
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Similar to Lemma \color{red}ef{lem1} and its proof, the following lemma can be given without proof.
\color{blue}egin{lemma}\label{lem2}
For the open-loop Nash equilibrium $(u_{k}^{L,*},u_{k}^{R,*})$,
choose $\eta \in \mathbb{R}$, and for $k = 0, \cdots, N$, let $u_k^{R,\eta}= u_k^{R,*}+\eta\Delta u_k^{R}$, where $\Delta u_k^{R}\in \mathcal{U}_N^R$, $y_{k} = \frac{{x_{k}^\eta -{x_{k}}}}{\eta}$. Let ${x_{k}^\eta}$, $J_N^R(u_k^{L,*},u_k^{R,\eta})$ be the state and cost function associated with
$u_k^{R,\eta}$, $k =0, \cdots, N$, respectively. Then, we have
\color{blue}egin{align}\label{rbf1}
&J_N^{R}(u_k^{L,*},u_k^{R,\eta}) - J_N^{R}(u_k^{L,*},u_k^{R,*})\normalsizeotag\\
&={\eta ^2}\Delta J_N^{R}(\Delta u_k^{R})+2\eta \smallum\limits_{k =0}^N \mathbb{E}[[{(B^R)^T}\theta _k^R + {M^R}u_k^{R,*}]^T\Delta u_k^{R}],
\end{align}
where $\Delta J_N^{R}(\Delta u_k^{R})$ is given by
\color{blue}egin{align}\label{rbf2}
\Delta J_N^{R}(\Delta u_k^{R})=&\smallum\limits_{k = 0}^N \mathbb{E}[y_k^T{Q^R}{y_k} + {{(\Delta u_k^{R})}^T}M^R\Delta u_k^{R}]\normalsizeotag\\
&+ \mathbb{E}[y_{N + 1}^TP_{N + 1}^R{y_{N + 1}}].
\end{align}
And the costate $\theta_k^R(k =0, \cdots, N)$ satisfies
\color{blue}egin{align}\label{rbf4}
\theta _{k - 1}^R = {Q^R}{x_k} + \mathbb{E}[{A^T}{\theta _k^R}|{\mathcal F_k^L}],
\end{align}
with terminal condition $\theta_{N}^R=P_{N+1}^Rx_{N+1}$.
\end{lemma}
Using the results derived in Lemmas \color{red}ef{lem1}-\color{red}ef{lem2}, the main results of this section will be presented, and the necessary and sufficient conditions for the open-loop Nash equilibrium of Problem LRSNG will be derived.
\color{blue}egin{theorem}\label{th-01}
For system \eqref{ss1} and cost functions \eqref{cf1} and \eqref{cf2}, the open-loop Nash equilibrium $(u_k^{L,*}, u_k^{R,*})$ for Problem LRSNG is unique if and only if the following two conditions are satisfied
1) The convexity condition holds:
\color{blue}egin{align}\label{cc}
\inf \delta J_N^L(\delta u_k^{L})\geq 0,~~\text{and}~~
\inf \Delta J_N^R(\Delta u_k^{R})\geq 0,
\end{align}
in which $\delta J_N^L(\delta u_k^{L})$ and $\Delta J_N^R(\Delta u_k^{R})$ are given by \eqref{lbf2} and \eqref{rbf2}, respectively.
2) The stationary conditions can be uniquely solved:
\color{blue}egin{align}
0&=S^Lu_k^{L,*}+\mathbb{E}[(B^L)^T\theta _k^L|\mathcal{F}_k^L],\label{ec1}\\
0&=M^Ru_k^{R,*}+ \mathbb{E}[(B^R)^T\theta _k^R|\mathcal{F}_k^R],\label{ec2}
\end{align}
where $\theta_k^L$, $\theta _k^R$ satisfy \eqref{lbf4} and \eqref{rbf4}, respectively.
\end{theorem}
\color{blue}egin{proof}
`Necessity': Suppose the open-loop Nash equilibrium $(u_k^{L,*},u_k^{R,*})$ for Problem LRSNG is unique, we will show the two conditions 1)-2) are satisfied. In fact, for the open-loop Nash equilibrium $(u_k^{L,*},u_k^{R,*})$, from Lemmas \color{red}ef{lem1}-\color{red}ef{lem2}, we know that for arbitrary $\delta u_k^{L}\in \mathcal{U}_N^L$, $\varepsilon\in \mathbb{R}$, and arbitrary $\Delta u_k^{R}\in \mathcal{U}_N^R$, $\eta \in \mathbb{R}$,
there holds
\color{blue}egin{align}\label{diff1}
&{J_N^L}(u_k^{L,\varepsilon},u_k^{R,*}) - {J_N^L}(u_k^{L,*},u_k^{R,*})\normalsizeotag\\
= &2\varepsilon \smallum\limits_{k = 0}^N \mathbb{E}[[(B^L)^T\theta _k^L + S^Lu_k^{L,*}]^T\delta
u_k^{L}]+{\varepsilon ^2}\delta J_N^L(\delta u_k^{L}) \normalsizeotag\\
\geq& 0,
\end{align}
and
\color{blue}egin{align}\label{diff2}
&J_N^R(u_k^{L,*},u_k^{R,\eta}) - J_N^R(u_k^{L,*},u_k^{R,*})\normalsizeotag\\
=&2\eta \smallum\limits_{k = 0}^N \mathbb{E}[[(B^R)^T\theta _k^R + {M^R}u_k^{R,*}]^T\Delta u_k^{R}]+{\eta ^2}\Delta J_N^R(\Delta u_k^{R}) \normalsizeotag\\
\geq& 0.
\end{align}
On the one hand, suppose the convexity condition \eqref{cc} is not true, then there exists some $\delta u_k^{L}$, $k=0, \cdots, N$ such that
${J_N^L}(u_k^{L,\varepsilon} ,u_k^{R,*}) - {J_N^L}(u_k^{L,*},u_k^{R,*})=-\infty$ with
$\varepsilon\color{red}ightarrow\infty$. For the same reason, there exists some $\Delta u_k^{R}$, $k=0, \cdots, N$ satisfying
$J_N^R(u_k^{L,*},u_k^{R,\eta} )- J_N^R(u_k^{L,*},u_k^{R,*})=-\infty$ with $\eta\color{red}ightarrow\infty$.
This contradicts 1).
On the other hand, if 2) is not satisfied, then we can assume that
\color{blue}egin{align}
S^Lu_k^{L,*}+\mathbb{E}[(B^L)^T\theta _k^L|\mathcal{F}_k^L]&=\Theta_k^L \normalsizeeq 0,\label{u11}\\
M^Ru_k^{R,*}+\mathbb{E}[(B^R)^T\theta _k^R|\mathcal{F}_k^R]&=\Theta_k^R \normalsizeeq 0.\label{u12}
\end{align}
In this case, if we choose $\delta u_k^{L}=\Theta_k^L$ and $\Delta u_k^{R}=\Theta_k^R$, then
from \eqref{diff1} and \eqref{diff2} we have
\color{blue}egin{align*}
&{J_N^L}(u_k^{L,\varepsilon},u_k^{R,*}) - {J_N^L}(u_k^{L,*},u_k^{R,*})\\
&= 2\varepsilon\smallum_{k=0}^{N} (\Theta_k^L)^T\Theta_k^L +\varepsilon^2\delta J_N^L(\delta
u_k^{L}),\\
&J_N^R(u_k^{L,*},u_k^{R,\eta} ) - J_N^R(u_k^{L,*},u_k^{R,*})\\
&=2\eta\smallum_{k=0}^{N} (\Theta_k^R)^T\Theta_k^R +\eta^2\Delta J_N^R(\Delta u_k^{R}).
\end{align*}
Then, we can always find some $\varepsilon$ and $\eta$ $<0$ such that ${J_N^L}(u_k^{L,\varepsilon},u_k^{R,*}) -
{J_N^L}(u_k^{L,*},u_k^{R,*})$and $J_N^R(u_k^{L,*},u_k^{R,\eta} ) - J_N^R(u_k^{L,*},u_k^{R,*})$ $<0$,
which contradicts with \eqref{diff1}, \eqref{diff2}. Thus, $\Theta_k^L$, $\Theta_k^R=0$, i.e.,
\eqref{ec1}, \eqref{ec2} holds. This ends the necessity proof.
`Sufficiency': Suppose the two conditions 1)-2) hold, we need to prove that the open-loop Nash equilibrium $(u_k^{L,*}, u_k^{R,*})$ is unique.
Actually, it can be deduced from \eqref{lbf1} and \eqref{rbf1} that for any
$\varepsilon\in\mathbb{R}$, $\eta\in\mathbb{R}$ and $\delta u_k^{L}\in \mathcal{U}_N^L$, $\Delta u_k^{R}\in \mathcal{U}_N^R$, we have
\color{blue}egin{align*}
{J_N^L}(u_k^{L,\varepsilon},u_k^{R,*}) - {J_N^L}(u_k^{L,*},u_k^{R,*})&=\varepsilon^2\delta
J_N^L(\delta u_k^{L})\geq 0,\\
J_N^R(u_k^{L,*},u_k^{R,\eta} ) - J_N^R(u_k^{L,*},u_k^{R,*})&={\eta ^2}\Delta J_N^R(\Delta
u_k^{R})\geq 0,
\end{align*}
which means that open-loop Nash equilibria of Problem LRSNG is uniquely solvable. The proof is complete.
\end{proof}
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|
\color{blue}egin{remark}\label{rm2}
By using the variational method, the maximum principle for Problem LRSNG is derived. Furthermore, Theorem \color{red}ef{th-01} provides the necessary and sufficient solvability conditions for the open-loop Nash equilibrium of Problem LRSNG with inconsistent information structure for the first time.
\end{remark}
\color{blue}egin{remark}\label{rm3}
From Theorem \color{red}ef{th-01}, the forward and backward difference equations (FBSDEs) can be given as follows
\color{blue}egin{align}\label{fbsde}
\left\{ \color{blue}egin{array}{ll}
&x_{k + 1}= Ax_k + {B^L}u_k^L + B^Ru_k^R + w_k,\\
&\theta _{k-1}^{L}=Q^Lx_k+ \mathbb{E}\left[ A^T\theta_{k}^{L}|\mathcal{F}_k^L \color{red}ight],\\
&\theta _{k - 1}^R= {Q^R}{x_k} + \mathbb{E}[{A^T}{\theta _k^R}|{\mathcal F_k^L}],\\
&0=S^Lu_k^{L,*}+\mathbb{E}[(B^L)^T\theta _k^L|\mathcal{F}_k^L],\\
&0=M^Ru_k^{R,*}+ \mathbb{E}[(B^R)^T\theta _k^R|\mathcal{F}_k^R],\\
&\theta_N^L=P_{N+1}^{L}x_{N+1},\theta_N^R=P_{N+1}^{R}x_{N+1}.
\end{array} \color{red}ight.
\end{align}
Due to the different information structure caused by the unreliable uplink channel, the FBSDEs \eqref{fbsde} cannot be decoupled as the traditional consistent information case, see \cite{nglb2014,nb2012}. Hence the explicit feedback Nash equilibrium cannot be derived via solving FBSDEs \eqref{fbsde}, which is challenging.
\end{remark}
\smallection{Closed-loop Nash equilibrium}
In this section, the closed-loop Nash equilibrium for Problem LRSNG will be studied. As illustrated in Remark \color{red}ef{rm3}, in view of the existence of inconsistent information structure, the FBSDEs cannot be decoupled, which indicates the explicit Nash equilibrium cannot be derived via decoupling FBSDEs \eqref{fbsde} of Theorem \color{red}ef{th-01}. Alternatively, our aim is to obtain a feedback explicit Nash equilibrium (the closed-loop Nash equilibrium) by the use of the orthogonal decomposition and completing square approaches.
\smallubsection{Preliminaries}
To begin with, some preliminary results shall be introduced on the orthogonal decomposition method in deriving a feedback explicit Nash equilibrium (the closed-loop Nash equilibrium).
Without loss of generality, the following standard assumption will be made, see \cite{eln2013,y2013,ls1995}.
\color{blue}egin{assumption}\label{ass1}
The weighting matrices in \eqref{cf1}, \eqref{cf2} satisfy
$Q^L\ge 0$, $Q^R\ge 0$, $S^L>0$, $S^R>0$, $M^L>0$, $M^R>0$, and $P_{N+1}^L\ge 0$, $P_{N+1}^R\ge 0$.
\end{assumption}
For the sake of discussion, the following notations are introduced.
\color{blue}egin{align}\label{ncl}
&{\Lambda^L}=
\color{blue}egin{bmatrix} S^L & \\
& M^L \end{bmatrix},
{\Lambda^R}=
\color{blue}egin{bmatrix} S^R & \\
& M^R \end{bmatrix},
U_k=\color{blue}egin{bmatrix}
\hat{u}_k^L\\
u_k^R\\
\end{bmatrix},\normalsizeotag\\
&\mathcal {B}=\color{blue}egin{bmatrix} B^L&B^R \end{bmatrix},
\hat{u}_k^L=\mathbb{E}[u_k^L|\mathcal F_k^R],
\tilde u_k^L=u_k^L-\hat {u}_k^L.
\end{align}
Consequently, due to the existence of the unreliable uplink channel from the local player to the remote player, based on the disturbed state information, an estimator should be derived, which will be presented in the following lemma.
\color{blue}egin{lemma}\label{lem3}
For system \eqref{ss1} and the disturbed state \eqref{is1}, in the sense of minimizing the error covariance, using the notations introduced in \eqref{ncl}, the optimal estimator $\hat x_{k|k}=\mathbb{E}[x_k|\gamma_0x_0,\cdots, \gamma_kx_k]$ can be calculated as
\color{blue}egin{align}\label{kw6}
\hat{x}_{k|k}={\gamma _k}x_k+(1 - {\gamma _k})(A\hat x_{k-1|k-1}+\mathcal BU_{k-1}),
\end{align}
with initial condition $\hat x_{0|0}={\gamma _0}x_0+(1 - {\gamma _0})\mu$, and $\mu$ is the mean value of the initial state $x_0$. Moreover, the estimation error ${{\tilde x}_k} = {x_k} - {{\hat x}_{k|k}}$ satisfies
\color{blue}egin{align}\label{kw7}
\tilde{x}_k= (1 - {\gamma _k})(A{{\tilde x}_{k - 1}}
+ {B^L}{{\tilde u}_{k - 1}^L} + {w_{k - 1}}),
\end{align}
with initial condition $\tilde{x}_0=(1-\gamma _0)(x_0-\mu)$.
\end{lemma}
\color{blue}egin{proof}
The detailed proof can be found in \cite{qz2017a,qz2017b}, which is omitted here.
\end{proof}
In order to be consistent with the information structure introduced in \eqref{cs1}, the form of the feedback explicit Nash equilibrium of Problem LRSNG is assumed as follows, see \cite{sjz2012a,sjz2012b,ott2016,aon2019}. Specifically, based on \eqref{ncl} and Lemma \color{red}ef{lem3}, we assume:
\color{blue}egin{assumption}\label{ass2}
On the one hand, $U_k$ is the feedback of the optimal estimator $\hat{x}_{k|k}$, i.e., $U_k=\tilde{K}_k^{L}\hat{x}_{k|k}$. On the other hand, $\tilde u_k^{L}$ is the feedback of the optimal estimation error $\tilde x_k$, i.e., $\tilde u_k^{L}=\tilde{K}_k^R\tilde x_k$.
\end{assumption}
From Assumption \color{red}ef{ass2}, we know that the feedback explicit Nash equilibrium (the closed-loop Nash equilibrium) is of the following form:
\color{blue}egin{align}\label{form}
u_k^L & =[I_{m_1} ~ 0]\tilde{K}_k^{L}\hat{x}_{k|k}+\tilde{K}_k^R\tilde x_{k},~\text{and}~
u_k^{R} =[0 ~ I_{m_2}]\tilde{K}_k^{L}\hat{x}_{k|k}.
\end{align}
In the following, we will introduce the definition of closed-loop Nash equilibrium.
\color{blue}egin{definition}\label{def2}
Under Assumption \color{red}ef{ass2}, a pair $(K_k^{L},K_k^{R})$ is called the closed-loop Nash equilibrium of Problem LRSNG if for any $(\tilde{K}_k^{L},\tilde{K}_k^{R})\in \mathbb{R}^{(m_1+m_2) \times n}\times \mathbb{R}^{m_1 \times n}$, the following relationships hold:
\color{blue}egin{align}
J_N^L&([I_{m_1} ~~ 0]K_k^{L}\hat{x}_{k|k}+{K}_k^{R}\tilde x_{k},[0 ~~ I_{m_2}]K_k^{L}\hat{x}_{k|k})\label{clne1}\\
&\leq J_N^L([I_{m_1} ~~ 0]\tilde{K}_k^{L}\hat{x}_{k|k}+{K}_k^{R}\tilde x_{k},[0 ~~ I_{m_2}]\tilde{K}_k^L\hat{x}_{k|k}),\normalsizeotag\\
J_N^R&([I_{m_1} ~~ 0]K_k^{L}\hat{x}_{k|k}+{K}_k^{R}\tilde x_{k},[0 ~~ I_{m_2}]K_k^{L}\hat{x}_{k|k})\label{clne2}\\
&\leq J_N^R([I_{m_1} ~~ 0]K_k^{L}\hat{x}_{k|k}+\tilde{K}_k^R\tilde x_{k},[0 ~~ I_{m_2}]K_k^{L}\hat{x}_{k|k}).\normalsizeotag
\end{align}
\end{definition}
From Assumption \color{red}ef{ass2} and Definition \color{red}ef{def2}, the following lemma on the orthogonal property can be directly derived.
\color{blue}egin{lemma}\label{lem4}
Under Assumption \color{red}ef{ass2}, for arbitrary $(\tilde{K}_k^{L},\tilde{K}_k^{R})\in \mathbb{R}^{(m_1+m_2) \times n}\times \mathbb{R}^{m_1 \times n}$, $U_k$ and $\tilde u_k^{L}$ given in Assumption \color{red}ef{ass2} are orthogonal, i.e., $\mathbb{E}[U_k^TH\tilde u_k^{L}]=0$ for any constant matrix $H$ with compatible dimensions.
\end{lemma}
\color{blue}egin{proof}
In fact, from Assumption \color{red}ef{ass2}, we know that $U_k=\tilde{K}_k^L\hat{x}_{k|k}$ and $\tilde u_k^{L}=\tilde{K}_k^R\tilde x_k$, hence
\color{blue}egin{align}
\mathbb{E}[U_k^TH\tilde{u}_k^L]
=&\mathbb{E}[(\tilde{K}_k^L\hat{x}_{k|k})^TH\tilde{K}^R_k\tilde{x}_k]\normalsizeotag\\
=&\mathbb{E}[\hat{x}_{k|k}^T(\tilde{K}_k^L)^TH\tilde{K}^R_k\tilde{x}_k]\normalsizeotag\\
=&0,
\end{align}
in which the orthogonality of $\hat{x}_{k|k}$ and $\tilde{x}_k$ has been inserted.
\end{proof}
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In order to be consistent with the information structure introduced in \eqref{cs1}, the form of the feedback explicit Nash equilibrium of Problem LRSNG is assumed as follows, see \cite{sjz2012a,sjz2012b,ott2016,aon2019}. Specifically, based on \eqref{ncl} and Lemma \color{red}ef{lem3}, we assume:
\color{blue}egin{assumption}\label{ass2}
On the one hand, $U_k$ is the feedback of the optimal estimator $\hat{x}_{k|k}$, i.e., $U_k=\tilde{K}_k^{L}\hat{x}_{k|k}$. On the other hand, $\tilde u_k^{L}$ is the feedback of the optimal estimation error $\tilde x_k$, i.e., $\tilde u_k^{L}=\tilde{K}_k^R\tilde x_k$.
\end{assumption}
From Assumption \color{red}ef{ass2}, we know that the feedback explicit Nash equilibrium (the closed-loop Nash equilibrium) is of the following form:
\color{blue}egin{align}\label{form}
u_k^L & =[I_{m_1} ~ 0]\tilde{K}_k^{L}\hat{x}_{k|k}+\tilde{K}_k^R\tilde x_{k},~\text{and}~
u_k^{R} =[0 ~ I_{m_2}]\tilde{K}_k^{L}\hat{x}_{k|k}.
\end{align}
In the following, we will introduce the definition of closed-loop Nash equilibrium.
\color{blue}egin{definition}\label{def2}
Under Assumption \color{red}ef{ass2}, a pair $(K_k^{L},K_k^{R})$ is called the closed-loop Nash equilibrium of Problem LRSNG if for any $(\tilde{K}_k^{L},\tilde{K}_k^{R})\in \mathbb{R}^{(m_1+m_2) \times n}\times \mathbb{R}^{m_1 \times n}$, the following relationships hold:
\color{blue}egin{align}
J_N^L&([I_{m_1} ~~ 0]K_k^{L}\hat{x}_{k|k}+{K}_k^{R}\tilde x_{k},[0 ~~ I_{m_2}]K_k^{L}\hat{x}_{k|k})\label{clne1}\\
&\leq J_N^L([I_{m_1} ~~ 0]\tilde{K}_k^{L}\hat{x}_{k|k}+{K}_k^{R}\tilde x_{k},[0 ~~ I_{m_2}]\tilde{K}_k^L\hat{x}_{k|k}),\normalsizeotag\\
J_N^R&([I_{m_1} ~~ 0]K_k^{L}\hat{x}_{k|k}+{K}_k^{R}\tilde x_{k},[0 ~~ I_{m_2}]K_k^{L}\hat{x}_{k|k})\label{clne2}\\
&\leq J_N^R([I_{m_1} ~~ 0]K_k^{L}\hat{x}_{k|k}+\tilde{K}_k^R\tilde x_{k},[0 ~~ I_{m_2}]K_k^{L}\hat{x}_{k|k}).\normalsizeotag
\end{align}
\end{definition}
From Assumption \color{red}ef{ass2} and Definition \color{red}ef{def2}, the following lemma on the orthogonal property can be directly derived.
\color{blue}egin{lemma}\label{lem4}
Under Assumption \color{red}ef{ass2}, for arbitrary $(\tilde{K}_k^{L},\tilde{K}_k^{R})\in \mathbb{R}^{(m_1+m_2) \times n}\times \mathbb{R}^{m_1 \times n}$, $U_k$ and $\tilde u_k^{L}$ given in Assumption \color{red}ef{ass2} are orthogonal, i.e., $\mathbb{E}[U_k^TH\tilde u_k^{L}]=0$ for any constant matrix $H$ with compatible dimensions.
\end{lemma}
\color{blue}egin{proof}
In fact, from Assumption \color{red}ef{ass2}, we know that $U_k=\tilde{K}_k^L\hat{x}_{k|k}$ and $\tilde u_k^{L}=\tilde{K}_k^R\tilde x_k$, hence
\color{blue}egin{align}
\mathbb{E}[U_k^TH\tilde{u}_k^L]
=&\mathbb{E}[(\tilde{K}_k^L\hat{x}_{k|k})^TH\tilde{K}^R_k\tilde{x}_k]\normalsizeotag\\
=&\mathbb{E}[\hat{x}_{k|k}^T(\tilde{K}_k^L)^TH\tilde{K}^R_k\tilde{x}_k]\normalsizeotag\\
=&0,
\end{align}
in which the orthogonality of $\hat{x}_{k|k}$ and $\tilde{x}_k$ has been inserted.
\end{proof}
\color{blue}egin{remark}
In fact, noting that $u_k^L$ is $\mathcal{F}_{k}^L$-adapted, $u_k^R$ is $\mathcal{F}_{k}^R$-adapted, and $\mathcal{F}_k^R \smallubseteq \mathcal{F}_{k}^L$. Based on this basic property, hence Assumption \color{red}ef{ass2} is given, the orthogonal of $U_k$ and $\tilde u_k^{L}$ is shown in Lemma \color{red}ef{lem4}. The above method is called the orthogonal decomposition method, which is important in deriving the closed-loop Nash equilibrium.
\end{remark}
Using the results of Lemma \color{red}ef{lem4}, the cost functions \eqref{cf1}-\eqref{cf2} can be equivalently rewritten as follows:
\color{blue}egin{align}
J_N^L(u_k^L, u_k^R)\normalsizeotag
&= \smallum\limits_{k = 0}^N \mathbb{E}[x_k^T{Q^L}{x_k} + U_k^T{\Lambda^L}{U_k} + (\tilde u_k^L)^T{S^L}{\tilde
u_k^L}]\normalsizeotag\\
&+\mathbb{E}[ x_{N + 1}^T{P_{N + 1}^L}{x_{N + 1}}],\label{kw2}\\
J_N^R({u_k^L},u_k^R)\normalsizeotag
&= \smallum\limits_{k = 0}^N \mathbb{E}[x_k^T{Q^R}{x_k} + U_k^T{\Lambda^R}{U_k} + (\tilde u_k^L)^T{S^R}{\tilde
u_k^L}]\normalsizeotag\\
&+\mathbb{E}[ x_{N + 1}^T{P_{N + 1}^R}{x_{N + 1}}]\label{kw3}.
\end{align}
Besides, for simplicity, we can always rewrite system dynamics \eqref{ss1} as:
\color{blue}egin{align}\label{kw1}
x_{k+1}=Ax_k+\mathcal BU_k+B^L\tilde{u}_k^L+w_k.
\end{align}
\smallubsection{The closed-loop Nash equilibrium}
Before presenting the main results of closed-loop Nash equilibrium, we will introduce the following coupled Riccati equations in the first place.
{\smallmall\color{blue}egin{align}\label{rere}
\left\{ \color{blue}egin{array}{ll}
P_k^L&= {A^T} P_{k+1}^LA+ Q^L-(K_k^L)^T(\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal {B})K_k^L,\\
P_k^R&={A^T} P_{k+1}^RA + Q^R-(K_k^R)^T[{S^R}+(B^L)^TP_{k+1}^RB^L]K_k^R\\
&+p[(A + B^LK_k^R)^T \Omega_{k+1}^R(A + B^LK_k^R)\\
&- {(A + B^LK_k^R)^T} P_{k+1}^R(A + B^LK_k^R)],\\
\Omega_k^L&=p[(A + B^LK_k^R)^TP_{k+1}^L(A + B^LK_k^R)]\\
&+(1-p)[(A + B^LK_k^R)^T\Omega_{k+1}^L(A + B^LK_k^R)]\\
&+{Q^L}+ (K_k^R)^T{S^L}K_k^R,\\
\Omega_k^R&={(A + \mathcal {B}K_k^L)^T} \Omega_{k+1}^R(A + \mathcal {B}K_k^L)\\
&+ {Q^R}+ (K_k^L)^T\Lambda^{R}K_k^L,\\
K_k^{L}&=-(\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal B)^{-1}\mathcal {B}^TP_{k+1}^LA,\\
K_k^{R}&=-[S^R+(B^L)^TP_{k+1}^RB^L]^{-1}(B^L)^TP_{k+1}^RA,\\
&S^R+(B^L)^TP_{k+1}^RB^L>0,
\end{array} \color{red}ight.
\end{align}}
with terminal conditions $\Omega_{N+1}^L=P_{N+1}^L$, $\Omega_{N+1}^R=P_{N+1}^R$ and $P_{N+1}^L, P_{N+1}^R$ are given in \eqref{cf1}-\eqref{cf2}, respectively.
\color{blue}egin{theorem}\label{th-02}
Under Assumptions \color{red}ef{ass1} and \color{red}ef{ass2}, if the coupled Riccati equation \eqref{rere} is solvable, then the closed-loop Nash equilibrium of Problem LRSNG is unique, and the closed-loop optimal Nash equilibrium is derived as
\color{blue}egin{align}
u_k^{L,*} & =[I_{m_1} ~~ 0]K_k^{L}\hat{x}_{k|k}+{K}_k^{R}\tilde x_{k}\label{occ1},\\
u_k^{R,*} & =[0 ~~ I_{m_2}]K_k^{L}\hat{x}_{k|k}\label{occ2},
\end{align}
where $\hat{x}_{k/k}$ and $\tilde x_{k}$ are the optimal estimator and estimation error given in Lemma \color{red}ef{lem3}, and the gain matrices
$K_k^L,K_k^R$ can be calculated via \eqref{rere}.
With the closed-loop Nash equilibrium \eqref{occ1}-\eqref{occ2}, the corresponding optimal cost functions can be respectively calculated as follows.
\color{blue}egin{align}
&J_N^L(u_k^{L,*}, u_k^{R,*})
= \mathbb{E}[\hat x_{0|0}^TP_0^L\hat x_{0|0}+\tilde x_0^T \Omega_0^L\tilde {x}_0]\normalsizeotag\\
&~~~~~+p\smallum\limits_{k = 0}^N Tr(\Sigma _wP_{k+1}^L)
+(1 - p)\smallum\limits_{k = 0}^N Tr(\Sigma _w\Omega_{k+1}^L)\label{ocf1},\\
&J_N^R(u_k^{L,*}, u_k^{R,*})
=\mathbb{E}[{{\hat x}_{0|0}}^T\Omega_0^R{\hat x}_{0|0} + \tilde {x}_0^TP_0^R\tilde {x}_0]\normalsizeotag\\
&~~~~~+p\smallum\limits_{k = 0}^NTr(\Sigma _w\Omega_{k+1}^R)
+(1 - p)\smallum\limits_{k = 0}^NTr(\Sigma _wP_{k+1}^R)\label{ocf2}.
\end{align}
\end{theorem}
Before we give the proof of Theorem \color{red}ef{th-02}, we will show the following propositions, which will be useful in deriving the main results.
\color{blue}egin{proposition}\label{prop1}
Under Assumption \color{red}ef{ass1}, $\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal B$ is positive definite for $k=0, \cdots, N$.
\end{proposition}
\color{blue}egin{proof}
The backward induction method will be adopted to show that $\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal B>0$ for $k=0, \cdots, N$.
Actually, from Assumption \color{red}ef{ass1}, we know that $P_{N+1}^L \geq 0$, and $\Lambda^L = \color{blue}egin{bmatrix} S^L & \\
& M^L \end{bmatrix} > 0$. Then, $\Lambda^L+\mathcal {B}^TP_{N+1}^L\mathcal B >0$ can be derived, and \eqref{rere} is solvable for $k=N$, that is
\color{blue}egin{align}\label{pr1}
P_N^L&=Q^L+A^TP_{N+1}^LA-(K_N^L)^T(\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal {B})K_N^L\normalsizeotag\\
&=Q^L+A^TP_{N+1}^LA+(K_N^L)^T\mathcal {B}^TP_{N+1}^LA\normalsizeotag\\
&+A^TP_{N+1}^L\mathcal {B}K_N^L+(K_N^L)^T(\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal {B})K_N^L\normalsizeotag\\
&=Q^L+(K_N^L)^T\Lambda^LK_N^L\normalsizeotag\\
&+(A+\mathcal {B}K_N^L)^TP_{N+1}^L(A+\mathcal {B}K_N^L)
\end{align}
where $K_N^L$ is given as in \eqref{rere} for $k=N$.
Notice that $\Lambda^L>0$, $Q^L\geq0$, $P_{N+1}^L \geq 0$, then $P_N^L\geq 0$ can be obtained from \eqref{pr1}.
By repeating the above procedures step by step backwardly, we can conclude that $P_k^L\geq0$, which indicates that $\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal B>0$ for any $0\leq k\leq N$.
\end{proof}
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0.135.6
|
From the definition of the closed-loop Nash equilibrium (Definition \color{red}ef{def2}), the following two propositions are to be shown.
\color{blue}egin{proposition}\label{prop2}
Under Assumptions \color{red}ef{ass1}-\color{red}ef{ass2}, for the closed-loop Nash equilibrium $(K_k^{L}, K_k^{R})$, which minimizes $J_N^L(u_k^L,u_k^R)$ (see Definition \color{red}ef{def2}), if $\tilde u_k^{L,*}=K_k^{R}\tilde{x}_{k}$ is given in advance, then $U_k^*$ can be calculated as
\color{blue}egin{align}\label{mathu}
U_k^*=K_k^L\hat{x}_{k|k},
\end{align}
in which $K_k^L$ is given by
\color{blue}egin{align}\label{kkl}
K_k^{L}&=-(\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal {B})^{-1}\mathcal {B}^T{P}_{k+1}^LA,
\end{align}
with $P_k^L$ satisfying
\color{blue}egin{align}
P_k^L&= {A^T} P_{k+1}^LA-(K_k^L)^T(\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal {B})K_k^L\normalsizeotag\\
&+ {Q^L},~ P_{N+1}^L,\label{pkl}\\
\Omega_k^L&=p[(A + B^LK_k^R)^TP_{k+1}^L(A + {B}^LK_k^R)]\normalsizeotag\\
&+(1-p)[(A + B^LK_k^R)^T\Omega_{k+1}^L(A + {B}^LK_k^R)]\normalsizeotag\\
&+{Q^L}+ (K_k^R)^TS^LK_k^R,~~\Omega_{N+1}^L=P_{N+1}^L.\label{okl}
\end{align}
In this case, the optimal $J_N^L(u_k^{L,*}, u_k^{R,*})$ is given by \eqref{ocf1}.
\end{proposition}
\color{blue}egin{proof}
For the sake of discussion, we denote $V_N^L(\hat{x}_{k|k},\tilde{x}_k)$ as follows:
\color{blue}egin{align}\label{vnlk}
V_N^L(\hat{x}_{k|k},\tilde{x}_k)&=\mathbb{E}[\hat x_{k|k}^T P_k^L\hat {x}_{k|k}+\tilde x_k^T \Omega_k^L\tilde {x}_k],
\end{align}
where $P_k^L$, $\Omega_k^L$ satisfy the equations \eqref{pkl}-\eqref{okl}.
Noting that $K_k^{R}$ is given, i.e., $\tilde{u}_k^{L,*}=K_k^{R}\tilde{x}_{k}$, hence we have,
\color{blue}egin{align}\label{lzhs1}
&V_N^L(\hat{x}_{k|k},\tilde{x}_k)- V_N^L(\hat{x}_{k+1|k+1},\tilde{x}_{k+1})\normalsizeotag\\
&= \mathbb{E}[\hat x_{k|k}^TP_k^L{{\hat x}_{k|k}}+\tilde x_k^T \Omega_k^L\tilde {x}_k]\normalsizeotag\\
&- \mathbb{E}[\hat x_{k + 1|k + 1}^T P_{k+1}^L{{\hat x}_{k + 1|k + 1}}]-\mathbb{E}[\tilde x_{k + 1}^T \Omega_{k+1}^L\tilde {x}_{k + 1}]\normalsizeotag\\
&= \mathbb{E}[\hat x_{k|k}^TP_k^L\hat {x}_{k|k}+\tilde {x}_k^T \Omega_k^L\tilde {x}_k]\normalsizeotag\\
&-\mathbb{E}[[\gamma _{k + 1}(A\hspace{-1mm}+\hspace{-1mm} B^LK_k^{R})\tilde {x}_k \hspace{-0.5mm}+\hspace{-0.5mm} \gamma _{k + 1}w_k\hspace{-1mm} +\hspace{-1mm} A\hat {x}_{k|k}
\hspace{-1mm}+\hspace{-1mm}\mathcal {B}U_k]^TP_{k+1}^L\normalsizeotag\\
&\times [\gamma _{k + 1}(A\hspace{-1mm}+\hspace{-1mm} B^LK_k^{R})\tilde {x}_k \hspace{-0.5mm}+ \hspace{-0.5mm}\gamma _{k + 1}w_k\hspace{-1mm} +\hspace{-1mm} A\hat {x}_{k|k}
\hspace{-1mm}+\hspace{-1mm}\mathcal {B}U_k]]\normalsizeotag\\
&- \mathbb{E}[[(1 - \gamma _{k + 1})(A + B^LK_k^{R})\tilde {x}_k +(1 - \gamma _{k + 1}) w_k]^T\Omega_{k+1}^L\normalsizeotag\\
&\times[(1 - \gamma _{k + 1})(A + B^LK_k^{R})\tilde {x}_k +(1 - \gamma _{k + 1}) w_k]]\normalsizeotag\\
&= \mathbb{E}[-(U_k - K_k^L\hat{x}_{k|k})^T(\Lambda^L + \mathcal {B}^TP_{k+1}^L\mathcal {B})\normalsizeotag\\
&\times(U_k - K_k^L\hat{x}_{k|k})+ U_k^T\Lambda^LU_k]\normalsizeotag\\
&+\mathbb{E}[\tilde x_k^T[\Omega_k^L \hspace{-0.5mm}- \hspace{-0.5mm}(1-p)(A\hspace{-0.5mm} +\hspace{-0.5mm} B^LK_k^{R})^T \Omega_{k+1}^L(A \hspace{-0.5mm}+\hspace{-0.5mm} B^LK_k^{R})\normalsizeotag\\
&-p(A + B^LK_k^{R})^T P_{k+1}^L(A + B^LK_k^{R})]\tilde {x}_k]\normalsizeotag\\
&+\mathbb{E}[\hat{x}_{k|k}^T [P_k^L-A^TP_{k+1}^LA\normalsizeotag\\
&+(K_k^L)^T(\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal {B})K_k^L]\hat{x}_{k|k}]\normalsizeotag\\
&-(1-p)Tr(\Sigma _w\Omega_{k+1}^L)-pTr(\Sigma _wP_{k+1}^L).
\end{align}
Furthermore, from \eqref{kkl}-\eqref{okl}, it can be derived from \eqref{lzhs1} that
\color{blue}egin{align}\label{lzhs2}
&V_N^L(\hat{x}_{k|k},\tilde{x}_k)- V_N^L(\hat{x}_{k+1|k+1},\tilde{x}_{k+1})\normalsizeotag\\
&= \mathbb{E}[-(U_k - K_k^L\hat{x}_{k|k})^T(\Lambda^L + \mathcal {B}^TP_{k+1}^L\mathcal {B})\normalsizeotag\\
&\times(U_k - K_k^L\hat{x}_{k|k})+ U_k^T\Lambda^LU_k]\normalsizeotag\\
&+\mathbb{E}[\hat x_{k|k}^TQ^L\hat {x}_{k|k}]+\mathbb{E}[\tilde{x}_k^T[Q^L+(K_k^{R})^TS^LK_k^{R}]\tilde{x}_k]\normalsizeotag\\
&-(1 - p)Tr(\Sigma _w\Omega_{k+1}^L)-pTr(\Sigma _wP_{k+1}^L).
\end{align}
Taking summation on both sides of \eqref{lzhs2} from $k=0$ to $k=N$, there holds
\color{blue}egin{align}\label{lzhs3}
&V_N^L(\hat{x}_{0|0},\tilde{x}_0)- V_N^L(\hat{x}_{N+1|N+1},\tilde{x}_{N+1})\normalsizeotag\\
&= \mathbb{E}[{\hat x_{0|0}}^TP_0^L\hat x_{0|0}+\tilde x_0^T\Omega_0^L\tilde {x}_0]\normalsizeotag\\
&- \mathbb{E}[\hat x_{N+1|N+1}^TP_{N+1}^L\hat {x}_{N+1|N+1}+\tilde x_{N+1}^T\Omega_{N+1}^L\tilde {x}_{N + 1}]\normalsizeotag\\
&= \smallum\limits_{k = 0}^N \mathbb{E}[-(U_k - K_k^L\hat{x}_{k|k})^T(\Lambda^L + \mathcal {B}^TP_{k+1}^L\mathcal {B})\normalsizeotag\\
&\times(U_k - K_k^L\hat{x}_{k|k})+ U_k^T\Lambda^LU_k]\normalsizeotag\\
&+ \mathbb{E}[\hat x_{k|k}^TQ^L\hat {x}_{k|k}] + \mathbb{E}[\tilde x_k^T[Q^L+(K_k^{R})^TS^LK_k^{R}]\tilde
{x}_k]\normalsizeotag\\
&-(1 - p)\smallum\limits_{k = 0}^N Tr(\Sigma _w \Omega_{k+1}^L)- p\smallum\limits_{k = 0}^N Tr(\Sigma _wP_{k+1}^L).
\end{align}
Then, from \eqref{kw2} we have that
\color{blue}egin{align}\label{ocf3}
&J_N^L(u_k^L, u_k^R)=\smallum\limits_{k = 0}^N \mathbb{E}[\hat x_{k|k}^T{Q^L}{{\hat x}_{k|k}}+U_k^T\Lambda^LU_k\normalsizeotag\\
& + \tilde x_k^T[Q^L+(K_k^{R})^TS^LK_k^{R}]\tilde {x}_k]+\mathbb{E}[x_{N + 1}^TP_{N+1}^Lx_{N + 1}]\normalsizeotag\\
&= \smallum\limits_{k = 0}^N \mathbb{E}[(U_k - K_k^L\hat{x}_{k|k})^T(\Lambda^L + \mathcal {B}^TP_{k+1}^L\mathcal {B})\normalsizeotag\\
&\times(U_k - K_k^L\hat{x}_{k|k})]+ \mathbb{E}[\hat x_{0|0}^TP_0^L\hat x_{0|0}+\tilde x_0^T\Omega_0^L\tilde {x}_0]\normalsizeotag\\
&+ (1 - p)\smallum\limits_{k = 0}^N Tr(\Sigma _w \Omega_{k+1}^L)+ p\smallum\limits_{k = 0}^NTr(\Sigma _wP_{k+1}^L).
\end{align}
As shown in Proposition \color{red}ef{prop1}, $\Lambda^L+\mathcal{B}^TP_{k+1}^L\mathcal{B}>0$, therefore, $J_N^L(u_k^L, u_k^R)$ can be minimized by \eqref{mathu} with $\tilde{u}_k^{L,*}=K_k^{R}\tilde{x}_{k}$ given in advance. This completes the proof.
\end{proof}
| 3,116 | 30,909 |
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|
Similarly, the following proposition can be shown.
\color{blue}egin{proposition}\label{prop3}
Suppose Assumptions \color{red}ef{ass1}-\color{red}ef{ass2} hold, and denote $(K_k^{L}, K_k^{R})$ as the closed-loop Nash equilibrium of optimizing $J_N^R(u_k^L,u_k^R)$, if we set $ U_k^*=K_k^L\hat{x}_{k|k}$ in advance, then $ \tilde{u}_k^{L,*}$ is given by
\color{blue}egin{align}\label{tildeu}
\tilde{u}_k^{L,*}&=K_k^{R}\tilde{x}_{k}.
\end{align}
In the above, $K_k^R$ satisfies
{\smallmall\color{blue}egin{align}\label{kkr}
\left\{ \color{blue}egin{array}{ll}
K_k^{R}&\hspace{-2mm}=-[S^R+(B^L)^T{P}_{k+1}^RB^L]^{-1}(B^L)^T{P}_{k+1}^RA,\\
P_k^R&\hspace{-2mm}={A^T} P_{k+1}^RA + Q^R-(K_k^R)^T[S^R+(B^L)^TP_{k+1}^RB^L]K_k^R\\
&+p[(A + B^LK_k^R)^T \Omega_{k+1}^R(A + B^LK_k^R)\\
&- {(A + B^LK_k^R)^T} P_{k+1}^R(A + B^LK_k^R)],\\
\Omega_k^R&\hspace{-2mm}=(A + \mathcal {B}K_k^L)^T \Omega_{k+1}^R(A + \mathcal {B}K_k^L)\\
&+ Q^R+ (K_k^L)^T\Lambda^RK_k^L
\end{array} \color{red}ight.
\end{align}}
Furthermore, $J_N^R(u_k^{L}, u_k^{R})$ can be minimized as \eqref{ocf2}.
\end{proposition}
\color{blue}egin{proof}
By following Proposition \color{red}ef{prop2} and its proof, we define
\color{blue}egin{align*}
V_N^R(\hat{x}_{k|k},\tilde{x}_k) = \mathbb{E}[\hat x_{k|k}^T \Omega_k^R\hat {x}_{k|k} + \tilde x_k^TP_k^R\tilde {x}_k],
\end{align*}
in which $P_k^R$, $\Omega_k^R$ satisfy \eqref{kkr}.
Hence, it can be derived that
\color{blue}egin{align}\label{rzhs1}
&V_N^R(\hat{x}_{k|k},\tilde{x}_k)-V_N^R(\hat{x}_{k+1|k+1},\tilde{x}_{k+1})\normalsizeotag\\
&= \mathbb{E}[\hat x_{k|k}^T \Omega_k^R\hat {x}_{k|k} + \tilde x_k^TP_k^R\tilde {x}_k]\normalsizeotag\\
&- \mathbb{E}[\hat x_{k + 1|k + 1}^T\Omega_{k+1}^R\hat x_{k + 1|k + 1}]- \mathbb{E}[\tilde x_{k +1}^TP_{k+1}^R\tilde {x}_{k + 1}]\normalsizeotag\\
&= \mathbb{E}[\hat x_{k|k}^T \Omega_k^R{{\hat x}_{k|k}} + \tilde x_k^TP_k^R\tilde {x}_k]\normalsizeotag\\
&- \mathbb{E}[[\gamma _{k + 1}(A\tilde {x}_k\hspace{-0.5mm} +\hspace{-0.5mm} B^L\tilde {u}_k^L\hspace{-0.5mm} +\hspace{-0.5mm} w_k)\hspace{-0.5mm} +\hspace{-0.5mm} (A+\mathcal BK_k^{L})\hat {x}_{k|k}]^T\Omega_{k+1}^R\normalsizeotag\\
&\times [{\gamma _{k + 1}}(A\tilde {x}_k\hspace{-0.5mm} +\hspace{-0.5mm} B^L\tilde {u}_k^L\hspace{-0.5mm} +\hspace{-0.5mm} w_k)\hspace{-0.5mm} +\hspace{-0.5mm} (A+\mathcal BK_k^{L})\hat {x}_{k|k}]]\normalsizeotag\\
&- \mathbb{E}[[(1 - \gamma _{k + 1})(A\tilde {x}_k + B^L\tilde {u}_k^L+
{w_k})]^TP_{k+1}^R\normalsizeotag\\
&\times[(1 - \gamma _{k + 1})(A\tilde {x}_k + {B^L}\tilde {u}_k^L + w_k)]]\normalsizeotag\\
&= \mathbb{E}[\hat x_{k|k}^T \Omega_k^R\hat {x}_{k|k} + \tilde x_k^TP_k^R\tilde {x}_k]-\mathbb{E}[\tilde{x}_k^TA^TP_{k+1}^RA\tilde{x}_k]\normalsizeotag\\
&-\mathbb{E}[\hat{x}_{k|k}^T(A+\mathcal {B}K_k^{L})^T\Omega_{k+1}^R(A+\mathcal {B}K_k^{L})\hat{x}_{k|k}]\normalsizeotag\\
&-\mathbb{E}[(\tilde{u}_k^L-K_k^R\tilde{x}_k)^T[S^R+(B^L)^TP_{k+1}^RB^L](\tilde{u}_k^L-K_k^R\tilde{x}_k)]\normalsizeotag\\
&+\mathbb{E}[(\tilde{u}_k^L)^TS^R\tilde{u}_k^L]\hspace{-1mm}+\hspace{-1mm}\mathbb{E}[p(A\tilde{x}_k\hspace{-1mm}+\hspace{-1mm}B^L\tilde{u}_k^L)^TP_{k+1}^R(A\tilde{x}_k\hspace{-1mm}+\hspace{-1mm}B^L\tilde{u}_k^L)\normalsizeotag\\
&-p(A\tilde{x}_k+B^L\tilde{u}_k^L)^T\Omega_{k+1}^R(A\tilde{x}_k+B^L\tilde{u}_k^L)]\normalsizeotag\\
&+\mathbb{E}[\tilde{x}_k^T(K_k^R)^T[S^R+(B^L)^TP_{k+1}^RB^L]K_k^R\tilde{x}_k]\normalsizeotag\\
&-pTr(\Sigma _w\Omega_{k+1}^R) - (1 - p)Tr(\Sigma _wP_{k+1}^R).
\end{align}
Next, by using \eqref{rere}, we obtain
\color{blue}egin{align}\label{rzhs2}
&V_N^R(\hat{x}_{k|k},\tilde{x}_k)-V_N^R(\hat{x}_{k+1|k+1},\tilde{x}_{k+1})\normalsizeotag\\
&=-\mathbb{E}[(\tilde{u}_k^L\hspace{-0.5mm}-\hspace{-0.5mm}K_k^R\tilde{x}_k)^T[S^R\hspace{-0.5mm}+\hspace{-0.5mm}(B^L)^TP_{k+1}^RB^L](\tilde{u}_k^L\hspace{-0.5mm}-\hspace{-0.5mm}K_k^R\tilde{x}_k)]\normalsizeotag\\
&+\mathbb{E}[(\tilde{u}_k^L)^TS^R\tilde{u}_k^L]+ \mathbb{E}[\hat x_{k|k}^T[{Q^R} + (K_k^{L})^T\Lambda^RK_k^{L}]{\hat
x}_{k|k}]\normalsizeotag\\
&+\mathbb{E}[\tilde {x}_k^TQ^R\tilde {x}_k]-pTr(\Sigma _w\Omega_{k+1}^R) - (1 - p)Tr(\Sigma _wP_{k+1}^R).
\end{align}
Then, by adding from $k=0$ to $k=N$ of \eqref{rzhs2}, there holds
\color{blue}egin{align}\label{rzhs3}
&V_N^R(\hat{x}_{0|0},\tilde{x}_0)-V_N^R(\hat{x}_{N+1|N+1},\tilde{x}_{N+1})\normalsizeotag\\
&= \mathbb{E}[\hat x_{0|0}^T\Omega_0^R\hat {x}_{0|0} + \tilde x_0^TP_0^R\tilde {x}_0]\normalsizeotag\\
&- \mathbb{E}[\hat x_{N + 1|N + 1}^T\Omega_{N+1}^R{\hat x_{N + 1|N + 1}}- \tilde x_{N + 1}^TP_{N+1}^R\tilde {x}_{N + 1}]\normalsizeotag\\
&= \smallum\limits_{k = 0}^N-\mathbb{E}[(\tilde{u}_k^L-K_k^R\tilde{x}_k)^T[S^R+(B^L)^TP_{k+1}^RB^L]\normalsizeotag\\
&\times(\tilde{u}_k^L-K_k^R\tilde{x}_k)]+\mathbb{E}[(\tilde{u}_k^L)^TS^R\tilde{u}_k^L+\tilde x_k^TQ^R\tilde {x}_k]\normalsizeotag\\
&+ \mathbb{E}[\hat x_{k|k}^T[Q^R + (K_k^{L})^T\Lambda^RK_k^{L}]{{\hat
x}_{k|k}}]\normalsizeotag\\
&-pTr(\Sigma _w\Omega_{k+1}^R) - (1 - p)Tr(\Sigma _wP_{k+1}^R).
\end{align}
Finally, from \eqref{kw3}, we have
\color{blue}egin{align}\label{ocf4}
&J_N^R(u_k^{L}, u_k^{R})= \smallum\limits_{k = 0}^N \mathbb{E}[\tilde x_k^TQ^R\tilde {x}_k+(\tilde{u}_k^L)^TS^R\tilde{u}_k^L] \normalsizeotag\\
&+\mathbb{E}[ \hat x_{k|k}^T[Q^R + (K_k^R)^T{S^L}K_k^R]\hat {x}_{k|k}]+ \mathbb{E}[x_{N + 1}^TP_{N+1}^Rx_{N + 1}] \normalsizeotag\\
&= \smallum\limits_{k = 0}^N \mathbb{E}[(\tilde{u}_k^L-K_k^R\tilde{x}_k)^T[S^R+(B^L)^TP_{k+1}^RB^L]\normalsizeotag\\
&\times(\tilde{u}_k^L-K_k^R\tilde{x}_k)]+ \mathbb{E}[\hat x_{0|0}^T\Omega_0^R\hat {x}_{0|0} + \tilde
x_0^TP_0^R\tilde {x}_0]\normalsizeotag\\
&+p\smallum\limits_{k = 0}^N Tr(\Sigma _w\Omega_{k+1}^R) + (1 - p)\smallum\limits_{k = 0}^N Tr(\Sigma _wP_{k+1}^R).
\end{align}
Since $S^R+(B^L)^T{P}_{k+1}^RB^L>0$ given in \eqref{rere}, thus $J_N^R(u_k^{L}, u_k^{R})$ can be minimized by \eqref{tildeu}. The proof is complete.
\end{proof}
In the following, the proof of Theorem \color{red}ef{th-02} shall be given.\\
\color{blue}egin{proof}
\textbf{Proof of Theorem \color{red}ef{th-02}.} Combining Propositions \color{red}ef{prop2}-\color{red}ef{prop3}, we can conclude that if the coupled Riccati equations \eqref{rere} is solvable, then $(K_k^L, K_k^R)$ given in \eqref{mathu} and \eqref{tildeu} is the unique closed-loop Nash equilibrium, as given in \eqref{clne1}-\eqref{clne2} of Definition \color{red}ef{def2}). Therefore, from \eqref{ncl}, we know that the optimal action of $(u_k^{L,*},u_k^{R,*})$ can be given as \eqref{occ1}-\eqref{occ2}. Moreover, the optimal $J_N^L(u_k^{L,*}, u_k^{R,*}), J_N^R(u_k^{L,*}, u_k^{R,*})$ are given as \eqref{ocf1}-\eqref{ocf2}, this ends the proof.
\end{proof}
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\color{blue}egin{remark}
In Theorem \color{red}ef{th-02}, the closed-loop Nash equilibrium of Problem LRSNG is derived in the feedback form, the obtained results are new to the best of our knowledge. For the closed-loop Nash equilibrium of Problem LRSNG, on the one hand, the local player $u_k^{L}$ should take the closed-loop optimal Nash equilibrium $u_k^{L,*}$ if and only if the remote player takes the closed-loop optimal feedback Nash equilibrium $u_k^{R,*}$. Meanwhile, the optimal cost function is given as the value of \eqref{ocf1}. On the other hand, the remote player $u_k^{R}$ should take the closed-loop optimal Nash equilibrium $u_k^{R,*}$ if and only if the local player takes the closed-loop optimal feedback Nash equilibrium $u_k^{L,*}$, and the optimal cost function \eqref{ocf2} is also derived.
Besides, the calculation of closed-loop Nash equilibrium is based on the coupled Riccati equations.
\end{remark}
\smallection{Numerical Example}
\color{blue}egin{figure}
\centering
\includegraphics[width=0.38\textwidth]{Figure2.pdf}\\
\caption{Closed-loop Nash equilibrium: $K_k^L(1,i), i=1,\cdots,4$, the first column value of $K_k^L$.}\label{Figure2}
\end{figure}
\color{blue}egin{figure}
\centering
\includegraphics[width=0.38\textwidth]{Figure3.pdf}\\
\caption{Closed-loop Nash equilibrium: $K_k^L(2,i), i=1,\cdots,4$, the second column value of $K_k^L$.}\label{Figure3}
\end{figure}
\color{blue}egin{figure}
\centering
\includegraphics[width=0.38\textwidth]{Figure4.pdf}\\
\caption{Closed-loop Nash equilibrium: $K_k^R(1,i), i=1,2$, the first column value of $K_k^R$.}\label{Figure4}
\end{figure}
\color{blue}egin{figure}
\centering
\includegraphics[width=0.38\textwidth]{Figure5.pdf}\\
\caption{Closed-loop Nash equilibrium: $K_k^R(2,i), i=1,2$, the second column value of $K_k^R$.}\label{Figure5}
\end{figure}
In order to illustrate the main results obtained in Theorem \color{red}ef{th-02}, a numerical example is provided in this section.
Without loss of generallity, we shall consider the system dynamics \eqref{ss1} and the cost functions \eqref{cf1}-\eqref{cf2} with the following coefficients:
\color{blue}egin{align}\label{coes}
&N=50, p=0.5, \mu=0,A= \color{blue}egin{bmatrix}
{1.2}&{0}\\
{0}&{1.1}
\end{bmatrix};\normalsizeotag\\
&B^L= \color{blue}egin{bmatrix}
{0.3}&{0.2}\\
{0.4}&{-0.1}
\end{bmatrix},
B^R= \color{blue}egin{bmatrix}
{0.1}&{0.2}\\
{0}&{0.1}
\end{bmatrix};\normalsizeotag\\
&Q^L=Q^R=S^L=S^R=M^L=M^R=I_2;\normalsizeotag\\
&\Lambda^L=\color{blue}egin{bmatrix} S^L & \\
& M^L \end{bmatrix}=I_4, P^L_{N+1}=P^R_{N+1}=I_2.
\end{align}
From the above coefficients in \eqref{coes}, obviously, Assumption \color{red}ef{ass1} is satisfied. Consequently, $P_k$ can be calculated from \eqref{rere} backwardly using \eqref{coes}, then it can be verified that $\Lambda^L+\mathcal {B}^TP_{k+1}^L\mathcal {B}$ and $S^R+(B^L)^TP_{k+1}^RB^L$ are positive definite for $k=1, \cdots, 50$. Using Theorem \color{red}ef{th-02}, we can conclude that the closed-loop Nash equilibrium of Problem LRSNG is unique, and $(K_k^L, K_k^R)$, $k=0, \cdots, 50$ can be calculated from \eqref{rere}, which are shown in Figures \color{red}ef{Figure2}-\color{red}ef{Figure5}, backwardly.
As can be seen from Figures \color{red}ef{Figure2}-\color{red}ef{Figure5}, it is apparent that the closed-loop Nash equilibrium $(K_k^L,K_k^R)$ would be convergent with $N$ becomes large.
\smallection{Conclusion}
In this paper, we have discussed the open-loop and closed-loop local and remote Nash equilibrium for LRSNG problem for discrete-time stochastic systems with inconsistent information structure. This paper extends the existing works on LQ stochastic games with consistent information structure to the inconsistent information structure case. Both the open-loop and closed-loop Nash equilibrium have been derived in this paper, and a numerical example is given to illustrate the main results. We believe the proposed methods and results would shed a light in solving other kinds of stochastic game problem with inconsistent information structure.
\ifCLASSOPTIONcaptionsoff
\normalsizeewpage
\fi
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\end{document}
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\begin{document}
\begin{center}{\Large \bf
Hafnian point processes and quasi-free states on the CCR algebra}
\end{center}
{\large Maryam Gharamah Ali Alshehri}\\ Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, KSA; \\
e-mail: \texttt{[email protected]}
{\large Eugene Lytvynov\\ Department of Mathematics, Swansea University, Swansea, UK;\\
e-mail: \texttt{[email protected]}
{\small
\begin{center}
{\bf Abstract}
\end{center}
\noindent Let $X$ be a locally compact Polish space and $\sigma$ a nonatomic reference measure on $X$ (typically $X=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). Let $X^2\ni(x,y)\mapsto\mathbb K(x,y)\in\mathbb C^{2\times 2}$ be a $2\times 2$-matrix-valued kernel that satisfies $\mathbb K^T(x,y)=\mathbb K(y,x)$. We say that a point process $\mu$ in $X$ is hafnian with correlation kernel $\mathbb K(x,y)$ if, for each $n\in\mathbb N$, the $n$th correlation function of $\mu$ (with respect to $\sigma^{\otimes n}$) exists and is given by $k^{(n)}(x_1,\dots,x_n)=\operatorname{haf}\big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}\,$. Here $\operatorname{haf}(C)$ denotes the hafnian of a symmetric matrix $C$. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process $\Pi_R$ is a Poisson point process in $X$ with random intensity $R(x)$. Let $G(x)$ be a complex Gaussian field on $X$ satisfying $\int_{\Delta}\mathbb E(|G(x)|^2)\sigma(dx)<\infty$ for each compact $\Delta\subset X$. Then the Cox process $\Pi_R$ with $R(x)=|G(x)|^2$ is a hafnian point process. The main result of the paper is that each such process $\Pi_R$ is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCR), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free.
}
{\bf Keywords:} Hafnian point process, Cox process, permanental point process; quasi-free state on CCR algebra
\noindent
{\it Mathematics Subject Classification (2020):} Primary 60G55; 46L30 Secondary 60G15
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\section{Introduction}
\subsection{Hafnian point processes}
Let $X$ be a locally compact Polish space, let $\mathcal B(X)$ denote the Borel $\sigma$-algebra on $X$, and let $\mathcal B_0(X)$ denote the algebra of all pre-compact sets from $\mathcal B(X)$. Let $\sigma$ be a reference measure on $(X,\mathcal B(X))$ which is non-atomic (i.e., $\sigma(\{x\})=0$ for all $x\in X$) and Radon (i.e., $\sigma(\Delta)<\infty$ for all $\Delta\in\mathcal B_0(X)$). For applications, the most important example is $X=\R^d$ $\sigma(dx)=dx$ is the Lebesgue measure.
A {\it (simple) configuration} $\gamma$ in $X$ is a Radon measure on $X$ of the form $\gamma=\sum_i\delta_{x_i}$, where $\delta_{x_i}$ denotes the Dirac measure with mass at $x_i$ and $x_i\ne x_j$ if $i\ne j$. Note that, since $\gamma$ is a Radon measure, it has a finite number of atoms in each compact set in $X$. Let $\Gamma(X)$ denote the set of all configurations $\gamma$ in $X$. Let $\mathcal C(\Gamma(X))$ denote the minimal $\sigma$-algebra on $\Gamma(X)$ such that, for each $\Delta\in\mathcal B_0(X)$, the mapping $\Gamma(X)\ni\gamma\mapsto\gamma(\Delta)$ is measurable. A {\it (simple) point process in $X$} is a probability measure on $(\Gamma(X),\mathcal C(\Gamma(X)))$.
Denote $X^{(n)}:=\{(x_1,\dots,x_n)\in X^n\mid x_i\ne x_j\text{ if }i\ne j\}$. A measure on $X^{(n)}$ is called symmetric if it remains invariant under the natural action of the symmetric group $\mathfrak S_n$ on $X^{(n)}$.
For each $\gamma=\sum_i\delta_{x_i}\in\Gamma(X)$, the {\it spatial falling factorial} $(\gamma)_n$ is the symmetric measure on $X^{(n)}$ of the form
\begin{equation}\label{ctrsw5ywus}
(\gamma)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}\delta_{(x_{i_1}, x_{i_2}, \dots,x_{i_n})}.\end{equation}
Let $\mu$ be a point process in $X$. The {\it $n$-th correlation measure of $\mu$} is the symmetric measure $\theta^{(n)}$ on $X^{(n)}$ defined by
\begin{equation}\label{rqay45qy4q}
\theta^{(n)}(dx_1\dotsm dx_n):=\frac1{n!}\int_{\Gamma(X)}(\gamma)_n(dx_1\dotsm dx_n)\,\mu(d\gamma).
\end{equation}
If each measure $\theta^{(n)}$ is absolutely continuous with respect to $\sigma^{\otimes n}$, then the symmetric functions $k^{(n)}:X^{(n)}\to[0,\infty)$ satisfying
\begin{equation}\label{5w738}
d\theta^{(n)}=\frac1{n!}\,k^{(n)}d\sigma^{\otimes n}\end{equation} are called the {\it correlation functions of the point process $\mu$}. Under a very weak assumption, the correlations functions (or correlation measures) uniquely identify a point process, see \cite{Lenard}.
Let $C=[c_{ij}]_{i,j=1,\dots,2n}$ be a symmetric $2n\times2n$-matrix. The {\it hafnian of $C$} is defined by
$$\operatorname{haf}(C):=\frac1{n!\,2^n}\sum_{\pi\in\mathfrak S_{2n}}\prod_{i=1}^n c_{\pi(2i-1)\pi(i)}, $$
see e.g.\ \cite[Section~4.1]{Barvinok}.
(Note the the value of the hafnian of $C$ does not depend on the diagonal elements of the matrix $C$.)
The hafnian can also be written as
\begin{equation}\label{cxtseewu645}
\operatorname{haf}(C)=\sum c_{i_1j_1}\dotsm c_{i_nj_n},\end{equation}
where the summation is over all (unordered) partitions $\{i_1,j_1\},\dots,\{i_n,j_n\}$ of $\{1,\dots,2n\}$.
Hafnians were introduced by physicist Edoardo Caianiello in the 1950's, while visiting Niels Bohr's group in Copenhagen (whose latin name is Hafnia), as a Boson analogue of the formula expressing the
correlations of a quasi-free Fermi state.\footnote{We are grateful to the referee for sharing with us this historical fact.}
By analogy with the definition of a pfaffian point process (see e.g.\ \cite[Section~10]{Borodin} and the references therein), we now define a hafnian point process. Let $X^2\ni(x,y)\mapsto\mathbb K(x,y)\in\mathbb C^{2\times 2}$ be a $2\times 2$-matrix-valued kernel that satisfies $\mathbb K^T(x,y)=\mathbb K(y,x)$.
We will say that a point process $\mu$ is {\it hafnian with correlation kernel $\mathbb K(x,y)$} if, for each $n\in\mathbb N$, the $n$th correlation function of $\mu$ exists and is given by
\begin{equation}\label{vytds6}
k^{(n)}(x_1,\dots,x_n)=\operatorname{haf}\big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}\,.\end{equation}
Note that the matrix
$$ \big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}=\left[\begin{matrix}
\mathbb K(x_1,x_1)&\mathbb K(x_1,x_2)&\dotsm&\mathbb K(x_1,x_n)\\
\mathbb K(x_2,x_1)&\mathbb K(x_2,x_2)&\dotsm&\mathbb K(x_2,x_n)\\
\vdots&\vdots&\vdots&\vdots\\
\mathbb K(x_n,x_1)&\mathbb K(x_n,x_2)&\dotsm&\mathbb K(x_n,x_n)
\end{matrix}\right]$$
is built upon $2\times2$-blocks $\mathbb K(x_i,x_j)$, hence it has dimension $2n\times 2n$. Furthermore, the condition $\mathbb K^T(x,y)=\mathbb K(y,x)$ ensures that the matrix $\big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}$ is symmetric, and so its hafnian is a well-defined number.
Since
$$X^2=\{(x,x)\mid x\in X\}\sqcup X^{(2)},$$
for the definition of a hafnian point process, it is sufficient to assume that $\mathbb K(x,x)$ is defined for $\sigma$-a.a.\ $x\in X$, and the restriction of $\mathbb K(x,y)$ to $X^{(2)}$ is defined for $\sigma^{\otimes 2}$-a.a.\ $(x,y)\in X^{(2)}$.
Note that, for the hafnian point process $\mu$, the correlation kernel $\mathbb K(x,y)$ is not uniquely determined by $\mu$. Indeed, since the hafnian of a matrix does not depend on its diagonal elements, formula \eqref{vytds6} implies that the correlation functions $k^{(n)}(x_1,\dots,x_n)$ do not depend on the diagonal elements of the $2\times 2$-matrices $\mathbb K(x,x)$ for $x\in X$. Hence, these elements can be chosen arbitrarily.
Let $\alpha\in\mathbb R$ and let $B=[b_{ij}]_{i,j=1,\dots,n}$ be an $n\times n$ matrix. The {\it $\alpha$-determinant of $B$} is defined by
\begin{equation}\label{tera5yw3}
\operatorname{det}_\alpha (B):=\sum_{\pi\in \mathfrak S_n}\prod_{i=1}^n\alpha^{n-\nu(\pi)}\,b_{i\,\pi(i)},\end{equation}
see \cite{VJ,ST}. In formula \eqref{tera5yw3}, for $\pi\in \mathfrak S_n$, $\nu(\pi)$ denotes the number of cycles in the permutation $\pi$. In particular, for $\alpha=1$, $\operatorname{det}_1(B)$ is the usual permanent of $B$.
A point process $\mu$ is called {\it $\alpha$-permanental (or $\alpha$-determinantal) with correlation kernel $K:X^2\to\mathbb C$} if, for each $n\in\mathbb N$, the $n$th correlation function of $\mu$ exists and is given by
$$k^{(n)}(x_1,\dots,x_n)=\operatorname{det}_\alpha [K(x_i,x_j)]_{i,j=1,\dots,n},$$
\cite{ST}, see also \cite{E2}. For $\alpha=1$, one calls $\mu$ a {\it permanental point process}.
As easily follows from \cite[Section~4.1]{Barvinok} a permanental point process with correlation kernel $K(x,y)$ is hafnian with correlation kernel
$$\mathbb K(x,y)=\left[\begin{matrix}0&K(x,y)\\K(y,x)&0\end{matrix}\right].$$
Furthermore, similarly to \cite[Proposition~1.1]{Frenkel}, we see that a $2$-permanental point process with a symmetric correlation kernel $K(x,y)=K(y,x)$ is hafnian with the correlation kernel
$$\mathbb K(x,y)=\left[\begin{matrix}K(x,y)&K(x,y)\\K(x,y)&K(x,y)\end{matrix}\right].$$
For studies of permanental, and more generally $\alpha$-permanental point processes, we refer to \cite{BM,KMR,Macchi1,Macchi2,ST}.
Recall that a {\it Cox process $\Pi_R$} is a Poisson point process with a random intensity $R(x)$. Here $R(x)$ is a random field defined for $\sigma$-a.a\ $x\in X$ and taking a.s.\ non-negative values. The correlation functions of the Cox process $\Pi_R$ are given by
\begin{equation}\label{xreew5u}
k^{(n)}(x_1,\dots,x_n)=\mathbb E\big(R(x_1)\dotsm R(x_n)\big).\end{equation}
Let $G(x)$ be a mean-zero, complex Gaussian field defined for $\sigma$-a.a.\ $x\in X$. Assume additionally that $\int_\Delta\mathbb E(|G(x)|^2)\sigma(dx)<\infty$ for each $\Delta\in\mathcal B_0(X)$. Let $R(x):=|G(x)|^2=G(x)\overline{G(x)}$.
Comparing the classical moment formula for Gaussian random variables with formula \eqref{cxtseewu645}, we immediately see that
\begin{equation}\label{dr6e6u43wq}
\mathbb E\big(R(x_1)\dotsm R(x_n)\big)=\operatorname{haf}\big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}\, , \end{equation}
where
\begin{equation}\label{d6e6ie4}
\mathbb K(x,y)=\left[\begin{matrix}
\mathbb E(G(X)G(y))&\mathbb E(G(x)\overline{G(y)})\\\mathbb E(\overline{G(x)}G(y))&\mathbb E(\overline{G(x)G(y)})\end{matrix}\right]=\left[\begin{matrix}
\mathcal K_2(x,y)&\mathcal K_1(x,y)\\\overline{\mathcal K_1(x,y)}&\overline{\mathcal K_2(x,y)}\end{matrix}\right].\end{equation}
Here $\mathcal K_1(x,y):=\mathbb E(G(x)\overline{G(y)})$ is the {\it covariance} of the Gaussian field and $\mathcal K_2(x,y):=\mathbb E(G(x)G(y))$ is the {\it pseudo-covariance} of the Gaussian field. By \eqref{xreew5u}--\eqref{d6e6ie4}, the corresponding Cox process $\Pi_R$ is hafnian with the correlation kernel \eqref{d6e6ie4}.
In the case where the Gaussian field $G(x)$ is real-valued, the moments
of $R(x)$ are given by the $2$-determinants built upon the kernel $K(x,y):=\mathcal K_1(x,y)=\mathcal K_2(x,y)$, hence $R(x)$ is a 2-permanental process. For studies of $\alpha$-permanental processes, we refer e.g.\ to \cite{E1,E2,E3,E4,KMR,MR1,MR2,MR3} and the references therein.
Obviously, in this case, $\Pi_R$ is a $2$-permanental point process with the correlation kernel $K(x,y)$, compare with \cite[Subsection~6.4]{ST}.
A Gaussian random field is called {\it proper} if $\mathcal K_2(x,y)=0$ for all $x$ and $y$. Since the moments of the random field $R(x)$ are given by permanents built upon the kernel $K(x,y):=\mathcal K_1(x,y)$, $R(x)$ is a permanental process, compare with \cite{BM,Macchi1,Macchi2}. We note, however, that the available studies of $\alpha$-permanental processes usually discuss only the case where the kernel is real-valued. In the case of $R(x)$, the correlation kernel is, of course, complex-valued.
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\subsection{Aim of the paper}
Quasi-free states play a central role in studies of operator algebras related to quantum statistical mechanics, see e.g.\ \cite{A1,A2,A3,A4,BR,DG,MV}.
Let $\mathcal H=L^2(X,\sigma)$ be the $L^2$-space of $\sigma$-square-integrable functions $h:X\to\mathbb C$. Let $\mathfrak F$ be a separable complex Hilbert spaces. Let $A^+(h)$, $A^-(h)$ ($h\in\mathcal H$) be linear operators in $\mathfrak F$ that satisfy the following assumptions:
\begin{itemize}
\item[(i)] $ A^+(h)$ and $A^-(h)$ depend linearly on $h\in\mathcal H$;
\item[(ii)] for each $h\in\mathcal H$, $A^-(\overline h)$ is (the restriction of) the adjoint operator of $A^+(h)$, where $\bar h$ is the complex conjugate of $h$;
\item[(iii)] the operators $A^+(h)$, $A^-(h)$ satisfy the canonical commutation relations (CCR).
\end{itemize}
See Section~\ref{xrwaq4q2} for details.
Let $\mathbb A$ be the unital $*$-algebra generated by the operators $A^+(h)$, $A^-(h)$. If we additionally assume that $\mathfrak F$ is a certain symmetric Fock space, then we can define the vacuum state $\tau$ on $\mathbb A$. If $\tau$ appears to be a quasi-free state, one says that the operators $A^+(h)$ and $A^-(h)$ form a {\it quasi-free representation of the CCR}.
We define operator-valued distributions $A^+(x)$ and $A^-(x)$
($x\in X$) through the equalities
\begin{equation}\label{ray45}
A^+(h)=\int_X h(x)A^+(x)\sigma(dx),\quad A^-(h)=\int_X h(x)A^-(x)\sigma(dx),\end{equation}
holding for all $h\in\mathcal H$.
Then the {\it particle density $\rho(x)$} is formally defined as
$$\rho(x):=A^+(x)A^-(x),\quad x\in X.$$
We called this definition {\it formal} since it requires to take product of two operator-valued distributions, and {\it a priori\/} it is not clear if this product indeed makes sense. Nevertheless, in all the examples below, we will be able to rigorously define $\rho(x)$ as an operator-valued distribution.
The CCR imply the commutation $[\rho(x),\rho(y)]=0$ ($x,y\in X$), where $[\cdot,\cdot]$ denotes the commutator. For each $\Delta\in\mathcal B_0(X)$, we denote
\begin{equation}\label{raq5wu}
\rho(\Delta):=\int_\Delta\rho(x)\sigma(dx)=\int_\Delta A^+(x)A^-(x)\sigma(dx),\end{equation}
which is a family of Hermitian commuting operators in the Fock space $\mathfrak F$. In view of the spectral theorem, one can expect that the operators $(\rho_\Delta)_{\Delta\in\mathcal B_0(X)}$ can be realized as operators of multiplication in $L^2(\Gamma_X,\mu)$, where $\mu$ is the joint spectral measure of this family of operators at the vacuum.
Let $G(x)$ be a complex-valued Gaussian field and $R(x)=|G(x)|^2$. The main aim of the paper is show that the Cox process $\Pi_R$ is the joint spectral measure of
a (rigorously defined) particle density $(\rho_\Delta)_{\Delta\in\mathcal B_0(X)}$ for a certain quasi-free representation of the CCR. As a by-product, we obtain a unitary isomorphism between a subspace of a Fock space and $L^2(\Gamma_X,\Pi_R)$.
In the special case where $\Pi_R$ is a permanental point process (with a real-valued correlation kernel), such a statement was proved in \cite{LM} (see also \cite{Ly}).
In that case, the corresponding quasi-free state has an additional property of being gauge-invariant, so one could use the gauge-invariant quasi-free representation of the CCR by Araki and Woods \cite{ArWoods}.
We stress that, even in the case of a gauge-invariant quasi-free state, the representation of the CCR that we use in this paper has a different form as compared to the one by Araki and Woods \cite{ArWoods}. Nevertheless, since both gauge-invariant quasi-free representations have the same $n$-point functions, one can show that these representations are unitarily equivalent.
We note that, in \cite{Ly,LM}, it was also shown that each determinantal point process ($\alpha=-1$) arises as the joint spectral measure of the particle density of a quasi-free representation of the Canonical Anticommutation Relations (CAR). In that case, the state is also gauge-invariant, so one can use the Araki--Wyss representation of the CAR from \cite{ArWyss}.
It is worth to compare our result with the main result of Koshida \cite{Koshida}. In the latter paper, it is proven that, when the underlying space $X$ is {\it discrete}, every pfaffian point process on $X$ arises as the particle density of a quasi-free representation of the CAR. As noted in \cite{Koshida}, a similar statement in the case of a continuous space $X$ is still an open problem.
\subsection{Organization of the paper}
The starting point of our considerations is the observation that the Poisson point process with (deterministic) intensity $|\lambda(x)|^2$ arises from the trivial (quasi-free) representation of the CCR with
\begin{equation}\label{teraw5uw}
A^+(x)=a^+(x)+\overline{\lambda(x)},\quad A^-(x)=a^-(x)+\lambda(x),
\end{equation}
where $a^+(x)$, $a^-(x)$ are the creation and annihilation operators at point $x$, acting in the symmetric Fock space $\mathcal F(\mathcal H)$ over $\mathcal H$, compare with \cite{GGPS}. We then proceed as follows:
\begin{itemize}
\item We realize a Gaussian field $G(x)$ as a family of operators $\Phi(x)$
acting in a Fock space $\mathcal F(\mathcal G)$ over a Hilbert space $\mathcal G$ (typically $\mathcal G=\mathcal H$ or $\mathcal G=\mathcal H\oplus\mathcal H$).
\item We consider a quasi-free representation of the CCR with
\begin{equation}\label{fyr6sw6u}
A^+(x)=a^+(x)+\Phi^*(x),\quad A^-(x)=a^-(x)+\Phi(x)
\end{equation}
acting in the Fock space $\mathcal F(\mathcal H\otimes\mathcal G)=\mathcal F(\mathcal H)\otimes\mathcal F(\mathcal G)$.
\item We prove that the corresponding particle density $(\rho_\Delta)_{\Delta\in\mathcal B_0(X)}$ is well-defined and has the joint spectral measure $\Pi_R$.
\end{itemize}
The paper is organized as follows. In Section \ref{ew56u3wu}, we discuss complex-valued Gaussian fields on $X$ realized in a symmetric Fock space $\mathcal F(\mathcal G)$ over a separable Hilbert space $\mathcal G$. We start with a $\mathcal G^2$-valued function $(L_1(x),L_2(x))$ that is defined for $\sigma$-a.a.\ $x\in X$ and satisfies the assumptions \eqref{ydxdrdxdrg}, \eqref{xrsa5yw4} below. We then define operators $\Phi(x)$ in the Fock space $\mathcal F(\mathcal G)$ by formula \eqref{waaq4yq5y}. Theorem~\ref{reaq5y43wu} states that the operators $\Phi(x)$ form a Fock-space realization of a Gaussian field $G(x)$ that is defined for $\sigma$-a.a.\ $x\in X$. (Note, however, that the set of those $x\in X$ for which $G(x)$ is defined can be smaller than the set of those $x\in X$ for which the function $(L_1(x),L_2(x))$ was defined.) The covariance and pseudo-covariance of the Gaussian field $G(x)$ are given by formulas \eqref{w909i8u7y689} and \eqref{ftsqw43qd}, respectively.
As a consequence of our considerations, in Example~\ref{vcrtw5y3}, we derive a Fock-space realization of a proper Gaussian field. The operators $\Phi(x)$ in this case resemble the classical Fock-space realization of a real-valued Gaussian field. The main difference is that, in the case of a real-valued Gaussian field, the creation and annihilation operators use same real vectors, whereas in the case of a proper Gaussian field, the creation and annihilation operators use orthogonal copies of same complex vectors.
In Section~\ref{xrwaq4q2}, we briefly recall the definition of a quasi-free state on the CCR algebra and a quasi-free representation of the CCR.
Next, in Section~\ref{tew532w}, we recall in Theorem~\ref{tes56uwe4u6} a result from \cite{LM} which gives sufficient conditions for a family of commuting Hermitian operators, $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$, in a separable complex Hilbert space, to be essentially self-adjoint and have a point process $\mu$ in $X$ as their joint spectral measure. The key condition of Theorem~\ref{tes56uwe4u6} is that the family of operators, $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$, should possess certain correlation measures $\theta^{(n)}$, whose definition is given in Section~\ref{tew532w}. These measures $\theta^{(n)}$ are then also the correlation measures of the point process $\mu$. We also present formal considerations about the form of the correlation measures $\theta^{(n)}$ when $\rho(\Delta)$ is a particle density given by \eqref{raq5wu}.
In Section~\ref{xeeraq54q}, we apply Theorem~\ref{tes56uwe4u6} to show that a Poisson point process is the joint spectral measure of the operators $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$, where $\rho(\Delta)$ is the particle density of the trivial quasi-free representation of the CCR in which the creation and annihilation operators are given by \eqref{teraw5uw}.
The main results of the paper are in Section~\ref{yd6w6wdd}. Using the $\mathcal G^2$-valued function $(L_1(x),L_2(x))$ from Section~\ref{ew56u3wu}, we construct a quasi-free representation of the CCR in the symmetric Fock space $\mathcal F(\mathcal H\oplus\mathcal G)$. We prove that the corresponding particle density is well defined as a family of commuting Hermitian operators, $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$ (Corollary~\ref{3rtlgpr}). Theorem~\ref{due6uew4} states that these operators satisfy the assumptions of Theorem~\ref{tes56uwe4u6} and their joint spectral measure $\mu$ is the Cox process $\Pi_R$, where $R(x)=|G(x)|^2$ and $G(x)$ is the Gaussian field as in in Theorem~\ref{reaq5y43wu}. In particular, $\mu$ is a hafnian point process.
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\section{Fock-space realization of complex Gaussian fields}\label{ew56u3wu}
Let $\mathcal G$ be a separable Hilbert space with an antilinear involution $\mathcal J$ satisfying $(\mathcal Jf,\mathcal Jg)_{\mathcal G}=(g,f)_{\mathcal G}$ for all $f,g\in\mathcal G$. Let $\mathcal G^{\odot n}$ denote the $n$th symmetric tensor power of $\mathcal G$. For $n\in\mathbb N$, let $\mathcal F_n(\mathcal G):=\mathcal G^{\odot n}n!$, i.e., $\mathcal F_n(\mathcal G)$ coincides with $\mathcal G^{\odot n}$ as a set and the inner product in $\mathcal F_n(\mathcal G)$ is equal to $n!$ times the inner product in $\mathcal G^{\odot n}$. Let also $\mathcal F_0(\mathcal G):=\mathbb C$. Then $\mathcal F(\mathcal G)=\bigoplus_{n=0}^\infty\mathcal F_n(\mathcal G)$ is called the {\it symmetric Fock space over $\mathcal G$}. The vector $\Omega=(1,0,0,\dots)\in\mathcal F(\mathcal G)$ is called the {\it vacuum}.
Let $\mathcal F_{\mathrm{fin}}(\mathcal G)$ denote the (dense) subspace of $\mathcal F(\mathcal G)$ consisting of all finite vectors $f=(f^{(0)},f^{(1)},\dots,f^{(n)},0,0,\dots)$ ($n\in\mathbb N$). We equip $\mathcal F_{\mathrm{fin}}(\mathcal G)$ with the topology of the topological direct sum of the $\mathcal F_n(\mathcal G)$ spaces.
For topological vector spaces $V$ and $W$, we denote by $\mathcal L(V,W)$ the space of all linear continuous operators $A:V\to W$. We also denote $\mathcal L(V):=\mathcal L(V,V)$.
Let $g\in\mathcal G$. We define a {\it creation operator} $a^+(g)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal G))$ by $a^+(g)\Omega:=g$, and for $f^{(n)}\in\mathcal F_n(\mathcal G)$ ($n\in\mathbb N$),
$a^+(g)f^{(n)}:=g\odot f^{(n)}\in\mathcal F_{n+1}(\mathcal G)$ . Next, we define an {\it annihilation operator} $a^-(g)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal G))$ that satisfies $a^-(g)\Omega:=0$ and for any $f_1,\dots,f_n\in\mathcal G$,
$$a^-(g)f_1\odot\dotsm\odot f_n=\sum_{i=1}^n(f_i,\mathcal Jg)_{\mathcal G}\,f_1\odot\dots\odot f_{i-1}\odot f_{i+1}\odot\dots\odot f_n.$$
We have $a^+(g)^*\restriction_{\mathcal F_{\mathrm{fin}}(\mathcal G)}=a^-(\mathcal Jg)$ and the operators $a^+(g)$, $a^-(g)$ satisfy the CCR:
\begin{equation}\label{tsw654w}
[a^+(f),a^+(g)]=[a^-(f),a^-(g)]=0,\quad [a^-(f),a^+(g)]=(g,\mathcal Jf)_{\mathcal G}\end{equation}
for all $f,g\in\mathcal G$.
Let $D\in\mathcal B(X)$ be such that $\sigma(X\setminus D)=0$. Let $D\ni x\mapsto (L_1(x),L_2(x))\in\mathcal G^2$ be a measurable mapping. We assume that
\begin{align}
&(L_1(x),\mathcal JL_2(y))_{\mathcal G}=(L_1(y),\mathcal JL_2(x))_{\mathcal G},\label{ydxdrdxdrg}\\
&(L_1(x),L_1(y))_{\mathcal G}=(L_2(x),L_2(y))_{\mathcal G}\quad \text{for all }x,y\in D.\label{xrsa5yw4}
\end{align}
Define
\begin{equation}\label{waaq4yq5y}
\Phi(x):=a^+(L_1(x))+a^-(L_2(x)).\end{equation}
Let $\Psi(x):=\Phi(x)^*\restriction_{\mathcal F_{\mathrm{fin}}(\mathcal G)}$.
Then
\begin{equation}\label{ytd7r57}
\Psi(x)=a^+(\mathcal JL_2(x))+a^-(\mathcal JL_1(x)).\end{equation}
It follows from \eqref{tsw654w} that conditions \eqref{ydxdrdxdrg}, \eqref{xrsa5yw4} are necessary and sufficient in order that $[\Phi(x),\Phi(y)]=[\Psi(x),\Psi(y)]=[\Phi(x),\Psi(y)]=0$ for all $x,y\in D$.
Below, for each $\Lambda\subset D$, we denote by $\mathbb F_\Lambda$ the subspace of the Fock space $\mathcal F(\mathcal G)$ that is the closed linear span of the set
\begin{multline*}
\big\{\Psi(x_1)^{k_1}\dotsm\Psi(x_m)^{k_m}\Phi(y_1)^{l_1}\dotsm \Phi(y_n)^{l_n}\Omega\mid x_1,\dots,x_m,y_1,\dots,y_n\in \Lambda,\\ k_1,\dots,k_m,l_1,\dots,l_n\in\mathbb N_0,\ m,n\in\mathbb N\big\}.\end{multline*}
Here $\mathbb N_0:=\{0,1,2,3,\dots\}$.
\begin{theorem}\label{reaq5y43wu}
There exists a measurable subset $\Lambda\subset D$ with $\sigma(X\setminus\Lambda)=0$ and a mean-zero complex-valued Gaussian field $\{G(x)\}_{x\in \Lambda}$ on a probability space $(\Xi,\mathfrak A,P)$ such that:
\begin{itemize}
\item[(i)] The Gaussian field $\{G(x)\}_{x\in \Lambda}$ has the covariance
\begin{equation}\label{w909i8u7y689}
\mathcal K_1(x,y)=(L_1(x),L_1(y))_{\mathcal G},\quad x,y\in \Lambda,\end{equation}
and the pseudo-covariance
\begin{equation}\label{ftsqw43qd}
\mathcal K_2(x,y)=(L_1(x),\mathcal JL_2(y))_{\mathcal G},\quad x,y\in \Lambda.\end{equation}
\item[(ii)] There exists a unique unitary operator $\mathcal I:\mathbb F_\Lambda\to L^2(\Xi,P)$ that satisfies
\begin{align}
&\mathcal I\Psi(x_1)^{k_1}\dotsm\Psi(x_m)^{k_m}\Phi(y_1)^{l_1}\dotsm \Phi(y_n)^{l_n}\Omega\notag\\
&\quad= \overline{G(x_1)}^{\,k_1}\dotsm\overline{G(x_m)}^{\,k_m}G(y_1)^{l_1}\dotsm G(y_n)^{l_n}\label{cy6e64u43}\end{align}
for all $x_1,\dots,x_m,y_1,\dots,y_n\in \Lambda$, $k_1,\dots,k_m,l_1,\dots,l_n\in\mathbb N_0$, $m,n\in\mathbb N$.
\end{itemize}
\end{theorem}
\begin{proof} We define $L_1(x)=L_2(x)=0$ for all $x\in X\setminus D$. Then
\begin{equation}\label{tes5w53w}
X\ni x\mapsto(L_1(x),L_2(x))\in\mathcal G^2
\end{equation}
is measurable and satisfies \eqref{ydxdrdxdrg}, \eqref{xrsa5yw4} for all $x,y\in X$. By Lusin's theorem (see e.g.\ \cite[26.7~Theorem]{Bauer}), there exists a sequence of mutually disjoint compact sets $(\Lambda_n)_{n=1}^\infty$ such that $\sigma\big(X\setminus\bigcup_{n=1}^\infty\Lambda_n\big)=0$, and the restriction of the mapping \eqref{tes5w53w} to each $\Lambda_n$ is continuous. Denote $\Lambda:=\bigcup_{n=1}^\infty\Lambda_n$ and choose a countable subset $\Lambda'\subset\Lambda$ such that, for each $n\in\mathbb N$, the set $\Lambda'\cap\Lambda_n$ is dense in $\Lambda_n$. As easily seen by approximation, $\mathbb F_{\Lambda'}=\mathbb F_\Lambda$.
Let us consider the real and imaginary parts of the operators $\Phi(x)$:
\begin{align}
\Re(\Phi(x))&:=\frac12(\Phi(x)+\Psi(x))=\frac12\left(a^+\big(L_1(x)+\mathcal JL_2(x)\big)+a^-\big(\mathcal JL_1(x)+L_2(x)\big)\right),\notag\\
\Im(\Phi(x))&:=\frac1{2i}(\Phi(x)-\Psi(x))=\frac1{2i}\left(a^+\big(L_1(x)-\mathcal JL_2(x)\big)-a^-\big(\mathcal JL_1(x)-L_2(x)\big)\right).\label{se5w5u33w}
\end{align}
These operators are Hermitian and commuting.
It is a standard fact that, for each $g\in\mathcal G$,
\begin{equation}\label{cr6sw6u4e}
\|a^+(g)\|_{\mathcal L(\mathcal F_k(\mathcal G),\mathcal F_{k+1}(\mathcal G))}=\|a^-(g)\|_{\mathcal L(\mathcal F_{k+1}(\mathcal G),\mathcal F_{k}(\mathcal G))}=\sqrt{k+1}\,\|g\|_{\mathcal G}. \end{equation}
From here it easily follows that each $f\in\mathcal F_{\mathrm{fin}}(\mathcal G)$ is an analytic vector for each $\Re(\Phi(x))$ and $\Im(\Phi(x))$ ($x\in X$), and the projection-valued measures of the closures of all these operators commute, see \cite[Chapter~5, Theorem~1.15]{BK}.
We now apply the projection spectral theorem \cite[Chapter~3, Theorems~2.6 and 3.9 and Section~3.1]{BK} to the closures of the operators $\Re(\Phi(x))$ and $\Im(\Phi(x))$ with $x\in\Lambda'$. This implies the existence of a probability space $(\Xi,\mathfrak A,P)$, real-valued random variables $G_1(x)$ and $G_2(x)$ ($x\in\Lambda'$) and a unique unitary operator $\mathcal I:\mathbb F_\Lambda\to L^2(\Xi,P)$ that satisfies
\begin{align}
&\mathcal I\,\Re(\Phi(x_1))^{k_1}\dotsm\Re(\Phi(x_m))^{k_m}\Im(\Phi(y_1))^{l_1}\dotsm \Im(\Phi(y_n))^{l_n}\Omega\notag\\
&\quad= G_1(x_1)^{\,k_1}\dotsm G_1(x_m)^{\,k_m}G_2(y_1)^{l_1}\dotsm G_2(y_n)^{l_n}
\label{xrq5y3q2u}
\end{align}
for all $x_1,\dots,x_m,y_1,\dots,y_n\in \Lambda'$, $k_1,\dots,k_m,l_1,\dots,l_n\in\mathbb N_0$, $m,n\in\mathbb N$.
\begin{remark} In fact, $\Xi=\{\omega:\Lambda'\to\R^2\}$, $\mathfrak A$ is the cylinder $\sigma$-algebra on $\Xi$ (equivalently the countable product of the Borel $\sigma$-algebras $\mathcal B(\R)$), and $P(\cdot)=(E(\cdot)\Omega,\Omega)_{\mathcal F(\mathcal G)}$. Here, $E$ is the projection-valued measure on $(\Xi,\mathfrak A)$ that is constructed as the countable product of the projection-valued measures of the closures of the operators $\Re(\Phi(x))$ and $\Im(\Phi(x))$ with $x\in\Lambda'$. Furthermore, for each $x\in\Lambda'$, $(G_i(x))(\omega)=\omega_i(x)$ for $\omega=(\omega_1,\omega_2)\in\Xi$.
\end{remark}
Next, let $n\in\mathbb N$ and $x\in\Lambda_n\setminus\Lambda'$. Then we can find a sequence $(x_k)_{k=1}^\infty$ in $\Lambda'\cap \Lambda_n$ such that $x_k\to x$, hence, by continuity, $(L_1(x_k),L_2(x_k))\to (L_1(x),L_2(x))$ in $\mathcal G^2$ . It follows from \eqref{xrq5y3q2u} that $(G_i(x_k))_{k=1}^\infty$ is a Cauchy sequence in $L^2(\Xi,P)$ ($i=1,2$), so we define $G_i(x):=\lim_{k\to\infty}G_i(x_k)$. Then we easily see by approximation that \eqref{xrq5y3q2u} remains true for all $x_1,\dots,x_m,y_1,\dots,y_n\in \Lambda$.
Let $Z$ be an arbitrary finite linear combination (with real coefficients) of random variables from $\{G_1(x),\,G_2(x)\mid x\in\Lambda\}$. Then it follows from \eqref{se5w5u33w} that the moments of $Z$ are given by
$$\mathbb E(Z^k)=\big((a^+(g)+a^-(\mathcal Jg))^k\,\Omega,\Omega\big)_{\mathcal F(\mathcal G)}$$
for some $g\in\mathcal G$. But this implies that the random variable $Z$ has a Gaussian distribution, see e.g.\ \cite[Chapter~3, Subsection~3.8]{BK}. Hence, $\{G_1(x),\,G_2(x)\mid x\in\Lambda\}$ is a Gaussian field.
Finally, for each $x\in\Lambda$, we define $G(x):=G_1(x)+iG_2(x)$. Then $\{G(x)\}_{x\in \Lambda}$ is a complex-valued Gaussian field. Formula \eqref{xrq5y3q2u} implies \eqref{cy6e64u43}. This, in turn, gives us the covariance and the pseudo-covariance of the Gaussian field $\{G(x)\}_{x\in \Lambda}$.
\end{proof}
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Let $\mathcal H=L^2(X,\sigma)$ and define an antilinear involution $J:\mathcal H\to \mathcal H$ by $(Jh)(x):=\overline{h(x)}$.
Let us consider a measurable mapping $x\mapsto L(x)\in\mathcal H$ defined $\sigma$-a.e.\ on $X$, and let $K(x,y):=(L(x),L(y))_{\mathcal H}$. We will now consider two examples of complex-valued Gaussian fields with covariance
$K(x,y)$.
\begin{example}\label{fyte6i47}
Let $\mathcal G=\mathcal H$, $\mathcal J=J$, and let $L_1(x)=L_2(x)=L(x)$. Obviously, conditions \eqref{ydxdrdxdrg}, \eqref{xrsa5yw4} are satisfied. Then,
$$\Phi(x)=a^+(L(x))+a^-(L(x)),\quad \Psi(x)=a^+(JL(x))+a^-(JL(x)).$$
By Theorem~\ref{reaq5y43wu}, the corresponding Gaussian field $G(x)$ has the covariance $\mathcal K_1(x,y)=K(x,y)$ and the pseudo-covariance $\mathcal K_2(x,y)=(L(x),JL(y))_{\mathcal H}$. If $L(x)$ is a real-valued function for $\sigma$-a.a.\ $x\in X$, $\mathcal K_1(x,y)=\mathcal K_2(x,y)=K(x,y)$, while the function $K(x,y)$ is symmetric. Hence, as discussed in Introduction, $R(x):=|G(x)|^2$ is a 2-permanental process, defined for $\sigma$-a.a.\ $x\in X$. If $L(x)$ is not real-valued on a set of positive $\sigma$ measure, then the moments of $R(x)$
are given by \eqref{dr6e6u43wq}, \eqref{d6e6ie4} with $\mathcal K_1(x,y)$, $\mathcal K_2(x,y)$ as above.
\end{example}
\begin{example}\label{vcrtw5y3}
Let $\mathcal G=\mathcal H\oplus\mathcal H$, $\mathcal J=J\oplus J$, and let $L_1(x)=(L(x),0)$, $L_2(x)=(0,L(x
))$. As easily seen, conditions \eqref{ydxdrdxdrg}, \eqref{xrsa5yw4} are satisfied.
We define, for each $h\in\mathcal H$, $a_1^+(h):=a^+(h,0)$, $a_2^+(h):=a^+(0,h)$ and similarly $a_1^-(h)$, $a_2^-(h)$. Then
$$\Phi(x)=a_1^+(L(x))+a_2^-(L(x)),\quad \Psi(x)=a_2^+(JL(x))+a_1^-(JL(x)).$$
For the corresponding Gaussian field $G(x)$, $\mathcal K_1(x,y)=K(x,y)$, while $\mathcal K_2(x,y)=0$, i.e., $G(x)$ is a proper Gaussian field.
Hence, as discussed in Introduction, $R(x):=|G(x)|^2$ is a permanental process, defined for $\sigma$-a.a.\ $x\in X$.
\end{example}
\begin{remark}
Let $G_1(x)$ and $G_2(x)$ be two independent copies of the Gaussian field from Example~\ref{fyte6i47}. Then, the Gaussian field $G(x)$ from Example~\ref{vcrtw5y3} can be constructed as $G(x)=\frac1{\sqrt2}\big(G_1(x)+iG_2(x)\big)$.
\end{remark}
The following example generalizes the constructions in Examples~\ref{fyte6i47} and \ref{vcrtw5y3}.
\begin{example}\label{xerwu56}
Let $\mathcal G=\mathcal H\oplus\mathcal H$ be as in Example~\ref{vcrtw5y3} and consider a measurable mapping $x\mapsto(\alpha(x),\beta(x))\in\mathcal G^2$ defined $\sigma$-a.e.\ on $X$. Let
$$L_1(x):=\left(\frac{\alpha(x)+\beta(x)}2\,,\frac{\alpha(x)-\beta(x)}2\right),\quad L_2(x):=\left(\frac{\alpha(x)-\beta(x)}2\,,\frac{\alpha(x)+\beta(x)}2\right).$$
As easily seen, conditions \eqref{ydxdrdxdrg} and \eqref{xrsa5yw4} are satisfied. For the corresponding Gaussian field $G(x)$,
\begin{align}
\mathcal K_1(x,y)&=\frac12\big((\alpha(x),\alpha(y))_{\mathcal H}+(\beta(x),\beta(y))_{\mathcal H}\big)\notag,\\
\mathcal K_2(x,y)&=\frac12\big((\alpha(x), J\alpha(y))_{\mathcal H}-(\beta(x),J\beta(y))_{\mathcal H}\big).\notag
\end{align}
In the special case where $L(x)=\alpha(x)=\beta(x)$, this is just the construction from Example~\ref{vcrtw5y3}. When choosing $\alpha(x)=\sqrt 2\, L(x)$ and $\beta(x)=0$, the corresponding Gaussian field $G(x)$ has the same finite-dimensional distributions as the Gaussian field from Example~\ref{fyte6i47}.
\end{example}
Let the conditions of Theorem~\ref{reaq5y43wu} be satisfied and $R(x)=|G(x)|^2$. To construct the Cox process $\Pi_R$ with correlation functions given by \eqref{xreew5u}, we further assume that, for each $\Delta\in\mathcal B_0(X)$, $\int_\Delta\mathbb E(R(x))\sigma(dx)<\infty$. By \eqref{xrsa5yw4} and \eqref{w909i8u7y689}, this is equivalent to the condition
\begin{equation}\label{vd5yw573}
\int_\Delta\|L_1(x)\|_{\mathcal G}^2\,\sigma(dx)=\int_\Delta\|L_2(x)\|_{\mathcal G}^2\,\sigma(dx)<\infty.\end{equation}
\setcounter{theorem}{2}
\begin{example}[continued] Since $\mathcal G=\mathcal H$, we define $L(x,y):=(L(x))(y)$. By \eqref{vd5yw573},
$$\int_{\Delta\times X} |L(x,y)|^2\sigma^{\otimes 2}(dx\,dy)<\infty.$$
Hence, for each $\Delta\in\mathcal B_0(X)$, we can define a Hilbert--Schmidt operator $L^\Delta$ in $\mathcal H$ with integral kernel $\chi_\Delta(x)L(x,y)$. Here $\chi_\Delta$ denotes the indicator function of the set $\Delta$. Define $K^\Delta:=L^\Delta(L^\Delta)^*$. This operator is nonnegative ($K^\Delta\ge0$), trace-class, and has integral kernel $K^\Delta(x,y)=(L(x),L(y))_{\mathcal H}$ for $x,y\in\Delta$. (Note that $K^\Delta(x,y)$ vanishes outside $\Delta^2$). Thus, for $x,y\in\Delta$, the covariance $\mathcal K_1(x,y)$ of the Gaussian $G(x)$ is equal to $K^\Delta(x,y)$.
Next, for a bounded linear operator $A\in\mathcal H$, we define the {\it transposed} of $A$ by $A^T:=JA^*J$. If $A$ is an integral operator with integral kernel $A(x,y)$, then $A^T$ is the integral operator with integral kernel $A^T(x,y)=A(y,x)$. Hence, for all $x,y\in\Delta$, the pseudo-covariance $\mathcal K_2(x,y)$ of the Gaussian $G(x)$ is equal to the integral kernel $Q^\Delta(x,y)$
of the operator $Q^\Delta:=L^\Delta(L^\Delta)^T$.
In the case where $L(x,y)$ is an integral kernel of a bounded linear operator $L$ in $\mathcal H$, we can define $K:=LL^*$ and $Q:=LL^T$, and $\mathcal K_1(x,y)=K(x,y)$, $\mathcal K_2(x,y)=Q(x,y)$, where $K(x,y)$ and $Q(x,y)$ are the integral kernels of the operators $K$ and $Q$, respectively.
\end{example}
\begin{example}[continued] We proceed similarly to Example~\ref{fyte6i47}. However, in this case, the moments of the Gaussian field $G(x)$ are determined by the covariance $\mathcal K_1(x,y)$ only. Hence, if $L(x,y)$ is an integral kernel of a bounded linear operator $L$ in $\mathcal H$, the moments are determined by (the integral kernel of) the operator $K:=LL^*$. Hence, without loss of generality, we may assume that $L=\sqrt K$, equivalently the operator $L$ is self-adjoint.
\end{example}
\setcounter{theorem}{5}
\begin{example}[continued] Since $\mathcal G=\mathcal H$, we define $\alpha(x,y):=(\alpha(x))(y)$ and similarly $\beta(x,y)$.
In this case, condition \eqref{vd5yw573} means that, for each $\Delta\in\mathcal B_0(X)$,
$$\int_{\Delta\times X}(|\alpha(x,y)|^2+|\beta(x,y)|^2)\sigma^{\otimes 2}(dx\,dy)<\infty, $$
and we can proceed similarly to Example~\ref{fyte6i47}. Assume additionally that $\alpha(x,y)$ and $\beta(x,y)$ are integral kernels of operators $A,B\in\mathcal L(\mathcal H)$, respectively. Then the covariance $\mathcal K_1(x,y)$ of the Gaussian field $G(x)$ is the integral kernel of the operator $\frac12(AA^*+BB^*)$, while the pseudo-covariance $\mathcal K_2(x,y)$ is the integral kernel of the operator $\frac12(AA^T-BB^T)$.
\end{example}
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\section{Quasi-free states on the CCR algebra}\label{xrwaq4q2}
In this section, we assume that $\mathcal H$ is a separable complex Hilbert space with an antilinear involution $J$ in $\mathcal H$
satisfying $(Jf,Jh)_{\mathcal H}=(h,f)_{\mathcal H}$ for all $f,h\in\mathcal H$.
Let $\mathcal V$ be a dense subspace of $\mathcal H$ that is invariant for $J$.
Let $\mathfrak F$ be a separable Hilbert space and $\mathfrak D$ a dense subspace of $\mathfrak F$. For each $h\in\mathcal V$, let $A^+(h),\, A^-(h):\mathfrak D\to\mathfrak D$ be linear operators satisfying the following assumptions:
\begin{itemize}
\item[(i)] $A^+(h)$ and $A^-(h)$ depend linearly on $h\in\mathcal V$;
\item[(ii)] the domain of the adjoint operator of $A^+(h)$ in $\mathfrak F$ contains $\mathfrak D$ and $A^+(h)^*\restriction\mathfrak D=A^-(Jf)$;
\item[(iii)] the operators $A^+(h)$, $A^-(h)$ satisfy the CCR:
\begin{equation}\label{ydey7e76}
[A^+(f),A^+(h)]=[A^-(f),A^-(h)]=0,\quad [A^-(f),A^+(h)]=(h,Jf)_{\mathcal H}\end{equation}
for all $f,h\in\mathcal V$.
\end{itemize}
Let $\mathbb A$ denote the complex unital $*$-algebra generated by the operators $A^+(h)$, $A^-(h)$ ($h\in\mathcal V$). Let $\tau:\mathbb A\to\mathbb C$ be a state on $\mathbb A$, i.e., $\tau$ is linear, $\tau(\mathbf 1)=1$ and $\tau(a^*a)\ge0$ for all $a\in\mathbb A$. For each $h\in\mathcal V$, we define a Hermitian operator
\begin{equation}\label{e5w7w37w2} B(h):=A^+(h)+A^-(Jh).
\end{equation}
These operators satisfy the commutation relation
\begin{equation}\label{vctsdtu6}
[B(f),B(h)]=2\Im(h,f)_{\mathcal H},\quad h,f\in\mathcal H.
\end{equation}
Note that
$$A^+(h)=\frac12(B(h)-iB(ih)),\quad A^-(h)=\frac12(B(Jh)+iB(Jh)).$$
Therefore, we can think of the algebra $\mathbb A$ as generated by the operators $B(h)$ ($h\in\mathcal V$), subject to the commutation relation \eqref{vctsdtu6}. Hence, the state $\tau$ is completely determined by the functionals $T^{(n)}:\mathcal V^n\to\mathbb C$ ($n\ge1$), where $T^{(1)}(h):=\tau(B(h))$ and
$$ T^{(n)}(h_1,\dots,h_n):=\tau\big((B(h_1)-T^{(1)}(h_1))\dotsm (B(h_n)-T(h_n))\big),\quad n\ge2.$$
The state $\tau$ is called {\it quasi-free} if, for each $n\in\mathbb N$, $T^{(2n+1)}=0$ and
$$T^{(2n)}(h_1,\dots,h_{2n})=\sum T^{(2)}(h_{i_1},h_{j_1})\dotsm T^{(2)}(h_{i_n},h_{j_n}) $$
where the summation is over all partitions $\{i_1,j_1\},\dots,\{i_n,j_n\}$ of $\{1,\dots,2n\}$ with $i_k<j_k$ ($k=1,\dots,n$), see e.g.\ \cite[Section~5.2]{BR}.
\begin{remark} \label{vtw53u}
Let $\phi:\mathcal V\to\mathbb C$ be a linear functional.
For each $h\in\mathcal V$, we define operators $\mathbf A^+(h):=A^+(h)+\phi'(h)$ and $\mathbf A^-(h):=A^-(h)+\phi(h)$, where $\phi'(h):=\overline{\phi(Jh)}$. The operators $\mathbf A^+(h)$, $\mathbf A^-(h)$ also satisfy the conditions (i)--(iii) discussed above. Obviously, the algebra generated by the operators $\mathbf A^+(h)$, $\mathbf A^-(h)$ coincides with $\mathbb A$.
If $\tau:\mathbb A\to\mathbb C$ is a quasi-free state with respect to the operators $A^+(h)$, $A^-(h)$, then $\tau$ is also quasi-free with respect to the operators $\mathbf A^+(h)$, $\mathbf A^-(h)$.
\end{remark}
Let us now present an explicit construction of a representation of the CCR algebra $\mathbb A$ and a quasi-free state $\tau$ on it. This construction resembles the {\it Bogoliubov transformation}, see e.g.\ \cite[Subsection 5.2.2.2]{BR} or \cite[Section~4]{Berezin}\footnote{In \cite{Berezin}, a Bogoliubov transformation is called a {\it linear canonical transformation}.}.
Let $\mathcal E$ be a separable Hilbert space with an antilinear involution $\mathcal J$ satisfying $(\mathcal Jf,\mathcal Jg)_{\mathcal E}=(g,f)_{\mathcal E}$ for all $f,g\in\mathcal E$. Let $K_i\in\mathcal L(\mathcal H,\mathcal E)$ ($i=1,2$). Denote
\begin{equation}\label{rdtsw5u4w3wq}K_i':=\mathcal JK_iJ\in\mathcal L(\mathcal H,\mathcal E)
\end{equation}
and assume that
\begin{align}
(K'_2)^*K_1-(K_1')^*K_2&=0,\notag\\
K_2^*K_2-K_1^*K_1=1.\label{cdtwe64u53e}
\end{align}
For each $h\in\mathcal H$, we define, in the symmetric Fock space $\mathcal F(\mathcal E)$, the operators
\begin{equation}\label{tsw53w3}
A^+(h):=a^+(K_2h)+a^-(K_1h),\quad A^-(h):=a^-(K_2'h)+a^+(K_1'h)
\end{equation}
with domain $\mathcal F_{\mathrm{fin}}(\mathcal E)$. Here $a^+(\cdot)$ and $a^-(\cdot)$ are the creation and annihilation operators in $\mathcal F(\mathcal E)$, respectively.
It follows from \eqref{tsw654w}, \eqref{cdtwe64u53e}, and \eqref{tsw53w3} that $A^+(h)$ and $A^-(h)$ satisfy the conditions (i)--(iii) with
$\mathcal V=\mathcal H$, $\mathfrak F=\mathcal F(\mathcal E)$, and $\mathfrak D=\mathcal F_{\mathrm{fin}}(\mathcal E)$.
Let $\mathbb A$ denote the corresponding CCR algebra and let $\tau:
\mathbb A\to\mathbb C$ be the {\it vacuum state on $\mathbb A$}, i.e., $\tau(a)$$:=(a\Omega,\Omega)_{\mathcal F(\mathcal E)}$\,. For each $h\in\mathcal H$,
\begin{equation}\label{vcreaw53y}
B(h)=A^+(h)+A^-(Jh)=a^+((K_2+\mathcal JK_1)h)+a^-((K_1+\mathcal JK_2)h). \end{equation}
In particular, $\tau(B(h))=0$. Hence, it easily follows from \eqref{vcreaw53y} that $\tau$ is a quasi-free state with
\begin{equation}\label{rtwe64ue3}
T^{(2)} (f,h)=\big((K_1+\mathcal JK_2)f,(K_1+\mathcal JK_2)h\big)_{\mathcal E}\,.
\end{equation}
\begin{remark}Note that, in the classical Bogoliubov transformation, one chooses $\mathcal E=\mathcal H$.
\end{remark}
\begin{remark}\label{vrtw52}
Choosing $\mathcal E=\mathcal H$, $K_1=1$ and $K_2=0$, we get $A^+(h)=a^+(h)$, $A^-(h)=a^-(h)$. In this case, the vacuum state is quasi-free, with $T^{(1)}=0$ and $T^{(2)}(f,h)=(f,Jh)_{\mathcal H}$.
\end{remark}
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\section{Particle density and correlation functions}\label{tew532w}
Let $\mathfrak F$ be a separable Hilbert space and let $\mathfrak D$ be a dense subspace of $\mathfrak F$. For each $\Delta\in\mathcal B_0(X)$, let $\rho(\Delta):\mathfrak D\to\mathfrak D$ be a linear Hermitian operator in $\mathfrak F$. We further assume that the operators $\rho(\Delta)$ commute, i.e.,
$[\rho(\Delta_1),\rho(\Delta_2)]=0$ and for any $\Delta_1,\Delta_2\in\mathcal B_0(X)$.
Let $\mathcal A$ denote
the complex unital (commutative) $*$-algebra generated by $\rho(\Delta)$ ($\Delta\in\mathcal B_0(X)$). Let $\Omega$ be a fixed vector in $\mathfrak F$ with $\|\Omega\|_{\mathfrak F}=1$, and let a state $\tau:\mathcal A\to\mathbb C$ be defined by $\tau(a):=(a\Omega,\Omega)_{\mathfrak F}$ for $a\in\mathcal A$.
We define {\it Wick polynomials in $\mathcal A$} by the following recurrence formula:
\begin{align}
{:}\rho(\Delta){:}&=\rho(\Delta),\notag\\
{:}\rho(\Delta_1)\dotsm \rho(\Delta_{n+1}){:}&=\rho(\Delta_{n+1})\,
{:}\rho(\Delta_1)\dotsm \rho(\Delta_{n}){:}\notag\\
&\quad-\sum_{i=1}^n
{:}\rho(\Delta_1)\dotsm \rho(\Delta_{i-1})\rho(\Delta_i\cap\Delta_{n+1})\rho(\Delta_{i+1})\dotsm \rho(\Delta_n){:}\,,\label{dsaea78}
\end{align}
where $\Delta,\Delta_1,\dots,\Delta_{n+1}\in\mathcal B_0(X)$ and $n\in\mathbb N$.
It is easy to see that, for each permutation $\pi\in \mathfrak S_n$,
$${:}\rho(\Delta_1)\cdots \rho(\Delta_n){:} = {:}\rho(\Delta_{\pi(1)})\cdots \rho(\Delta_{\pi(n)}){:}\, .$$
We assume that, for each $n\in\mathbb N$, there exists a symmetric measure $\theta^{(n)}$ on $X^n$ that is concentrated on $X^{(n)}$ (i.e., $\theta^{(n)}(X^n\setminus X^{(n)})=0$) and satisfies\footnote{In view of formulas \eqref{dsaea78}, \eqref{6esuw6u61d}, it is natural to call $\theta^{(n)}$ the {\it $n$-th correlation measure of the family of operators} $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$.}
\begin{equation}\label{6esuw6u61d}
\theta^{(n)}\big(\Delta_1\times\dots\times\Delta_n\big)=\frac1{n!}\,
\tau\big({:}\rho(\Delta_1)\dotsm \rho(\Delta_{n}){:}\big),\quad \Delta_1,\dots,\Delta_{n}\in\mathcal B_0(X).\end{equation}
Furthermore, we assume that, for each $\Delta\in\mathcal B_0(X)$, there exists a constant $C_\Delta>0$ such that
\begin{equation}\label{fte76i4}
\theta^{(n)}(\Delta^n)\le C_\Delta^n,\quad n\in\mathbb N,
\end{equation}
and for any sequence
$\{\Delta_{l}\}_{l\in\mathbb{N}}\subset\mathcal{B}_{0}(X)$ such
that $\Delta_{l}\downarrow\varnothing$ (i.e., $\Delta_1\supset\Delta_2\supset\Delta_3\supset\cdots$ and $\bigcap_{l=1}^\infty\Delta_l=\varnothing$), we have $C_{\Delta_{l}}\rightarrow
0$ as $l\rightarrow\infty$.
\begin{theorem}[\!\!\cite{LM}] {\rm (i)} Under the above assumptions, there exists a unique point process $\mu$ in $X$ whose correlation measures are $(\theta^{(n)})_{n=1}^\infty$.
{\rm (ii)} Let $\mathfrak D':=\{a\Omega\mid a\in\mathcal A\}$ and let $\mathfrak F'$ denote the closure of $\mathfrak D'$ in $\mathfrak F$. Then each operator $(\rho(\Delta),\mathfrak D')$ is essentially self-adjoint in $\mathfrak F'$, and the operator-valued measures of the closures of the operators $(\rho(\Delta),\mathfrak D')$ commute. Furthermore, there exists a unique unitary operator $U:\mathfrak F'\to L^2(\Gamma(X),\mu)$ satisfying $U\Omega=1$ and
\begin{equation}\label{te5w6u3e}
U(\rho(\Delta_1)\dotsm \rho(\Delta_{n})\Omega)=\gamma(\Delta_1)\dotsm\gamma(\Delta_n)\end{equation}
for any $\Delta_1,\dots,\Delta_{n}\in\mathcal B_0(X)$ ($n\in\mathbb N$). In particular,
\begin{equation}\label{cts6wu4w5}
\tau\big(\rho(\Delta_1)\dotsm \rho(\Delta_{n})\big)= \int_{\Gamma(X)}\gamma(\Delta_1)\dotsm\gamma(\Delta_n)\,\mu(d\gamma).\end{equation}
\label{tes56uwe4u6}
\end{theorem}
We finish this section with a formal observation.
Let again $\mathcal H=L^2(X,\sigma)$ and the antilinear involution $J$ in $\mathcal H$ be given by $(Jf)(x):=\overline{f(x)}$. Let $A^+(h)$ and $A^-(h)$ ($h\in\mathcal V$) be operators satisfying the CCR, and let the corresponding operators $A^+(x)$, $A^-(x)$ ($x\in X$) be derfined by \eqref{ray45}. For each $\Delta\in\mathcal B_0(X)$, let $\rho(\Delta)$ be given by \eqref{raq5wu}.
The CCR \eqref{ydey7e76} and formulas \eqref{raq5wu}, \eqref{dsaea78} imply that, for any $\Delta_1,\dots,\Delta_{n}\in\mathcal B_0(X)$,
\begin{equation}\label{terw5yw35}
{:}\rho(\Delta_1)\dotsm \rho(\Delta_{n}){:}=\int_{\Delta_1\times\dots\times \Delta_n}A^+(x_n)\dotsm A^+(x_1)A^-(x_1)\dotsm A^-(x_n)\sigma^{\otimes n}(dx_1\dotsm dx_n).\end{equation}
Thus, the Wick polynomials correspond to the Wick normal ordering, in which all the creation operators are to the left of all the annihilation operators.
Hence, by \eqref{6esuw6u61d} and~\eqref{terw5yw35}, we formally obtain
\begin{align*}
&\theta^{(n)}\big(\Delta_1\times\dots\times\Delta_n\big)\notag\\
&\quad =\frac1{n!}\int_{\Delta_1\times\dots\times \Delta_n}\tau\big(A^+(x_n)\dotsm A^+(x_1)A^-(x_1)\dotsm A^-(x_n)\big)\sigma^{\otimes n}(dx_1\dotsm dx_n).\end{align*}
Therefore, by \eqref{5w738}, the point process $\mu$ from Theorem~\ref{tes56uwe4u6} has the correlation functions
$$k^{(n)}(x_1,\dots,x_n)=\tau\big(A^+(x_n)\dotsm A^+(x_1)A^-(x_1)\dotsm A^-(x_n)\big).$$
Below we will see that, in the case of a Cox process $\Pi_R$, where $R(x)=|G(x)|^2$ and $G(x)$ is a complex-valued Gaussian field from Section~\ref{ew56u3wu}, the above formal calculations can be given a rigorous meaning. We will start, however, with the simpler case of a Poisson point process.
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\section{Application of Theorem~\ref{tes56uwe4u6} to Poisson point processes}\label{xeeraq54q}
Recall Remark~\ref{vtw53u}. Let $\mathcal V$ denote the (dense) subspace of $\mathcal H=L^2(X,\sigma)$ consisting of all measurable bounded (versions of) functions $h:X:\to\mathbb C$ with compact support.
Let us fix a function $\lambda\in L^2_{\mathrm{loc}}(X,\sigma)$ and define a functional
$\phi:\mathcal V\to\mathbb C$ by
\begin{equation}\label{s5w5as}
\phi(h):=\int_X h(x)\lambda(x)\sigma(dx).
\end{equation}
Note that
\begin{equation}\label{xse5w64ue}
\phi'(h):=\int_X h(x)\overline{\lambda(x)}\,\sigma(dx).
\end{equation}
For each $h\in\mathcal V$, we define operators $A^+(h),A^-(h)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H))$ by
\begin{equation}\label{er4q53}
A^+(h):=a^+(h)+\phi'(h),\quad A^-(h):=a^-(h)+\phi(h).\end{equation}
Here $a^+(h)$ and $a^-(h)$ are the creation and annihilation operators in $\mathcal F(\mathcal H)$. By Remarks~\ref{vtw53u} and~\ref{vrtw52}, the vacuum state $\tau$ on the CCR algebra generated by $A^+(h)$, $A^-(h)$ ($h\in\mathcal V$) is quasi-free.
Let $a^+(x)$ and $a^-(x)$ be the operator-valued distributions corresponding to $a^+(h)$ and $a^-(h)$, respectively. Then $A^+(x)=a^+(x)+\overline{\lambda(x)}$ and $A^-(x)=a^-(x)+\lambda(x)$. Hence, the corresponding particle density takes the form
$$\rho(x)=\lambda(x)a^+(x)+\overline{\lambda(x)}\,a^-(x)+a^+(x)a^-(x)+|\lambda(x)|^2.$$
Our next aim is to rigorously define, for each $\Delta\in\mathcal B_0(X)$, an operator $\rho(\Delta)=\int_\Delta A^+(x)A^-(x)\sigma(dx)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H))$.
We clearly have, for each $\Delta\in\mathcal B_0(X)$,
$$\int_\Delta \lambda(x)a^+(x)\,\sigma(dx)=\int_X\chi_\Delta(x)\lambda(x)a^+(x)\,\sigma(dx)=a^+(\chi_\Delta \lambda) $$
and similarly
$$\int_\Delta \overline{\lambda(x)}\,a^-(x)\,\sigma(dx)=a^-(\chi_\Delta \overline{\lambda}).$$
(Note that $\chi_\Delta \lambda\in\mathcal H$.) Next, for $h\in\mathcal V$ and
$f^{(n)}\in\mathcal F_n(\mathcal H)$,
$$\big(a^-(h)f^{(n)}\big)(x_1,\dots,x_{n-1})=n\int_X h(x)f^{(n)}(x,x_1,\dots,x_{n-1})\sigma(dx).$$
Hence
\begin{equation}\label{cyd64uew6wqa}
(a^-(x)f^{(n)})(x_1,\dots,x_{n-1})=nf^{(n)}(x,x_1,\dots,x_{n-1}),\end{equation}
which implies
\begin{equation}\label{w4q53aestyp}
\bigg(\int_\Delta a^+(x)a^-(x)\sigma(dx)f^{(n)}\bigg)(x_1,\dots,x_n)=\big(\chi_\Delta(x_1)+\dots+\chi_\Delta(x_n)\big)f^{(n)}(x_1,\dots,x_n).
\end{equation}
Hence, $a^0(\chi_\Delta):=\int_\Delta a^+(x)a^-(x)\sigma(dx)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H))$. The $a^0(\chi_\Delta)$ is called a {\it neutral operator}.
Thus, for each $\Delta\in\mathcal B_0(X)$, we have rigorously defined
\begin{equation}\label{rtw52qy}
\rho(\Delta)=a^+(\chi_\Delta\lambda)+a^-(\chi_\Delta\overline{\lambda})+a^0(\chi_\Delta)+\int_\Delta|\lambda(x)|^2\sigma(dx)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H)).\end{equation}
Obviously, $\rho(\Delta)$ is a Hermitian operator in $\mathcal F(\mathcal H)$. To construct a state on the corresponding $*$-algebra, we use the vacuum $\Omega$
in the Fock space $\mathcal F(\mathcal H)$.
\begin{proposition}\label{t7re6}
The operators $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$ defined by \eqref{rtw52qy} and the vacuum state $\tau$ satisfy the assumptions of Theorem~\ref{tes56uwe4u6}. In this case, $\theta^{(n)}=\frac1{n!}(|\lambda|^2\sigma)^{\otimes n}$, so that $\mu$ is the Poisson point process with intensity $|\lambda(x)|^2$. Furthermore, we have $\mathfrak F'=\mathcal F(\mathcal H)$.
\end{proposition}
\begin{remark}
For the Poisson point process $\mu$ with intensity $|\lambda|^2$, the existence of the unitary isomorphism $U:\mathcal F(\mathcal H)\to L^2(\Gamma(X),\mu)$ that satisfies \eqref{te5w6u3e}, \eqref{cts6wu4w5} is a well-known fact, see e.g.\ \cite{Surgailis}. Our approach to the construction of the isomorphism $U$ may be compared with paper \cite{GGPS}.
\end{remark}
\begin{proof}[Proof of Proposition \ref{t7re6}] For any $\Delta_1,\Delta_2\in\mathcal B_0(X)$, the commutation $[\rho(\Delta_1),\rho(\Delta_2)]=0$ follows from the CCR and the commutation relations
$$\big[a^0(\chi_{\Delta_1}),a^+(\chi_{\Delta_2}\lambda)\big]=a^+(\chi_{\Delta_1\cap\Delta_2}\lambda),\quad \big[a^0(\chi_{\Delta_1}),a^-(\chi_{\Delta_2}\overline{\lambda})\big]=-a^-(\chi_{\Delta_1\cap\Delta_2}\overline{\lambda}).$$
Next, let $C\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H))$. Similarly to \eqref{cyd64uew6wqa}, \eqref{w4q53aestyp}, we see that, for each $\Delta\in\mathcal B_0(X)$, $\int_\Delta a^+(x)Ca^-(x)\sigma(dx) $ determines an operator from $\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H))$. In particular, for $f\in\mathcal H$ and $n\in\mathbb N$,
\begin{equation}\label{dr6wu3}
\int_\Delta a^+(x)Ca^-(x)\sigma(dx)f^{\otimes n}=n(\chi_\Delta f)\odot(Cf^{\otimes(n-1)}).\end{equation}
Therefore,
\begin{align*}
&\int_\Delta A^+(x)CA^-(x)\sigma(dx)\\
&\quad=\int_\Delta a^+(x)Ca^-(x)\sigma(dx)+a^+(\chi_\Delta\lambda)C+Ca^-(\chi_\Delta\overline{\lambda})+\int_\Delta|\lambda(x)|^2\sigma(dx)\,C\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H)).
\end{align*}
Hence, we may define, for any $\Delta_1,\dots,\Delta_n\in\mathcal B_0(X)$,
\begin{align*}
&\int_{\Delta_1\times\dots\times \Delta_n}A^+(x_n)\dotsm A^+(x_1)A^-(x_1)\dotsm A^-(x_n)\,\sigma^{\otimes n}(dx_1\dotsm dx_n)\\
&\quad:=\int_{\Delta_n}A^+(x_n)\bigg(\int_{\Delta_{n-1}}A^+(x_{n-1})\bigg(\dotsm\int_{\Delta_1}A^+(x_1)A^-(x_1)\sigma(dx_1)\bigg)\\
&\qquad\dotsm A^-(x_{n-1})\sigma(dx_{n-1})\bigg)A^-(x_n)\sigma(dx_n)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H)).\end{align*}
We next state that formula \eqref{terw5yw35} now holds rigorously. Indeed, a direct calculation shows that, for any $\Delta_1,\Delta_2 \in\mathcal B_0(X)$ and $C\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal H))$,
\begin{align}
&\rho(\Delta_1)\int_{\Delta_2}A^+(x)CA^-(x)\,\sigma(dx)\notag\\
&\quad=
\int_{\Delta_2}A^+(x)\rho(\Delta_1)CA^-(x)\,\sigma(dx)+\int_{\Delta_1\cap\Delta_2}A^+(x)CA^-(x)\,\sigma(dx).
\label{cx4t3y7}\end{align}
Now formula \eqref{terw5yw35} follows by induction from \eqref{dsaea78} and \eqref{cx4t3y7}.
Applying the vacuum state $\tau$ to \eqref{terw5yw35} , we get
\begin{equation}\label{ctea5ywq}
\tau\big({:}\rho(\Delta_1)\dotsm \rho(\Delta_{n}){:}\big)=\int_{\Delta_1\times\dots\times\Delta_n}|\lambda(x_1)|^2\dotsm|\lambda(x_n)|^2\sigma^{\otimes n}(dx_1\dotsm dx_n). \end{equation}
Since the measure $\sigma$ is non-atomic, $\sigma^{\otimes n}$ is concentrated on $X^{(n)}$. By \eqref{6esuw6u61d} and \eqref{ctea5ywq},
estimate \eqref{fte76i4} holds with $C_\Delta=\int_\Delta |\lambda(x)|^2\sigma(dx)$. Hence, the assumptions of Theorem~\ref{tes56uwe4u6} are satisfied. The form of the correlation measures implies that $\mu$ is the Poisson point process with intensity $|\lambda(x)|^2$.
Finally, the proof of the equality $\mathfrak F'=\mathcal F(\mathcal H)$ is standard and we leave it to the interested reader. \end{proof}
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\section{Application of Theorem~\ref{tes56uwe4u6} to hafnian point processes}\label{yd6w6wdd}
We will use below the notations from Section~\ref{ew56u3wu}. We assume that conditions \eqref{ydxdrdxdrg}, \eqref{xrsa5yw4}, and \eqref{vd5yw573} are satisfied. Let also the subspace $\mathcal V$ of $\mathcal H$ be as in Section~\ref{xeeraq54q}.
Let $h\in\mathcal V$. By the Cauchy inequality,
$$\int_X |h(x)|\,\|L_i(x)\|_{\mathcal G}\,\sigma(dx)<\infty,\quad i=1,2.$$
Hence, by using e.g.\ \cite[Chapter 10, Theorem~3.1]{BUS},
we can define
\begin{equation*}
\int_X hL_i\,d\sigma, \int_X h\,\mathcal JL_i\,d\sigma\in\mathcal G\end{equation*}
as Bochner integrals.
\begin{example}
Recall Examples \ref{fyte6i47} and \ref{vcrtw5y3}. As easily seen, for each $h\in\mathcal V$,
\begin{equation}\label{eaw4ayaqwy}
\int_X hL\,d\sigma =(L^\Delta)^Th,\ \int_X h\, JL\,d\sigma =(L^\Delta)^*h\in\mathcal H,
\end{equation}
where $\Delta\in\mathcal B_0(X)$ is chosen so that $h$ vanishes outside $\Delta$. In particular, if $L(x,y)$ is the integral kernel of an operator $L\in\mathcal L(\mathcal H)$, then we can replace $L^\Delta$ in formula \eqref{eaw4ayaqwy} with $L$. Furthermore, in the latter case, we could set $\mathcal V=\mathcal H$.
\end{example}
Denote $\mathcal E:=\mathcal H\oplus \mathcal G$. We recall the well-known unitary isomorphism between\linebreak $\mathcal F(\mathcal H)\otimes\mathcal F(\mathcal G)$ and $\mathcal F(\mathcal E)$. In view of our considerations in Sections~\ref{ew56u3wu} and \ref{xeeraq54q}, see, in particular, formulas \eqref{waaq4yq5y}, \eqref{ytd7r57}, and \eqref{s5w5as}--\eqref{er4q53},
we consider in $\mathcal F(\mathcal E)$ the following linear operators with domain $\mathcal F_{\mathrm{fin}}(\mathcal E)$,
\begin{align}
A^+(h):=&a^+\bigg(h,\int_X h\,\mathcal JL_2\,d\sigma\bigg)+a^-\bigg(0,\int_X h\,\mathcal JL_1\,d\sigma\bigg),\notag\\
A^-(h):=&a^+\bigg(0,\int_X h L_1\,d\sigma\bigg)+a^-\bigg(h,\int_X hL_2\,d\sigma\bigg),\quad h\in\mathcal V.\label{tdr65eS}
\end{align}
\begin{proposition}
The operators $A^+(h)$, $A^-(h)$ defined by \eqref{tdr65eS} satisfy the conditions (i)--(iii) from Section~\ref{xrwaq4q2} with $\mathfrak F=\mathcal F(\mathcal E)=\mathcal F(H\oplus \mathcal G)$ and $\mathfrak D=\mathcal F_{\mathrm{fin}}(\mathcal E)$. The vacuum state on the corresponding CCR algebra is quasi-free with $T^{(1)}=0$ and
\begin{align}
T^{(2)}(f,h)&=\int_X \overline{f(x)}\, h(x)\sigma(dx)\notag\\
&\quad+2\int_{X^2}\Re\left(f(x)h(y)\overline{\mathcal K_2(x,y)}+\overline{f(x)}\,h(y)\mathcal K_1(x,y)\right)\sigma^{\otimes 2}(dx\,dy).
\end{align}
Here $\mathcal K_1$ and $\mathcal K_2$ are defined by \eqref{w909i8u7y689} and \eqref{ftsqw43qd}, respectively.
\end{proposition}
\begin{proof} The first statement of the proposition is obvious in view of the commutation of the operators $\Phi(x)$, $\Psi(x)$. By \eqref{e5w7w37w2} and \eqref{tdr65eS},
\begin{align*}
B(h)&=a^+\bigg(h,\int_X (h\,\mathcal JL_2+(Jh)\, L_1)d\sigma\bigg)+a^-\bigg(Jh, \int_X(h\,\mathcal JL_1+(Jh)\, L_2)d\sigma\bigg).
\end{align*}
Hence, by \eqref{ydxdrdxdrg}, \eqref{xrsa5yw4}, \eqref{w909i8u7y689}, and \eqref{ftsqw43qd}, the second statement of the proposition also follows.
\end{proof}
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\begin{remark}\label{yre7i645i7r45e}
Assume that, for $i=1,2$, the map $\mathcal V\ni h\mapsto \int_X hL_i\,d\sigma\in\mathcal G$ extends by continuity to a bounded linear operator $\mathbb L_i\in\mathcal L(\mathcal H,\mathcal G)$. Then, by \eqref{rdtsw5u4w3wq} and \eqref{tdr65eS},
$$A^+(h)=a^+(h,\mathbb L_2'h)+a^-(0,\mathbb L_1'h),\quad A^-(h)=a^+(0,\mathbb L_1h)+a^-(h,\mathbb L_2h).$$
This quasi-free representation of the CCR is a special case of \eqref{cdtwe64u53e}, \eqref{tsw53w3}. By \eqref{rtwe64ue3},
$$ T^{(2)}(f,h)=(h,f)_{\mathcal H}+\big((\mathcal J\mathbb L_1+\mathbb L_2)Jf,(\mathcal J\mathbb L_1+\mathbb L_2)Jh\big)_{\mathcal G}.$$
\end{remark}
By \eqref{tdr65eS}, the corresponding operator-valued distributions $A^+(x)$ and $A^-(x)$ are given by
\begin{align}
A^+(x)&=a_1^+(x)+a_2^+(\mathcal JL_2(x))+a_2^-(\mathcal JL_1(x)),\notag\\
A^-(x)&=a_1^-(x)+a_2^+(L_1(x))+a_2^-(L_2(x)), \label{ytyuryr}\end{align}
compare with \eqref{waaq4yq5y}, \eqref{ytd7r57}. The operators $a_i^\pm(\cdot)$ ($i=1,2$) are defined similarly to Example~\ref{vcrtw5y3}.
\begin{proposition}\label{cyse6ui5i} Let $A^+(x)$, $A^-(x)$ be given by \eqref{ytyuryr}, and let $C\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal E))$. For each $\Delta\in\mathcal B_0(X)$, $\int_\Delta A^+(x)CA^-(x)\,\sigma(dx) $ determines an operator from $\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal E))$, in the sense explained in the proof. \end{proposition}
\begin{proof} It is sufficient to prove the statement when $C\in\mathcal L(\mathcal F_n(\mathcal E),\mathcal F_m(\mathcal E))$. We also fix $\Delta\in\mathcal B_0(X)$.
By \eqref{cr6sw6u4e} and \eqref{vd5yw573},
\begin{align}
&\int_\Delta \|a_2^-(\mathcal JL_1(x))Ca_2^+(L_1(x))\|_{\mathcal L(\mathcal F_{n-1}(\mathcal E),\mathcal F_{m-1}(\mathcal E))}\,\sigma(dx)\notag\\
&\quad\le\|C\|_{\mathcal L(\mathcal F_{n}(\mathcal E),\mathcal F_{m}(\mathcal E))}\sqrt{nm}\,\int_{\Delta}\|L_1(x)\|_{\mathcal G}^2 \,\sigma(dx)<\infty.\notag\end{align}
Hence, by \cite[Chapter 10, Theorem~3.1]{BUS}, the following Bochner integral is well-defined:
\begin{equation*}
\int_\Delta a_2^-(\mathcal JL_1(x))C a_2^+(L_1(x))\sigma(dx)\in \mathcal L(\mathcal F_{n-1}(\mathcal E),\mathcal F_{m-1}(\mathcal E)).\end{equation*}
Note that, by e.g.\ \cite[Chapter 10, Theorem~3.2]{BUS}, for each $f^{(n-1)}\in\mathcal F_{n-1}(\mathcal E)$,
$$\bigg(\int_\Delta a_2^-(\mathcal JL_1(x))C a_2^+(L_1(x))\sigma(dx)\bigg)f^{(n-1)}=\int_\Delta a_2^-(\mathcal JL_1(x))C a_2^+(L_1(x))f^{(n-1)}\sigma(dx),$$
where the right hand side is a Bochner integral with values in $\mathcal F_{m-1}(\mathcal E)$.
The proof of existence of the other Bochner integrals of the type
$\int_\Delta a_2^\pm(\mathcal JL_i(x))Ca_2^\pm(L_j(x))\sigma(dx)$ ($i,j\in\{1,2\}$) is similar.
Next, we define
$$\int_\Delta a_1^+(x)Ca_1^-(x)\sigma(dx)\in\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))$$
by analogy \eqref{dr6wu3}.
Let $(e_i)_{i\ge1}$ be an orthonormal basis in $\mathcal H$ such that $Je_i=e_i$ for all $i\ge1$. As easily seen,
\begin{equation}\label{r5w5w32}
\int_\Delta a_1^+(x)Ca_1^-(x)\sigma(dx)=\sum_{i,j\ge1}(\chi_\Delta e_i,e_j)_{\mathcal H}\,a_1^+(e_j)Ca_1^-(e_i),\end{equation}
where the series converges strongly in $\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))$.
By \eqref{vd5yw573}, we can define a linear operator $L_2^\Delta\in\mathcal L(\mathcal G,\mathcal H)$ by
$$(L_2^\Delta g)(x):=\chi_\Delta(x)(L_2(x),\mathcal Jg)_{\mathcal G}\,.$$
By analogy with \eqref{dr6wu3}, we define
\begin{equation}\label{cw5uu5}
\int_\Delta a_1^+(x)Ca_2^-(L_2(x))\sigma(dx)\in\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))\end{equation}
that satisfies, for each $f=(h,g)\in\mathcal E$,
\begin{equation}\label{vctsw5w3}
\int_\Delta a_1^+(x)Ca_2^-(L_2(x))\sigma(dx)f^{\otimes (n+1)}=(n+1)\big(L_2^\Delta g\big)\odot(Cf^{\otimes n}).\end{equation}
Let $(u_j)_{j\ge1}$ be an orthonormal basis in $\mathcal G$ such that $\mathcal Ju_j=u_j$ for all $j\ge1$. Then, similarly to \eqref{r5w5w32}, we obtain
\begin{equation}\label{fys6w6u4}
\int_\Delta a_1^+(x)Ca_2^-(L_2(x))\sigma(dx)=\sum_{i,j\ge1}(L_2^\Delta u_j,e_i)_{\mathcal H}\,a_1^+(e_i)Ca_2^-(u_j),
\end{equation}
where the series converges strongly in $\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))$.
Next, we note that $\mathcal H\otimes\mathcal G=L^2(X\to\mathcal G,\sigma)$. Hence, by \eqref{vd5yw573}, $\chi_\Delta(\cdot)L_1(\cdot)\in \mathcal H\otimes\mathcal G$.
Note also that $\mathcal H\otimes\mathcal G$ is a subspace of $\mathcal E^{\otimes 2}=(\mathcal H\oplus\mathcal G)^{\otimes 2}$. For each $m\in\mathbb N$, we denote by $P_m:\mathcal E^{\otimes m}\to\mathcal E^{\odot m}$ the symmetrization operator. We naturally set, for each $f^{(k)}\in\mathcal F_{k}(\mathcal E)$,
$$\int_\Delta a_1^+(x)a_2^+(L_1(x))\sigma(dx)f^{(k)}=P_{k+2}\big((\chi_\Delta(\cdot)L_1(\cdot))\otimes f^{(k)}\big).$$
Hence, we define
$$\int_\Delta a_1^+(x)Ca_2^+(L_1(x))\sigma(dx)\in\mathcal L(\mathcal F_{n-1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))$$
by
$$\int_\Delta a_1^+(x)Ca_2^+(L_1(x))\sigma(dx)f^{(n-1)}:=P_{m+1}(1_{\mathcal E}\otimes (CP_n))\big(
(\chi_\Delta(\cdot)L_1(\cdot))
\otimes f^{(n-1)}\big) $$
for $f^{(n-1)}\in\mathcal F_{n-1}(\mathcal E)$. Here $1_\mathcal E$ is the identity operator in $\mathcal E$. Therefore,
\begin{equation}\label{s4qy}
\int_\Delta a_1^+(x)Ca_2^+(L_1(x))\sigma(dx)=\sum_{i,j\ge1}(\chi_\Delta(\cdot)L_1(\cdot)
,e_i\otimes u_j)_{\mathcal H\otimes\mathcal G}\,a_1^+(e_i)Ca_2^+(u_j),
\end{equation}
where the series converges strongly in $\mathcal L(\mathcal F_{n-1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))$.
Similarly to Remark \ref{yre7i645i7r45e}, for $i=1,2$, we define $\mathbb L_i^\Delta\in\mathcal L(\mathcal H,\mathcal G)$ by
$$\mathbb L_i^\Delta h:=\int_\Delta h(x)L_i(x)\sigma(dx),\quad h\in\mathcal H$$ (in the sense of Bochner integration).
Similarly to \eqref{cw5uu5}, \eqref{vctsw5w3}, we define
$$\int_\Delta a_2^+ (\mathcal JL_2(x))Ca_1^-(x)\sigma(dx)\in\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))$$
by
$$\int_\Delta a_2^+ (\mathcal JL_2(x))Ca_1^-(x)\sigma(dx) f^{\otimes(n+1)}:=(n+1)\big((\mathbb L_2^\Delta)' h\big)
\odot (Cf^{\otimes n}),\quad f=(h,g)\in\mathcal E. $$
Similarly to \eqref{fys6w6u4},
\begin{equation}\label{vctesw5y}
\int_\Delta a_2^+ (\mathcal JL_2(x))Ca_1^-(x)\sigma(dx)=\sum_{i,j\ge1}\big((\mathbb L_2^\Delta)' e_i,u_j\big)_{\mathcal G}\,
a_2^+(u_j)Ca_1^-(e_i),
\end{equation}
where the series converges strongly in $\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m+1}(\mathcal E))$.
Finally, we define
$$\int_\Delta a_2^- (\mathcal JL_1(x))Ca_1^-(x)\sigma(dx)\in\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m-1}(\mathcal E))$$
by
$$\int_\Delta a_2^- (\mathcal JL_1(x))Ca_1^-(x)\sigma(dx) f^{\otimes(n+1)}
:=(n+1)a^-_2\big((\mathbb L_1^\Delta)'h\big)(Cf^{\otimes n}),\quad f=(h,g)\in\mathcal E.$$
Hence,
\begin{equation}\label{cxrw53ew}
\int_\Delta a_2^- (\mathcal JL_1(x))Ca_1^-(x)\sigma(dx)=\sum_{i,j\ge1}
\big((\mathbb L_1^\Delta)'e_i,u_j\big)_{\mathcal G}
\,a_2^-(u_j)Ca_1^-(e_i),
\end{equation}
where the series converges strongly in $\mathcal L(\mathcal F_{n+1}(\mathcal E),\mathcal F_{m-1}(\mathcal E))$.
\end{proof}
\begin{proposition}\label{3rtlgpr}
For each $\Delta\in\mathcal B_0(X)$, the particle density $\rho(\Delta)=\int_\Delta A^+(x)A^-(x)\,\sigma(dx)$ is a well-defined Hermitian operator in $\mathcal F(\mathcal E)$ with domain $\mathcal F_{\mathrm{fin}}(\mathcal E)$. Furthermore, for any $\Delta_1,\Delta_2\in\mathcal B_0(X)$, $[\rho(\Delta_1)$, $\rho(\Delta_2)]=0$. \end{proposition}
\begin{proof} By Proposition \ref{cyse6ui5i}, we have $\rho(\Delta)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal E))$. The fact that $\rho(\Delta)$ is a Hermitian operator in $\mathcal F(\mathcal E)$ easily follows from the proof of Proposition~ \ref{cyse6ui5i}.
To prove the commutation, one uses the corresponding Bochner integrals, formulas
\eqref{r5w5w32}, \eqref{fys6w6u4}--\eqref{cxrw53ew}. In doing so, one uses the fact that every strongly convergent sequence of bounded linear operators is norm-bounded. Hence, for every strongly convergent sums of bounded linear operators, $A=\sum_{i\ge1}^\infty A_i$ and $B=\sum_{j\ge1}^\infty B_j$, one has $AB=\sum_{i,j\ge 1}A_iB_j$, where the latter double series converges strongly.
\end{proof}
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\begin{proposition}\label{3rtlgpr}
For each $\Delta\in\mathcal B_0(X)$, the particle density $\rho(\Delta)=\int_\Delta A^+(x)A^-(x)\,\sigma(dx)$ is a well-defined Hermitian operator in $\mathcal F(\mathcal E)$ with domain $\mathcal F_{\mathrm{fin}}(\mathcal E)$. Furthermore, for any $\Delta_1,\Delta_2\in\mathcal B_0(X)$, $[\rho(\Delta_1)$, $\rho(\Delta_2)]=0$. \end{proposition}
\begin{proof} By Proposition \ref{cyse6ui5i}, we have $\rho(\Delta)\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal E))$. The fact that $\rho(\Delta)$ is a Hermitian operator in $\mathcal F(\mathcal E)$ easily follows from the proof of Proposition~ \ref{cyse6ui5i}.
To prove the commutation, one uses the corresponding Bochner integrals, formulas
\eqref{r5w5w32}, \eqref{fys6w6u4}--\eqref{cxrw53ew}. In doing so, one uses the fact that every strongly convergent sequence of bounded linear operators is norm-bounded. Hence, for every strongly convergent sums of bounded linear operators, $A=\sum_{i\ge1}^\infty A_i$ and $B=\sum_{j\ge1}^\infty B_j$, one has $AB=\sum_{i,j\ge 1}A_iB_j$, where the latter double series converges strongly.
\end{proof}
\begin{theorem}\label{due6uew4}
The operators $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$ defined by Propositions~\ref{cyse6ui5i}, \ref{3rtlgpr} and the state $\tau$ defined by the vacuum vector $\Omega$ satisfy the assumptions of Theorem~\ref{tes56uwe4u6}. The corresponding point process $\mu$ is the Cox process $\Pi_R$, where $R(x)=|G(x)|^2$, and $G(s)$ is the Gaussian field from Theorem~\ref{reaq5y43wu}. The point process $\Pi_R$ is hafnian with the correlation kernel $\mathbb K(x,y)$ given by \eqref{d6e6ie4}, where $\mathcal K_1(x,y)$ and $\mathcal K_2(x,y)$ are given by \eqref{w909i8u7y689} and \eqref{ftsqw43qd}, respectively.
\end{theorem}
\begin{proof} Direct calculations show that, for any $\Delta_1,\Delta_2 \in\mathcal B_0(X)$ and $C\in\mathcal L(\mathcal F_{\mathrm{fin}}(\mathcal E))$, formula \eqref{cx4t3y7} holds, which implies formula \eqref{terw5yw35}. We apply the vacuum state $\tau$ to \eqref{terw5yw35}. Using formulas \eqref{dr6e6u43wq}, \eqref{d6e6ie4}, \eqref{waaq4yq5y}, \eqref{ytd7r57}, \eqref{ytyuryr}, Theorem~\ref{reaq5y43wu}, and Proposition~\ref{cyse6ui5i}, we conclude that the measure $\theta^{(n)}$ is given by
\begin{align}
\theta^{(n)}(dx_1\dotsm dx_n)&=\frac1{n!}\,\tau\big(\Psi(x_n)\dotsm\Psi(x_1)\Phi(x_1)\dotsm\Phi(x_n)\big)
\sigma^{\otimes n}(dx_1\dotsm dx_n)\notag\\
&=\frac1{n!}\,\mathbb E\big(\overline{G(x_n)}\dotsm \overline{G(x_1)}G(x_1)\dotsm G(x_n)\big)\sigma^{\otimes n}(dx_1\dotsm dx_n)\label{buyftd7s}\\
&=\frac1{n!}\,\mathbb E\big(|G(x_1)|^2\dotsm|G(x_n)|^2\big)\sigma^{\otimes n}(dx_1\dotsm dx_n)\label{vud6ws5}\\
&=\frac1{n!}\,\operatorname{haf}[\mathbb K(x_i,x_j)]_{i,j=1,\dots,n}\,\sigma^{\otimes n}(dx_1\dotsm dx_n).\label{uyfy7de6}
\end{align}
In particular, $\theta^{(n)}$ is a positive measure that is concentrated on $X^{(n)}$.
If $\mathcal Y=G(x)$ or $\overline{G(x)}$ and $\mathcal Z=G(y)$ or $\overline{G(y)}$, then
$$|\mathbb E(\mathcal Y\mathcal Z)|\le\big(\mathbb E(|\mathcal Y|^2)\mathbb E(|\mathcal Z|^2)\big)^{1/2}=\big(\mathbb E(|G(x)|^2)\,\mathbb E(|G(y)|^2)\big)^{1/2}=\|L_1(x)\|_{\mathcal G}\,\|L_1(y)\|_{\mathcal G}. $$ The number of all partitions $\{i_1,j_1\},\dots,\{i_n,j_n\}$ of $\{1,\dots,2n\}$ is $\frac{(2n)!}{(n!)2^n}\le 2^nn!$\,. Hence, by \eqref{buyftd7s} and the formula for the moments of Gaussian random variables,
$$\theta^{(n)}(\Delta^n)\le\bigg(2\int_\Delta\|L_1(x)\|_{\mathcal G}^2\,\sigma(dx)\bigg)^n,\quad\Delta\in\mathcal B_0(X).$$
Thus, the operators $(\rho(\Delta))_{\Delta\in\mathcal B_0(X)}$ satisfy the assumptions of Theorem~\ref{tes56uwe4u6}.
The statement of the theorem about the arising point process $\mu$ follows immediately from \eqref{vud6ws5} and \eqref{uyfy7de6}.
\end{proof}
\subsection*{Acknowledgments}
The authors are grateful to the referee for valuable comments that improved the manuscript.
\end{document}
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\begin{document}
\title{Bootstrapping partition regularity of linear systems}
\author{\tsname}
\address{\tsaddress}
\email{\tsemail}
\begin{abstract}
Suppose that $A$ is a $k \times d$ matrix of integers and write $\mathfrak{R}_A:\N \rightarrow \N\cup \{ \infty\}$ for the function taking $r$ to the largest $N$ such that there is an $r$-colouring $\mathcal{C}$ of $[N]$ with $\bigcup_{C \in \mathcal{C}}{C^d}\cap \ker A =\emptyset$. We show that if $\mathfrak{R}_A(r)<\infty$ for all $r \in \N$ then $\mathfrak{R}_A(r) \leq \exp (\exp(r^{O_{A}(1)}))$ for all $r \geq 2$.
When the kernel of $A$ consists only of Brauer configurations -- that is vectors of the form $(y,x,x+y,\dots,x+(d-2)y)$ -- the above has been proved by Chapman and Prendiville with good bounds on the $O_A(1)$ term.
\end{abstract}
\maketitle
\section{Introduction}
Our work concerns colourings. For a set $X$ and natural $r$ we say that $\mathcal{C}$ is an \textbf{$r$-colouring of $X$} if $\mathcal{C}$ is a cover of $X$ \emph{i.e.} $X \subset \bigcup_{C \in \mathcal{C}}{C}$, and $\mathcal{C}$ has size $r$. In particular we shall not need our colours to be disjoint, though such colourings are included.
Suppose that $A$ is a $k\times d$ matrix of integers. We write $\mathfrak{R}_A:\N \rightarrow \N \cup \{\infty\}$ for the function taking $r$ to the largest $N$ such that there is an $r$-colouring $\mathcal{C}$ of $[N]:=\{1,\dots,N\}$ with $\bigcup_{C \in \mathcal{C}}{C^d}\cap \ker A =\emptyset$ -- in words, such that there are no monochromatic solutions to $Ax=0$. Note that the function $\mathfrak{R}_A$ is monotonically increasing.
Not all matrices $A$ have $\mathfrak{R}_A(r)<\infty$ for all $r \in \N$ (\emph{e.g.} if all the non-zero terms in $A$ are positive), but those that do we call \textbf{partition regular}. There are matrices $A$ such that van der Waerden's theorem \cite[Exercise 6.3.7]{taovu::} (first proved in \cite{van::0}) is implied by the partition regularity of $A$ (see \cite[Satz I]{rad::1}), and similarly for Schur's Theorem \cite[6.12]{taovu::} (first proved in \cite{sch::4}). Schur's theorem actually gives the stronger fact that\footnote{It is a result of Abbott and Moser \cite{abbmos::} that we cannot do much better.} $\mathfrak{R}_A(r) \leq \lfloor e r!\rfloor$, and since the celebrated work of Gowers \cite{gow::4,gow::0} we know that van der Waerden's theorem also has reasonable bounds in terms of the number of colours.
It is the purpose of this paper to use Gowers' work to show the following.
\begin{theorem}\label{thm.mn}
Suppose that $A$ is a $k\times d$ integer-valued partition regular matrix and $r\geq 2$ is natural. Then there is some $N \leq \exp(\exp(r^{O_{A}(1)}))$ such that any $r$-colouring of $[N]$ contains a colour class $C$ and some $x \in C^d$ such that $Ax=0$.
\end{theorem}
The basic method is expounded in the model setting\footnote{The model setting has proved very fruitful for distilling the important aspects of arguments in additive combinatorics. See the paper \cite{gre::9} and the sequel \cite{wol::3}.} of $\F_2^n$ by Shkredov in \cite[Theorem 24]{shk::7} for the purpose of illustrating how analytic techniques can be applied to colouring results. Chapman and Prendiville in \cite{chapre::} independently discovered the argument given in \cite[Theorem 24]{shk::7} (though with some technical differences around expansion vs large Fourier coefficients) and importantly showed how it could be applied to provide good bounds in colouring problems in the integers where none were previously known. Specifically in \cite[Theorem 1.1]{chapre::} they prove Theorem \ref{thm.mn} for Brauer configurations, meaning for a matrix $A$ whose kernel is the set of vectors of the form $(y,x,x+y,\dots,x+(d-2)y)$ for some fixed $d \geq 3$, with a doubly exponential bound on $d$ in place of the $O_A(1)$ term. (They also show in \cite[Theorem 1.2]{chapre::} that one may replace the $O_A(1)$ term by $1+o(1)$ for Brauer configurations with $d=4$.)
It is the purpose of this note to extend the arguments of Chapman and Prendiville to partition regular linear systems. This entails a large notational burden and as a result, while they are able to give rather good estimates for the $O_A(1)$-term when $A$ is a matrix corresponding to a Brauer configuration, we give no meaningful estimates.\footnote{Though see the remark after the proof of Theorem \ref{thm.main}.}
The above work comes on the back of a wave of investigations using analytic techniques for colouring problems. This really took off with the paper \cite{cwasch::0} of Cwalina and Schoen, and was followed by the work of Green and collaborators \cite{grelin::,gresan::1}, then Chow, Lindqvist and Prendiville \cite{cholinpre::}, and most recently Chapman \cite{cha::6} which inspired this particular paper.
One would often like to insist that the $x$ found in Theorem \ref{thm.mn} is in a certain sense non-degenerate. The extent to which this is possible varies, but the question has been dealt with comprehensively by Hindman and Leader in \cite{hinlea::}. See also \cite{fragrarod::0} for a related supersaturated formulation.
\subsection*{Existing bounds on the Rado numbers $\mathfrak{R}_A(r)$} Other than the aforementioned \cite[Theorems 1.1 \& 1.2]{chapre::} most work has focused on the case where $A$ has one row \emph{i.e.} systems with one equation which for clarity we write in the comma-delimited form $A=(a_1,\dots,a_k)$. In this case Rado's theorem \cite[Theorem 9.5]{lanrob::} tells us that if (and only if) $A$ is partition regular then there is $\emptyset \neq I \subset [k]$ such that $\sum_{i \in I}{a_i}=0$.
Schur's theorem itself gives rather good bounds on $\mathfrak{R}_A(r)$ when $A=(1,1,-1)$, and more generally \cite[Theorem 1.3]{cwasch::0} gives singly exponential bounds when $A$ is a partition regular row. Stronger results when the equation satisfies additional properties are given in \cite[Theorems 1.4 \& 1.5]{cwasch::0} and \cite[Theorem 4.7]{gasmortum::}.
When $A=(1,\dots,1,-1)$ the numbers $\mathfrak{R}_A(r)$ are sometimes called the generalised diagonal Schur numbers (although they are just called Schur numbers in \cite{beubre::}). These have been computed for many values of $r$, being completely known for $r=2$ \cite[Theorem 1.3]{beubre::}, and for $r \geq 3$ the reader is directed to \cite[Table 1]{ahmsch::} for recent calculations. Note that the bounds in Theorem \ref{thm.mn} as ineffective so, for example, when $r=2$ our result says nothing more than $\mathfrak{R}_A(2)<\infty$.
When $A$ has just one row there is a large body of work computing the exact value of $\mathfrak{R}_A(2)$ using arguments which are much more combinatorial than those in the present paper. This work has many extensions covering things such as Rado numbers for inhomogenous equations \cite[p259]{lanrob::}; off-diagonal Rado numbers \cite[p280]{lanrob::}; and Rado numbers for disjunctive equations \cite[p293]{lanrob::}. We restrict ourselves to recording those results which ask for bounds on $\mathfrak{R}_A(2)$ under the same hypotheses as Theorem \ref{thm.mn}.
When $A=(a_1,a_2, -a_2)$ for $a_1,a_2 \in \N$ the value of $\mathfrak{R}_A(2)$ is computed in \cite[Theorem 9.17]{lanrob::}; when $A=(a_1, a_2, -(a_1+a_2))$ for $a_1,a_2 \in \N$ the value of $\mathfrak{R}_A(2)$ is computed in \cite[Theorem 1.1]{gupthutri::}; when $A=(1, 1, a_3, -a_4)$ for $a_3,a_4 \in \N$ (where partition regularity of $A$ ensures that $a_4 \in \{1,2,a_3,a_3+1,a_3+2\}$) the value of $\mathfrak{R}_A(2)$ is computed in \cite[Theorems 3, 4 \& 8]{robmye::} for $a_4=a_3$, $a_4=a_3+1$ and $a_4=2$ respectively, with the case $a_4=a_3+2$ being trivial; and when
\begin{equation}\label{eqn.study}
A=(\overbrace{1,\dots , 1}^{n \text{ times}}, -a_{n+1},\dots , -a_k)
\end{equation}
with $a_{n+1},\dots,a_k \in \N$ and $n \geq a_{n+1}+\cdots +a_k$, the value of $\mathfrak{R}_A(2)$ is computed in \cite[Theorem 3]{sar::2}. Note that the work of \cite{lanrob::}, \cite{robmye::} and \cite{sar::2} goes further and computes $\mathfrak{R}_A(2)$ for some $A$ which are not partition regular. (This makes sense since we may have $\mathfrak{R}_A(2)<\infty$ without $\mathfrak{R}_A(r)<\infty$ for all $r \in \N$. See \cite[Theorem 9.2]{lanrob::} for conditions on a single row $A$ such that $\mathfrak{R}_A(2)<\infty$.)
Finally, \cite[Theorem 1.8]{cwasch::0} shows that $\mathfrak{R}_A(r)=o_{r \rightarrow \infty}(r!)$ when $A$ is as in (\ref{eqn.study}) with $n=3$, $l=2$ and $a_1,a_2=1$, beating the bound following from Schur's argument.
\subsection*{Variable conventions} There are some conflicts between standard uses for certain symbols in different areas. $m$, $p$ and $c$ are the parameters of an $(m,p,c)$-set in Deuber's sense (as defined in \S\ref{ssec.mpc}), so that $c$ need not be an absolute constant, and $p$ need not be prime. $\mathcal{C}$ usually denotes a colouring and $\mathfrak{C}$ the conjugation operator (see \S\ref{ssec.gn}). $C$ then typically denotes a colour class in $\mathcal{C}$, rather than an absolute constant.
\subsection*{Big-$O$ notation} We use big-$O$ notation in the usual way, see \emph{e.g.} \cite[p11]{taovu::}. The constants behind the big-$O$ and $\Omega$ expressions may depend in peculiar ways on other parameters, and we shall sometimes need some control. We capture this in the same way as \cite[p17]{gresan::1}:
Some big-$O$ expressions will be replaced by `universal functions' of the form $f:D_1 \times \cdots \times D_k \rightarrow D_0$ where each $D_i$ is one of the sets $(0,1]$, $\N_0$, or $\N$. If $D_i=(0,1]$ then we write $x \preceq_i y$ if and only if $y \leq x$; otherwise we write $x \preceq_i y$ if and only if $x \leq y$. We say that $f$ is \textbf{monotone} if $f(x) \preceq_0 f(y)$ whenever $x_i \preceq_i y_i$ for all $1 \leq i \leq k$.
Note that the above is the usual order on $\N$ and $\N_0$ and the opposite of the usual order on $(0,1]$. This reflects the fact that we shall want bounds on, say, the size of an interval which do not get too much worse as, say, a the number of colours grows -- that would be a natural number parameter -- and also as the density of some related set does not get too small -- that would be a $(0,1]$ parameter. Our notation of monotone aligns these different notions of large and small to point in the same direction.
It is useful to note that if $f(x)=O_{a}(g(x))$ where $a \in \N_0^d$ then there is a monotone function $F:\N_0^d \rightarrow \N$ such that
\begin{equation*}
|f(x)| \leq F(a)g(x) \text{ for all }x.
\end{equation*}
This can be shown by letting $F(a)$ be the max of the constants behind the $O_{a'}$ term as $a' \preceq a$ -- a finite set.
The universal functions mapping into $\N$ or $\N_0$ will usually be denoted by $F$s with various decorations \emph{e.g.} subscripts and superscripts, while those mapping into $(0,1]$ will usually be denoted by $\eta$s with various decorations. To avoid too many different functions, we shall often use the same functions in situations where the optimal functions are almost certainly different but where there is little cost to doing so.
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\section{Setup and tools}\label{sec.deuber}
In this section we record the tools we need. First, in \S\ref{ssec.mpc}, we explain Deuber's framework \cite{deu::} for understanding colouring problems. This will reduce the problem to proving Theorem \ref{thm.main}. The key tools to prove this are recorded in \S\ref{ssec.gn}. Finally we gather a few more technical facts in \S\ref{ssec.tool}.
\subsection{Deuber's Theorem}\label{ssec.mpc} In \cite[Satz 3.1]{deu::} Deuber proved a conjecture of Rado, and we shall use Deuber's ideas here too. We follow the exposition and definitions of \cite{gun::}: as in \cite[Definition 2.5]{gun::}\footnote{Which Gunderson notes is slightly different to Deuber's original.}, given $m,p,c \in \N$, a set $S \subset \N$ is an \textbf{$(m,p,c)$-set} if there is some $s=(s_0,\dots,s_{m+1}) \in \N^{m+1}$ such that
\begin{equation*}
S=\bigcup_{j=0}^m{\{cs_{m-j}+i_{m-j+1}s_{m-j+1}+ \cdots + i_{m}s_{m}: -p \leq i_{m-j+1},\dots,i_{m} \leq p\}}.
\end{equation*}
For example, if $m=2$ then
\begin{equation*}
S=\{cs_2\} \cup \{cs_1+i_2s_2: -p \leq i_2 \leq p\} \cup \{cs_0+i_1s_1+i_2s_2: -p \leq i_1,i_2\leq p\},
\end{equation*}
and even more concretely, the set $\{s_1\}\cup \{s_0-s_1,s_0,s_0+s_1\}$ -- which is a three-term arithmetic progression union its common difference -- is a $(1,1,1)$-set.
We shall give a little more motivation for these sets in a moment but first we state Deuber's Theorem.
\begin{theorem}[Deuber's Theorem, {\cite[Theorem 2.8]{gun::}}]
Suppose that $m,p,c,r \in \N$. Then there are $M,P,C \in \N$ such that any $r$-colouring of an $(M,P,C)$-set contains a monochromatic $(m,p,c)$-set.
\end{theorem}
The main result of this paper is the following.
\begin{theorem}\label{thm.main} Suppose that $m,p,c,r \in \N$. Then there is $N \leq \exp(\exp(r^{O_{m,p,c}(1)}))$ such that any $r$-colouring of $[N]$ contains a monochromatic $(m,p,c)$-set.
\end{theorem}
Qualitatively this is a special case of Deuber's Theorem since $[N]$ is a $(1,N,N+1)$-set.
One of the reasons $(m,p,c)$-sets are important is their relationship with solutions of equations, which we now explain. In \cite[Satz IV]{rad::1} Rado famously proved that partition regularity of a system is equivalent to something called the columns condition: we say that a $k \times d$ matrix $A$ satisfies the \textbf{columns condition} if there is a $d \times t$ matrix of rationals $\alpha$ and a partition $[d]=I_1 \sqcup \cdots \sqcup I_t$, such that writing $a_1,\dots,a_d$ for the columns of $A$ in their given order we have
\begin{equation*}
\sum_{i \in I_{j+1}}{a_i} =\sum_{i \in I_1\cup \cdots \cup I_j}{\alpha_{ij}a_i} \text{ for all }0 \leq j < t,
\end{equation*}
with the usual convention that the empty sum is $0$. When we need to refer to a specific $\alpha$ we shall call it a \textbf{witness} for the columns condition. It is natural to assume that $\alpha_{ij}=0$ for all $i \in I_{j+1}\cup \cdots \cup I_t$ and we shall always do so without remark. In view of this we see that $t=1+\rk \alpha$.
\begin{theorem}[Rado's theorem, {\cite[Theorem 2.3]{gun::}}]\label{thm.rad} Suppose that $A$ is a $k \times d$ integer-valued matrix. Then $A$ is partition regular if and only if $A$ satisfies the columns condition.
\end{theorem}
Deuber connected the columns condition to $(m,p,c)$-sets through the following.
\begin{theorem}[{\cite[Theorem 2.6(i)]{gun::}}]\label{thm.gun} Suppose that $A$ is a $k \times d$ integer-valued matrix satisfying the columns condition as witnessed by $\alpha$. Then, writing $c$ for the least common multiple of the denominators of the rationals in $\alpha$, every $(1+\rk \alpha, \max_{i,j}{|c\alpha_{ij}|},c)$-set $S$ has some $x \in S^d$ such that $Ax=0$.
\end{theorem}
This is not \cite[Theorem 2.6(i)]{gun::} as stated, but a quick look at the proof shows that this is what is proved.
\begin{proof}[Proof of Theorem \ref{thm.mn} given Theorem \ref{thm.main}]
By Theorem \ref{thm.rad} and Theorem \ref{thm.gun}, we see that if $A$ is partition regular then there are naturals $m,p,c=O_A(1)$ such that any $(m,p,c)$-set $S$ contains some $x \in S^d$ with $Ax=0$. By Theorem \ref{thm.main} we see that for $N \leq \exp(\exp(r^{O_{m,p,c}(1)}))=\exp(\exp(r^{O_A(1)}))$ any $r$-colouring of $[N]$ has a colour class $C$ containing a set $S$ that is an $(m,p,c)$-set, and hence there is some $x \in S^d \subset C^d$ with $Ax=0$ as required.
\end{proof}
\subsection{Gowers norms}\label{ssec.gn} The Gowers norms are defined in \cite[Lemma 3.9]{gow::0}, though they are not given that name, and while they can be defined more generally for finite Abelian groups we shall restrict attention to cyclic groups of prime order (in line with \cite{gow::0}). We use \cite{tao::10} as our basic reference though admittedly many of the result there are left as exercises. The material is developed in considerable generality in \cite{gretao::7}; the generality we need is closer to that discussed in \cite[\S2]{gowwol::0}. (Other introductions may be found in many places including \cite[\S4]{gretao:::}, \cite[\S\S2\&3]{hatlov::}, \cite[Appendix A]{wal::2}, and \cite[\S1]{man::3}. All these, including \cite{gowwol::0}, ultimately refer to \cite{gretao::7} for details, though the paper \cite{wal::2} does expand on the details somewhat in \S4.)
For $N \in \N$ (which will be prime though need not be right now), $k \in \N$ and $f:\Z/N\Z \rightarrow \C$ we put
\begin{equation*}
\|f\|_{U^k(\Z/N\Z)}:=\left(\E_{x,h_1,\dots,h_k \in \Z/N\Z}{\prod_{\omega \in \{0,1\}^k}{\mathfrak{C}^{|\omega|}f(x+\omega\cdot h)}}\right)^{2^{-k}},
\end{equation*}
where $\mathfrak{C}$ denotes the operation of complex conjugation.
The map $\|\cdot \|_{U^k(\Z/N\Z)}$ defines a norm \cite[Exercise 1.3.19]{tao::10} for $k \geq 2$, and enjoys the nesting property $\|\cdot \|_{U^k(\Z/N\Z)} \leq \|\cdot \|_{U^{k+1}(\Z/N\Z)}$ for $k \in \N$ \cite[Exercise 1.3.19]{tao::10}. (Proofs of these two facts are given explicitly on \cite[p466]{taovu::} and in \cite[(11.7)]{taovu::}.)
One of the reasons these norms are important is that they control counts of various linear configurations. Specifically, suppose that $\Psi:\Z^d \rightarrow \Z^l$ is a homomorphism and $f$ is a vector of $l$ functions $\Z/N\Z \rightarrow \C$. We define
\begin{equation}\label{eqn.dlp}
\Lambda_\Psi(f):=\E_{x \in [N]^d}{\prod_{i=1}^l{f_i(\Psi_i(x)+N\Z)}}.
\end{equation}
The following is the `generalised von Neumann Theorem' we need. It is a special case of \cite[Theorem 4.1]{gretao:::} once the notation has been unpacked, and also of \cite[Exercise 1.3.23]{tao::10} combined with \cite[Exercise 1.3.14]{tao::10}.
\begin{theorem}\label{thm.gvn}
Suppose that $\Psi:\Z^d \rightarrow \Z^l$ is a homomorphism and for every $i \neq j$, $(\Psi_i,\Psi_j)$ is a pair of independent vectors (\emph{i.e.} if $z\Psi_i + w\Psi_j\equiv 0$ for some $z,w \in \Z$ then $z=w=0$). Then there are naturals $N_0(\Psi)$ and $k(\Psi)$ such that if $N \geq N_0(\Psi)$ is a prime and $f$ is a vector of $l$ functions $\Z/N\Z \rightarrow \C$ bounded by $1$ we have
\begin{equation*}
|\Lambda_\Psi(f)| \leq \inf_{1 \leq i \leq l}{\|f_i\|_{U^k(\Z/N\Z)}}.
\end{equation*}
\end{theorem}
We shall use the above to count $(m,p,c)$-sets. This was already done by L{\^e} in \cite{Le::} for the purpose of transferring the partition regularity of Brauer configurations to the sets $\{p-1:p \text{ is prime}\}$ and $\{p+1:p \text{ is prime}\}$, itself answering a question of Li and Pan \cite{lipan::}.
We also need Gowers' inverse theorem. The following result is what is proved in \cite[Theorem 18.1]{gow::0}, though it is not stated in precisely this way.
\begin{theorem}\label{thm.gi} There is a monotone function $F_1:\N \rightarrow \N$ such that the following holds. Suppose that $N$ is prime, $\epsilon \leq \frac{1}{2}$ and $f:\Z/N\Z\rightarrow \C$ is bounded in magnitude by $1$ with $\|f\|_{U^k(\Z/N\Z)} \geq \epsilon$. Then there is a partition of $[N]$ into arithmetic progressions $P_1,\dots,P_M$ of average size at least $N^{\epsilon^{F_1(k)}}$ such that
\begin{equation*}
\sum_{j=1}^M{\left|\sum_{s \in P_j}{f(s+N\Z)}\right|} \geq \epsilon^{F_1(k)}N.
\end{equation*}
\end{theorem}
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\subsection{Gowers norms}\label{ssec.gn} The Gowers norms are defined in \cite[Lemma 3.9]{gow::0}, though they are not given that name, and while they can be defined more generally for finite Abelian groups we shall restrict attention to cyclic groups of prime order (in line with \cite{gow::0}). We use \cite{tao::10} as our basic reference though admittedly many of the result there are left as exercises. The material is developed in considerable generality in \cite{gretao::7}; the generality we need is closer to that discussed in \cite[\S2]{gowwol::0}. (Other introductions may be found in many places including \cite[\S4]{gretao:::}, \cite[\S\S2\&3]{hatlov::}, \cite[Appendix A]{wal::2}, and \cite[\S1]{man::3}. All these, including \cite{gowwol::0}, ultimately refer to \cite{gretao::7} for details, though the paper \cite{wal::2} does expand on the details somewhat in \S4.)
For $N \in \N$ (which will be prime though need not be right now), $k \in \N$ and $f:\Z/N\Z \rightarrow \C$ we put
\begin{equation*}
\|f\|_{U^k(\Z/N\Z)}:=\left(\E_{x,h_1,\dots,h_k \in \Z/N\Z}{\prod_{\omega \in \{0,1\}^k}{\mathfrak{C}^{|\omega|}f(x+\omega\cdot h)}}\right)^{2^{-k}},
\end{equation*}
where $\mathfrak{C}$ denotes the operation of complex conjugation.
The map $\|\cdot \|_{U^k(\Z/N\Z)}$ defines a norm \cite[Exercise 1.3.19]{tao::10} for $k \geq 2$, and enjoys the nesting property $\|\cdot \|_{U^k(\Z/N\Z)} \leq \|\cdot \|_{U^{k+1}(\Z/N\Z)}$ for $k \in \N$ \cite[Exercise 1.3.19]{tao::10}. (Proofs of these two facts are given explicitly on \cite[p466]{taovu::} and in \cite[(11.7)]{taovu::}.)
One of the reasons these norms are important is that they control counts of various linear configurations. Specifically, suppose that $\Psi:\Z^d \rightarrow \Z^l$ is a homomorphism and $f$ is a vector of $l$ functions $\Z/N\Z \rightarrow \C$. We define
\begin{equation}\label{eqn.dlp}
\Lambda_\Psi(f):=\E_{x \in [N]^d}{\prod_{i=1}^l{f_i(\Psi_i(x)+N\Z)}}.
\end{equation}
The following is the `generalised von Neumann Theorem' we need. It is a special case of \cite[Theorem 4.1]{gretao:::} once the notation has been unpacked, and also of \cite[Exercise 1.3.23]{tao::10} combined with \cite[Exercise 1.3.14]{tao::10}.
\begin{theorem}\label{thm.gvn}
Suppose that $\Psi:\Z^d \rightarrow \Z^l$ is a homomorphism and for every $i \neq j$, $(\Psi_i,\Psi_j)$ is a pair of independent vectors (\emph{i.e.} if $z\Psi_i + w\Psi_j\equiv 0$ for some $z,w \in \Z$ then $z=w=0$). Then there are naturals $N_0(\Psi)$ and $k(\Psi)$ such that if $N \geq N_0(\Psi)$ is a prime and $f$ is a vector of $l$ functions $\Z/N\Z \rightarrow \C$ bounded by $1$ we have
\begin{equation*}
|\Lambda_\Psi(f)| \leq \inf_{1 \leq i \leq l}{\|f_i\|_{U^k(\Z/N\Z)}}.
\end{equation*}
\end{theorem}
We shall use the above to count $(m,p,c)$-sets. This was already done by L{\^e} in \cite{Le::} for the purpose of transferring the partition regularity of Brauer configurations to the sets $\{p-1:p \text{ is prime}\}$ and $\{p+1:p \text{ is prime}\}$, itself answering a question of Li and Pan \cite{lipan::}.
We also need Gowers' inverse theorem. The following result is what is proved in \cite[Theorem 18.1]{gow::0}, though it is not stated in precisely this way.
\begin{theorem}\label{thm.gi} There is a monotone function $F_1:\N \rightarrow \N$ such that the following holds. Suppose that $N$ is prime, $\epsilon \leq \frac{1}{2}$ and $f:\Z/N\Z\rightarrow \C$ is bounded in magnitude by $1$ with $\|f\|_{U^k(\Z/N\Z)} \geq \epsilon$. Then there is a partition of $[N]$ into arithmetic progressions $P_1,\dots,P_M$ of average size at least $N^{\epsilon^{F_1(k)}}$ such that
\begin{equation*}
\sum_{j=1}^M{\left|\sum_{s \in P_j}{f(s+N\Z)}\right|} \geq \epsilon^{F_1(k)}N.
\end{equation*}
\end{theorem}
\subsection{Convolution, dilation and progressions}\label{ssec.tool}
First we record notation for dilation and translation: given $x,y \in \Z$ we write $\lambda_x(y):=xy$; further, given $f:\Z \rightarrow \C$ we write $\tau_x(f)(y):=f(y+x)$.
Suppose that $P\subset \Z$ is an arithmetic progression of odd length. Then
\begin{equation*}
P=x_P+d_P \cdot \{-N_P,\dots,N_P\}
\end{equation*}
for some $x_P \in \Z$ called the \textbf{centre}; $d_P \in \N_0$ called the \textbf{common difference}; and $N_P \in \N_0$ called the \textbf{radius}. Technically the common difference and radius need not be uniquely defined but this only becomes a problem for arithmetic progressions of size $1$ where the necessary adaptations of any argument are trivial and omitted for clarity.
We work with arithmetic progressions of odd length for convenience not because of any important difference. We say that $P$ is a \textbf{centred arithmetic progression} if $x_P=0$, so in particular a centred arithmetic progression is of odd length.
For $\delta \geq 0$ we shall define
\begin{equation*}
I_\delta(P):=d_P \cdot \{-\lfloor \delta N_P\rfloor,\dots,\lfloor \delta N_P\rfloor\},
\end{equation*}
and below record some basic properties of these `fractional dilates' of progressions. In many cases these properties are special cases of properties of Bohr sets (see \cite[\S4.4]{taovu::}). We do not require the generality of Bohr sets here because the result of Gowers' inverse theorem (Theorem \ref{thm.gi}) is a decomposition in terms of progressions. This has the additional benefit of meaning we do not need to deal with the problem of finding regular Bohr sets (see \cite[Lemma 4.24]{taovu::}), since all progressions are regular in a suitable sense. This is captured in the last three properties below.
\begin{lemma}[Basic properties]\label{lem.triv}
Suppose that $P$ and $P'$ are arithmetic progressions of odd length; $c \in \N$; $x \in \Z$; and $\delta, \delta' \in (0,1]$.
\begin{enumerate}
\item \label{pt.trivsym} \emph{(Symmetry)} $I_\delta(P)$ is a centred progression of size at least $\frac{1}{3}\delta |P|$;
\item \label{pt.monpar} \emph{(Monotonicity in radius)} $I_{\delta'}(P) \subset I_{\delta}(P)$ whenever $\delta' \leq \delta$;
\item \label{pt.monprog} \emph{(Monotonicity in progression)} $I_{\delta}(P') \subset I_{\delta}(P)$ whenever $P' \subset P$;
\item \label{pt.cen} \emph{(Translation)} $x+P$ is a progression of odd length and $I_\delta(P)=I_\delta(x+P)$;
\item \label{pt.trivdil} \emph{(Dilations)} $c\cdot P$ is an arithmetic progression of odd length and $I_{\delta}(c\cdot P) = c\cdot I_\delta(P)$;
\item \label{pt.trivsa} \emph{(Sub-additivity)} $I_{\delta}(P)+I_{\delta'}(P) \subset I_{\delta+\delta'}(P)$;
\item \label{pt.comp} \emph{(Composition)} $I_{\delta}(I_{\delta'}(P))\subset I_{\delta\delta'}(P)$;
\item \label{pt.trivint}\emph{(Interiors)} there is an arithmetic progression of odd length, $\Int_\delta(P)$, such that
\begin{equation*}
\Int_\delta(P)+I_\delta(P) \subset P \text{ and } |\Int_\delta(P)| \geq (1-\delta)|P|;
\end{equation*}
\item \label{pt.trivclos}\emph{(Closures)} $P+I_\delta(P)$ is an arithmetic progression of odd length and
\begin{equation*}
|P + I_\delta(P)|\leq (1+\delta)|P|;
\end{equation*}
\item \label{pt.trivinv} \emph{(Invariance)} for all $f:\Z \rightarrow \C$ and $y \in I_\delta(P)$ we have
\begin{equation*}
|\E_{x \in P}{\tau_y(f)(x)} - \E_{x \in P}{f(x)}| \leq 2\delta \|f\|_{L_\infty}.
\end{equation*}
\end{enumerate}
\end{lemma}
\begin{proof}
(\ref{pt.trivsym}) is trivial on noting that $|I_\delta(P)| = 2\lfloor \delta N_P\rfloor +1 \geq \frac{1}{3}\delta (2N_P+1)$. (\ref{pt.monpar}), (\ref{pt.monprog}), (\ref{pt.cen}), and (\ref{pt.trivdil}) are immediate. (\ref{pt.trivsa}) follows since $\lfloor \delta N_P\rfloor + \lfloor \delta' N_P\rfloor \leq \lfloor (\delta+\delta') N_P\rfloor$; and (\ref{pt.comp}) since $\lfloor \delta \lfloor \delta' N\rfloor \rfloor \leq \lfloor \delta\delta'N\rfloor$. For (\ref{pt.trivint}) set
\begin{equation*}
\Int_\delta(P):=x_P+d_P\cdot \{-(N_P-\lfloor \delta N_P\rfloor),\dots,N_P-\lfloor \delta N_P\rfloor\},
\end{equation*}
so that $\Int_\delta(P)$ is an arithmetic progression of odd length, $\Int_\delta(P) + I_\delta(P) \subset P$, and
\begin{equation*}
|\Int_\delta(P)| \geq 2(N_P-\lfloor \delta N_P\rfloor) +1 = 2N_P +1 - 2\lfloor \delta N_P\rfloor \geq (1-\delta)|P|.
\end{equation*}
For (\ref{pt.trivclos}) we note that
\begin{equation*}
P+I_\delta(P) = x_P+d_P\cdot \{-N_P,\dots,N_P\} + d_P\cdot \{ -\lfloor \delta N_P\rfloor,\dots,\lfloor \delta N_P\rfloor\},
\end{equation*}
and so
\begin{equation*}
|P+I_\delta(P)| = 2(N_P+\lfloor \delta N_P\rfloor) +1 \leq (1+\delta)(2N_P+1).
\end{equation*}
For (\ref{pt.trivinv}) note if $s' \in I_\delta(P)$ then we have
\begin{align*}
& \left|\E_{s \in P}{\tau_{s'}(f)(s)} - \E_{s \in P}{f(s)}\right|\\
& \qquad = \left|\E_{s \in P}{(1-1_{\Int_\delta(P)})(s+s') f(s+s')} + \E_{s \in P}{1_{\Int_\delta(P)}(s+s')f(s+s')}\right.\\
&\qquad \qquad \left.- \E_{s \in P}{1_{\Int_\delta(P)}(s)f(s)} - \E_{s \in P}{(1-1_{\Int_\delta(P)})(s)f(s)}\right|\\
&\qquad \leq \left|\E_{s \in P}{(1-1_{\Int_\delta(P)})(s+s') f(s+s')}\right|+\left|\E_{s \in P}{(1-1_{\Int_\delta(P)})(s)f(s)}\right| \leq 2\delta \|f\|_{L_\infty}.
\end{align*}
The result is proved.
\end{proof}
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\section{An example}
The notation in the final proof is quite heavy, so before turning to this we present an example case kindly suggested by one of the referees.
The arguments of \cite{chapre::} work to deal with $(1,p,c)$-sets (see \cite[Theorem 5.1]{chapre::}) and are similar to ours in terms of how these sorts of sets are dealt with. The additional complexity we encounter is in dealing with $(m,p,c)$-sets for $m>1$. Our approach is inductive on $m$, and our intention is that that by motivating the $m=2$ case the general argument will become clear.
We consider the problem of finding monochromatic septuples
\begin{equation}\label{eqn.config}
(x;y,x+y; z,z+x,z+y,z+x+y)
\end{equation}
in $r$-colourings of $\{1,\dots,N\}$. Such septuples do not correspond to an $(m,p,c)$-set, but for our purposes it behaves rather like a $(2,1,1)$-set. In fact looking for configurations of this type is a special case of Folkman's theorem \cite[Theorem 11, \S3.4]{grarotspe::0} (an explanation of the name may also be found in that reference), which was discovered independently by Folkman\footnote{Folkman's proof was unpublished, but a record of the fact he proved is found in \cite[Corollary 4]{grarot::}.}, Rado \cite{rad::4}, and Sanders \cite[Theorem 2]{san::31}.
We treat the three sets of terms in (\ref{eqn.config}) separated by semi-colons at three different scales. In particular, we shall find arithmetic progressions $P_1$, $P_2$, and $P_3$ iteratively by using the Gowers inverse theorem (Theorem \ref{thm.gi}) to give a density increment for a colour class on a certain progression. This increment translates to an increment to the sum over all colour classes of their maximum density on the translate of a progression. Importantly the translate may be different for different colour classes; and the process terminates since the sum of the maximal densities is bounded above by $r$. On termination we have control of some localised Gowers norms like
\begin{equation*}
\|f\|_{U^3(P_i;P_3)}:=\E_{z \in P_3}{\|f1_{z+P_i}\|_{U^3}} \text{ for }i\in \{1,2\}.
\end{equation*}
Localised Gowers norms of this type are defined in \cite[(2.12)]{pre::0} amongst other places and the additional discussion around that definition may be of interest.
The Generalised von Neumann Theorem (Theorem \ref{thm.gvn}) ensures that control of these localised Gowers norms of a colour class $C$ on the translate $a_C+P_3$ on which $C$ has maximal density $\delta$ leads to
\begin{align*}
&\E_{x \in P_1,y \in P_2, z \in a_C+P_3}{1_C(x)1_C(y)1_C(x+y)1_C(z)1_C(x+z)1_C(y+z)1_C(x+y+z)}\\
& \qquad \qquad \approx \delta^4\E_{x \in P_1,y \in P_2}{1_C(x)1_C(y)1_C(x+y)}.
\end{align*}
Inductively we can ensure that this second term is large for some colour class $C$, and the largeness of that colour class in turn ensures that $\delta$ is large. This gives a large count of septuples in one colour class as required.
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\section{The proof}
We begin by recording some notation for linear forms associated with $(m,p,c)$-sets. For $p,c \in \N$ and $t \in \N_0$ put
\begin{equation*}
\mathcal{D}_{p,c;t}:=\left\{ i \in \Z^{\N_0}: i_j \in \{-p,\dots,p\} \text{ for all }j<t; i_{t} =c;\text{ and } i_{j}=0 \text{ for all }j >t\right\}.
\end{equation*}
The sets $\mathcal{D}_{p,c;t}$ as $t$ ranges $\N_0$ are disjoint. For $m \in \N_0$ put
\begin{equation*}
\mathcal{U}_{m,p,c}:=\bigcup_{t=0}^m{\mathcal{D}_{p,c;t}}.
\end{equation*}
Then for $i \in \mathcal{U}_{m,p,c}$ write
\begin{equation}\label{eqn.lis}
L_i:\Z^{\N_0}\rightarrow \Z; s \mapsto \sum_{j:i_j \neq 0}{s_ji_j},
\end{equation}
which is well-defined since the set of $j$ such that $i_j \neq 0$ has size at most $m+1$ (and in particular is finite) for $i \in \mathcal{U}_{m,p,c}$.
The unique element of $\mathcal{D}_{p,c;t}$ with support of size $1$ is particularly important: we put
\begin{equation*}
i^*(c,t):=(\overbrace{0,\dots,0}^{t \text{ times}},c,0,\dots) \text{ and } \mathcal{D}_{p,c;t}^*:=\mathcal{D}_{p,c;t}\setminus \{i^*(c,t)\}.
\end{equation*}
Suppose that $P_0,\dots,P_m$ are arithmetic progressions. We are interested in the count
\begin{equation}\label{eqn.qdef}
Q_{m,p,c}(A;P_0,\dots,P_m):=\E_{s_0 \in P_0,\dots,s_m \in P_m}{\prod_{i \in \mathcal{U}_{m,p,c}}{1_A(L_i(s))}},
\end{equation}
since if $P_0,\dots,P_m \subset \N$ then this quantity is non-zero only if $A$ contains an $(m,p,c)$-set.
The following is our key counting/density-increment dichotomy.
\begin{lemma}\label{lem.count}
There are monotone functions $F_2:\N^3 \rightarrow \N$ and $\eta_0:\N^3 \rightarrow (0,1]$ such that the following holds. Suppose that $m,p,c \in \N$, $\delta\in (0,1]$ and $P_0,\dots , P_m\subset \Z$ are arithmetic progressions of odd length with
\begin{equation}\label{eqn.hyp1}
P_i \subset I_\delta(c\cdot P_m) \text{ for all }0 \leq i \leq m-1;
\end{equation}
$P'' \subset \Z$ is a centred arithmetic progression with
\begin{equation}\label{eqn.hyp2}
P'' \subset I_\delta(P_i) \text{ for all }0 \leq i \leq m;
\end{equation}
$P_m \subset \N$ and $A \subset \Z$ has $\alpha:=\E_{x \in c\cdot P_m}{1_A(x)}>0$. Then at least one of the following holds:
\begin{enumerate}
\item \label{pt.smallN} $|P''| \leq \exp(\delta^{-F_2(m,p,c)})$;
\item \label{pt.ldelta} $\delta \geq \eta_0(m,p,c)$;
\item \label{pt.harder} there is an arithmetic progression of odd length, $P'''\subset \N$, with $I_1(c\cdot P''')\subset I_{F_2(m,p,c)\delta^2}(c\cdot P_m)$, such that
\begin{equation*}
|P'''| \geq |P''|^{\delta^{F_2(m,p,c)}} \text{ and } \E_{x \in c \cdot P'''}{1_A(x)} \geq \alpha+\delta.
\end{equation*}
\item \label{pt.count} or
\begin{align*}
& \left|Q_{m,p,c}(A;P_0,\dots,P_m)\right.\\
& \qquad \qquad \left. - \alpha^{|\mathcal{D}_{p,c;m}|}Q_{m-1,p,c}(A;P_0,\dots,P_{m-1})\right|\leq F_2(m,p,c)\delta^{\eta_0(m,p,c)};
\end{align*}
\end{enumerate}
\end{lemma}
The proof below is long but not at all conceptually difficult. The length is a result of taking care with technicalities and somewhat licentious notation. The basic idea is to use Theorem \ref{thm.gvn} to control the $Q$s by suitable uniformity norms and then Theorem \ref{thm.gi} to show that if that error is not small then there is a density increment. There are two types of density increment, one is the expected increment resulting from large $U^k$ norm in Theorem \ref{thm.gi}. The other results from ensuring that the density of $A$ is the same on two progressions, one of which is a small dilate of the other. This second increment is common to arguments where groups are replaced by approximate groups -- in this case progressions -- and they perhaps originate in the work of Bourgain \cite{bou::5}. (See \cite[(10.16)]{taovu::} and the definition of the function $G$ there.)
\begin{proof}
With $L_i$s defined as in (\ref{eqn.lis}) let $\Psi:=(L_i)_{i \in \mathcal{U}_{m,p,c}}$ so that $\Psi:\Z^{m+1} \rightarrow \Z^{\mathcal{U}_{m,p,c}}$ is a homomorphism. Every $(L_i,L_j)$ with $i \neq j$ is a pair of independent vectors, so by Theorem \ref{thm.gvn} applied to $\Psi$ there is $k=k(\Psi)=O_{m,p,c}(1)$ such that if $N \geq N_0(\Psi)$ is prime and $h$ is a vector of functions $\Z/N\Z \rightarrow \C$ (indexed by $\mathcal{U}_{m,p,c}$) then
\begin{equation}\label{eqn.cnt}
\left|\E_{x \in [N]^{m+1}}{\prod_{i \in \mathcal{U}_{m,p,c}}{h_i(L_i(x)+N\Z)}}\right|\leq \inf_{i \in \mathcal{U}_{m,p,c}}{\|h_i\|_{U^{k}(\Z/N\Z)}}.
\end{equation}
Note that this will be applied with a vector $h$ to be determined, but which will be a combination of translates of the set $A$ suitably restricted, and also translates of the balanced function of $A$. The particular choice is made in (\ref{eqn.hdef}) (which itself depends on (\ref{eqn.gdef}) and (\ref{eqn.fdef})).
As remarked in the subsection on big-$O$ notation, since $m$, $p$ and $c$ are in $\N$ we see that there is a monotone function $F:\N^3 \rightarrow \N$ such that $F(m,p,c) \geq N_0(\Psi)$. (We shall take $F_2 \geq F$, but there are other functions later which will determine exactly what $F_2$ needs to be.)
Since $P''$ is a centred arithmetic progression there are natural numbers $d''$ and $N''$ such that $P''=d''\cdot \{-N'',\dots,N''\}$. By Bertrand's postulate there is a prime $N$ such that
\begin{equation*}
\max\{(mp+c)N'',F(m,p,c)\} <N =O_{m,p,c}(N'').
\end{equation*}
The reason for this choice will become clear just before (\ref{eqn.newrhs}) below. Before that we record the following claim.
\begin{claim*}
There is $\delta'=O_{m,p,c}(\delta)$ such that if $s_0 \in P_0, \dots ,s_{m-1} \in P_{m-1}$, $x \in d''\cdot \{-N,\dots,N\}$ and $i \in \mathcal{D}_{p,c;m}$ then
\begin{equation*}
L_i(s)+x-cs_m \in c\cdot I_{\delta'}(P_m)
\end{equation*}
and if additionally $s_m \in \Int_{\delta'}(P_m)$ then
\begin{equation*}
L_i(s)+x \in c \cdot P_m.
\end{equation*}
\end{claim*}
\begin{proof}
Write $l=\left\lceil \frac{N}{N''}\right\rceil$ so by (\ref{eqn.hyp1}), (\ref{eqn.hyp2}) and Lemma \ref{lem.triv} we have
\begin{align*}
d''\cdot \{-N,\dots,N\} &\subset {\overbrace{(d''\cdot \{-N'',\dots,N''\})+\cdots + (d''\cdot \{-N'',\dots,N''\})}^{l\text{ times}}}\\ & = lP'' \subset I_{l\delta}(P_0) \subset I_{l\delta}(I_\delta(c\cdot P_m)) \subset I_{l\delta^2}(c\cdot P_m).
\end{align*}
If follows that if $i \in \mathcal{D}_{p,c;m}$, $s_0\in P_0,\dots,s_{m-1} \in P_{m-1}$, and $x \in d''\cdot \{-N,\dots,N\}$, then by (\ref{eqn.hyp1}) and Lemma \ref{lem.triv} we have
\begin{equation}\label{eqn.kul}
i_0s_0 + \cdots + i_{m-1}s_{m-1} + x \in mpI_\delta(c\cdot P_m)+I_{l\delta^2}(c\cdot P_m)\subset I_{(mp+l\delta)\delta}(c\cdot P_m).
\end{equation}
Let $\delta':=(mp+l\delta)\delta=O_{m,p,c}(\delta)$. If $i \in \mathcal{D}_{p,c;m}$ we have $i_m=c$ and then by (\ref{eqn.kul}) and Lemma \ref{lem.triv} we have
\begin{equation*}
L_i(s)+x -cs_m= i_0s_0 + \cdots + i_{m-1}s_{m-1} +x \in I_{\delta'}(c\cdot P_m) = c\cdot I_{\delta'}(P_m)
\end{equation*}
giving the first conclusion. Finally, if $s_m \in \Int_{\delta'}(P_m)$ then by Lemma \ref{lem.triv} again
\begin{equation*}
L_i(s)+x \in cs_m + c\cdot I_{\delta'}(P_m) \subset c\cdot (\Int_{\delta'}(P_m) +I_{\delta'}(P_m) ) \subset c\cdot P_m.
\end{equation*}
The claim is proved.
\end{proof}
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0.137.5
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Let $\epsilon>0$ be a further constant to be optimised later and suppose (using $\tau$ for translation as defined in \S\ref{ssec.tool}) that for some $j \in \mathcal{D}_{p,c;m}$ we have
\begin{equation}\label{eqn.small}
\E_{s_0 \in P_0,\dots,s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\left|\E_{x \in d''\cdot [N]}{\tau_{L_j(s)}(1_A - \alpha 1_{c \cdot P_m})(x)}\right|} > \epsilon.
\end{equation}
By interchanging order of summation we have
\begin{align}
\label{eqn.split} & \left|\E_{s_0 \in P_0,\dots,s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\E_{x \in d''\cdot [N]}{\tau_{L_j(s)}(1_A - \alpha 1_{c \cdot P_m})(x)}}\right|\\
\nonumber& \qquad = \left|\E_{s_0 \in P_0,\dots,s_{m-1} \in P_{m-1},x \in d''\cdot [N]}{}\right.\\
\nonumber & \qquad \qquad \qquad \quad\left(\E_{s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\tau_{L_j(s)+x-cs_m}1_A (cs_m)}\right.\\
\nonumber & \qquad \qquad \qquad \qquad \qquad \qquad \left.\left.-\alpha\E_{s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\tau_{L_j(s)+x-cs_m}(1_{c\cdot P_m})(cs_m )}\right)\right|.
\end{align}
Suppose that $s_0 \in P_0,\dots,s_{m-1} \in P_{m-1}$ and $x \in d''\cdot [N]$. The claim and Lemma \ref{lem.triv} tell us that
\begin{align*}
\E_{s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\tau_{L_j(s)+x-cs_m}1_A (cs_m)} & \leq\E_{s_m \in P_m}{\tau_{L_j(s)+x-cs_m}1_A (cs_m)}\\
& \leq \E_{s_m \in P_m}{1_A(cs_m)}+ 2\delta' = \alpha+2\delta';
\end{align*}
and
\begin{align*}
\E_{s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\tau_{L_j(s)+x-cs_m}1_A (cs_m)} & \geq\E_{s_m \in P_m}{\tau_{L_j(s)+x-cs_m}1_A (cs_m)} -\delta'\\
& \geq \E_{s_m \in P_m}{1_A(c\cdot s_m)}-3\delta' = \alpha-3\delta'.
\end{align*}
We conclude that
\begin{equation*}
|\E_{s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\tau_{L_j(s)+x-cs_m}1_{A} (cs_m)}- \alpha| = O(\delta')=O_{m,p,c}(\delta),
\end{equation*}
and similarly
\begin{equation*}
|\E_{s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\tau_{L_j(s)+x-cs_m}1_{c\cdot P_m} (cs_m)}- 1| = O(\delta')=O_{m,p,c}(\delta)
\end{equation*}
for all $s_0 \in P_0,\dots,s_{m-1} \in P_{m-1}$ and $x \in d''\cdot [N]$. It follows from these and (\ref{eqn.split}) that
\begin{equation*}
\left|\E_{s_0 \in P_0,\dots,s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)\E_{x \in d''\cdot [N]}{\tau_{L_j(s)}(1_A - \alpha 1_{c \cdot P_m})(x)}}\right|=O_{m,p,c}(\delta).
\end{equation*}
Combining this with (\ref{eqn.small}) and averaging we see that there are elements $s_0 \in P_0,\dots,s_{m-1} \in P_{m-1},s_{m} \in \Int_{\delta'}(P_m)$ such that
\begin{equation}\label{eqn.cd}
\E_{x \in d''\cdot [N]}{\tau_{L_j(s)}(1_A - \alpha 1_{c \cdot P_m})(x)}> \frac{1}{2}\epsilon - O_{m,p,c}(\delta).
\end{equation}
Since $s_m \in \Int_{\delta'}(P_m)$ the claim tells us
\begin{equation*}
\tau_{L_j(s)}(1_{c\cdot P_m})(x) = 1_{c\cdot P_m}(L_j(s)+x) =1 \text{ for all }x \in d''\cdot \{-N,\dots,N\},
\end{equation*}
and so $L_j(s)-d''\cdot \{-N,\dots,N\} \subset c\cdot P_m \subset \N$. It follows from this that $c$ divides $L_j(s)$ and of course $c$ divides $d''$ (since $P'' \subset I_1(P_0) \subset I_1(c\cdot P_m)=c\cdot I_1(P_m)$ by Lemma \ref{lem.triv}). From (\ref{eqn.cd}) we then have
\begin{equation*}
\E_{x \in c\cdot (c^{-1}L_j(s)-(c^{-1}d'')\cdot [N])}{1_A(x)} >\alpha + \frac{1}{2}\epsilon - O_{m,p,c}(\delta).
\end{equation*}
We can take $\epsilon \in [\delta,O_{m,p,c}(\delta)]$ such that the right hand side is at least $\alpha+\delta$, and we are in case (\ref{pt.harder}) of the lemma (with $P''':=c^{-1}L_j(s)-(c^{-1}d'') \cdot [N]$ so $|P'''|=N \geq N''=|P''|$ where $|P'''|$ is odd since it is prime, and
\begin{equation*}
I_1(c\cdot P''') =I_1(d''\cdot [N])\subset I_{O_{m,p,c}(1)}(P'') \subset I_{O_{m,p,c}(\delta^2)}(c\cdot P_m)
\end{equation*}
by Lemma \ref{lem.triv}. Again by Lemma \ref{lem.triv} and the discussion in the section on big-$O$ notation there is a monotone function $F':\N^3 \rightarrow \N$ such that $I_1(c\cdot P''') \subset I_{F(m,p,c)\delta^2}(c\cdot P_m)$. And, again, we shall take $F_2 \geq F$.)\footnote{This is much stronger than the conclusion offered in case (\ref{pt.harder}) but this is because this is the easy density increment mentioned at the end of the discussion before the proof of this lemma.}
In view of the above we may suppose that (\ref{eqn.small}) does not happen for any $j \in \mathcal{D}_{p,c;m}$; we are in the main case.
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0.137.6
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In view of the above we may suppose that (\ref{eqn.small}) does not happen for any $j \in \mathcal{D}_{p,c;m}$; we are in the main case.
We split the integrand in (\ref{eqn.qdef}) into two factors
\begin{equation*}
\prod_{i \in \mathcal{U}_{m,p,c}}{1_A(L_i(s))}=\left(\prod_{t=0}^{m-1}{\prod_{i \in \mathcal{D}_{p,c;t}}{1_A(L_i(s))}}\right)\cdot \left(\prod_{i \in \mathcal{D}_{p,c;m}}{1_A(L_i(s))}\right).
\end{equation*}
The first term on the right is independent of $s_m$; we decompose the second through an arbitrary fixed total order on $\mathcal{D}_{p,c;m}^*$. For $i,j \in \mathcal{D}_{p,c;m}^*$ put
\begin{equation}\label{eqn.fdef}
f_{j,i}:=\begin{cases} 1_A & \text{ if }i<j\\
1_A - \alpha 1_{c\cdot P_m} & \text{ if } i=j\\
\alpha 1_{c\cdot P_m} & \text{ if }i>j\end{cases};
\end{equation}
so for all $x \in \Z^{\mathcal{D}_{p,c;m}}$ we have
\begin{align*}
& \sum_{j \in \mathcal{D}_{p,c;m}^*}{1_A(x_{i^*(c,m)})\prod_{i \in \mathcal{D}_{p,c;m}^*}{f_{j,i}(x_i)}}\\ & \qquad \qquad = \prod_{i \in \mathcal{D}_{p,c;m}}{1_A(x_i)} -\alpha^{|\mathcal{D}_{p,c;m}^*|}1_A(x_{i^*(c,m)})\prod_{i \in \mathcal{D}_{p,c;m}^*}{1_{c\cdot P_m}(x_i)}.
\end{align*}
It follows that
\begin{align}
\label{eqn.i}&Q_{m,p,c}(A;P_0,\dots,P_m)\\
\nonumber & \qquad - \alpha^{|\mathcal{D}_{p,c;m}^*|}\E_{s_0 \in P_0,\dots,s_{m-1} \in P_{m-1}}{\left(\prod_{t=0}^{m-1}{\prod_{i \in \mathcal{D}_{p,c;t}}{1_A(L_i(s))}}\right)}\\
\nonumber & \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left(\E_{s_m \in P_m}{1_A(L_{i^*(c,m)}(s))\prod_{i \in \mathcal{D}_{p,c;m}^*}{1_{c\cdot P_m}(L_i(s))}}\right)\\
\nonumber &\qquad \qquad = \sum_{j \in \mathcal{D}_{p,c;m}^*}{\E_{s_0 \in P_0,\dots,s_{m} \in P_{m}}{\left(\prod_{t=0}^{m-1}{\prod_{i \in \mathcal{D}_{p,c;t}}{1_A(L_i(s))}}\right)}}\\
\nonumber &\qquad \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad \times \left(1_A(L_{i^*(c,m)}(s))\prod_{i \in \mathcal{D}_{p,c;m}^*}{f_{j,i}(L_i(s))}\right).
\end{align}
On the other hand for all $i \in \mathcal{D}_{p,c;m}$ and $s \in P_0 \times \cdots \times P_{m-1} \times \Int_{\delta'}(P_m)$ we have from the claim (with $x=0$) that $L_i(s) \in c\cdot P_m$. Hence for all $s_0 \in P_0,\dots,s_{m-1} \in P_{m-1}$ we have
\begin{align*}
& \E_{s_m \in P_m}{1_A(L_{i^*(c,m)}(s))\prod_{i \in \mathcal{D}_{p,c;m}^*}{1_{c\cdot P_m}(L_i(s))}}\\
&\qquad \qquad \geq \E_{s_m \in P_m}{1_A(cs_m)1_{c\cdot \Int_{\delta'}(P_m)}(cs_m)\prod_{i \in \mathcal{D}_{p,c;m}^*}{1_{c\cdot P_m}(L_i(s))}}\\
& \qquad \qquad = \E_{s_m \in P_m}{1_A(cs_m)1_{c\cdot \Int_{\delta'}(P_m)}(cs_m)}\geq \alpha-O_{m,p,c}(\delta).
\end{align*}
On the other hand
\begin{equation*}
\E_{s_m \in P_m}{1_A(L_{i^*(c,m)}(s))\prod_{i \in \mathcal{D}_{p,c;m}^*}{1_{c\cdot P_m}(L_i(s))}}\leq \E_{s_m \in P_m}{1_A(L_{i^*(c,m)}(s))} = \alpha,
\end{equation*}
and so
\begin{equation*}
\E_{s_m \in P_m}{1_A(L_{i^*(c,m)}(s))\prod_{i \in \mathcal{D}_{p,c;m}^*}{1_{c\cdot P_m}(L_i(s))}}= \alpha+O_{m,p,c}(\delta).
\end{equation*}
Moreover,
\begin{equation*}
Q_{m-1,p,c}(A;P_0,\dots,P_{m-1})=\E_{s_0 \in P_0,\dots,s_{m-1} \in P_{m-1}}{\left(\prod_{t=0}^{m-1}{\prod_{i \in \mathcal{D}_{p,c;t}}{1_A(L_i(s))}}\right)}
\end{equation*}
since the sets $\mathcal{D}_{p,c;t}$ are disjoint over $0 \leq t <m$. We conclude that the left hand side of (\ref{eqn.i}) is equal to
\begin{equation*}
Q_{m,p,c}(A;P_0,\dots,P_m) - \alpha^{|\mathcal{D}_{p,c;m}|}Q_{m-1,p,c}(A;P_0,\dots,P_m)+O_{m,p,c}(\delta).
\end{equation*}
To estimate the right hand side of (\ref{eqn.i}) first note (by (\ref{eqn.hyp2}) and Lemma \ref{lem.triv}) that for $j \in \mathcal{D}_{p,c;m}^*$ the summand equals
\begin{align}
\label{eqn.innerexpect}&\E_{s_0 \in P_0,\dots,s_{m} \in P_{m}}{\E_{s_0',\dots,s_m' \in d''\cdot [N'']}{\left(\prod_{t=0}^{m-1}{\prod_{i \in \mathcal{D}_{p,c;t}}{1_A(L_i(s+s'))}}\right)}}\\
\nonumber &\qquad \qquad\qquad \qquad\qquad \qquad\qquad \times \left(1_A(L_{i^*(c,m)}(s+s'))\prod_{i \in \mathcal{D}_{p,c;m}^*}{f_{j,i}(L_i(s+s'))}\right)\\
\nonumber & \qquad \qquad\qquad \qquad\qquad \qquad\qquad\qquad \qquad\qquad+ O_{m}(\delta).
\end{align}
We shall look at the inner expectation of the first term here for which it will be useful to introduce some more notation. We shall use $\lambda$ for dilation in the way defined in \S\ref{ssec.tool}, and then for $s \in P_0 \times \cdots \times P_m$ and $y \in \Z^{m+1}$ put
\begin{equation}\label{eqn.gdef}
g_i^{(s)}(y)=\begin{cases}
\tau_{L_i(s)}(1_A) \circ \lambda_{d''}(y) & \text{ if }i \in \mathcal{U}_{m-1,p,c} \text{ or }i=i^*(c,m)\\
\tau_{L_i(s)}(f_{j,i})\circ \lambda_{d''}(y) & \text{ if } i \in \mathcal{D}_{p,c;m}^*.
\end{cases}
\end{equation}
With this notation, the inner expectation in (\ref{eqn.innerexpect}) equals
\begin{align*}
& \E_{y_0,\dots,y_m \in [N'']}{\prod_{i \in \mathcal{U}_{m,p,c}}{g_i^{(s)}(L_i(y))}}\\
& \qquad = \frac{1}{(N'')^{m+1}} \cdot \sum_{y \in \Z^{m+1}}{\left(\prod_{t=0}^m{1_{[N'']}(y_t)}\right)\left(\prod_{t=0}^m{\prod_{i \in \mathcal{D}_{p,c;t}^*}{g_i^{(s)}(L_i(y))}}\right)\left(\prod_{t=0}^m{g_{i^*(c,t)}^{(s)}(L_{i^*(c,t)}(y))}\right)}\\
& \qquad = \frac{1}{(N'')^{m+1}} \cdot \sum_{y \in \Z^{m+1}}{\left(\prod_{t=0}^m{\prod_{i \in \mathcal{D}_{p,c;t}^*}{g_i^{(s)}(L_i(y))}}\right)\left(\prod_{t=0}^m{g_{i^*(c,t)}^{(s)}|_{c\cdot [N'']}(L_{i^*(c,t)}(y))}\right)},
\end{align*}
since $L_{i^*(c,t)}(y)=cy_t$ for all $0 \leq t \leq m$. The notation is potentially a little confusing here: $g_{i^*(c,t)}^{(s)}|_{c\cdot [N'']}$ denotes the function $g_{i^*(c,t)}^{(s)}$ restricted to the set $c\cdot [N'']$.
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0.137.7
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For $x \in [N]$ write
\begin{equation}\label{eqn.hdef}
h_i^{(s)}(x+N\Z)=\begin{cases}
g_i^{(s)}(x) & \text{ if }i \in \bigcup_{t=0}^m{\mathcal{D}_{p,c;t}^*} \\
g_{i}^{(s)}|_{c \cdot [N'']}(x) & \text{ if }i \in \{i^*(c,t):0 \leq t \leq m\}
\end{cases}.
\end{equation}
In view of this definition, for $x \in [N]^{m+1}$, the product
\begin{equation*}
\prod_{i \in \mathcal{U}_{m,p,c}}{h_i^{(s)}(L_i(x)+N\Z)}
\end{equation*}
is non-zero only if $x \in ((c\cdot [N'']+N\Z)^{m+1})\cap ([N]^{m+1})$. This set equals $(c\cdot [N''])^{m+1}$ since $N >cN''$. Now, if $x \in (c\cdot [N''])^{m+1}$ then $L_i(x) \in [N]$ since $N > (mp+c)N''$ and so $h_i^{(s)}(x+N\Z)=g_i^{(s)}(x)$. It follows that
\begin{equation}\label{eqn.newrhs}
\sum_{x \in [N]^{m+1}}{\prod_{i \in \mathcal{U}_{m,p,c}}{h_i^{(s)}(L_i(x)+N\Z)}} = (N'')^{m+1}\cdot \E_{x_0,\dots,x_m \in [N'']}{\prod_{i \in \mathcal{U}_{m,p,c}}{g_i^{(s)}(L_i(x))}}.
\end{equation}
Apply (\ref{eqn.cnt}) to the above and conclude that the right hand side of (\ref{eqn.i}) is at most
\begin{align*}
&\sum_{j \in \mathcal{D}_{p,c;m}^*}{\E_{s_0 \in P_0,\dots,s_{m} \in P_{m}}{\left(\frac{N}{N''}\right)^{m+1}\left\|h_j^{(s)}\right\|_{U^k(\Z/N\Z)}}}+ O_{m,p,c}(\delta).
\end{align*}
Let $\eta:=(4\sqrt{\epsilon})^{\frac{1}{F_1(k)}} + \sqrt{\epsilon} + \delta'$ (where $F_1$ is as in Theorem \ref{thm.gi}) and suppose
\begin{equation}\label{eqn.earear}
\sum_{j \in \mathcal{D}_{p,c;m}^*}{\E_{s_0 \in P_0,\dots,s_m \in P_m}{\left\|h_j^{(s)}\right\|_{U^k(\Z/N\Z)}}} < \eta |\mathcal{D}_{p,c;m}^*|.
\end{equation}
Then it follows that
\begin{equation*}
\left|Q_{m,p,c}(A;P_0,\dots,P_m) - \alpha^{|\mathcal{D}_{p,c;m}|}Q_{m-1,p,c}(A;P_0,\dots,P_m)\right| = O_{m,p,c}(\delta) + O_{m,p,c}(\eta),
\end{equation*}
and we will find ourselves in case (\ref{pt.count}) in view of the definition of $\eta$ and choice of $\epsilon$ earlier. On the other hand suppose that (\ref{eqn.earear}) does not hold, so that by averaging there is some $j \in \mathcal{D}_{p,c;m}^*$ such that
\begin{equation}\label{eqn.upit}
\E_{s_0 \in P_0,\dots,s_m \in P_m}{\left\|h_j^{(s)}\right\|_{U^k(\Z/N\Z)}} \geq \eta.
\end{equation}
Since (\ref{eqn.small}) does not happen, writing
\begin{equation*}
\mathcal{S}:=\{s \in P_0 \times \cdots \times P_m: s_m \in \Int_{\delta'}(P_m) \text{ and } \left|\E_{x \in d''\cdot [N]}{\tau_{L_j(s)}(1_A - \alpha 1_{c \cdot P_m})(x)}\right|>\sqrt{\epsilon} \}
\end{equation*}
we have
\begin{equation*}
\E_{s_0 \in P_0,\dots,s_m \in P_m}{1_{\mathcal{S}}(s)\sqrt{\epsilon}} \leq \epsilon.
\end{equation*}
By (\ref{eqn.upit}), Lemma \ref{lem.triv}, the value of $\eta$ and the triangle inequality we see that
\begin{equation*}
\E_{s_0 \in P_0,\dots,s_m \in P_m}{1_{\Int_{\delta'}(P_m)}(s_m)1_{\mathcal{S}^c}(s)\left\|h_j^{(s)}\right\|_{U^k(\Z/N\Z)}} \geq \eta - \sqrt{\epsilon} - \delta' \geq (4\sqrt{\epsilon})^{\frac{1}{F_1(k)}}.
\end{equation*}
By averaging there is some $s \in (P_0\times \cdots \times P_{m-1}\times \Int_{\delta'}(P_m))\setminus \mathcal{S}$ such that
\begin{equation*}
\left\|h_j^{(s)}\right\|_{U^k(\Z/ N\Z)}\geq (4\sqrt{\epsilon})^{\frac{1}{F_1(k)}}.
\end{equation*}
By Theorem \ref{thm.gi} (applicable since $4\sqrt{\epsilon} \leq 2^{-F_1(k)}$ as otherwise we are in case (\ref{pt.ldelta}) of the Lemma since $\epsilon \geq \delta$) there is a partition of $[N]$ into arithmetic progressions $Q_1,\dots,Q_M$ of average size at least $N^{\epsilon^{F_1(k)}}$ such that
\begin{equation}\label{eqn.yy}
\sum_{l=1}^M{\left|\sum_{x \in Q_l}{\tau_{L_j(s)}(f_{j,j}) \circ \lambda_{d''}(x)}\right|} =\sum_{l=1}^M{\left|\sum_{x \in Q_l}{g_j^{(s)}(x)}\right|} =\sum_{l=1}^M{\left|\sum_{x \in Q_l}{h_j^{(s)}(x+N\Z)}\right|} \geq 4\sqrt{\epsilon}N.
\end{equation}
Of course
\begin{align*}
\sum_{l=1}^M{\sum_{x \in Q_l}{\tau_{L_j(s)}(f_{j,j})\circ \lambda_{d''}(x)}} & = \sum_{x \in d''\cdot [N]}{\tau_{L_j(s)}(f_{j,j})(x)}\\
& = \sum_{x \in d''\cdot [N]}{\tau_{L_j(s)}(1_A - \alpha 1_{c \cdot P_m})(x)},
\end{align*}
and so (since $s \not \in \mathcal{S}$)
\begin{equation*}
\left|\sum_{l=1}^M{\sum_{x \in Q_l}{\tau_{L_j(s)}(f_{j,j})\circ \lambda_{d''}(x)}}\right| \leq \sqrt{\epsilon }N.
\end{equation*}
Moreover, since the average size of $Q_l$ is at least $N^{\epsilon^{F_1(k)}}$, we have
\begin{equation*}
\sum_{\substack{1 \leq l \leq M\\|Q_l| \leq N^{\epsilon^{F_1(k)/2}}}}{\left|\sum_{x \in Q_l}{\tau_{L_j(s)}(f_{j,j}) \circ \lambda_{d''}(x)}\right|} \leq N^{\epsilon^{F_1(k)/2}}M \leq N^{-\epsilon^{F_1(k)/2}}N \leq \sqrt{\epsilon }N,
\end{equation*}
since we may assume $N^{-\epsilon^{F_1(k)/2}}\leq \sqrt{\epsilon}$ (or else we are in case (\ref{pt.smallN}) of the Lemma since $\epsilon \geq \delta$ and $N \geq N''=|P''|$). We may assume all the $Q_l$s are of odd size by removing at most one point from each at a cost of at most $M$ in (\ref{eqn.yy}). (Again $M\leq \sqrt{\epsilon}N$ or else we are in case (\ref{pt.smallN}) of the Lemma.) It follows by the triangle inequality and averaging that there is some $1 \leq l \leq M$ with
\begin{equation*}
\sum_{x \in Q_l}{\tau_{L_j(s)}(f_{j,j}) \circ \lambda_{d''}(x)} \geq \frac{1}{2}\sqrt{\epsilon }|Q_l| \text{ and } |Q_l| > N^{\epsilon^{F_1(k)}/2}.
\end{equation*}
Rewriting the first expression we get that
\begin{align*}
\E_{x \in d''\cdot Q_l}{1_A(L_j(s) + x)} & \geq \alpha \E_{x \in d''\cdot Q_l}{1_{c\cdot P_m}(L_j(s)+x)} + \frac{1}{2}\sqrt{\epsilon} = \alpha + \frac{1}{2}\sqrt{\epsilon}
\end{align*}
by the claim. First, $\frac{1}{2}\sqrt{\epsilon }\geq 2\delta$ (or else we are in case (\ref{pt.ldelta}) of the Lemma in view of the fact that $\epsilon \geq \delta$). Secondly, as noted after (\ref{eqn.cd}), $L_j(s)-d''\cdot \{-N,\dots,N\} \subset c\cdot P_m \subset \N$, $c$ divides $L_j(s)$ and $c$ divides $d''$, and so
\begin{equation*}
\E_{x \in c^{-1}L_j(s)-(c^{-1}d'')\cdot Q_l}{1_A( x)} \geq \alpha + \delta,
\end{equation*}
and putting $P''':=c^{-1}L_j(s)-(c^{-1}d'')\cdot Q_l \subset \N$ we are in case (\ref{pt.harder}) and hence are done. (Indeed, $|P'''|=|Q_l| \geq N^{\epsilon^{F_1(k)}/2} \geq N^{\delta^{F_1(k)}/2}$, and $|Q_l|$ is odd. As before
\begin{equation*}
I_1(c\cdot P''') =I_1(d''\cdot [N])\subset I_{O_{m,p,c}(1)}(P'') \subset I_{O_{m,p,c}(\delta^2)}(c\cdot P_m)
\end{equation*}
by Lemma \ref{lem.triv}. Again by Lemma \ref{lem.triv} and the discussion in the section on big-$O$ notation there is a monotone function $F':\N^3\rightarrow \N$ such that $I_1(c\cdot P''') \subset I_{F(m,p,c)\delta^2}(c\cdot P_m)$. And, again, we shall take $F_2 \geq F$.)
The lemma is proved.
\end{proof}
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0.137.8
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Our main result is the following theorem.
\begin{theorem}\label{thm.key}
Suppose that $m,p,c,r \in \N$, $\delta \in (0,1]$; $P \subset \N$ is an arithmetic progression of odd length $N$; and $\mathcal{C}$ is an $r$-colouring of $I_1(P)\cap \N$. Then at least one of the following holds.
\begin{enumerate}
\item \label{cs.n3b} $N \leq \exp(\exp(\delta^{-O_{m,p,c}(1)}))$;
\item \label{cs.d2b} $\delta \geq (2r)^{-O_{m,p,c}(1)}$;
\item \label{cs.conc} there are progressions $P_0,\dots, P_m \subset I_\delta(c\cdot P)\cap \N$ of odd length with
\begin{equation*}
P_i \subset I_1(P_{i+1}) \text{ for all }0 \leq i \leq m-1 \text{ and } |P_0| \geq N^{\exp(-\delta^{-O_{m,p,c}(1)})},
\end{equation*}
and some $C \in \mathcal{C}$ such that
\begin{equation*}
Q_{m,p,c}(C;P_0,\dots,P_m)\geq (2r)^{-O_{m,p,c}(1)}.
\end{equation*}
\end{enumerate}
\end{theorem}
We shall proceed by a double induction. The outer induction will be on $m$ and the inner is a density increment argument.
\begin{lemma}[Iteration Lemma]\label{lem.iil}
Suppose that Theorem \ref{thm.key} holds for some $m \in \N_0$, \emph{i.e.}
\begin{center}\fbox{\begin{minipage}{35em}There are monotone functions $F^{(m)}:\N^2\times (0,1] \rightarrow \N$, $\eta^{(m)}_0:\N^3\rightarrow (0,1]$ and $\eta^{(m)}_1:\N^2 \times (0,1] \rightarrow (0,1]$ such that the following holds. For any $p,c,r \in \N$, $\delta \in (0,1]$, $P\subset \N$ an arithmetic progressions of odd length $N$, and $r$-colouring $\mathcal{C}$ of $I_1(P)\cap \N$ at least one of the following holds.
\begin{enumerate}
\item \label{cs.n3} $N \leq F^{(m)}(p,c,\delta)$;
\item \label{cs.d2} $\delta \geq \eta_0^{(m)}(p,c,r)$;
\item there are progressions $P_0,\dots, P_m \subset I_\delta(c\cdot P)\cap \N$ of odd length with
\begin{equation*}
P_i \subset I_1(P_{i+1}) \text{ for all }0 \leq i \leq m-1 \text{ and } |P_0| \geq N^{\eta_1^{(m)}(p,c,\delta)},
\end{equation*}
and some $C \in \mathcal{C}$ such that
\begin{equation*}
Q_{m,p,c}(C;P_0,\dots,P_m)\geq \eta_0^{(m)}(p,c,r).
\end{equation*}
\end{enumerate}\end{minipage}}\end{center}
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0.137.9
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\begin{center}\fbox{\begin{minipage}{35em}There are monotone functions $F^{(m)}:\N^2\times (0,1] \rightarrow \N$, $\eta^{(m)}_0:\N^3\rightarrow (0,1]$ and $\eta^{(m)}_1:\N^2 \times (0,1] \rightarrow (0,1]$ such that the following holds. For any $p,c,r \in \N$, $\delta \in (0,1]$, $P\subset \N$ an arithmetic progressions of odd length $N$, and $r$-colouring $\mathcal{C}$ of $I_1(P)\cap \N$ at least one of the following holds.
\begin{enumerate}
\item \label{cs.n3} $N \leq F^{(m)}(p,c,\delta)$;
\item \label{cs.d2} $\delta \geq \eta_0^{(m)}(p,c,r)$;
\item there are progressions $P_0,\dots, P_m \subset I_\delta(c\cdot P)\cap \N$ of odd length with
\begin{equation*}
P_i \subset I_1(P_{i+1}) \text{ for all }0 \leq i \leq m-1 \text{ and } |P_0| \geq N^{\eta_1^{(m)}(p,c,\delta)},
\end{equation*}
and some $C \in \mathcal{C}$ such that
\begin{equation*}
Q_{m,p,c}(C;P_0,\dots,P_m)\geq \eta_0^{(m)}(p,c,r).
\end{equation*}
\end{enumerate}\end{minipage}}\end{center}
Then for any $p,c,r \in \N$, $\delta\in (0,1]$, $P\subset \N$ an arithmetic progressions of odd length $N$, and $r$-colouring $\mathcal{C}$ of $I_1(P)\cap \N$ at least one of the following holds.
\begin{enumerate}
\item \label{cs.n4}
\begin{equation*}
N \leq \min\left\{F^{(m)}(p,c,\delta),\exp(\delta^{-O_{m,p,c}(1)}\eta_1^{(m)}(p,c,\delta)^{-1})\right\}
\end{equation*}
\item \label{cs.d5}
\begin{equation*}
\delta \geq\left(\frac{\eta_0^{(m)}(p,c,r)}{2r}\right)^{O_{m,p,c}(1)};
\end{equation*}
\item \label{cs.inc} or there is a progression $P'''$ of odd length with
\begin{equation*}
I_1(P''') \subset I_1(P) \text{ and } |P'''| \geq N^{\delta^{O_{m,p,c}(1)} \eta_1^{(m)}(p,c,\delta)}
\end{equation*}
such that
\begin{equation*}
\sum_{C \in \mathcal{C}}{\max_{y: y+ P''' \subset \N}{\E_{x \in c\cdot (y+P''')}{1_C(x)}}} \geq \sum_{C \in \mathcal{C}}{\max_{y: y+ P \subset \N}{\E_{x \in c\cdot (y+P)}{1_C(x)}}}+ \frac{1}{2}\delta;
\end{equation*}
\item \label{cs.ct} there are progressions $P_0,\dots, P_{m+1} \subset I_\delta(c\cdot P)\cap \N$ of odd length with
\begin{equation*}
P_i \subset I_1(P_{i+1}) \text{ for all }0 \leq i \leq m \text{ and } |P_0| \geq N^{\eta_1^{(m)}(p,c,\delta)},
\end{equation*}
and some $C \in \mathcal{C}$ such that
\begin{equation*}
Q_{m+1,p,c}(C;P_0,\dots,P_{m+1})\geq \left(\frac{\eta_0^{(m)}(p,c,r)}{2}\right)^{O_{m,p,c}(1)}.
\end{equation*}
\end{enumerate}
\end{lemma}
\begin{proof}
Apply the content of the box to $P$ to get that either we are in case (\ref{cs.n3}) or (\ref{cs.d2}) of the hypothesis and so in case (\ref{cs.n4}) and (\ref{cs.d5}) respectively of the present lemma, or else there are progressions $P_0,\dots,P_m \subset I_\delta(c\cdot P)\cap \N$ with $P_i \subset I_1(P_{i+1})$ for all $0 \leq i \leq m-1$, and some $C \in \mathcal{C}$ with
\begin{equation*}
Q_{m,p,c}(C;P_0,\dots,P_m) \geq \eta_0^{(m)}(p,c,r) \text{ and }|P_0| \geq N^{\eta_1^{(m)}(p,c,\delta)}.
\end{equation*}
Let $y_0 \in \Z$ be such that $y_0+P \subset \N$ and
\begin{equation}\label{eqn.max}
\E_{x \in c\cdot (y_0+P)}{1_C(x)}=\max_{y: y+ P \subset \N}{\E_{x \in c\cdot (y+P)}{1_C(x)}},
\end{equation}
and let $P_{m+1}=y_0+P$. Then $P_0,\dots,P_{m+1}$ are arithmetic progressions of odd length. Furthermore, by Lemma \ref{lem.triv} (and assuming we are not in case (\ref{cs.d5})) we have
\begin{equation*}
P_i \subset I_\delta(c\cdot P_m) \subset I_\delta(c\cdot I_\delta(c\cdot P)) \subset I_\delta(c\cdot P_{m+1}) \text{ for all } 0 \leq i \leq m-1,
\end{equation*}
and
\begin{equation*}
P_m \subset I_\delta(c\cdot P) = I_\delta(c\cdot P_{m+1}).
\end{equation*}
It follows that we can apply Lemma \ref{lem.count} with parameters $m+1, p, c \in \N$ and $\delta\in (0,1]$, set $C$, and odd length arithmetic progressions $P_0,\dots,P_{m+1} \subset \Z$, and
\begin{equation*}
P'':=I_\delta(P_0) \subset I_\delta(P_i) \text{ for all }0 \leq i \leq m+1.
\end{equation*}
We have four cases.
\begin{enumerate}
\item \emph{(Case ({\ref{pt.smallN}}))} Then
\begin{equation*}
\delta N^{\eta_1^{(m)}(p,c,\delta)} \leq |I_\delta(P_0)| = |P''| \leq \exp(\delta^{-F_2(m+1,p,c)}),
\end{equation*}
and we are in case (\ref{cs.n4}) of this lemma.
\item \emph{(Case ({\ref{pt.ldelta}}))}
\begin{equation*}
\delta\geq \eta_0(m+1,p,c)
\end{equation*}
and we are in case (\ref{cs.d5}) of the lemma.
\item \emph{(Case ({\ref{pt.harder}})} Then there is an arithmetic progression $P''' \subset \N$ of odd length with $I_1(c\cdot P''') \subset I_{F_2(m+1,p,c)\delta^2}(c\cdot P)$ such that
\begin{equation*}
|P'''| \geq N^{\eta_1^{(m)}(p,c,\delta)\delta^{O_{m,p,c}(1)}} \text{ and } \E_{x \in c\cdot P'''}{1_C(x)} \geq \E_{x \in c\cdot P_{m+1}}{1_C(x)} + \delta.
\end{equation*}
(Then either we are in case (\ref{cs.d5}) of the lemma or else $I_1(P''') \subset I_1(P)$.) In view of the choice of $y_0$ (\ref{eqn.max}) and the definition of $P_{m+1}$ the second expression tells us that
\begin{equation*}
\max_{w:w+P''' \subset \N}{\E_{z \in c\cdot (w+P''')}{1_{C}(x)}}\geq \E_{x \in c\cdot P'''}{1_C(x)} \geq \max_{y:y+P\subset \N}{\E_{x \in c\cdot (y+P)}{1_C(x)}} + \delta.
\end{equation*}
For (the other) $C' \in \mathcal{C}$ and $y \in \Z$ such that $y+P \subset \N$ we have, by Lemma \ref{lem.triv}, that
\begin{align*}
\max_{w:w+P''' \subset \N}{\E_{z \in c\cdot (w+P''')}{1_{C'}(z)}}& \geq \E_{x \in c\cdot P}{\E_{z \in c\cdot (x+y+P''')}{1_{C'}(z)}}\\ & =\E_{u \in c\cdot (y+P)}{\E_{z \in c\cdot P'''}{\tau_z(1_{C'})(u)}}\\ &\geq \E_{x \in c\cdot (y+P)}{1_{C'}(x)} - F_2(m+1,p,c)\delta^2.
\end{align*}
Taking the maximum over $y$ such that $y+P \subset \N$ and summing it follows that
\begin{align*}
& \sum_{C' \in \mathcal{C}}{ \max_{w:w+P''' \subset \N}{\E_{z \in c\cdot (w+P''')}{1_{C'}(z)}}}\\
&\qquad \qquad \geq\sum_{C' \in \mathcal{C}}{ \max_{w:w+P \subset \N}{\E_{z \in c\cdot (w+P)}{1_{C'}(z)}}} +\delta - (r-1)F_2(m+1),p,c)\delta^2.
\end{align*}
So we are either in case (\ref{cs.d5}) of the lemma or (\ref{cs.inc}) of the lemma.
\item \emph{(Case ({\ref{pt.count}}))} Then
\begin{align}
\label{eqn.ss}& \left|Q_{m+1,p,c}(C;P_0,\dots,P_{m+1})\right.\\
\nonumber & \qquad \qquad \left. - \alpha^{|\mathcal{D}_{m+1,p,c}|}Q_{m,p,c}(C;P_0,\dots,P_{m})\right|\leq F_2(m+1,p,c)\delta^{\eta_0(m+1,p,c)},
\end{align}
where $\alpha:=\E_{x \in c\cdot P_{m+1}}{1_C(x)}$. First, suppose that
\begin{equation}\label{eqn.falssup}
\max_{y:y+P_m \subset \N}{\E_{x \in c\cdot (y+P_m)}{1_C(x)}}>\alpha + ((r-1)F_2(m+1,p,c)+1)\delta
\end{equation}
and so by (\ref{eqn.max}) and the definition of $P_{m+1}$ we have
\begin{align*}
\max_{y:y+P_m \subset \N}{\E_{x \in c\cdot (y+P_m)}{1_C(x)}}&>\max_{y:y+P \subset \N}{\E_{x \in y+P}{1_C(x)}}\\ &\qquad \qquad+((r-1)F_2(m+1,p,c)+1)\delta.
\end{align*}
For the other $C' \in \mathcal{C}$, we use that $P_m \subset I_\delta(c\cdot P_{m+1})=I_\delta(c\cdot P)$, and Lemma \ref{lem.triv} to give that for any $y+P \subset \N$ we have
\begin{align*}
\max_{w:w+P_m \subset \N}{\E_{z \in c\cdot (w+P_m)}{1_{C'}(z)}}& \geq \E_{x \in c\cdot (y+P)}{\E_{z \in c\cdot (x+y+P_m)}{1_{C'}(z)}}\\ & =\E_{u \in c\cdot (y+P)}{\E_{z \in c\cdot P'''}{\tau_z(1_{C'})(u)}}\\ &\geq \E_{x \in c\cdot (y+P)}{1_{C'}(x)} - F_2(m+1,p,c)\delta.
\end{align*}
Taking the maximum over $y$ such that $y+P \subset \N$ and summing it follows that
\begin{align*}
& \sum_{C' \in \mathcal{C}}{ \max_{w:w+P_m \subset \N}{\E_{z \in c\cdot (w+P_m)}{1_{C'}(z)}}}\\
&\qquad \qquad \geq\sum_{C' \in \mathcal{C}}{ \max_{w:w+P \subset \N}{\E_{z \in c\cdot (w+P)}{1_{C'}(z)}}} +\delta.
\end{align*}
We are in case (\ref{cs.inc}). (We should also note that we are in case (\ref{cs.d5}) of the lemma or else $I_1(P_m) \subset I_1(P)$.) We conclude that (\ref{eqn.falssup}) does not hold and so
\begin{align*}
& \alpha+ ((r-1)F_2(m+1,p,c)+1)\delta\\ &\qquad \qquad \geq \max_{y:y+P_m \subset \N}{\E_{x \in c\cdot (y+P_m)}{1_C(x)}}\\
& \qquad \qquad \geq \E_{x \in c\cdot P_m}{1_C(x)} \geq Q_{m,p,c}(C;P_0,\dots,P_m) \geq \eta_0^{(m)}(p,c,r).
\end{align*}
Either
\begin{equation*}
\delta \geq \frac{\eta_0^{(m)}(p,c,r)}{2((r-1)F_2((m+1),p,c)+1)},
\end{equation*}
and we are in case (\ref{cs.d5}) of the lemma; or from (\ref{eqn.ss}) we have
\begin{equation*}
F_2(m+1,p,c)\delta^{\eta_0(m+1,p,c)} \geq \frac{1}{2}\left(\frac{1}{2}\eta_0^{(m)}(p,c,r)\right)^{|\mathcal{D}_{m+1,p,c}|+1}
\end{equation*}
and we are in case (\ref{cs.d5}) of the lemma; or we are in case (\ref{cs.ct}) of the lemma.
\end{enumerate}
The lemma is proved.
\end{proof}
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0.137.10
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\begin{proof}[Proof of Theorem \ref{thm.key}]
We proceed by induction on $m$ to show that the content of the box in Lemma \ref{lem.iil} holds. This gives the theorem. The result holds for $m=0$ and it is convenient to use that as the base case. To see this take $P_0:=I_\delta(c\cdot P)\cap \N$. If $\delta N <2$ then we are in case (\ref{cs.n3}) of the box. If not then $|P_0| =\Omega(\delta N)$, and again we are either in case (\ref{cs.n3}), or $|P_0|$ satisfies the required lower bound. Finally, we are in case (\ref{cs.d2}) or else $I_\delta(c^2\cdot P) \subset I_1(P)$ and
\begin{equation*}
\sum_{C \in \mathcal{C}}{Q_{0,p,c}(C;P_0)} = \sum_{C \in \mathcal{C}}{\E_{s_0 \in P_0}{1_C(cs_0)}} \geq \E_{s_0 \in P_0}{1_{I_1(P)\cap \N}(cs_0)} =1.
\end{equation*}
The result follows by averaging. Now, suppose we have proved that the content of the box holds for some $m$.
We proceed iteratively defining progressions $P^{(0)},P^{(1)},\dots$ with $I_{1}( P^{(j+1)}) \subset I_1(P^{(j)})$ for all $j \geq 0$. Begin with $P^{(0)}:=I_1(P)\cap \N$ and define
\begin{equation*}
\mu_j:=\sum_{C \in \mathcal{C}}\max_{y:y+P^{(j)} \subset \N}{\E_{z \in y+P^{(j)}}{1_C(z)}}.
\end{equation*}
By hypothesis we have $\mu_0 \geq 1$ and we also have $\mu_j \leq r$ for all $j$. At stage $j \in \N_0$ we apply Lemma \ref{lem.iil} to $P^{(j)}$ and unless we are in case (\ref{cs.inc}) we terminate. If we are in case (\ref{cs.inc}) then we let $P^{(j+1)}$ be the progression given, which has
\begin{equation*}
I_1(P^{(j+1)}) \subset I_1(P^{(j)}), |P^{(j+1)}| \geq |P^{(j)}|^{\delta^{O_{m,p,c}(1)}\eta_1^{(m)}(p,c,\delta)} \text{ and }\mu_{j+1} \geq \mu_j + \frac{1}{2}\delta.
\end{equation*}
In view of the last fact this iteration can proceed for at most $2\delta^{-1}$ steps before terminating. When it terminates we have either
\begin{equation*}
N^{\delta^{-O_{m,p,c}(\delta^{-1})}(\eta_1^{(m)}(p,c,\delta))^{2\delta^{-1}}} \leq \min \left\{F^{(m)}(p,c,\delta),\exp(\delta^{-O_{m,p,c}(1)}\eta_1^{(m)}(p,c,\delta)^{-1})\right\};
\end{equation*}
or
\begin{equation*}
\delta \geq \left(\frac{\eta_0^{(m)}(p,c,r)}{2r}\right)^{O_{m,p,c}(1)};
\end{equation*}
or there is some $C \in \mathcal{C}$ such that
\begin{equation*}
Q_{m+1,p,c}(C;P_0,\dots,P_{m+1}) \geq \left(\frac{\eta_0^{(m)}(p,c,r)}{2}\right)^{O_{m,p,c}(1)} \text{ and } |P_0| \geq N^{\delta^{-O_{m,p,c}(\delta^{-1})}(\eta_1^{(m)}(p,c,\delta))^{2\delta^{-1}}}.
\end{equation*}
It follows that we can take
\begin{equation*}
F^{(m+1)}(p,c,\delta) \leq (2F^{(m)}(p,c,\delta))^{\eta_1^{(m)}(p,c,\delta)^{-O_{m,p,c}(\delta^{-1})}};
\end{equation*}
\begin{equation*}
\eta_1^{(m+1)}(p,c,\delta) \geq \left(\frac{\eta_1^{(m)}(p,c,\delta)}{2}\right)^{O_{m,p,c}(\delta^{-1})};
\end{equation*}
and
\begin{equation*}
\eta_0^{(m+1)}(p,c,r) \geq \left(\frac{\eta_0^{(m)}(p,c,r)}{2r}\right)^{O_{m,p,c}(1)}.
\end{equation*}
These recursions give the claimed bounds.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm.main}]
We apply Theorem \ref{thm.key} with $P=[N]$ (or $P=[N-1]$ if $N$ is even) with $\frac{1}{2c} \geq \delta = r^{-O_{m,p,c}(1)}$ such that case (\ref{cs.d2b}) never holds. If the colouring contains no $(m,p,c)$-set then $Q_{m,p,c}(C;P_0,\dots,P_m)=0$ for any $P_0,\dots,P_m$ of form described in case (\ref{cs.conc}) and so that does not happen. We conclude that $N$ is bounded in a way that yields the result.
\end{proof}
As a final remark, although we have made no effort to track the $m$, $p$, and $c$ dependencies they should also not be too bad given the known bounds in Theorem \ref{thm.gvn} and Theorem \ref{thm.gi}.
\end{document}
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\begin{document}
\centerline{\Large \bf Performance of Equal Phase-Shift}
\centerline{\Large
\bf Search for One Iteration}
\footnote{
The paper was supported by NSFC(Grants No. 60433050 and 60673034), the basic
research fund of Tsinghua university NO: JC2003043.}
\centerline{Dafa Li$^{a}$\footnote{email address:[email protected]},
Jianping Chen$^a$, Xiangrong Li$^{b}$, Hongtao Huang$^{c}$, Xinxin Li$^{d}$ }
\centerline{$^a$ Dept of mathematical sciences, Tsinghua University, Beijing
100084 CHINA}
\centerline{$^b$ Department of Mathematics, University of California, Irvine, CA
92697-3875, USA}
\centerline{$^c$ Electrical Engineering and Computer Science Department}
\centerline{ University of Michigan, Ann Arbor, MI 48109, USA}
\centerline{$^d$ Dept. of computer science, Wayne State University, Detroit, MI 48202,
USA}
Abstract
Grover presented the phase-shift search by replacing the selective
inversions by\ selective phase shifts of $\pi /3$. In this paper, we
investigate the phase-shift search with general equal phase shifts. We show
that for small uncertainties, the failure probability of the Phase-$\pi /3$
search is smaller than the general phase-shift search and for large
uncertainties, the success probability of the large phase-shift search is
larger than the Phase-$\pi /3$ search. Therefore, the large phase-shift
search is suitable for large-size of databases.
PACS number: 03.67.Lx
Keywords: Amplitude amplification, the \ phase-shift search, quantum
computing.
\section{Introduction}
Grover's quantum search algorithm is used to find a target state in an
unsorted database of size $N$\cite{Grover98}\cite{Grover05}. Grover's
quantum search algorithm can be considered as a rotation of the state
vectors in two-dimensional Hilbert space generated by the initial ($s$) and
target ($t$) vectors\cite{Grover98}. The amplitude of the desired state
increases monotonically towards its maximum and decreases monotonically\
after reaching the maximum \cite{LDF05}. As mentioned in \cite{Grover05}
\cite{Brassard97}, unless we stop when it is right at the target state, it
will drift away. A new search algorithm was presented in \cite{Grover05} to
avoid drifting away from the target state. Grover proposed the new algorithm
by replacing the selective inversions by\ selective phase shifts of $\pi /3$
, the algorithm converges to the target state irrespective of the number of
iterations. In his paper, Grover demonstrated the power of his algorithm by
calculating its success probability when only a single query into the
database was allowed. It turned out that if the success probability for a
random item in the database was $1-\epsilon $, where $\epsilon $ is known to
randomly lie somewhere in the range $\left( 0,\epsilon _{0}\right) $, after
a single quantum query into the database, Grover's new \ phase-shift
algorithm was able to increase the success probability to $1-\epsilon
_{0}^{3}.$ This was shown to be superior to existing algorithms and later
shown to be optimal\cite{Grover06}\cite{Tulsi}.
In \cite{Farhi}\cite{Roland}, adiabatic quantum computation provides an
alternative scheme for amplitude amplification\ that also does not drift
away from the solution. In \cite{Tulsi}, an algorithm for obtaining fixed
points in iterative quantum transformations was presented and the average
number of oracle queries for the fixed -point search algorithm was
discussed. In \cite{Boyer}, Boyer et al. described an algorithm that
succeeds with probability approaching to 1. In \cite{LDF06}, \ we discussed
the \ phase-shift search algorithm with different phase shifts.
As discussed below, the implementation of the general \ phase-shift search\
relies on selective phase shifts. In this paper, we investigate the \
phase-shift search with general but equal phase shifts. We are able to
considerably improve the algorithm by varying the phase-shift away from $\pi
/3$ when $\epsilon $ is large. As well known, the smaller deviation makes
the algorithm converge to the target state more rapidly. The deviation for
the Phase-$\pi /3$ search is $\epsilon ^{3}$\cite{Grover05}. For the large
size of database, we investigate that the deviation for any phase shifts of $
\theta >\pi /3$ is smaller than $\epsilon ^{3}$ and the closer to $\pi $ the
phase shifts are, the smaller the deviation is. In this paper, we study the
performance of the general \ phase-shift search for only one iteration. This
also determines the failure probability and success probability of the
general \ phase-shift search after recursively applying the single iteration
for $n$ times. Note that we neglect the effects\ of decoherence completely
in this paper.
This paper is organized as follows. In section 3, we give the necessary and
sufficient conditions for the smaller deviation than $\epsilon ^{3}$. In
section 4, we show that the Phase-$\pi /3$ search algorithm performs well
for the small $\epsilon $. In section 6, we demonstrate that the closer to $
\pi $ the phase shifts are, the smaller\ the deviation is. In section 7, we
propose the ratio measurement of the behavior of the Phase-$\theta $ search
algorithm for one query.
\section{Grover's \ phase-shift search and the reduction of the deviation}
The standard amplitude amplification algorithm would overshoot the target
state. To avoid drifting away from the target state, Grover presented the \
phase-shift search\cite{Grover05}.
In \cite{Grover05} the transformation $UR_{s}^{\pi /3}U^{+}R_{t}^{\pi /3}U$
was applied to the initial state $|s\rangle $,
\begin{eqnarray}
R_{s}^{\pi /3} &=&I-[1-e^{i\frac{\pi }{3}}]|s\rangle \langle s|, \nonumber
\\
R_{t}^{\pi /3} &=&I-[1-e^{i\frac{\pi }{3}}]|t\rangle \langle t|,
\label{grover1}
\end{eqnarray}
\noindent where $|t\rangle $ stands for the target state. The transformation
$UR_{s}^{\pi /3}U^{+}R_{t}^{\pi /3}U$ is denoted as Grover's the Phase-$\pi
/3$ search algorithm in \cite{Tulsi}.
Grover let $\theta $ denote $\pi /3$. Then
\begin{eqnarray}
R_{s}^{\theta } &=&I-[1-e^{i\theta }]|s\rangle \langle s|, \nonumber \\
R_{t}^{\theta } &=&I-[1-e^{i\theta }]|t\rangle \langle t|. \label{grover2}
\end{eqnarray}
\noindent The transformation $UR_{s}^{\theta }U^{+}R_{t}^{\theta }U$ is
called as the Phase-$\theta $ search algorithm in this paper. As indicated
in \cite{Grover05}, when $\theta =\pi $, this becomes one iteration of the
amplitude amplification algorithm\cite{Grover98}\cite{Brassard97}. Note that
if we apply $U$ to the initial state $|s\rangle $, then the amplitude of
reaching the target state $|t\rangle $ is $U_{ts}$\cite{Grover98}.\ Applying
the transformation $UR_{s}^{\theta }U^{+}R_{t}^{\theta }U$ to the start
state $|s\rangle $, Grover derived the following,
\begin{equation}
UR_{s}^{\theta }U^{+}R_{t}^{\theta }U|s\rangle =U|s\rangle \lbrack
e^{i\theta }+\left\vert U_{ts}\right\vert ^{2}(e^{i\theta
}-1)^{2}]+|t\rangle U_{ts}(e^{i\theta }-1). \label{grover3}
\end{equation}
Let $D(\theta )$ be the deviation from the $t$ state for any phase shifts of
$\theta $. Then from (\ref{grover3}) the following was obtained in \cite
{Grover05},
\begin{equation}
D(\theta )=(1-\left\vert U_{ts}\right\vert ^{2})|e^{i\theta }+\left\vert
U_{ts}\right\vert ^{2}(e^{i\theta }-1)^{2}|^{2}. \label{groverdev}
\end{equation}
\noindent Grover chose $\pi /3$ as phase shifts and let $\left\vert
U_{ts}\right\vert ^{2}=1-\epsilon $, where $0<\epsilon <1$. Substituting $
\left\vert U_{ts}\right\vert ^{2}=1-\epsilon $, the deviation from the $t$
state becomes $D(\pi /3)=\epsilon ^{3}$\cite{Grover05}.
Deviation $D(\theta )$ in (\ref{groverdev}) can be reduced as follows. For
any $\theta $,
\begin{equation}
e^{i\theta }+\left\vert U_{ts}\right\vert ^{2}(e^{i\theta
}-1)^{2}=e^{i\theta }+2(\cos \theta -1)e^{i\theta }(1-\epsilon )=e^{i\theta
}[1+2(\cos \theta -1)(1-\epsilon )]. \label{reduction}
\end{equation}
\noindent So by (\ref{reduction}), we obtain
\begin{equation}
D(\theta )=\epsilon \lbrack 1+2(\cos \theta -1)(1-\epsilon )]^{2}.
\label{dev1}
\end{equation}
In this paper, we study the \ phase-shift search algorithm with two equal
phase shifts. It is clear that it is enough to consider $\theta $ in $[0,\pi
]$. It can be shown that the maximum and minimum of deviation $D(\theta )$
are $1$ and $0$. That is,
\begin{equation}
0\leq D(\theta )\leq 1. \label{maxmin}
\end{equation}
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\section{Grover's \ phase-shift search and the reduction of the deviation}
The standard amplitude amplification algorithm would overshoot the target
state. To avoid drifting away from the target state, Grover presented the \
phase-shift search\cite{Grover05}.
In \cite{Grover05} the transformation $UR_{s}^{\pi /3}U^{+}R_{t}^{\pi /3}U$
was applied to the initial state $|s\rangle $,
\begin{eqnarray}
R_{s}^{\pi /3} &=&I-[1-e^{i\frac{\pi }{3}}]|s\rangle \langle s|, \nonumber
\\
R_{t}^{\pi /3} &=&I-[1-e^{i\frac{\pi }{3}}]|t\rangle \langle t|,
\label{grover1}
\end{eqnarray}
\noindent where $|t\rangle $ stands for the target state. The transformation
$UR_{s}^{\pi /3}U^{+}R_{t}^{\pi /3}U$ is denoted as Grover's the Phase-$\pi
/3$ search algorithm in \cite{Tulsi}.
Grover let $\theta $ denote $\pi /3$. Then
\begin{eqnarray}
R_{s}^{\theta } &=&I-[1-e^{i\theta }]|s\rangle \langle s|, \nonumber \\
R_{t}^{\theta } &=&I-[1-e^{i\theta }]|t\rangle \langle t|. \label{grover2}
\end{eqnarray}
\noindent The transformation $UR_{s}^{\theta }U^{+}R_{t}^{\theta }U$ is
called as the Phase-$\theta $ search algorithm in this paper. As indicated
in \cite{Grover05}, when $\theta =\pi $, this becomes one iteration of the
amplitude amplification algorithm\cite{Grover98}\cite{Brassard97}. Note that
if we apply $U$ to the initial state $|s\rangle $, then the amplitude of
reaching the target state $|t\rangle $ is $U_{ts}$\cite{Grover98}.\ Applying
the transformation $UR_{s}^{\theta }U^{+}R_{t}^{\theta }U$ to the start
state $|s\rangle $, Grover derived the following,
\begin{equation}
UR_{s}^{\theta }U^{+}R_{t}^{\theta }U|s\rangle =U|s\rangle \lbrack
e^{i\theta }+\left\vert U_{ts}\right\vert ^{2}(e^{i\theta
}-1)^{2}]+|t\rangle U_{ts}(e^{i\theta }-1). \label{grover3}
\end{equation}
Let $D(\theta )$ be the deviation from the $t$ state for any phase shifts of
$\theta $. Then from (\ref{grover3}) the following was obtained in \cite
{Grover05},
\begin{equation}
D(\theta )=(1-\left\vert U_{ts}\right\vert ^{2})|e^{i\theta }+\left\vert
U_{ts}\right\vert ^{2}(e^{i\theta }-1)^{2}|^{2}. \label{groverdev}
\end{equation}
\noindent Grover chose $\pi /3$ as phase shifts and let $\left\vert
U_{ts}\right\vert ^{2}=1-\epsilon $, where $0<\epsilon <1$. Substituting $
\left\vert U_{ts}\right\vert ^{2}=1-\epsilon $, the deviation from the $t$
state becomes $D(\pi /3)=\epsilon ^{3}$\cite{Grover05}.
Deviation $D(\theta )$ in (\ref{groverdev}) can be reduced as follows. For
any $\theta $,
\begin{equation}
e^{i\theta }+\left\vert U_{ts}\right\vert ^{2}(e^{i\theta
}-1)^{2}=e^{i\theta }+2(\cos \theta -1)e^{i\theta }(1-\epsilon )=e^{i\theta
}[1+2(\cos \theta -1)(1-\epsilon )]. \label{reduction}
\end{equation}
\noindent So by (\ref{reduction}), we obtain
\begin{equation}
D(\theta )=\epsilon \lbrack 1+2(\cos \theta -1)(1-\epsilon )]^{2}.
\label{dev1}
\end{equation}
In this paper, we study the \ phase-shift search algorithm with two equal
phase shifts. It is clear that it is enough to consider $\theta $ in $[0,\pi
]$. It can be shown that the maximum and minimum of deviation $D(\theta )$
are $1$ and $0$. That is,
\begin{equation}
0\leq D(\theta )\leq 1. \label{maxmin}
\end{equation}
\section{The phase shifts for smaller deviation}
As indicated in \cite{Grover98}, in the case of database search, $|U_{ts}|$
is almost $1/\sqrt{N}$, where $N$ is the size of the database. Thus, $
\epsilon $ is almost $1-1/N$ and $\epsilon $ is close to $1$ for the large
size of database. It is known that the deviation for Grover's the Phase-$\pi
/3$ search is $\epsilon ^{3}$. In this section, we give the phase shifts for
smaller deviation than $\epsilon ^{3}$.
\subsection{Necessary and sufficient conditions}
From (\ref{dev1}) let us calculate
\begin{eqnarray}
D(\theta )-\epsilon ^{3} &=&\epsilon \lbrack 1+2(\cos \theta -1)(1-\epsilon
)]^{2}-\epsilon ^{3} \nonumber \\
&=&\allowbreak \allowbreak \epsilon (1-\epsilon )(2\cos \theta -1)[2+(2\cos
\theta -3)(1-\epsilon )]. \label{dev2}
\end{eqnarray}
See Figs. 1 and 4. From (\ref{dev2}), we have the following statement.
Lemma 1. Deviation $D(\theta )$ in (\ref{dev1}) for any phase shifts of $
\theta $ in $[0,\pi /3)$ is greater than $\epsilon ^{3}$ for any $\epsilon $
. That is, $D(\theta )>\epsilon ^{3}$ for any $\theta $ in $[0,\pi /3)$ and
for any $\epsilon $. See table 1.
The argument is as follows.
When $0\leq \theta <\pi /3$, $0<2\cos \theta -1\leq 1$ and $2\epsilon
<2+(2\cos \theta -3)(1-\epsilon )\leq 1+\epsilon $ for any $\epsilon $.
Therefore when $0\leq \theta <\pi /3$, it follows (\ref{dev2})\ that $
D(\theta )>\epsilon ^{3}$ for any $\epsilon $.
\ From (\ref{dev2}) and Lemma 1,\ the following lemma holds immediately. See
table 1.
Lemma 2. $D(\theta )<\epsilon ^{3}$ if and only if
\begin{eqnarray}
\theta >\pi /3\wedge \epsilon >1-2/(3-2\cos \theta ). \label{cond2}
\end{eqnarray}
\noindent The following remark is used to describe the monotonicity of $
1-2/(3-2\cos \theta )$ in (\ \ref{cond2}). The monotonicity is used to find
smaller deviation than $\epsilon ^{3}$ below.
Remark 1. $1-2/(3-2\cos \theta )$ increases from $-1$ to $3/5$ as $\theta $
increases from $0$ to $\pi $. Thus,
\begin{eqnarray}
-1\leq 1-2/(3-2\cos \theta )\leq 3/5. \label{remark1}
\end{eqnarray}
Table 1. The phase shifts for deviations
\begin{tabular}{|c|c|c|}
\hline
$\theta $ & & $\epsilon $ \\ \hline
When $\theta >\pi /3$ & $D(\theta )<\epsilon ^{3}$ & for $\epsilon
>1-2/(3-2\cos \theta )$ \\ \hline
When $\theta <\pi /3$ & $D(\theta )>\epsilon ^{3}$ & for any $\epsilon $ \\
\hline
When $\theta >\pi /3$ & $D(\theta )>\epsilon ^{3}$ & for $\epsilon
<1-2/(3-2\cos \theta )$ \\ \hline
\end{tabular}
\subsection{The phase shifts for smaller deviation}
In this subsection, we give the phase shifts for which the deviations are
smaller than $\epsilon ^{3}$. \
Corollary 1. Deviation $D(\theta )$ for any phase shifts of $\theta $ in $
(\pi /3,\alpha ]$ is smaller than $\epsilon ^{3}$ whenever $\epsilon >$ $
1-2/(3-2\cos \alpha )$.
Proof. By Remark 1, $1-2/(3-2\cos \theta )$ increases from $0$ to $
1-2/(3-2\cos \alpha )$ as $\theta $ increases from $\pi /3$ to $\alpha $.
Thus, $0<1-2/(3-2\cos \theta )\leq $ $1-2/(3-2\cos \alpha )$\ whenever $\pi
/3<\theta \leq \alpha $. Therefore, when $\epsilon >$ $1-2/(3-2\cos \alpha )$
, always $\epsilon $ $>1-2/(3-2\cos \theta )$. Hence, this corollary follows
Lemma 2.
When $\alpha =\pi $, $2\pi /3$, $\pi /2$\ and $\arccos \frac{1-3\delta }{
2(1-\delta )}$, from Corollary 1 we have the following phase shifts for
smaller deviations than $\epsilon ^{3}$. See table 2.
Result 1. For any phase shifts of $\theta >\pi /3$, deviation $D(\theta
)<\epsilon ^{3}$ for $\epsilon >$ $3/5$. \ See Fig. 2 (a).
Result 2. For any phase shifts of $\theta $ in $(\pi /3,2\pi /3]$, deviation
$D(\theta )<\epsilon ^{3}$ for $\epsilon >1/2$. See Fig. 2 (b).
Result 3. For any phase shifts of $\theta $ in $(\pi /3,\pi /2]$, deviation $
D(\theta )<\epsilon ^{3}$ for $\epsilon >1/3$. See Fig. 2 (c).
Result 4. When $\epsilon >$ $\delta $, for any phase shifts of $\theta $ in $
(\pi /3$ ,$\arccos \frac{1-3\delta }{2(1-\delta )}]$, deviation $D(\theta
)<\epsilon ^{3}$.
Note that $\lim_{\delta \rightarrow 0}\arccos \frac{1-3\delta }{2(1-\delta )}
=\pi /3$ .
Our conclusion is when we search large database, i.e., $\epsilon $ is large,
for any phase shifts of $\theta >\pi /3$ the deviation is smaller than $
\epsilon ^{3}$.
Table 2. The phase shifts for $D(\theta )<\epsilon ^{3}$
\begin{tabular}{|c|c|c|}
\hline
$\theta $ & & $\epsilon $ \\ \hline
When $\theta >\pi /3$ & $D(\theta )<\epsilon ^{3}$ & for $\epsilon >3/5$ \\
\hline
When $\pi /3<\theta \leq 2\pi /3$ & $D(\theta )<\epsilon ^{3}$ & for $
\epsilon >1/2$ \\ \hline
When $\pi /3<\theta \leq \pi /2$ & $D(\theta )<\epsilon ^{3}$ & for $
\epsilon >1/3$ \\ \hline
When $\pi /3<\theta \leq \arccos \frac{1-3\delta }{2(1-\delta )}$ & $
D(\theta )<\epsilon ^{3}$ & for $\epsilon >\delta $ \\ \hline
\end{tabular}
\section{The Phase-$\protect\pi /3$ search is optimal for small
uncertainties.}
As indicated in \cite{Grover98}, the size of the database is very large,
i.e., $\epsilon $ is large. However, it is interesting to investigate the
performances of the Phase-$\pi /3$ search and the Phase-$\theta $ search for
small $\epsilon $.
\subsection{The Phase-$\protect\pi /3$ search possesses smaller failure
probability}
As said in \cite{Grover05}, $\epsilon ^{3}$ and $D(\theta )$ are the failure
probabilities of the Phase-$\pi /3$ search and the Phase-$\theta $ search,
respectively. Let us consider the ratio of the two failure probabilities. It
is easy to see that $\lim_{\epsilon \rightarrow 0}\epsilon ^{3}/D(\theta )=0$
for any $\theta \neq \pi /3$.\ That is, $\epsilon ^{3}=o(D(\theta ))$. In
other words, $\epsilon ^{3}$ is smaller than $D(\theta )$ for small $
\epsilon $. It means that $\epsilon ^{3}$ approaches $0$ more rapidly than $
D(\theta )$ as $\epsilon $ approaches $0$.
\subsection{The conditions under which the Phase-$\protect\pi /3$ search
behaves well \ \ \ \ \ \ \ \ \ \ }
Here, we discuss what $\epsilon $ satisfies $\epsilon ^{3}<$ $D(\theta )$.
From (\ref{dev2}) and Lemma 1, we have the following lemma.
Lemma 3. $D(\theta )>\epsilon ^{3}$ if and only if $\theta >\pi /3$ and $
\epsilon <1-2/(3-2\cos \theta )$ or $0\leq \theta <\pi /3$. See table 1.
The following corollary follows Lemma 3.
Corollary 2. When $\pi /3<\alpha \leq \theta $ and $\epsilon <1-2/(3-2\cos
\alpha )$, $D(\theta )>\epsilon ^{3}$.
The argument is as follows. By Remark 1, $1-2/(3-2\cos \theta )$ increases
from $1-2/(3-2\cos \alpha )$ to $3/5$ as $\theta $ increases from $\alpha $
to $\pi $. Thus, when $\pi /3<\alpha \leq \theta $, $1-2/(3-2\cos \alpha
)\leq 1-2/(3-2\cos \theta )$. Consequently, this corollary holds by Lemma 3.
From Corollary 2 we have the following results.
Result 5. When $\theta \geq \pi /2$, $D(\theta )>\epsilon ^{3}$ for $
\epsilon <1/3$.
Result 6. When $\theta \geq 2\pi /3$, $D(\theta )>\epsilon ^{3}$ for $
\epsilon <1/2$.
Result 7. When $\theta \geq \arccos \frac{1-3\delta }{2(1-\delta )}$, $
D(\theta )>\epsilon ^{3}$ for $\epsilon <\delta $.
Our conclusion is that for small $\epsilon $,\ the search algorithm performs
optimal for $\theta =\pi /3$. By means of the performance the Phase-$\pi /3$
search algorithm can be applied to quantum error corrections.
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\section{The Phase-$\protect\pi /3$ search is optimal for small
uncertainties.}
As indicated in \cite{Grover98}, the size of the database is very large,
i.e., $\epsilon $ is large. However, it is interesting to investigate the
performances of the Phase-$\pi /3$ search and the Phase-$\theta $ search for
small $\epsilon $.
\subsection{The Phase-$\protect\pi /3$ search possesses smaller failure
probability}
As said in \cite{Grover05}, $\epsilon ^{3}$ and $D(\theta )$ are the failure
probabilities of the Phase-$\pi /3$ search and the Phase-$\theta $ search,
respectively. Let us consider the ratio of the two failure probabilities. It
is easy to see that $\lim_{\epsilon \rightarrow 0}\epsilon ^{3}/D(\theta )=0$
for any $\theta \neq \pi /3$.\ That is, $\epsilon ^{3}=o(D(\theta ))$. In
other words, $\epsilon ^{3}$ is smaller than $D(\theta )$ for small $
\epsilon $. It means that $\epsilon ^{3}$ approaches $0$ more rapidly than $
D(\theta )$ as $\epsilon $ approaches $0$.
\subsection{The conditions under which the Phase-$\protect\pi /3$ search
behaves well \ \ \ \ \ \ \ \ \ \ }
Here, we discuss what $\epsilon $ satisfies $\epsilon ^{3}<$ $D(\theta )$.
From (\ref{dev2}) and Lemma 1, we have the following lemma.
Lemma 3. $D(\theta )>\epsilon ^{3}$ if and only if $\theta >\pi /3$ and $
\epsilon <1-2/(3-2\cos \theta )$ or $0\leq \theta <\pi /3$. See table 1.
The following corollary follows Lemma 3.
Corollary 2. When $\pi /3<\alpha \leq \theta $ and $\epsilon <1-2/(3-2\cos
\alpha )$, $D(\theta )>\epsilon ^{3}$.
The argument is as follows. By Remark 1, $1-2/(3-2\cos \theta )$ increases
from $1-2/(3-2\cos \alpha )$ to $3/5$ as $\theta $ increases from $\alpha $
to $\pi $. Thus, when $\pi /3<\alpha \leq \theta $, $1-2/(3-2\cos \alpha
)\leq 1-2/(3-2\cos \theta )$. Consequently, this corollary holds by Lemma 3.
From Corollary 2 we have the following results.
Result 5. When $\theta \geq \pi /2$, $D(\theta )>\epsilon ^{3}$ for $
\epsilon <1/3$.
Result 6. When $\theta \geq 2\pi /3$, $D(\theta )>\epsilon ^{3}$ for $
\epsilon <1/2$.
Result 7. When $\theta \geq \arccos \frac{1-3\delta }{2(1-\delta )}$, $
D(\theta )>\epsilon ^{3}$ for $\epsilon <\delta $.
Our conclusion is that for small $\epsilon $,\ the search algorithm performs
optimal for $\theta =\pi /3$. By means of the performance the Phase-$\pi /3$
search algorithm can be applied to quantum error corrections.
\section{Zero deviation and average zero deviation points}
\subsection{Zero deviation}
Let $d=1+2(\cos \theta -1)(1-\epsilon )$. Then, deviation $D(\theta )$ in (
\ref{dev1})\ can be rewritten as $D(\theta )=\epsilon d^{2}$. Let $d=0$.
Then we obtain $\cos \theta =1-\frac{1}{2(1-\epsilon )}$, where $0<\epsilon
\leq \frac{3}{4}$ to make $\left\vert 1-\frac{1}{2(1-\epsilon )}\right\vert
\leq 1$. Conclusively, if $U_{ts}$ is given, that is, $\epsilon $ is fixed,
then we choose $\theta =\arccos [1-\frac{1}{2(1-\epsilon )}],$ which is in $
(\pi /3$ ,$\pi ]$, as phase shifts. $\arccos [1-\frac{1}{2(1-\epsilon )}]$
will obviously make the deviation vanish and is called as a zero deviation
point. It means that one iteration will reach $t$ state with certainty if
the zero deviation point is chosen as phase shifts . Note that $
\lim_{\epsilon \rightarrow 0}\arccos [1-\frac{1}{2(1-\epsilon )}]=\pi /3$ .
This says that $\pi /3$ is the limit of the zero deviation points $\theta $
though it is not a zero deviation point.
\subsection{Average zero deviation points}
When $0<\epsilon \leq \frac{3}{4}$, $\arccos [1-\frac{1}{2(1-\epsilon )}]$
is called as a zero deviation point. Since $\epsilon $ is not given, the
zero deviation point is unknown. However, if we know the range of $\epsilon $
, then in terms of mean-value theorem for integrals, we can find the average
value $\bar{\theta}$ of the zero deviation points $\theta $. Here, we assume
that $\epsilon $ is uniformly distributed in the interval\ $(\beta ,\alpha
)\subseteq $ $(0,3/4]$.
Let $\epsilon $ be in the range $(\beta ,\alpha )$, where $(\beta ,\alpha
)\subseteq $ $(0,3/4]$. Then we calculate the average value of $1-\frac{1}{
2(1-\epsilon )}$ over the range $(\beta ,\alpha )$ as follows.
\begin{eqnarray}
\frac{1}{\alpha -\beta }\int_{\beta }^{\alpha }[1-\frac{1}{2(1-\epsilon )}
]d\epsilon =1+\frac{1}{2(\alpha -\beta )}\ln \frac{1-\alpha }{1-\beta }.
\end{eqnarray}
It can be argued that $-1\leq 1+\frac{1}{2(\alpha -\beta )}\ln \frac{
1-\alpha }{1-\beta }<\frac{1}{2}$. Thus, it is reasonable to define
\begin{eqnarray}
\bar{\theta}=\arccos [1+\frac{1}{2(\alpha -\beta )}\ln \frac{1-\alpha }{
1-\beta }]. \label{zeropoint}
\end{eqnarray}
\noindent $\bar{\theta}$ can be considered as the average value of the zero
deviation points $\theta $ and is called as the average zero deviation
point. It can be seen that $\pi /3<\bar{\theta}\leq \pi $.
When $\bar{\theta}$ is chosen as phase shift, we obtain the following
deviation
\begin{eqnarray}
D(\bar{\theta})=\epsilon (1+\frac{1-\epsilon }{\alpha -\beta }\ln \frac{
1-\alpha }{1-\beta })^{2}. \label{zpdev}
\end{eqnarray}
Let us compute $D(\bar{\theta})-\epsilon ^{3}$ \ as follows.
\begin{eqnarray}
D(\bar{\theta})-\epsilon ^{3}=\epsilon (1+\frac{1}{\alpha -\beta }\ln \frac{
1-\alpha }{1-\beta })(1-\epsilon )(1+\frac{1-\epsilon }{\alpha -\beta }\ln
\frac{1-\alpha }{1-\beta }+\epsilon ).
\end{eqnarray}
\noindent Notice that $1+\frac{1}{\alpha -\beta }\ln \frac{1-\alpha }{
1-\beta }<0$ and $1-\frac{1}{\alpha -\beta }\ln \frac{1-\alpha }{1-\beta }>0$
. Let $\kappa =1-2/(1-\frac{1}{\alpha -\beta }\ln \frac{1-\alpha }{1-\beta }
) $. It can be proven that $0<\kappa <1$. We can conclude when $\epsilon
>\kappa $, $D(\bar{\theta})<\epsilon ^{3}$.
We will find the average zero deviation point $\bar{\theta}$ for the ranges $
($ $0,1/2)$ and $(0,3/4)$ of $\epsilon $, respectively, as follows.
Example 1. Let $\epsilon $ lie in the range $(0,1/2]$.\ By (\ref{zeropoint}
),\ the average zero deviation point $\bar{\theta}_{1}=\arccos (1-\ln
2)=72^{\circ }30^{\prime }$. Taking $\bar{\theta}_{1}$ as phase shifts, by (
\ref{zpdev}) deviation $D(\bar{\theta}_{1})=\epsilon \lbrack 1-2(1-\epsilon
)\ln 2]^{2}$. Deviation $D(\bar{\theta}_{1})$ for phase shifts of $\bar{
\theta}_{1}$ is smaller than $\epsilon ^{3}$, i.e., $D(\bar{\theta}
_{1})<\epsilon ^{3}$, if and only if $\epsilon >\frac{2\ln 2-1}{2\ln 2+1}
=\allowbreak 0.16$.
Example 2. Let $(0,3/4]$ be the range of $\epsilon $. Then by (\ref
{zeropoint}), the average zero deviation point $\bar{\theta}_{2}$\ $=$ $
\arccos (1-\frac{4}{3}\ln 2)=86^{\circ }$. Choosing $\bar{\theta}_{2}$ as
phase shifts,\ by (\ref{zpdev})\ deviation\ $D(\bar{\theta}_{2})={\small
\epsilon \lbrack 1-}\frac{8}{3}{\small (1-\epsilon )}\ln {\small 2]}^{2}$
and $D(\bar{\theta}_{2})$\ is smaller than $\epsilon ^{3}$ when $\epsilon
>\allowbreak 0.30$. \
\section{Monotonicity of the deviation for large $\protect\epsilon $\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ }
As discussed above, when $\epsilon $\ is fixed and lies in the range $
(0,3/4] $ and $\arccos (1-\frac{1}{2(1-\epsilon )})$ is chosen as phase
shifts, the deviation vanishes. When $\epsilon >3/4$ ,\ since $\left\vert 1-
\frac{1}{2(1-\epsilon )}\right\vert >1$, deviation $D(\theta )$\ does not
vanish for any phase shifts of $\theta $ in $[0,\pi ]$.
When $\epsilon \geq \frac{3}{4}$,
\begin{eqnarray}
-1\leq 2(\cos \theta -1)(1-\epsilon )\leq 0. \label{cond1}
\end{eqnarray}
\noindent and $0\leq d\leq 1$. When $U_{ts}$ is given, that is, $\epsilon $
is fixed, by using (\ref{cond1}) it can be shown that deviation $D(\theta )$
monotonically decreases\ from $\epsilon $ to $\epsilon (4\epsilon -3)^{2}$
as $\theta $ increases from $0$ to $\pi $. See Fig. 3. for the monotonicity
of $D(\theta )$. When $\theta =\pi $, the deviation gets its minimum $
\epsilon (4\epsilon -3)^{2}$. That is,
\begin{equation}
\epsilon (4\epsilon -3)^{2}\leq D(\theta ) \label{inequality}
\end{equation}
\noindent for any phase shifts of $\theta $ in $[0,\pi ]$, whenever $
\epsilon \geq 3/4$.
Peculiarly,\textbf{\ }the deviation $\epsilon (4\epsilon -3)^{2}<$ $\epsilon
^{3}$ whenever $\epsilon >3/5$. The inequality in ({\ref{inequality})} also
follows that $(4\epsilon -3)\leq d$ for any phase shifts of $\theta $ in $
[0,\pi ]$\ whenever $\epsilon \geq 3/4$.
See table 3 for the deviations $D(\theta )$\ for $\theta =\pi /2,2\pi
/3,3\pi /4,5\pi /6$,$\pi $. Also see Fig. 4.
Table 3. The deviations for $\epsilon >3/4$
\begin{tabular}{|c|c|c|c|c|}
\hline
${\small \theta }$ & ${\small \pi /2}$ & ${\small 2\pi /3}$ & ${\small 3\pi
/4}$ & ${\small 5\pi /6}$ \\ \hline
${\tiny D(\theta )}$ & ${\tiny \epsilon (2\epsilon -1)}^{2}$ & ${\tiny
\epsilon (3\epsilon -2)}^{2}$ & ${\tiny \epsilon ((}\sqrt{2}{\tiny
+2)\epsilon -(}\sqrt{2}{\tiny +1))}^{2}$ & ${\tiny \epsilon ((}\sqrt{3}
{\tiny +2)\epsilon -(}\sqrt{3}{\tiny +1))}^{2}$ \\ \hline
\end{tabular}
Remark 2.
From the discussion above, it is easy to see that the closer to $\pi $ the
phase shifts are, the smaller\ the deviation is, when $\epsilon \geq \frac{3
}{4}$. By means of the inequality in ({\ref{inequality}) we can discuss the
lower bound of the number of iterations to find the $t$ state. }
Note that when the selective phase shift $\theta $ becomes $\pi $, the phase-
$\pi $ search is the amplitude amplification search.
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\section{Monotonicity of the deviation for large $\protect\epsilon $\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ }
As discussed above, when $\epsilon $\ is fixed and lies in the range $
(0,3/4] $ and $\arccos (1-\frac{1}{2(1-\epsilon )})$ is chosen as phase
shifts, the deviation vanishes. When $\epsilon >3/4$ ,\ since $\left\vert 1-
\frac{1}{2(1-\epsilon )}\right\vert >1$, deviation $D(\theta )$\ does not
vanish for any phase shifts of $\theta $ in $[0,\pi ]$.
When $\epsilon \geq \frac{3}{4}$,
\begin{eqnarray}
-1\leq 2(\cos \theta -1)(1-\epsilon )\leq 0. \label{cond1}
\end{eqnarray}
\noindent and $0\leq d\leq 1$. When $U_{ts}$ is given, that is, $\epsilon $
is fixed, by using (\ref{cond1}) it can be shown that deviation $D(\theta )$
monotonically decreases\ from $\epsilon $ to $\epsilon (4\epsilon -3)^{2}$
as $\theta $ increases from $0$ to $\pi $. See Fig. 3. for the monotonicity
of $D(\theta )$. When $\theta =\pi $, the deviation gets its minimum $
\epsilon (4\epsilon -3)^{2}$. That is,
\begin{equation}
\epsilon (4\epsilon -3)^{2}\leq D(\theta ) \label{inequality}
\end{equation}
\noindent for any phase shifts of $\theta $ in $[0,\pi ]$, whenever $
\epsilon \geq 3/4$.
Peculiarly,\textbf{\ }the deviation $\epsilon (4\epsilon -3)^{2}<$ $\epsilon
^{3}$ whenever $\epsilon >3/5$. The inequality in ({\ref{inequality})} also
follows that $(4\epsilon -3)\leq d$ for any phase shifts of $\theta $ in $
[0,\pi ]$\ whenever $\epsilon \geq 3/4$.
See table 3 for the deviations $D(\theta )$\ for $\theta =\pi /2,2\pi
/3,3\pi /4,5\pi /6$,$\pi $. Also see Fig. 4.
Table 3. The deviations for $\epsilon >3/4$
\begin{tabular}{|c|c|c|c|c|}
\hline
${\small \theta }$ & ${\small \pi /2}$ & ${\small 2\pi /3}$ & ${\small 3\pi
/4}$ & ${\small 5\pi /6}$ \\ \hline
${\tiny D(\theta )}$ & ${\tiny \epsilon (2\epsilon -1)}^{2}$ & ${\tiny
\epsilon (3\epsilon -2)}^{2}$ & ${\tiny \epsilon ((}\sqrt{2}{\tiny
+2)\epsilon -(}\sqrt{2}{\tiny +1))}^{2}$ & ${\tiny \epsilon ((}\sqrt{3}
{\tiny +2)\epsilon -(}\sqrt{3}{\tiny +1))}^{2}$ \\ \hline
\end{tabular}
Remark 2.
From the discussion above, it is easy to see that the closer to $\pi $ the
phase shifts are, the smaller\ the deviation is, when $\epsilon \geq \frac{3
}{4}$. By means of the inequality in ({\ref{inequality}) we can discuss the
lower bound of the number of iterations to find the $t$ state. }
Note that when the selective phase shift $\theta $ becomes $\pi $, the phase-
$\pi $ search is the amplitude amplification search.
\section{The ratio measurement of the success probabilities for one query}
\subsection{The ratio of the success probabilities}
Clearly, the greater the success probability is, the better the algorithm
performs. In other words, the more rapidly the algorithm converges. In this
section, it is demonstrated that the limit of the ratio of success
probabilities of the Phase-$\theta $ and the Phase-$\pi /3$ search
algorithms is used to quantify the performance of the Phase-$\theta $ search
algorithm.
From (\ref{maxmin}), let $\Delta (\theta )$ $=$ $1-D(\theta )$. Then $\Delta
(\theta )$ is the success probability with which the transformation $
UR_{s}^{\theta }U^{+}R_{t}^{\theta }U$ in (\ref{grover3}) drives the start
state to the target state. For instance, $\Delta (\pi /3)=1-D(\pi
/3)=1-\epsilon ^{3}$, which is the success probability of the Phase-$\pi /3$
search algorithm for one query. See Page 1 in \cite{Grover05}. Explicitly, $
\Delta (\theta )$ is not the desired measurement free of $\epsilon $ for the
Phase-$\theta $ search algorithm because $\Delta (\theta )$ is also a
function of $\epsilon $.
Let us compute the limit of $\Delta (\theta )$ as $\epsilon $ approaches 1
as follows.
$\lim_{\epsilon \rightarrow 1}\Delta (\theta )=\lim_{\epsilon \rightarrow
1}(1-\epsilon (1+2(\cos \theta -1)(1-\epsilon ))^{2}))=0$, for any $\theta $
in $[0,\pi ]$.
It is straightforward that the above limit can not be used to describe the
performance of the Phase-$\theta $ search algorithm for any phase shifts of $
\theta $ in $[0,\pi ]$\ because the limit always is zero for any $\theta $
in $[0,\pi ]$.
It is natural to consider and calculate $\frac{\Delta (\theta )}{\Delta (\pi
/3)}$ as follows.
\begin{eqnarray}
\frac{\Delta (\theta )}{\Delta (\pi /3)}=\frac{4\left( \cos ^{2}\theta
\right) \epsilon ^{2}-8\left( \cos \theta \right) \epsilon ^{2}+4\epsilon
^{2}+4\left( \cos \theta \right) \epsilon -4\left( \cos ^{2}\theta \right)
\epsilon +1}{\epsilon ^{2}+\epsilon +1}.
\end{eqnarray}
\noindent Then we obtain the following limit of $\frac{\Delta (\theta )}{
\Delta (\pi /3)}$ as $\epsilon $ approaches 1. Let
\begin{eqnarray}
\rho =\lim_{\epsilon \rightarrow 1}\frac{\Delta (\theta )}{\Delta (\pi /3)}
=\allowbreak \frac{5-4\cos \theta }{3}. \label{rate}
\end{eqnarray}
\noindent Then $\rho $ can be considered as the ratio of success
probabilities for the Phase-$\theta $ and the Phase-$\pi /3$ search
algorithms for large $\epsilon $.\ Notice that $\rho $ is free of $\epsilon $
and only depends on $\theta $. Hence, $\rho $ can be considered as a
measurement of performance of Phase-$\theta $ search algorithm for any phase
shifts of $\theta $\ in $[0,\pi ]$.
We can follow \cite{Grover05} to define by the recursion $U_{m+1}=$ $
U_{m}R_{s}^{\theta }U_{m}^{+}R_{t}^{\theta }U_{m}$, where $U_{0}=U$. For the
Phase-$\pi /3$ search, after recursive application of the basic iteration $m$
times, the success probability $\left\vert U_{m,ts}\right\vert =1-\epsilon
^{3^{m}}$\cite{Grover05}. For the Phase-$\theta $ search, as well we can
derive the success probability $\left\vert U_{m,ts}\right\vert $ and the
failure probability $1-\left\vert U_{m,ts}\right\vert $ after recursive
application of the basic iteration $m$ times. Fixed points of the Phase-$
\theta $ search algorithm are discussed in \cite{reviewer}.
\subsection{The larger phase shifts than $\protect\pi /3$ for larger size of
database}
It can be shown that $\rho $ increases from $1/3$ to $3$ as $\theta $
increases from 0 to $\pi $. In particular, $\rho $ increases from $1$ to $3$
as $\theta $ increases from $\pi /3$ to $\pi $. This also says that for
large databases, the larger the phase shifts are, the greater the success
probabilities are. For instance, $\rho =2.8$ for Phase-$5\pi /6$ search.
This means that for large $\epsilon $, the ratio of success probabilities
for the Phase-$5\pi /6$ and the Phase-$\pi /3$ search is 2.8. See table 4.
Table 4.\ $\rho $'s values for the Phase-$\theta $ search
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\theta $ & $\pi /2$ & $2\pi /3$ & $3\pi /4$ & $5\pi /6$ & $\pi $ \\ \hline
$\rho $ & $5/3$ & $7/3$ & $(5+2\sqrt{2})/3=2.6$ & $(5+2\sqrt{3})/3=2.8$ & $3$
\\ \hline
\end{tabular}
\ \ \ \ \ \ \ \ \ \ \
\section{Summary}
In this paper, we give the phase shifts for smaller deviation than $\epsilon
^{3}$. When $\epsilon \leq 3/4$ and $\epsilon $ is given, we choose the zero
deviation point as phase shifts to find the desired state for one iteration.
When $\epsilon \geq 3/4$, the deviation decreases from $\epsilon ^{3}$\ to $
\epsilon (4\epsilon -3)^{2}$ as $\theta $ increases from $\pi /3$ to $\pi $.
It is shown that for small $\epsilon $, the Phase-$\pi /3$ search behaves
better than the general Phase-$\theta $ search. Therefore the Phase-$\pi /3$
search can be applied to quantum error correction. We propose the limit of
the ratio of success probabilities of the Phase-$\theta $ and the Phase-$\pi
/3$ search algorithms as a measure of efficiency of a single Phase-$\theta $
iteration. The measure can help us find the optimal phase shifts for small
deviation and large success probability. Thus, there are more choices for
phase shifts to adjust an algorithm for large size of database and more
loose constraint opens a door for more feasible or robust realization.
Acknowledgement
We want to thank Lov K. Grover for his helpful discussions and
comments on the original manuscript ( in December, 2005) and the
reviewer for the helpful comments on this paper and useful
discussions about fixed points of the Phase-$\theta $ search.
\end{document}
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\begin{document}
\maketitle
\begin{abstract}
Geometric conditions on general polygons are given in \cite{GRB} in order to guarantee the error estimate for interpolants built from generalized barycentric coordinates, and the question about identifying sharp geometric restrictions in this setting is proposed. In this work, we address the question when the construction is made by using Wachspress coordinates. We basically show that the imposed conditions: {\it bounded aspect ratio property $(barp)$}, {\it maximum angle condition $(MAC)$} and {\it minimum edge length property $(melp)$} are actually equivalent to $[MAC,melp]$, and if any of these conditions is not satisfied, then there is no guarantee that the error estimate is valid. In this sense, $MAC$ and $melp$ can be regarded as sharp geometric requirements in the Wachspress interpolation error estimate.
\end{abstract}
\section{Introduction}
Many and different conditions on the geometry of finite elements were required in order to guarantee optimal convergence in the interpolation error estimate. Some of them deal with interior angles like the {\it maximum angle condition} (maximum interior angle bounded away from $\pi$) and the {\it minimum angle condition} (minimum interior angle bounded away from $0$), but others deal with some lengths of the element like the {\it minimum edge length property} (the diameter of the element is comparable to the length of the segment determined by any two vertices) and the {\it bounded aspect ratio property} often called {\it regularity condition} (the diameter of the element and the diameter of the largest ball inscribed are comparable).
Classical results on general Lagrange finite elements consider the regularity condition \cite{CR}. On triangular elements, the error estimate holds under the minimum angle condition \cite{Ze,Z}. However, on triangles, the minimum angle condition and the regularity condition are equivalent. From \cite{BA,BG,J} we know that the weakest sufficient condition on triangular elements is the maximum angle condition. Some examples can be constructed in order to show that if a family of triangles does not satisfy the maximum angle condition, then the error estimate on these elements does not hold.
Recently, it was proved \cite{AM:2} that, for quadrilaterals elements, the minimum angle condition ($mac$) is the weakest known geometric condition required to obtain the classical $W^{1,p}$-error estimate, when $1 \leq p < 3$, to any arbitrary order $k$ greater than 1. Moreover, in this case, $mac$ is also necessary. In \cite{AM,AM:2} it was proved that the {\emph{double angle condition}} (any interior angle bounded away from zero and $\pi$) is a sufficient requirement to obtain the error estimate for any order and any $p \geq 1$. When $k=1$ and $1 \leq p < 3$, a less restrictive condition ensures the error estimate \cite{AD,AM}: the {\it regular decomposition property} ($RDP$). Property $RDP$ requires that after dividing the quadrilateral into two triangles along one of its diagonals, each resultant triangle verifies the maximum angle condition and the quotient between the length of the diagonals is uniformly bounded.
This brief picture intends to show that study of sharp geometric restrictions on finite elements under which the optimal error estimate remains valid is an interesting and active field of research.
In \cite{GRB,GRB:2}, geometric conditions on general polygons are given in order to guarantee the error estimate for interpolants built from generalized barycentric coordinates, and the question about identifying sharp geometric restrictions in this setting is proposed. In this work, we address the question for the first-order Wachspress interpolation operator.
We show that the three sufficient conditions considered in \cite{GRB} ({\it regularity condition}, {\it maximum angle condition} and {\it minimum edge length property}) are actually equivalent to the last two since the regularity condition is a consequence of the maximum angle condition and the minimum edge length property. Then we exhibit families of polygons satisfying only one of these conditions and show that the interpolation error estimate does not hold to adequate functions. In this sense, the {\it maximum angle condition} and the {\it minimum edge length property} can be regarded as sharp geometric requirements to obtain the optimal error estimate.
This work is structured as follows: In Section \ref{geoimpl}, we introduce notation and exhibit some basic relationships between different geometric conditions on general convex polygons. Section \ref{wach} is devoted to recall Wachspress coordinates and some elementary results associated to them; a brief picture about error estimates for the first-order Wachspress interpolation operator is also given there. Finally, in Section \ref{sharp}, we present two counterexamples to show that $MAC$ and $melp$ are sharp geometric requirements under which the optimal error estimate is valid.
\section{Geometric conditions}
\label{geoimpl}
\setcounter{equation}{0}
In order to introduce notation and formalize the requirements of each geometric condition, we give the following definitions. From now on, $\Omega$ will refer to a general convex polygon.
\begin{enumerate}
\item[(i)] {\it (Bounded aspect ratio property)} We say that $\Omega$ satisfies the {\emph{bounded aspect ratio property}} (also called {\emph{regularity condition}}) if there exists a constant $\sigma>0$ such that
\begin{equation}
\label{barp}
\frac{diam(\Omega)}{\rho(\Omega)} \le \sigma,
\end{equation}
where $\rho(\Omega)$ is the diameter of the maximum ball inscribed in $\Omega$. In this case, we write $barp(\sigma)$.
\item[(ii)] {\it (Minimum edge length property)} We say that $\Omega$ satisfies the {\emph{minimum edge length property}} if there exists a constant $d_m>0$ such that
\begin{equation}
\label{mel}
0<d_m \leq \frac{\left\| {\bf v}_i-{\bf v}_j \right\|}{diam(\Omega)}
\end{equation}
for all $i \neq j$, where ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_n$ are the vertices of $\Omega$. In this case, we write $melp(d_m)$.
\item[(iii)] {\it (Maximum angle condition)} We say that $\Omega$ satisfies the {\emph{maximum angle condition}} if there exists a constant $\psi_M>0$ such that
\begin{equation}
\label{MAC}
\beta \leq \psi_M < \pi
\end{equation}
for all interior angle $\beta$ of $\Omega$. In this case, we write $MAC(\psi_M)$,
\item[(iv)] {\it (Minimum angle condition)} We say that $\Omega$ satisfies the {\emph{minimum angle condition}} if there exists a constant $\psi_m>0$ such that
\begin{equation}
\label{mac}
0 < \psi_m \leq \beta.
\end{equation}
for all interior angle $\beta$ of $\Omega$. In this case, we write $mac(\psi_m)$.
\end{enumerate}
All along this work, when we say {\it regular polygon}, we refer to a polygon satisfying the regularity condition given by \end{equation}ref{barp}.
\subsection{Some basic relationships}
It is well known that regularity assumption implies that the minimum interior angle is bounded away from zero. We state this result in the following lemma
\begin{lemma}
\label{lemma:regmac}
If $\Omega$ is a convex polygon satisfying $barp(\sigma)$, then $\Omega$ verifies $mac(\psi_m)$ where $\psi_m$ is a constant depending only on $\sigma$.
\end{lemma}
\proof See for instance \cite[Proposition 4 (i)]{GRB}. \qed
Considering the rectangle $R=[0,1] \times [0,s]$, where $0<s<1$, and taking $s \to 0^+$, we see that the converse statement of Lemma \ref{lemma:regmac} does not hold. Indeed, $R$ verifies the $mac(\pi/2)$ (independently of $s$), but, when $s$ tends to zero, $R$ is not regular in the sense given by \end{equation}ref{barp}. However, on triangular elements, $barp$ and $mac$ are equivalent. We use this fact to show that, on general polygons, the regularity condition is a consequence of the minimum edge length property and the maximum angle condition. To our knowledge, this elementary result has not been established or demonstrated previously.
\begin{figure}
\caption{(A): A polygon with its diameter attained as the length of the straight line joining two non-consecutive vertices. (B): A polygon with its diameter attained as the length of the straight line joining two consecutive vertices.}
\label{fig:macmel=reg}
\end{figure}
\begin{lemma}
\label{lemma:macmelpeqreg}
If $\Omega$ is a convex polygon satisfying $MAC(\psi_M)$ and $melp(d_m)$, then $\Omega$ verifies $barp(\sigma)$, where $\sigma=\sigma(\psi_M,d_m)$.
\end{lemma}
\proof We prove this by induction on the number $n$ of vertices of $\Omega$. If $n=3$, i.e., $\Omega$ is a triangle, the result follows from the law of sines. Indeed, we only have to prove that $\Omega$ has its minimum interior angle bounded away from zero. Let $\alpha$ be the minimum angle of $\Omega$ (if there is more than one choice, we choose it arbitrarily) and let $l$ be the length of its opposite side. Since $diam(\Omega)$ is attained on one side of $\Omega$, we can assume, without loss of generality, $l \neq diam(\Omega)$. We call $\beta$ the opposite angle to $diam(\Omega)$. It is clear that $\beta$ can not approach zero and since it is bounded above by $\psi_M$, we get that $1/\sin(\beta) \le C$ for some positive constant $C$. Then, from the law of sines and the assumption $melp(d_m)$, we have
$$\frac{\sin(\alpha)}{\sin(\beta)} = \frac{l}{diam(\Omega)} \geq d_m.$$
In consequence, $\sin(\alpha) \ge C^{-1} d_m$ which proves that $\alpha$ is bounded away from zero.
Let $n>3$. Since the diameter of $\Omega$ realizes as the length of its longest {\it diagonal}, i.e., the longest straight line joining two vertices of $\Omega$, we need to consider two cases depending if these vertices are consecutive or not.
Assume that $diam(\Omega)$ is attained as the length of the line joining two non-consecutive vertices (these may not be unique, in this case we choose them arbitrarily). We can divide $\Omega$ by this diagonal into two convex polygons $\Omega_1$ and $\Omega_2$ with less number of vertices (see Figure \ref{fig:macmel=reg} (A)). It is clear that both of them satisfy $MAC(\psi_M)$ and, since $diam(\Omega_i)=diam(\Omega)$ and the set of vertices of $\Omega_i$ is a subset of the vertices of $\Omega$, we conclude that $\Omega_i$ also verifies $melp(d_m)$. Therefore, by the inductive hypothesis, $\Omega_1$ and $\Omega_2$ verify $barp(\sigma_1)$ and $barp(\sigma_2)$, respectively, for some constants $\sigma_1, \sigma_2$ depending only on $\psi_M$ and $d_m$. Then, since $\rho(\Omega) \geq \rho(\Omega_i)$, $i=1,2$, we have
$$\displaystyle \frac{diam(\Omega)}{\rho(\Omega)} = \frac{diam(\Omega_i)}{\rho(\Omega)} \leq \frac{diam(\Omega_i)}{\rho(\Omega_i)} \leq \sigma_i.$$
Finally, if $diam(\Omega)$ is attained on a side of $\Omega$, i.e., is the length of the line joining two consecutive vertices ${\bf v}_{j-1}$ and ${\bf v}_j$ (these may not be unique, in this case we choose them arbitrarily), we divide $\Omega$ by the diagonal joining ${\bf v}_{j-1}$ and ${\bf v}_{j+1}$ into the triangle $T_1=\Delta({\bf v}_{j-1}{\bf v}_j{\bf v}_{j+1})$ and a convex polygon $\Omega_2$ (see Figure \ref{fig:macmel=reg} (B)). It is clear that $T_1$ verifies $melp(d_m)$ and $MAC(\psi_M)$, so (by the case $n=3$) we have that $T_1$ satisfies $barp(\sigma_1)$ for some positive constant $\sigma_1$. Then, since $diam(T_1)=diam(\Omega)$ and $\rho(\Omega) \geq \rho(T_1)$, we have
$$\displaystyle \frac{diam(\Omega)}{\rho(\Omega)} = \frac{diam(T_1)}{\rho(\Omega)} \leq \frac{diam(T_1)}{\rho(T_1)} \leq \sigma_1.$$ \qed
\begin{cor}
\label{cor:equiv}
$[MAC, melp]$ and $[barp, MAC, melp]$ are equivalent conditions.
\end{cor}
Finally, notice that reciprocal statement of Lemma \ref{lemma:macmelpeqreg} is false. Consider the following families of quadrilaterals: $\mathcal{F}_1=\{ K(1,1-s,s,1-s) \}_{0<s<1}$ where $K(1,1-s,s,1-s)$ denotes the convex quadrilateral with vertices $(0,0), (1,0), (s, 1-s)$ and $(0,1-s)$, and $\mathcal{F}_2=\{ K(1,1,s,s) \}_{1/2<s<1}$ where $K(1,1,s,s)$ denotes the convex quadrilateral with vertices $(0,0), (1,0), (s, s)$ and $(0,1)$. Clearly, any quadrilateral belonging to $\mathcal{F}_1 \cup \mathcal{F}_2$ is regular in the sense given by \end{equation}ref{barp}. Each element of $\mathcal{F}_1$ satisfies $MAC(3\pi/4)$, but taking $s \to 0^+$, we see that the minimum edge length property is violated. On the other hand, each element of $\mathcal{F}_2$ verifies $melp(1/2)$; but taking $s \to 1/2^+$, we see that the maximum angle condition is not satisfied.
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\section{Wachspress coordinates and the error estimate}
\label{wach}
\setcounter{equation}{0}
\subsection{Wachspress coordinates}
We start this section by remembering the definition of Wachspress coordinates and some of their main properties \cite{Fl:2, W}. Henceforth, we denote by ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_n$ the vertices of $\Omega$ enumerated in counterclockwise order starting in an arbitrary vertex. Let $\bf x$ denote an interior point of $\Omega$ and let $A_i(\bf x)$ denote the area of the triangle with vertices $\bf x$, ${\bf v}_i$ and ${\bf v}_{i+1}$, i.e., $A_i({\bf x})=|\Delta ({\bf x} {\bf v}_i {\bf v}_{i+1})|$, where, by convention, ${\bf v}_0:= {\bf v}_n$ and ${\bf v}_{n+1}:={\bf v}_1$. Let $B_i$ denote the area of the triangle with vertices ${\bf v}_{i-1}$, ${\bf v}_i$ and ${\bf v}_{i+1}$, i.e., $B_i=|\Delta ({\bf v}_{i-1} {\bf v}_i {\bf v}_{i+1})|$. We summarize the notation in Figure \ref{fig:notation}.
\begin{figure}
\caption{(A): Notation for $A_i({\bf x}
\label{fig:notation}
\end{figure}
Define the Wachspress weight function $w_i$ as the product of the area of the “boundary” triangle, formed by ${\bf v}_i$ and its two adjacent vertices, and the areas of the $n-2$ interior triangles, formed by the point ${\bf x}$ and the polygon's adjacent vertices (making sure to exclude the two interior triangles that contain the vertex ${\bf v}_i$), i.e.,
\begin{equation}
\label{wi}
\displaystyle w_i({\bf x}) = B_i \prod_{j \neq i,i-1} A_j(\bf x).
\end{equation}
After applying the standard normalization, Wachspress coordinates are then given by
\begin{equation}
\label{lambdai}
\displaystyle \lambda_i({\bf x}) = \frac{w_i({\bf x})}{\sum_{j=1}^n w_j({\bf x})}.
\end{equation}
An equivalent expression of \end{equation}ref{wi} for $w_i$ is given in \cite{Mey}; the main advantages of this alternative expression is that the result is easy to implement and it shows that only the edge $\overline{{\bf x} {\bf v}_i}$ and its two adjacent angles $\alpha_i$ and $\delta_i$ are needed (see Figure \ref{fig:notation} (A)). Indeed, $w_i$ can be written as
\begin{equation}
\label{weights}
w_i({\bf x}) = \frac{\cot(\alpha_i)+\cot(\delta_i)}{\left\| {\bf x}-{\bf v}_i \right\|^2}
\end{equation}
where $\alpha_i=\angle\ {\bf x} {\bf v}_i {\bf v}_{i+1}$ and $\delta_i=\beta_i-\alpha_i$ with $\beta_i$ being the inner angle of $\Omega$ associated to ${\bf v}_i$ (see Figure \ref{fig:notation}). The evaluation of the Wachspress basis function is carried out using elementary vector calculus operations. The angles $\alpha_i$ and $\delta_i$ are not explicitly computed, as suggested in \cite{Mey}, vector cross product and vector dot product formulas are used to find the cotangents.
Wachspress coordinates have the well-known following properties:
\begin{itemize}
\item[(I)] {\it (Non-negativeness)} $\lambda_i \geq 0$ on $\Omega$.
\item[(II)] {\it (Linear Completeness)} for any linear function $\ell :\Omega \to \mathbb R$, there holds $\ell = \sum_{i} \ell({\bf v}_i) \lambda_i$.
\item[] (Considering the linear map $\ell \end{equation}uiv 1$ yields $\sum_{i} \lambda_i = 1$; this property is usually named {\it partition of unity}).
\item[(III)] {\it (Invariance)} If $L:\mathbb R^2 \to \mathbb R^2$ is a linear map and $S:\mathbb R^2 \to \mathbb R^2$ is a composition of rotation, translation and uniform scaling transformations, then $\lambda_i({\bf x})=\lambda_i^L(L({\bf x}))=\lambda_i^S(S({\bf x}))$, where $\lambda_i^F(F({\bf x}))$ denotes a set of barycentric coordinates on $F(\Omega)$.
\item[(IV)] {\it (Linear precision)} $\sum_{i} {\bf v}_i \lambda_i({\bf x})={\bf x}$, i.e., every point on $\Omega$ can be written as a convex combination of the vertices ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_n$.
\item[(V)] {\it (Interpolation)} $\lambda_i({\bf v}_j)=\delta_{ij}$.
\end{itemize}
\subsection{Error estimate to the first-order Wachspress interpolation operator}
We only give a brief overview of some definitions and results which are of interest to us; for more details we refer to \cite{Das, GRB, Suk:2, Suk}.
Let $\{ \lambda_i \}$ be the Wachspress coordinates associated to $\Omega$ (see \end{equation}ref{lambdai}). Then, we can consider the first-order interpolation operator $I:H^2(\Omega) \to span \{ \lambda_i \} \subset H^1(\Omega)$ defined as
\begin{equation}
\label{defI}
\displaystyle I_{\Omega}u=Iu := \sum_{i} u({\bf v}_i) \lambda_i.
\end{equation}
Properties (I)-(V) of the Wachspress coordinates (more generally, generalized barycentric coordinates) guarantee that $I$ has the desirable properties of an interpolant. For this interpolant, called here the {\it first-order Wachspress interpolation operator}, the optimal convergence estimate
\begin{equation}
\label{errorestimate}
\left\| u-Iu \right\|_{H^1(\Omega)} \leq C diam(\Omega) |u|_{H^2(\Omega)}
\end{equation}
on polygons satisfying $[barp, MAC, melp]$ was proved \cite[Lemma 6]{GRB}.
\begin{rem}
\label{rem:red}
Thanks to {\rm Corollary \ref{cor:equiv}}, we can affirm that \end{equation}ref{errorestimate} holds on general convex polygons satisfying $[MAC, melp]$.
\end{rem}
\section{About sharpness on geometric restrictions}
\label{sharp}
\setcounter{equation}{0}
Since $[MAC, melp]$ are sufficient conditions to obtain \end{equation}ref{errorestimate}, we wonder if some of these requirements can be relaxed in order to obtain the error estimate. This question was partially answered in \cite{GRB}, where a counterexample, using pentagonal elements, is given in order to show that the $MAC$ can not be removed. For the sake of completeness, in Counterexample \ref{necmac}, we give a family of quadrilateral elements which does not satisfy $MAC$ but it verifies $melp$ and \end{equation}ref{errorestimate} does not hold. This example shows two things: $MAC$ is necessary in order to obtain the error estimate and, since every element in this family is regular in the sense given by \end{equation}ref{barp}, $barp$ is not enough to obtain \end{equation}ref{errorestimate}.
On the other hand, in Counterexample \ref{necmel}, we present a family of quadrilaterals which does not satisfy $melp$ but it verifies $MAC$ and \end{equation}ref{errorestimate} does not hold. Then, in order to obtain the interpolation error estimate, $melp$ is necessary.
In this sense, the question raised in \cite{GRB} about identifying sharp geometric restrictions under which the error estimates for the first-order Wachspress interpolation operator holds can be considered as answered.
\begin{figure}
\caption{Schematic picture of $K_s$ and $T_s$ (hatched area) considered in Counterexample \ref{necmac}
\label{fig:cex1}
\end{figure}
\begin{cex}
\label{necmac}
Consider the convex quadrilateral $K_s$ with the vertices ${\bf v}_1=(0,0), {\bf v}_2=(1,0), {\bf v}_3=(s,s)$ and ${\bf v}_4=(0,1)$, where $1/2<s<1$. We will be interested in the case when $s$ tends to $1/2$ since then the family of quadrilaterals $\{ K_s \}$ does not satisfy the maximum angle condition although it satisfies $melp(1/2)$.
Consider the function $u({\bf x})=x(1-x)$. Since $u({\bf v}_1)=0=u({\bf v}_2)=u({\bf v}_4)$, we have
$$Iu({\bf x})=u({\bf v}_3) \lambda_3({\bf x})= s(1-s) \lambda_3({\bf x}).$$
An straightforward computation yields
$$\displaystyle \lambda_3({\bf x}) = \frac{(2s-1)x}{s} \frac{y}{(s-1)(x+y)+s},$$
therefore
$$\displaystyle \frac{\partial \lambda_3}{\partial y} = \frac{(2s-1)x}{s} \frac{(s-1)x+s}{[(s-1)(x+y)+s]^2}.$$
Consider the triangle $T_s$ with vertices $(1/4,3/4)$, $(1/2,1/2)$ and $(1/2, (3s-1)/(2s))$ {\rm (see Figure \ref{fig:cex1})}. Then, on $T_s$, we have $1/4 \leq x \leq 1/2$, $1/2 \le y \le (3s-1)/(2s)$ and $x+y \geq 1$, so it follows that
$$0<(s-1)(x+y)+s \leq 2s-1
\quad \text{and} \quad
(s-1)x+s \geq (3s-1)/2$$
and hence
$$\displaystyle \frac{\partial \lambda_3}{\partial y} \geq \frac{(2s-1)}{4s} \frac{3s-1}{2(2s-1)^2}=\frac{3s-1}{8s(2s-1)}.$$
Then
$$|u-Iu|_{H^1(K_s)} \ge \left\| \frac{\partial (u-Iu)}{\partial y} \right\|_{L^2(K_s)} = \left\| \frac{\partial Iu}{\partial y} \right\|_{L^2(K_s)} = s(1-s)\left\| \frac{\partial \lambda_3}{\partial y} \right\|_{L^2(K_s)}$$
and, consequently,
$$|u-Iu|_{H^1(K_s)} \ge s(1-s)\left\| \frac{\partial \lambda_3}{\partial y} \right\|_{L^2(T_s)}.$$
Since $|T_s|=(2s-1)/(2^4s)$, we have
$$\left\| \frac{\partial \lambda_3}{\partial y} \right\|_{L^2(T_s)}^2 \geq \frac{(3s-1)^2}{(8s)^2(2s-1)^2}|T_s|=
\frac{(3s-1)^2}{2^{10}s^3(2s-1)} \to \infty$$
when $s \to 1/2^+$. Finally, as $|u|_{H^2(K_s)} = 2 |K_s|^{1/2} \leq 2$ and $diam(K_s)=\sqrt{2}$, we conclude that \end{equation}ref{errorestimate} can not hold.
\end{cex}
\begin{figure}
\caption{Schematic picture of $K_s$ and $D_s$ (hatched area) considered in Counterexample \ref{necmel}
\label{fig:cex2}
\end{figure}
\begin{cex}
\label{necmel}
Consider now the convex quadrilateral $K_s$ with the vertices ${\bf v}_1=(0,0), {\bf v}_2=(1,0), {\bf v}_3=(1-\sqrt[4]{s},s)$ and ${\bf v}_4=(0,s)$, where $0 < s < (1/2)^4$. Note that the family of quadrilaterals $\{ K_s \}$ satisfies $MAC(\pi/2+\tan^{-1}(2^3))$ $($independently of $s)$ but it does not satisfy the minimum edge length property when $s$ tends to zero since $\left\| {\bf v}_1-{\bf v}_4 \right\| = s \to 0^+$ and $diam(K_s) \sim 1$.
Consider the function $u({\bf x})=x^2$. Since $u({\bf v}_1)=0=u({\bf v}_4)$, we have, calling $a := 1-\sqrt[4]{s}$,
$$Iu({\bf x})=u({\bf v}_2) \lambda_2({\bf x})+u({\bf v}_3) \lambda_3({\bf x}) = \lambda_2({\bf x})+ a^2 \lambda_3({\bf x})$$
where
$$\lambda_2({\bf x})=\frac{x(s-y)}{s+y(a-1)} \quad \text{and} \quad
\lambda_3({\bf x})=\frac{xy}{s+y(a-1)}.$$
A simple computation yields
$$\frac{\partial (Iu-u)}{\partial y} = \frac{\partial Iu}{\partial y} = \frac{xsa(a-1)}{(s+y(a-1))^2}.$$
Let $D_s = K_s \cap \{ x \geq 1/2 \}$ $($see {\rm Figure \ref{fig:cex2})}. Since $a-1 <0$, we get $s+y(a-1) \leq s$ and then, on $D_s$, we have
$$\left| \frac{\partial (Iu-u)}{\partial y} \right| \geq \frac{xa(1-a)}{s} \geq \frac{a(1-a)}{2s}.$$
Therefore,
$$|Iu-u|_{H^1(K_s)}^2 \geq
\left\| \frac{\partial (Iu-u)}{\partial y} \right\|_{L^2(K_s)}^2 \geq
\left\| \frac{\partial (Iu-u)}{\partial y} \right\|_{L^2(D_s)}^2 \geq
\frac{a^2(1-a)^2}{4s^2} |D_s|,$$
and since $|D_s|=as/2$, we conclude that
$$|Iu-u|_{H^1(K_s)}^2 \geq
\frac{a^3(1-a)^2}{8s} =
\frac{(1-\sqrt[4]{s})^3}{8\sqrt{s}}$$
which tends to infinity when $s$ tends to zero. Finally, since $|u|_{H^2(K_s)} = 2 |K_s|^{1/2} \leq 2$ and $diam(K_s) \sim 1$, we conclude that \end{equation}ref{errorestimate} can not hold.
\end{cex}
\end{document}
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\begin{document}
\title{{Bounds for the first several prime character nonresidues}
\begin{abstract}\noindent Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial Dirichlet character modulo $m$, there are more than $m^{\kappa}$ primes $\ell \le m^{\frac{1}{4\sqrt{e}}+\varepsilon}$ with $\chi(\ell)\notin \{0,1\}$. The proof uses the fundamental lemma of the sieve, Norton's refinement of the Burgess bounds, and a result of Tenenbaum on the distribution of smooth numbers satisfying a coprimality condition. For quadratic characters, we demonstrate a somewhat weaker lower bound on the number of primes $\ell \le m^{\frac14+\epsilon}$ with $\chi(\ell)=1$.
\end{abstract}
\section{Introduction}
Let $\chi$ be a nonprincipal Dirichlet character. An integer $n$ is called a $\chi$-\emph{nonresidue} if $\chi(n) \notin \{0,1\}$. Problems about character nonresidues go back to the beginnings of modern number theory. Indeed, one can read out of Gauss's \emph{Disquisitiones} that for primes $p\equiv 1\pmod{8}$ and $\chi(\cdot) = \leg{p}{\cdot}$, the smallest $\chi$-nonresidue does not exceed $2\sqrt{p}+1$ \cite[Article 129]{gauss86}. This was an auxiliary result required for Gauss's first proof of the quadratic reciprocity law.
In the early 20th century, I.\,M. Vinogradov initiated the study of how the quadratic residues and nonresidues modulo a prime $p$ are distributed in the interval $[1,p-1]$. A particularly natural problem is to estimate the size of $n_p$, the smallest quadratic nonresidue modulo $p$. Vinogradov conjectured that $n_p \ll_{\varepsilon} p^{\varepsilon}$, for each $\varepsilon >0$. By means of a novel estimate for character sums (independently discovered by P\'olya), coupled with a clever sieving argument, he showed \cite{vinogradov18} that $n_p \ll_{\varepsilon} p^{\frac{1}{2\sqrt{e}} + \varepsilon}$. Burgess's character sum bounds \cite{burgess57}, in conjunction with Vinogradov's methods, yield the sharper estimate
\begin{equation}\label{eq:burgessquadratic} n_p \ll_{\varepsilon} p^{\frac{1}{4\sqrt{e}}+\varepsilon}. \end{equation}
Fifty years of subsequent research has not led to any improvement in the exponent $\frac{1}{4\sqrt{e}}$. But generalizing \eqref{eq:burgessquadratic}, Norton showed that if $\chi$ is any nontrivial character modulo $m$, then the least $\chi$-nonresidue is $O_{\varepsilon}(m^{1/4\sqrt{e} + \varepsilon})$. See \cite[Theorem 1.30]{norton98}.
Since $\chi$ is completely multiplicative, the smallest $\chi$-nonresidue is necessarily prime. In this note, we prove that there are actually many prime $\chi$-nonresidues satisfying the Burgess--Norton upper bound.
\begin{thm}\label{thm:main} For each $\varepsilon > 0$, there are numbers $m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon)> 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $m^{\frac{1}{4\sqrt{e}}+\varepsilon}$.
\end{thm}
The problem of obtaining an upper bound on the first several prime character nonresidues was considered already by Vinogradov. In \cite{vinogradov18}, he showed that for large $p$, there are at least $\frac{\log{p}}{7\log\log{p}}$ prime quadratic nonresidues modulo $p$ not exceeding \[ p^{\frac{1}{2}-\frac{1}{\log\log{p}}}. \]
For characters to prime moduli, a result resembling Theorem \ref{thm:main} was proved by Hudson in 1983 \cite{hudson83}. (See also Hudson's earlier investigations \cite{hudson73,hudson74,hudson74A}.) But even restricted to prime $m$, Theorem \ref{thm:main} improves on \cite{hudson83} in multiple respects. In \cite{hudson83}, the exponent on $p$ is $\frac{1}{4}+\varepsilon$ instead of $\frac{1}{4\sqrt{e}} +\varepsilon$, and the number of nonresidues produced is only $c_{\varepsilon} \frac{\log{p}}{\log\log{p}}$. Moreover, it is assumed in \cite{hudson83} that the order of $\chi$ is fixed. Stronger results than those of \cite{hudson83} were announced by Norton already in 1973 \cite{norton74}.\footnote{Norton claims in \cite{norton74}: \emph{Let $\varepsilon>0$ and $k_0 \ge 2$. If $m \ge 3$ and $[(\mathbf{Z}/m\mathbf{Z})^{\times}: {(\mathbf{Z}/m\mathbf{Z})^{\times}}^k] \ge k_0$, then each of the smallest $\lfloor \log{m}/\log\log{m}\rfloor$ primes not dividing $m$ that are $k$th power nonresidues modulo $m$ is $\ll_{\varepsilon,k_0}n^{1/4u_{k_0} + \varepsilon}$}. Here $u_{k_0}$ has the same meaning as in our introduction.} Unfortunately, a full account of Norton's work seems to have never appeared.
It becomes easier to produce small character nonresidues as the order of $\chi$ increases. This phenomenon was noticed by Vinogradov \cite{vinogradov27} and further investigated by Buchstab \cite{buchstab49} and Davenport and Erd\H{o}s \cite{DE52}. To explain their results requires us to first recall the rudiments of the theory of smooth numbers. For each positive integer $n$, let $P^{+}(n)$ denote the largest prime factor of $n$, with the convention that $P^{+}(1)=1$. A natural number $n$ is called \emph{$y$-smooth} (or \emph{$y$-friable}) if $P^{+}(n) \le y$. For $x \ge y \ge 2$, we let $\Psi(x,y)$ be the count of $y$-smooth numbers up to $x$. We let $\rho$ be Dickman's function, defined by
\[ \rho(u)=1\text{ for $0 \le u \le 1$}, \quad \text{and}\quad u \rho'(u) = -\rho(u-1) \quad\text{for $u > 1$}. \]
The functions $\Psi(x,y)$ and $\rho(u)$ are intimately connected; it is known that $\Psi(x,y) \sim x\rho(u)$, where $u:=\frac{\log{x}}{\log{y}}$, in a wide range of $x$ and $y$. In fact, Hildebrand \cite{hildebrand86} has shown that this asymptotic formula holds whenever $x\to\infty$, as long as
\[ y \ge \exp((\log\log{x})^{5/3+\lambda}) \]
for some fixed positive $\lambda$. For this estimate to be useful, one needs to understand the behavior of $\rho(u)$. It is not hard to show that $\rho$ is strictly decreasing for $u > 1$ and that $\rho(u) \le 1/\Gamma(u+1)$. So for any $k > 1$, there is a unique $u_k > 1$ with $\rho(u_k)=\frac{1}{k}$. Buchstab and, independently, Davenport and Erd\H{o}s (developing ideas implicit in \cite{vinogradov27}) showed that if $\chi$ mod $p$ has order $k \ge 2$, then the least $\chi$-nonresidue is $O_{\varepsilon,k}(p^{1/2u_k+\varepsilon})$. If in their argument Burgess's method (which was not available at the time) is used in place of the P\'olya--Vinogradov inequality, then $1/2u_k$ may be replaced by $1/4u_k$ \cite{wy64}. We prove the following:
\begin{thm}\label{thm:fixedprime} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$ of order $k \ge k_0$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $m^{\frac{1}{4u_{k_0}}+\varepsilon}$.\end{thm}
\begin{rmk}\mbox{ }
\begin{itemize}
\item It follows readily from the definition that $\rho(u) = 1-\log{u}$ for $1 \le u \le 2$, and so $u_2 = e^{1/2} = 1.6487\ldots$ and $u_3 = e^{2/3} = 1.9477\ldots$. For $k > 3$, it does not seem that $u_k$ has a simple closed form expression.
\item Theorem \ref{thm:main} is the special case $k_0=2$ of Theorem \ref{thm:fixedprime}.
\end{itemize}
\end{rmk}
One might compare Theorem \ref{thm:main} for the quadratic character modulo a prime $p$ with a result of Banks--Garaev--Heath-Brown--Shparlinski \cite{BGHBS08}. They show that for each fixed $\varepsilon > 0$, and each $N \ge p^{1/4\sqrt{e}+\varepsilon}$, the proportion of quadratic nonresidues modulo $p$ in $[1,N]$ is $\gg_{\varepsilon} 1$ for all primes $p > p_0(\varepsilon)$.
Our arguments use the ideas of Vinogradov and Davenport--Erd\H{o}s but take advantage of modern developments in sieve methods and the theory of smooth numbers. A variant of the Burgess bounds developed by Norton also plays an important role. We note that an application of the sieve that is similar in spirit to ours appears in work of Bourgain and Lindenstrauss \cite[Theorem 5.1]{BL03}.\footnote{A special case of their result: \emph{Given $\varepsilon >0$, there is an $\alpha>0$ such that $\sum_{\substack{p^{\alpha} \le \ell \le p^{1/4+\varepsilon} \\ \leg{\ell}{p}=-1}}\frac{1}{\ell} > \frac{1}{2}-\varepsilon$, for all $p > p_0(\varepsilon)$.}}
It is equally natural to ask for small prime character \emph{residues}, i.e., primes $\ell$ with $\chi(\ell)=1$. The most significant unconditional result in this direction is due to Linnik and A.\,I. Vinogradov \cite{VL66}. They showed that if $\chi$ is the quadratic character modulo a prime $p$, then the smallest prime $\ell$ with $\chi(\ell)=1$ satisfies $\ell \ll_{\varepsilon} p^{1/4+\varepsilon}$. More generally, Elliott \cite{elliott71} proved that when $\chi$ has order $k$, the least such $\ell$ is $O_{k,\varepsilon}(p^{\frac{k-1}{4}+\epsilon})$. As Elliott notes, this bound is only interesting for small values of $k$; otherwise, it is inferior to what follows from known forms of Linnik's theorem on primes in progressions. For extensions of the Linnik--Vinogradov method in a different direction, see \cite{pollack14B, pollack14}.
Our final result is a partial analogue of Theorem \ref{thm:main} for prime residues of quadratic characters. Regrettably, the number of primes produced falls short of a fixed power of $m$.
\begin{thm}\label{thm:smallresidue} Let $\varepsilon > 0$ and let $A >0$. There is an $m_0=m_0(\varepsilon,A)$ with the following property: If $m > m_0$, and $\chi$ is a quadratic character modulo $m$, then there are at least $(\log{m})^{A}$ primes $\ell \le m^{\frac{1}{4}+\varepsilon}$ with $\chi(\ell)=1$.
\end{thm}
Results of the sort proven here have direct consequences for prime splitting in cyclic extensions of $\mathbf{Q}$. For example, Theorem \ref{thm:main} (respectively Theorem \ref{thm:smallresidue}) implies that there are more than $|\Delta|^{\kappa}$ inert (respectively, more than $(\log|\Delta|)^{A}$ split) primes $p \le |\Delta|^{\frac{1}{4\sqrt{e}}+\varepsilon}$ (respectively, $p \le |\Delta|^{\frac{1}{4}+\varepsilon}$) in the quadratic field of discriminant $\Delta$, as soon as $|\Delta|$ is large enough in terms of $\varepsilon$ (and $A$).
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\section{Small prime nonresidues: Proofs of Theorems \ref{thm:main} and \ref{thm:fixedprime}}
\subsection{Preparation} As might be expected, the Burgess bounds play the key role in our analysis. The following version is due to Norton (see \cite[Theorem 1.6]{norton98}).
\begin{prop}\label{prop:norton} Let $\chi$ be a nontrivial character modulo $m$ of order dividing $k$. Let $r$ be a positive integer,
and let $\epsilon > 0$. For all $x > 0$,
\[ \sum_{n \le x} \chi(n) \ll_{\epsilon,r} R_k(m)^{1/r} x^{1-\frac{1}{r}} m^{\frac{r+1}{4r^2}+\epsilon}. \]
Here
\[ R_k(m) = \min\left\{M(m)^{3/4},Q(k)^{9/8}\right\}, \]
where
\[ M(m) = \prod_{p^e \parallel m,~e\ge 3} p^e \qquad\text{and}\quad Q(k) = \prod_{p^e \parallel k,~e\ge 2} p^e. \]
The factor of $R_k(m)^{1/r}$ can be omitted if $r \le 3$.
\end{prop}
Another crucial tool is a theorem of Tenenbaum concerning the distribution of smooth numbers satisfying a coprimality condition. For $x\ge y\ge 2$, let \[ \Psi_q(x,y) = \#\{n\le x: \gcd(n,q)=1, P^{+}(q) \le y\}. \]
\begin{prop}\label{prop:FT} For positive integers $q$ and real numbers $x, y$ satisfying
\[ P^{+}(q) \le y \le x \quad\text{and}\quad \omega(q) \le y^{1/\log(1+u)}, \]
we have
\[ \Psi_q(x,y) = \frac{\varphi(q)}{q} \Psi(x,y) \left(1+O\left(\frac{\log(1+u) \log(1+\omega(q))}{\log{y}}\right)\right). \]
As before, $u$ denotes the ratio $\log{x}/\log{y}$.
\end{prop}
\begin{proof} This is the main result of \cite{tenenbaum93} in the case $A=1$.\end{proof}
\begin{remark} If $q'$ is the largest divisor of $q$ supported on the primes not exceeding $y$, then $\Psi_{q}(x,y) = \Psi_{q'}(x,y)$. So the assumption in Proposition \ref{prop:FT} that $P^{+}(q) \le y$ does not entail any loss of generality.
\end{remark}
Theorem \ref{thm:fixedprime} will be deduced from two variant results claiming weaker upper bounds.
\begin{thm}\label{thm:fixedprime0} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$ of order $k \ge k_0$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $m^{\frac{1}{3u_{k_0}}+\varepsilon}$.\end{thm}
\begin{thm}\label{thm:fixedprime1} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$ of order $k \ge k_0$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $R_k(m) m^{\frac{1}{4u_{k_0}}+\varepsilon}$. Here $R_k(m)$ is as defined in Proposition \ref{prop:norton}.
\end{thm}
The proof of Theorem \ref{thm:fixedprime1} is given in detail in the next section. We include only a brief remark about the proof of Theorem \ref{thm:fixedprime0}, which is almost entirely analogous (but slightly simpler). We then present the derivation of Theorem \ref{thm:fixedprime} from Theorems \ref{thm:fixedprime0} and \ref{thm:fixedprime1}. We remind the reader that Theorem \ref{thm:main} is the special case $k_0=2$ of Theorem \ref{thm:fixedprime}.
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\subsection{Proof of Theorem \ref{thm:fixedprime1}}\label{sec:proofs} We let $\chi$ be a nontrivial character modulo $m$ of order $k \ge k_0$, where $k_0 \ge 2$ is fixed. With $\delta \in (0,\frac{1}{4})$, we set
\[ x= R_k(m) \cdot m^{\frac14 + \delta}, \quad y = x^{\frac{1}{u_{k_0}} + \delta}. \]
To prove Theorem \ref{thm:fixedprime1}, it suffices to show that for all large $m$ (depending only on $k_0$ and $\delta$), there are at least $x^{\kappa}$ prime $\chi$-nonresidues in $[1,y]$ for a certain constant $\kappa = \kappa(k_0,\delta) > 0$.
Let $q$ be the product of the prime $\chi$-nonresidues in $[1,y]$. Note that $\gcd(q,m)=1$, from the definition of a $\chi$-nonresidue.
Our strategy is to estimate
\begin{equation}\label{eq:different} \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \chi^2(n) + \dots + \chi^{k-1}(n)) \end{equation}
in two different ways.
We first derive a lower bound on \eqref{eq:different}, under the assumption that there are not so many prime $\chi$-nonresidues in $[1,y]$.
\begin{lem}\label{lem:lower} There are constants $\eta = \eta(\delta,k_0) > 0$, $\kappa =\kappa(\delta,k_0) > 0$, and $m_0 = m_0(\delta,k_0)$ with the following property: If $m > m_0$ and $\omega(q) \le x^{\kappa}$, then
\[ \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \dots +\chi(n)^{k-1}) \ge \left(1+\frac{2k}{3}\eta\right) \frac{\varphi(mq)}{mq} x. \]
\end{lem}
\begin{proof} Observe that
\[ \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \dots + \chi(n)^{k-1}) = k \sum_{\substack{n \le x \\ \gcd(n,q)=1,~\chi(n)=1}} 1 \ge k\sum_{\substack{n \le x\\ \gcd(n,mq)=1 \\ p \mid n \Rightarrow p \le y}} 1 = k \cdot \Psi_{mq}(x,y).\]
We estimate $\Psi_{mq}(x,y)$ using Proposition \ref{prop:FT} and the succeeding remark. We have $u \asymp_{k_0} 1$, or equivalently, $\log y\asymp_{k_0} \log{x}$. So if $\kappa$ is sufficiently small in terms of $k_0$, and $\omega(q) \le x^{\kappa}$, Proposition \ref{prop:FT} gives
\begin{align*} \Psi_{mq}(x,y) &= \bigg(\Psi(x,y)\prod_{\substack{p \mid mq \\ p \le y}}\left(1-\frac{1}{p}\right)\bigg) \left(1+O_{k_0}\left(\frac{\log(1+x^{\kappa})}{\log{x}}\right)\right)\\
&\ge \Psi(x,y) \frac{\varphi(mq)}{mq} \left(1+O_{k_0}\left(\frac{\log(1+x^{\kappa})}{\log{x}}\right)\right).\end{align*}
Now the result of Hildebrand quoted in the introduction (or a much more elementary theorem) shows that $\Psi(x,y) = \Psi(x, x^{\frac{1}{u_{k_0}}+\delta}) \ge (\frac{1}{k_0}+\eta) x$ for a certain $\eta= \eta(k_0,\delta) > 0$ and all large $x$. So if $\kappa$ is fixed sufficiently small, depending on $k_0$ and $\delta$, and $x$ is sufficiently large,
\[ \Psi_{mq}(x,y) > \left(\frac{1}{k_0} + \frac{2}{3}\eta\right) \frac{\varphi(mq)}{mq} x. \]
Hence,
\[ \sum_{\substack{n \le x \\ \gcd(n,q)=1}} (1+\chi(n) + \dots + \chi(n)^{k-1}) \ge \left(\frac{k}{k_0} + \frac{2k}{3}\eta\right)\frac{\varphi(mq)}{mq} x \ge \left(1+\frac{2k}{3}\eta\right) \frac{\varphi(mq)}{mq} x. \qedhere\]
\end{proof}
We turn next to an upper bound.
\begin{lem}\label{lem:upper} Let $\beta > 0$. There are numbers $\eta' = \eta'(\delta) > 0$, $\kappa' = \kappa'(\delta,\beta)>0$ and $m_0 = m_0(\delta,\beta)$ with the following property: If $m > m_0$ and $\omega(q) \le x^{\kappa'}$, then
\[ \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \chi(n)^2 + \dots + \chi(n)^{k-1}) \le (1+\beta) \frac{\varphi(mq)}{mq}x +O_{\delta}(k x^{1-\eta'}). \]
\end{lem}
\begin{proof} We let $\mathcal{A} = \{n \le x: \gcd(n,m)=1,~\chi(n)=1\}$ and observe that
\begin{equation}\label{eq:fundidentity} \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \chi(n)^2 + \dots + \chi(n)^{k-1}) = k \sum_{\substack{n \in \mathcal{A} \\ \gcd(n,q)=1}} 1. \end{equation}
We apply the fundamental lemma of the sieve to estimate the right-hand sum. (The precise form of the fundamental lemma is not so important, but we have in mind \cite[Theorem 4.1, p. 29]{diamond08}.) Let $d \in [1,x]$ be a squarefree integer dividing $q$. Then
\begin{align*} \sum_{\substack{n \in \mathcal{A} \\ d \mid n}} 1 &= \frac{1}{k} \sum_{\substack{n \le x \\\gcd(n,m)=1,~d\mid n}} (1+\chi(n) + \dots + \chi(n)^{k-1}).
\end{align*}
For each $j=0,1,2,\dots, k-1$,
\[ \sum_{\substack{n \le x \\ \gcd(n,m)=1,~d\mid n}}\chi^j(n) = \chi^j(d) \sum_{\substack{e \le x/d \\ \gcd(e,m)=1}} \chi^j(e). \]
When $j=0$, the right-hand side is $\frac{x}{d}\frac{\varphi(m)}{m} +O_{\epsilon}(m^{\epsilon})$, by a straightforward inclusion-exclusion. For $j \in \{1,2,\dots, k-1\}$, Proposition \ref{prop:norton} gives
\begin{align*} \sum_{\substack{e \le x/d \\ \gcd(e,m)=1}} \chi^j(e) = \sum_{e \le x/d} \chi^j(e) \sum_{\substack{f \mid e \\ f\mid m}} \mu(f) &= \sum_{f \mid m} \mu(f) \chi^j(f) \sum_{g \le x/df} \chi^j(g) \\
&\ll_{\epsilon,r} R_k(m)^{1/r} x^{1-\frac1r} d^{-1+\frac{1}{r}} m^{\frac{r+1}{4r^2}+\epsilon} \sum_{f \mid m} f^{-1+\frac{1}{r}} \\&\ll_{\epsilon} R_k(m)^{1/r} x^{1-\frac1r} d^{-1+\frac{1}{r}} m^{\frac{r+1}{4r^2}+2\epsilon};
\end{align*}
here $r \ge 2$ and $\epsilon > 0$ are parameters to be chosen. (We used in the last step that the sum on $f$ has only $O_{\epsilon}(m^{\epsilon})$ terms, each of which is $O(1)$.) Assembling the preceding estimates,
\[ \sum_{\substack{n \in \mathcal{A} \\ d \mid n}} 1 = \frac{x}{dk}\frac{\varphi(m)}{m} + r(d),
\quad\text{where}\quad r(d) \ll_{\epsilon,r} R_k(m)^{1/r} x^{1-\frac1r} d^{-1+\frac{1}{r}} m^{\frac{r+1}{4r^2}+2\epsilon}. \]
By the fundamental lemma, for any choices of real parameters $z\ge 2$ and $v\ge 1$ with $z^{2v} < x$,
\begin{multline*} \sum_{\substack{n \in \mathcal{A}\\\gcd(n,q)=1}} 1 \le \sum_{\substack{n \in \mathcal{A} \\ p \mid \gcd(n,q) \Rightarrow p \ge z}} 1 = \Bigg(\frac{x}{k}\frac{\varphi(m)}{m} \prod_{\substack{p \mid q \\ p< z}}\left(1-\frac{1}{p}\right)\Bigg)\left(1 + O(v^{-v})\right) \\ + O_{\epsilon,r}\Bigg(R_k(m)^{1/r} x^{1-\frac1r} m^{\frac{r+1}{4r^2}+2\epsilon} \sum_{\substack{d < z^{2v} \\ d\mid q}} \mu^2(d) 3^{\omega(d)} d^{-1+\frac1r}\Bigg).\end{multline*}
We now make a choice of parameters. Let $r = \lceil \frac{1}{2\delta}\rceil$ (so that $\delta \ge \frac{1}{2r}$). Since $x=R_k(m)\cdot m^{1/4+\delta}$, we have
\[ R_k(m)^{1/r} x^{1-\frac1r} m^{\frac{r+1}{4r^2}} = x \cdot m^{-\frac{1}{4r} -\delta/r} m^{\frac{r+1}{4r^2}} = x \cdot m^{\frac{1}{r}(\frac{1}{4r}-\delta)} \le x\cdot m^{-\frac{\delta}{4r^2}}. \]
We take $\epsilon = \frac{\delta}{16r^2}$, so that
\[ m^{2\epsilon} = m^{\frac{\delta}{8r^2}}. \]
Since $r\ge 2$ and $3^{\omega(d)} \ll d^{1/2}$, each term in the sum on $d$ is $O(1)$. Putting it all together, the $O$-term above is
\[ \ll_{\delta} x \cdot m^{-\frac{\delta}{4r^2}} \cdot m^{\frac{\delta}{8r^2}} \cdot z^{2v}. \]
Since $x = R_k(m) \cdot m^{1/4+\delta} \le m^{3/4} \cdot m^{1/4+\delta} < m^2$, this upper bound is $\ll_{\delta} x^{1-\frac{\delta}{16r^2}} z^{2v}$. Taking $z = x^{\frac{\delta}{64r^2 v}}$ gives a final upper bound on the $O$-term of
\[ \ll_{\delta} x^{1-\eta'},\quad\text{where}\quad \eta' = \frac{\delta}{32r^2}. \]
Turning attention to the main term, we fix $v$ large enough that the factor $1+O(v^{-v})$ is smaller than $1+\frac{1}{2}\beta$. Then our main term above does not exceed
\begin{align*} \frac{x}{k} \frac{\varphi(mq)}{mq} \left(1+\frac{1}{2}\beta\right) \prod_{\substack{p \mid q\\ p \ge z}} \left(1-\frac{1}{p}\right)^{-1} &\le \frac{x}{k} \frac{\varphi(mq)}{mq} \left(1+\frac{1}{2}\beta\right) \exp\bigg(2\sum_{\substack{p \mid q \\ p \ge z}}\frac{1}{p}\bigg) \\&\le \frac{x}{k} \frac{\varphi(mq)}{mq} \left(1+\frac{1}{2}\beta\right) \exp(2\omega(q) z^{-1}). \end{align*}
Take $\kappa' = \frac{\delta}{128r^2 v}$. Under the assumption that $\omega(q) \le x^{\kappa'}$, we have $2 \omega(q) z^{-1} \le 2x^{-\delta/128r^2 v}$, and $\exp(2\omega(q) z^{-1}) = 1 + O(x^{-\delta/128r^2 v})$. So once $x$ (or equivalently, $m$) is large enough, our main term is smaller than $\frac{x}{k}\frac{\varphi(mq)}{mq}(1+\beta)$. So we have shown that for large $m$,
\[ \sum_{\substack{n \in \mathcal{A} \\ \gcd(n,q)=1}} 1 \le \frac{x}{k}\frac{\varphi(mq)}{mq}(1+\beta) + O_{\delta}(x^{1-\eta'}). \]
Recalling \eqref{eq:fundidentity} finishes the proof.
\end{proof}
\begin{proof}[Completion of the proof of Theorem \ref{thm:fixedprime}] We keep the notation from earlier in this section. Let $\eta$, $\kappa$ be as specified in Lemma \ref{lem:lower}. With $\beta = \eta/2$, choose $\eta'$ and $\kappa'$ as in Lemma \ref{lem:upper}. If $m$ is large and we assume that
\[ \omega(q) \le x^{\kappa''}, \quad\text{where} \quad\kappa'' = \min\{\kappa,\kappa'\}, \]
then these lemmas imply that
\[ \bigg(1+\frac{2k}{3}\eta\bigg) \frac{\varphi(mq)}{mq} x \le \bigg(1+\frac{1}{2}\eta\bigg) \frac{\varphi(mq)}{mq} x + O_{\delta}(kx^{1-\eta'}). \]
Rearranging,
\[ k \eta \frac{\varphi(mq)}{mq} x \ll \frac{4k-3}{6} \eta \cdot \frac{\varphi(mq)}{mq}x \ll_{\delta} k x^{1-\eta'}, \]
and so
\[ \frac{mq}{\varphi(mq)} \gg_{k_0,\delta} x^{\eta'}. \]
Noting that $m < x^{4}$ and $q \le y^{\omega(q)}\le x^{\omega(q)}$, we see that for large $x$,
\[ \frac{mq}{\varphi(mq)} \ll \log\log(mq+2) \ll \log\log{x} + \log(\omega(q)+2) \ll \log{x}. \]
Comparing with the above lower bound, we see that $x$, and hence $m$, is bounded. Turning it around, for $m$ large enough, there are at least $x^{\kappa''}$ prime $\chi$-nonresidues in $[1,y]$.
\end{proof}
\begin{proof}[Sketch of the proof of Theorem \ref{thm:fixedprime0}] The proof of Theorem \ref{thm:fixedprime0} is quite similar, except that now we take $x = m^{1/3+\delta}$. With this choice of $x$, we can apply the Burgess bounds with $r=3$, which allows us to omit the factor of $R_k(m)$ in the resulting estimates.\end{proof}
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\subsection{Deduction of Theorem \ref{thm:fixedprime}} Let $\varepsilon> 0$ and $k_0 \ge 2$ be fixed. Let $\chi$ be a nonprincipal character mod $m$ of order $k$, where $k \ge k_0$. We would like to show that as long as $m$ is large enough there must be at least $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $x^{1/4u_{k_0}+\varepsilon}$, for a certain $\kappa = \kappa(\varepsilon,k_0) > 0$. Let $k_1$ be the smallest positive integer with $3u_{k_1} > 4u_{k_0}$. If $k \ge k_1$, apply Theorem \ref{thm:fixedprime0}: We find that for large $m$, there are at least $m^{\kappa_0}$ prime $\chi$-nonresidues
\[ \le m^{\frac{1}{3u_{k_1}} + \varepsilon} \le m^{\frac{1}{4u_{k_0}} + \varepsilon}, \]
where $\kappa_0 = \kappa(\varepsilon,k_1)$ in the notation of Theorem \ref{thm:fixedprime0}. Suppose instead that $k_0 \le k< k_1$. Then $R_k(m)$ is bounded in terms of $k_0$. Theorem \ref{thm:fixedprime1} thus shows that for large $m$, there are at least $m^{\kappa_1}$ prime $\chi$-nonresidues
\[ \le R_k(m) m^{\frac{1}{4u_{k_0}} + \varepsilon/2} \le m^{\frac{1}{4u_{k_0}}+\varepsilon}, \]
where $\kappa_1 = \kappa(\varepsilon/2,k_0)$ in the notation of Theorem \ref{thm:fixedprime1}. Theorem \ref{thm:fixedprime} follows with $\kappa = \min\{\kappa_0,\kappa_1\}$.
\begin{remark} By a minor modification of our proof, one can establish the following more general result. Theorem \ref{thm:fixedprime} corresponds to the case $H = \ker\chi$.
\begin{thm} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and every proper subgroup $H$ of $G=(\mathbf{Z}/m\mathbf{Z})^{\times}$ of index $k \ge k_0$, there are more than $m^{\kappa}$ primes $\ell$ not exceeding $m^{\frac{1}{4u_{k_0}}+\varepsilon}$ with $\ell \nmid m$ and $\ell \bmod{m}\notin H$.\end{thm}
\noindent This strengthens \cite[Theorem 1.20]{norton98}, where the bound $O_{k_0,\epsilon}(m^{\frac{1}{4u_{k_0}}+\varepsilon})$ is established for the first such prime $\ell$.
The main idea in the proof of the generalization is to replace $1+\chi(n)+\dots + \chi(n)^{k-1}$ with $\sum_{\chi\in \widehat{G/H}} \chi(n)$, where $\widehat{G/H}$ denotes the group of characters $\chi$ mod $m$ with $\ker \chi \supset H$. We leave the remaining details to the reader.
\end{remark}
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\section{Small prime residues of quadratic characters: Proof of Theorem \ref{thm:smallresidue}}
The next proposition is a variant of \cite[Theorem 2]{VL66}. Given a character $\chi$, we let $r_{\chi}(n) = \sum_{d \mid n} \chi(d)$. Since $\chi$ will be clear from context, we will suppress the subscript.
\begin{prop}\label{prop:LV} For each $\epsilon > 0$, there is a constant $\eta = \eta(\epsilon) >0$ for which the following holds: If $\chi$ is a quadratic character modulo $m$ and $x \ge m^{1/4+\epsilon}$, then
\[ \sum_{n \le x} r(n) = L(1,\chi) x + O_{\epsilon}(x^{1-\eta}). \] \end{prop}
\begin{proof} With $\upsilon = \frac{1/4+\epsilon/2}{1/4+\epsilon}$, put $y = x^{\upsilon}$, so that $y \ge m^{\frac{1}{4}+\frac{1}{2}\epsilon}$. Put $z=x/y$. By Dirichlet's hyperbola method,
\begin{equation}\label{eq:hyperbola} \sum_{n \le x} r(n) = \sum_{d \le y}\chi(d) \sum_{e \le x/d} 1 + \sum_{e \le z} \sum_{d \le x/e} \chi(d) - \sum_{d\le y}\chi(d)\sum_{e\le z}1. \end{equation}
By Proposition \ref{prop:norton} (with $k=2$, so that $R_k(m)^{1/r}=1$), there is an $\eta_0 =\eta_0(\epsilon) > 0$ with $\sum_{d \le T} \chi(d) \ll_{\epsilon} T^{1-\eta_0} \quad\text{for all}\quad T \ge y$. Thus, the second double sum on the right of \eqref{eq:hyperbola} is $\ll_{\delta} x^{1-\eta_0} \sum_{e \le z} e^{\eta_0-1} \ll_{\delta} x (z/x)^{\eta_0} = x y^{-\eta_0}$. Similarly, the third double sum is $\ll_{\epsilon} z y^{1-\eta_0} = x y^{-\eta_0}$. Finally,
\[ \sum_{d \le y}\chi(d) \sum_{e \le x/d} 1=\sum_{d\le y} \chi(d) \left(\frac{x}{d}+O(1)\right) = xL(1,\chi) - x\sum_{d > y} \frac{\chi(d)}{d} + O(y) = x L(1,\chi) + O_{\epsilon}(xy^{-\eta_0}) + O(y). \]
(Here the sum on $d>y$ has been handled by partial summation.)
Collecting our estimates and keeping in mind that $y=x^\upsilon$, we obtain the theorem with $\eta$ defined by $1-\eta = \max\{\upsilon,1-v\eta_0\}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:smallresidue}] Let $\varepsilon \in (0, \frac14)$ and let $\chi$ be a quadratic character modulo $m$. Let
\[ x = m^{\frac{1}{4}+\varepsilon}, \]
and let $q$ be the product of the primes $\ell \le x$ with $\chi(\ell)=1$. We suppose that $\omega(q) \le (\log{m})^{A}$, and we show this implies that $m$ is bounded by a constant depending on $\varepsilon$ and $A$. Throughout this proof, we suppress any dependence on $\varepsilon$ and $A$ in our $O$-notation.
By Proposition \ref{prop:LV},
\begin{equation} \sum_{n \le x} r(n) = L(1,\chi) \cdot x + O(x^{1-\eta}). \label{eq:ldub}\end{equation}
We can estimate the sum in a second way. Observe that
\begin{equation}\label{eq:rnexpression} r(n) = \prod_{\ell^e \parallel n} \left(1+\chi(\ell) + \dots + \chi(\ell^e)\right) \ge 0. \end{equation}
Hence, if the subset $\mathcal{S}$ of $[1,x]$ is chosen to contain the support of $r(n)$ on $[1,x]$, then
\[ 0 \le \sum_{n \le x} r(n) \le \#\mathcal{S} \cdot \left(\max_{n \in \mathcal{S}} r(n)\right). \]
Examining the expression in \eqref{eq:rnexpression} for $r(n)$, we see $\mathcal{S}$ can be chosen as the set of $n\le x$ where every prime that appears to the first power in the factorization of $n$ divides $mq$. For each $n \in \mathcal{S}$, we can write $n=n_1 n_2$, where $n_1$ is a squarefree divisor of $mq$ and $n_2$ is squarefull. The number of elements of $\mathcal{S}$ with $n_2 > x^{1/2}$ is $O(x^{3/4})$. For the remaining elements of $\mathcal{S}$, we have $n_1 \le x/n_2$ and $n_1$ is a squarefree product of primes dividing $mq$. There is a bijection
\[ \iota\colon \{\text{squarefree divisors of $mq$}\} \to \{\text{squarefrees composed of the first $\omega(mq)$ primes}\}
\]
with $\iota(r) \le r$ for all $r$. Hence, given $n_2$, the number of choices for $n_1$ is at most the number of integers in $[1,x/n_2]$ supported on the product of the first $\omega(mq)$ primes. By our assumption on $\omega(q)$, those primes all belong to the interval $[1, (\log{x})^{A+1}]$, once $x$ is large. Hence, given $n_2$, the number of possible values of $n_1$ is at most
\[ \Psi(x/n_2, (\log{x})^{A+1}). \]
For fixed $\theta \ge 1$, a classical theorem of de Bruijn \cite{dB66} asserts that $\Psi(X,(\log{X})^{\theta}) = X^{1-\frac{1}{\theta}+o(1)}$, as $X\to\infty$. Since $x/n_2 \ge x^{1/2}$, we deduce that
\[ \Psi(x/n_2, (\log{x})^{A+1}) \le (x/n_2)^{1-\frac{1}{A+2}} \]
if $x$ is large. Summing on squarefull $n_2 \le x^{1/4}$, we see that the number of elements of $\mathcal{S}$ arising in this way is $O(x^{1-\frac{1}{A+2}})$. Hence,
\[ \#\mathcal{S} \ll x^{3/4} + x^{1-\frac{1}{A+2}} \ll x^{1-\eta'}, \quad\text{where}\quad \eta'=\min\left\{\frac14,\frac{1}{A+2}\right\}. \]
Since $r(n) \le \tau(n) \ll x^{\eta'/2}$ for $n \le x$,
\begin{equation}\label{eq:udub} \sum_{n\le x} r(n) \ll \#\mathcal{S}\cdot x^{\eta'/2} \ll x^{1-\eta'/2}. \end{equation}
Comparing \eqref{eq:ldub} and \eqref{eq:udub} gives
\[ L(1,\chi) \ll x^{-\min\{\eta'/2,\eta\}}. \]
But for large $x$, this contradicts Siegel's theorem \cite[Theorem 11.14, p. 372]{MV07}.
\end{proof}
\begin{remark} Any improvement on Siegel's lower bound for $L(1,\chi)$ would boost the number of $\ell$ produced in Theorem \ref{thm:smallresidue}. Substantial improvements of this kind would have other closely related implications. For example, a simple modification of an argument of Wolke \cite{wolke69} shows that for any quadratic character $\chi$ mod $m$,
\[ \sum_{\substack{\ell \le m \\ \chi(\ell)= 1}}\frac{1}{\ell} \ge \frac{1}{2} \log\left(\frac{\varphi(m)}{m} L(1,\chi) \log{m}\right) + O(1), \]
where the $O(1)$ constant is absolute. {(Here is the short proof: By Proposition \ref{prop:LV}, $\frac{1}{m}\sum_{n \le m}r(n) \gg L(1,\chi)$. On the other hand, \cite[Theorem 5, p. 308]{tenenbaum95} yields $\frac{1}{m}\sum_{n \le m} r(n) \ll \frac{1}{\log{m}} \sum_{n \le m} \frac{r(n)}{n} \ll \frac{1}{\log{m}} \cdot \frac{m}{\varphi(m)} \cdot \exp\left(2 \sum_{\ell \le m,~\chi(\ell)=1}\frac{1}{\ell}\right)$.)}
\end{remark}
\section*{Acknowledgments} This work was motivated in part by observations made on \texttt{mathoverflow} by ``GH from MO'' \cite{52393}. The author is also grateful to ``Lucia'' for pointing out there the work of Bourgain--Lindenstrauss. He thanks Enrique Trevi\~no for useful feedback on an early draft. This research was supported by NSF award DMS-1402268.
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\begin{document}
\title{Measuring the quantum state of a single system with minimum state disturbance}
\author{Maximilian Schlosshauer}
\affiliation{Department of Physics, University of Portland, 5000 North Willamette Boulevard, Portland, Oregon 97203, USA}
\begin{abstract}
Conventionally, unknown quantum states are characterized using quantum-state tomography based on strong or weak measurements carried out on an ensemble of identically prepared systems. By contrast, the use of protective measurements offers the possibility of determining quantum states from a series of weak, long measurements performed on a single system. Because the fidelity of a protectively measured quantum state is determined by the amount of state disturbance incurred during each protective measurement, it is crucial that the initial quantum state of the system is disturbed as little as possible. Here we show how to systematically minimize the state disturbance in the course of a protective measurement, thus enabling the maximization of the fidelity of the quantum-state measurement. Our approach is based on a careful tuning of the time dependence of the measurement interaction and is shown to be dramatically more effective in reducing the state disturbance than the previously considered strategy of weakening the measurement strength and increasing the measurement time. We describe a method for designing the measurement interaction such that the state disturbance exhibits polynomial decay to arbitrary order in the inverse measurement time $1/T$. We also show how one can achieve even faster, subexponential decay, and we find that it represents the smallest possible state disturbance in a protective measurement. In this way, our results show how to optimally measure the state of a single quantum system using protective measurements. \\[-.1cm]
\noindent Journal reference: \emph{Phys.\ Rev.\ A\ }\textbf{93}, 012115 (2016), DOI: \href{http://dx.doi.org/10.1103/PhysRevA.93.012115}{10.1103/PhysRevA.93.012115}
\end{abstract}
\pacs{03.65.Ta, 03.65.Wj}
\maketitle
\section{Introduction}
The characterization of unknown quantum states is an important experimental task and of great significance to quantum information processing \cite{Vogel:1989:uu,Dunn:1995:oo,Smithey:1993:lm,Breitenbach:az:1997,White:1999:az,James:2001:uu,Haffner:2005:sc,Leibfried:2005:yy,Altepeter:2005:ll,Lvovsky:2009:zz}. In conventional quantum-state tomography \cite{Vogel:1989:uu,Paris:2004:uu,Altepeter:2005:ll}, the quantum state is reconstructed from expectation values obtained from strong measurements of different observables, performed on an ensemble of identically prepared systems. An alternative approach to quantum-state measurement \cite{Lundeen:2011:ii,Lundeen:2012:rr,Fischbach:2012:za,Bamber:2014:ee,Dressel:2011:au} uses a combination of weak and strong measurements on an ensemble of identically prepared systems, together with the concept of weak values \cite{Aharonov:1988:mz,Duck:1989:uu}. However, since both approaches require an ensemble of identically prepared systems, they can only be said to reconstruct the quantum state in the statistical sense of measurement averages over an ensemble of systems presumed to have been prepared in the same quantum state. This raises the question of whether it might be possible to determine the quantum state of an individual system from measurements carried out not on an ensemble but on this single system only. Such single-system state determination would not only offer a conceptually transparent and rigorous version of quantum-state measurement, but also avoid time-consuming postprocessing and error propagation associated with quantum-state tomography \cite{Altepeter:2005:ll,Maccone:2014:uu,Dressel:2011:au}.
As long as one demands perfect fidelity of the state reconstruction and possesses no prior knowledge of the initial quantum-state subspace, then it is well known that single-system state determination is impossible \cite{Wootters:1982:ww,Ariano:1996:om}. However, if one weakens these conditions, then it has been shown that one can, in principle, measure the quantum state of a single system by using the protective-measurement protocol \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp,Alter:1996:oo,Dass:1999:az,Vaidman:2009:po,Gao:2014:cu}. Protective measurement allows for a set of expectation values to be obtained from weak measurements performed on the same single system, provided the system is initially in an (potentially unknown) nondegenerate eigenstate of its (potentially unknown) Hamiltonian. A defining feature of a protective measurement is that the disturbance of the system's quantum state during the measurement can be made arbitrarily small by weakening the measurement interaction and increasing the measurement time \cite{Aharonov:1993:jm,Dass:1999:az,Vaidman:2009:po}. Thus, a series of expectation values can be measured on the same system while the system remains in its initial state with probability arbitrarily close to unity. In this sense, one can measure the quantum state of a single system with a fidelity arbitrarily close to unity \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp,Dass:1999:az,Vaidman:2009:po,Auletta:2014:yy,Diosi:2014:yy,Aharonov:2014:yy}, providing an important complementary approach to conventional quantum-state tomography based on ensembles. Recently, the possibility of using protective measurement for quantum-state determination has attracted renewed interest \cite{Gao:2014:cu}, and protective measurement has been shown to have many related applications, such as the determination of stationary states \cite{Diosi:2014:yy}, investigation of particle trajectories \cite{Aharonov:1996:ii,Aharonov:1999:uu}, translation of ergodicity into the quantum realm \cite{Aharonov:2014:yy}, studies of fundamental issues of quantum measurement \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp,Alter:1997:oo,Dass:1999:az,Gao:2014:cu}, and the complete description of two-state thermal ensembles \cite{Aharonov:2014:yy}.
The fact that each protective measurement has a nonzero probability of disturbing the quantum state of the measured system leads to error propagation and reduced fidelity over the course of the multiple measurements required to determine the set of expectation values \cite{Aharonov:1993:qa,Alter:1996:oo,Dass:1999:az,Schlosshauer:2014:tp,Schlosshauer:2014:pm,Schlosshauer:2015:pm}. Therefore, a chief goal when using protective measurement to characterize quantum states of single systems is the minimization of the state disturbance. However, the conventional approach of making the measurement interaction arbitrarily weak while allowing it to last for an arbitrarily long time \cite{Aharonov:1993:jm,Dass:1999:az,Vaidman:2009:po} is not only unlikely to be practical in an experimental setting but is also, as we will show in this paper, comparably ineffective. Here we will describe a dramatically more effective approach that allows one to minimize the state disturbance while keeping the strength and duration of the measurement interaction constant. In this way, we demonstrate how to optimally implement the measurement of an unknown quantum state of a single system using protective measurement.
Our approach consists of a systematic tuning of the time dependence of the measurement interaction, such that the state disturbance becomes dramatically reduced even for modestly weak and relatively short interactions. While early expositions of protective measurement \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp} had hinted at the role of the time dependence of the measurement interaction, this role had not been explicitly explored and was instead relegated to a reference to the quantum adiabatic theorem \cite{Born:1928:yf}, which, as we will see in this paper, provides a condition that is neither necessary nor sufficient for minimizing the state disturbance in a protective measurement. Issues of time dependence of the protective-measurement interaction were first considered explicitly in Ref.~\cite{Dass:1999:az}, which estimated the effect of the turn-on and turnoff of the measurement interaction on the adiabaticity of the interaction. Recently, the case of finite measurement times in a protective measurement and its influence on the reliability of the measurement were studied \cite{Auletta:2014:yy,Schlosshauer:2014:tp}, and a framework for the perturbative treatment of time-dependent measurement interactions in a protective measurement has been developed and applied to specific examples \cite{Schlosshauer:2014:pm,Schlosshauer:2015:pm}. None of these existing studies, however, have shown how to systematically minimize the state disturbance in a protective measurement for the physically and experimentally relevant case of finite measurement times and interaction strengths, such that the reliability of the protective measurement can be maximized.
Here we present a rigorous and comprehensive solution to this problem. Our results demonstrate how one can optimally measure the quantum state of an individual quantum system using protective measurements. In any future experimental implementation of protective quantum-state measurement, this will enable one to optimize the measurement interaction to produce a high fidelity of the quantum-state measurement. While our analysis is motivated by the goal of optimizing protective measurements, it also provides insights into the issue of state disturbance in any quantum measurement.
This paper is organized as follows. After a brief review of the basics of protective measurements involving time-dependent measurement interactions (Sec.~\ref{sec:prot-meas}), we first use a Fourier-like series approach to construct measurement interactions that achieve a state disturbance that decreases as $1/T^N$, where $T$ is the measurement time and $N$ can be made arbitrarily large by modifying the functional form of the time dependence of the measurement interaction using a systematic procedure (Sec.~\ref{sec:seri-appr-minim}). We also make precise the relationship between the smoothness of the measurement interaction and the dependence of the state disturbance on $T$. We then show that the measurement interaction can be further optimized, leading to an even faster, subexponential decay of the state disturbance with $T$, and we show that this constitutes the optimal choice (Sec.~\ref{sec:minim-state-dist}). These results are established by calculating the state disturbance from the perturbative transition amplitude to first order in the interaction strength. To justify this approach, we prove that this amplitude accurately represents the exact transition amplitude to leading order in $1/T$ (Sec.~\ref{sec:suff-first-order}).
\section{\label{sec:prot-meas}Protective measurement}
We begin by briefly reviewing protective measurements and their treatment with time-dependent perturbation theory. In a protective measurement \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp,Dass:1999:az,Vaidman:2009:po,Gao:2014:cu}, the interaction between system $S$ and apparatus $A$ is treated quantum mechanically and described by the interaction Hamiltonian $\op{H}_\text{int}(t) = g(t)\op{O} \otimes \op{P}$, where $\op{O}$ is an arbitrary observable of $S$, $\op{P}$ generates the shift of the pointer of $A$, and the coupling function $g(t)$ describes the time dependence of the interaction strength during the measurement interval $0 \le t \le T$, with $g(t)=0$ for $t <0$ and $t >T$. The function $g(t)$ is normalized, $\int_{0}^{T} \text{d} t\, g(t) =1$, which introduces an inverse relationship between the duration $T$ and the average strength of the interaction, so that the pointer shift depends neither on these two parameters nor on the functional form of $g(t)$. The spectrum $\{ E_n \}$ of $\op{H}_S$ is assumed to be nondegenerate and $S$ is assumed to be in an eigenstate $\ket{n}$ of $\op{H}_S$ at $t=0$. One can then show \cite{Aharonov:1993:qa,Aharonov:1993:jm,Dass:1999:az,Vaidman:2009:po,Gao:2014:cu} that for $T \rightarrow \infty$ the system remains in the state $\ket{n}$, while the apparatus pointer shifts by an amount proportional to $\bra{n}\op{O}\ket{n}$, thus providing partial information about $\ket{n}$. However, in the realistic case of finite $T$ and a corresponding non-infinitesimal average interaction strength, the system becomes entangled with the apparatus, disturbing the initial state \cite{Auletta:2014:yy,Schlosshauer:2014:tp,Schlosshauer:2014:pm,Schlosshauer:2015:pm}.
To quantify this state disturbance, we calculate the probability amplitude $A_m(T)$ for finding the system in an orthogonal state $\ket{m}\not=\ket{n}$ at the conclusion of the measurement. We write $A_m(T)$ as a perturbative series, $A_m(T) = \sum_{\ell=1}^\infty A_m^{(\ell)}(T)$. Here $A_m^{(1)}(T)$ is the transition amplitude to first order in the interaction strength and the amplitudes $A_m^{(\ell)}(T)$ for $\ell \ge 2$ are the $\ell$th-order corrections to $A_m^{(1)}(T)$, where \cite{Sakurai:1994:om,Schlosshauer:2014:pm}
\begin{align}\label{eq:g8fbvsv1}
A^{(\ell)}_{m}(T) &= \left(-\frac{\text{i}}{\hbar}\right)^\ell \sum_{k_1,\hdots,k_{\ell-1}} O_{mk_1}O_{k_1k_2}\cdots O_{k_{\ell-1}n}\notag \\ & \quad \times\int_{0}^{T} \text{d} t' \, \text{e}^{\text{i} \omega_{mk_1} t'} g(t') \cdots \notag \\ & \quad \times\int_{0}^{t^{(\ell-1)}} \text{d} t^{(\ell)} \,\text{e}^{\text{i} \omega_{k_{\ell-1} n} t^{(\ell)}} g(t^{(\ell)}).
\end{align}
Here $O_{ij}\equiv \bra{k_i} \op{O} \ket{k_j}$, and $\omega_{mn}\equiv (E_m-E_n)/\hbar$ is the frequency of the transition $\ket{n} \rightarrow \ket{m}$ \footnote{The full expression for $A^{(\ell)}_{m}(T)$ also contains contributions from the apparatus subspace \cite{Schlosshauer:2014:pm}. They are irrelevant to our present analysis.}. Of particular interest is the first-order transition amplitude $A_m^{(1)}(T)$,
\begin{align}\label{eq:8aadhj7gr7ss82}
A_m^{(1)}(T) &= -\frac{\text{i}}{\hbar} O_{mn} \int_{0}^{T} \text{d} t\, \text{e}^{\text{i} \omega_{mn} t} g(t).
\end{align}
The total state disturbance is measured by the probability $\abs{\sum_{m\not=n} A_m(T)}^2$ of a transition to the subspace orthogonal to the initial state $\ket{n}$. Our goal is now to determine coupling functions $g(t)$ that minimize this transition probability.
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\section{\label{sec:prot-meas}Protective measurement}
We begin by briefly reviewing protective measurements and their treatment with time-dependent perturbation theory. In a protective measurement \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp,Dass:1999:az,Vaidman:2009:po,Gao:2014:cu}, the interaction between system $S$ and apparatus $A$ is treated quantum mechanically and described by the interaction Hamiltonian $\op{H}_\text{int}(t) = g(t)\op{O} \otimes \op{P}$, where $\op{O}$ is an arbitrary observable of $S$, $\op{P}$ generates the shift of the pointer of $A$, and the coupling function $g(t)$ describes the time dependence of the interaction strength during the measurement interval $0 \le t \le T$, with $g(t)=0$ for $t <0$ and $t >T$. The function $g(t)$ is normalized, $\int_{0}^{T} \text{d} t\, g(t) =1$, which introduces an inverse relationship between the duration $T$ and the average strength of the interaction, so that the pointer shift depends neither on these two parameters nor on the functional form of $g(t)$. The spectrum $\{ E_n \}$ of $\op{H}_S$ is assumed to be nondegenerate and $S$ is assumed to be in an eigenstate $\ket{n}$ of $\op{H}_S$ at $t=0$. One can then show \cite{Aharonov:1993:qa,Aharonov:1993:jm,Dass:1999:az,Vaidman:2009:po,Gao:2014:cu} that for $T \rightarrow \infty$ the system remains in the state $\ket{n}$, while the apparatus pointer shifts by an amount proportional to $\bra{n}\op{O}\ket{n}$, thus providing partial information about $\ket{n}$. However, in the realistic case of finite $T$ and a corresponding non-infinitesimal average interaction strength, the system becomes entangled with the apparatus, disturbing the initial state \cite{Auletta:2014:yy,Schlosshauer:2014:tp,Schlosshauer:2014:pm,Schlosshauer:2015:pm}.
To quantify this state disturbance, we calculate the probability amplitude $A_m(T)$ for finding the system in an orthogonal state $\ket{m}\not=\ket{n}$ at the conclusion of the measurement. We write $A_m(T)$ as a perturbative series, $A_m(T) = \sum_{\ell=1}^\infty A_m^{(\ell)}(T)$. Here $A_m^{(1)}(T)$ is the transition amplitude to first order in the interaction strength and the amplitudes $A_m^{(\ell)}(T)$ for $\ell \ge 2$ are the $\ell$th-order corrections to $A_m^{(1)}(T)$, where \cite{Sakurai:1994:om,Schlosshauer:2014:pm}
\begin{align}\label{eq:g8fbvsv1}
A^{(\ell)}_{m}(T) &= \left(-\frac{\text{i}}{\hbar}\right)^\ell \sum_{k_1,\hdots,k_{\ell-1}} O_{mk_1}O_{k_1k_2}\cdots O_{k_{\ell-1}n}\notag \\ & \quad \times\int_{0}^{T} \text{d} t' \, \text{e}^{\text{i} \omega_{mk_1} t'} g(t') \cdots \notag \\ & \quad \times\int_{0}^{t^{(\ell-1)}} \text{d} t^{(\ell)} \,\text{e}^{\text{i} \omega_{k_{\ell-1} n} t^{(\ell)}} g(t^{(\ell)}).
\end{align}
Here $O_{ij}\equiv \bra{k_i} \op{O} \ket{k_j}$, and $\omega_{mn}\equiv (E_m-E_n)/\hbar$ is the frequency of the transition $\ket{n} \rightarrow \ket{m}$ \footnote{The full expression for $A^{(\ell)}_{m}(T)$ also contains contributions from the apparatus subspace \cite{Schlosshauer:2014:pm}. They are irrelevant to our present analysis.}. Of particular interest is the first-order transition amplitude $A_m^{(1)}(T)$,
\begin{align}\label{eq:8aadhj7gr7ss82}
A_m^{(1)}(T) &= -\frac{\text{i}}{\hbar} O_{mn} \int_{0}^{T} \text{d} t\, \text{e}^{\text{i} \omega_{mn} t} g(t).
\end{align}
The total state disturbance is measured by the probability $\abs{\sum_{m\not=n} A_m(T)}^2$ of a transition to the subspace orthogonal to the initial state $\ket{n}$. Our goal is now to determine coupling functions $g(t)$ that minimize this transition probability.
\section{\label{sec:seri-appr-minim}Series approach to minimization of state disturbance}
Our first approach will consist of building up coupling functions $g(t)$ from sinusoidal components such that the coupling functions become increasingly smooth (in a sense to be defined below). We take $g(t)$ to be symmetric about $t=T/2$ and expand it in terms of the functions
\begin{equation}\label{eq:bvdhkjb678678}
f_n(t)= (-1)^{n+1}\cos\left[\frac{2n\pi (t-T/2)}{T}\right], \quad n=1,2,3,\hdots,
\end{equation}
which form an orthogonal basis over the interval $[0,T]$ for functions symmetric about $t=T/2$. That is, we write $g(t)$ as
\begin{equation}\label{eq:bvdhkjbvd}
g(t) = \begin{cases} \frac{1}{T} \left( 1 + \sum_{n=1}^N a_n f_n(t)\right), & 0 \le t \le T, \\ 0, & \text{otherwise}, \end{cases}
\end{equation}
where the coefficients $a_n$ are dimensionless and do not depend on $T$. Since $\int_0^T \text{d} t \, f_n(t)=0$, the area under $g(t)$ is normalized as required. The dominant contribution comes from the $f_1(t)$ term describing a gradual increase and decrease. The terms $f_n(t)$ for $n \ge 2$ represent sinusoidal components with multiple peaks that we will now use to suitably shape the basic pulse represented by $f_1(t)$.
We will first consider the first-order transition amplitude $A_m^{(1)}(T)$ given by Eq.~\eqref{eq:8aadhj7gr7ss82}, and then subsequently justify this approach by showing that higher-order corrections $A_m^{(\ell\ge 2)}(T)$ do not modify the results. Equation~\eqref{eq:8aadhj7gr7ss82} shows that the coupling-dependent part of $A_m^{(1)}(T)$ is represented by the Fourier transform $G(\omega T) = \int_0^T \text{d} t\, \text{e}^{\text{i} \omega t} g(t)$ of $g(t)$, where $\omega\equiv \omega_{mn}$. Thus, to quantify the state disturbance we evaluate the Fourier transform of $g(t)$ given by Eq.~\eqref{eq:bvdhkjbvd},
\begin{align}\label{eq:bvdhkjbvd0}
G(\omega T) = \frac{2\text{e}^{\text{i} \omega T/2}}{\omega T} \sin\left( \omega T/2 \right) \left[ 1 - \sum_{n=1}^N \frac{a_n}{1-(2\pi n/\omega T)^2 }\right],
\end{align}
where $\omega T$ is a dimensionless quantity that measures the ratio of the measurement time to the internal timescale $\omega^{-1}$ associated with the transition $\ket{n}\rightarrow\ket{m}$. In physical situations, $\omega^{-1}$ typically represents atomic timescales and we may safely assume that $\omega T \gg N$. Then we can write Eq.~\eqref{eq:bvdhkjbvd0} as a power series in $1/\omega T$,
\begin{equation}\label{eq:uuuun}
G(\omega T) = \frac{2\text{e}^{\text{i} \omega T/2}}{\omega T} \sin\left( \omega T/2 \right)\left[ 1 - \sum_{k=0}^\infty \sum_{n=1}^N a_n \left(\frac{2\pi n}{\omega T}\right)^{2k} \right].
\end{equation}
To minimize the state disturbance, we want $G(\omega T)$ to decay quickly with $T$ from its initial value of 1 at $T=0$. For the constant-coupling function $g(t)=1/T$ (all $a_n=0$), which describes a sudden turn-on and turnoff, we obtain $A_m^{(1)}(T) \propto 1/\omega T$, where the $T$ dependence is due to the fact that the average interaction strength is proportional to $1/T$. Clearly, we must have $\omega T \gg 1$ to achieve small state disturbance. For arbitrary coefficients $a_n$, $A_m^{(1)}(T)$ is still of first order in $1/\omega T$. Equation~\eqref{eq:uuuun} shows that we may increase the order of the leading term in $1/\omega T$ by imposing the conditions
\begin{align}\label{eq:conds}
\sum_{n=1}^N a_n=1, \quad \sum_{n=1}^N a_n n^{2k}=0, \quad 1 \le k \le N-1,
\end{align}
which define a set of $N$ linearly independent coupled equations for $N$ coefficients $a_n$ with a unique solution $\bvec{a}_N=(a_1,\hdots,a_N)$; e.g., $\bvec{a}_1=\left(1 \right)$, $\bvec{a}_2=\left(\frac{4}{3},-\frac{1}{3}\right)$, $\bvec{a}_3=\left(\frac{3}{2},-\frac{3}{5},\frac{1}{10}\right)$, etc. Using the solution $\bvec{a}_N$, $A_m^{(1)}(T)$ to leading order in $1/\omega T$ becomes [see Eqs.~\eqref{eq:8aadhj7gr7ss82} and \eqref{eq:uuuun}]
\begin{align}\label{eq:uuuun2233}
\widetilde{A}_m^{(1)}(T)&= -\frac{2\text{i}}{\hbar} O_{mn} \text{e}^{\text{i} \omega T/2} \sin\left( \omega T/2 \right)\left(2\pi\right)^{2N} \notag \\ & \quad \times \left(\sum_{n=1}^N a_n n^{2N}\right) \left(\frac{1}{\omega T}\right)^{2N+1},
\end{align}
where the tilde indicates leading-order expressions. This amplitude is of order $(\omega T)^{-(2N+1)}$.
\begin{figure}
\caption{\label{fig:g}
\label{fig:g}
\end{figure}
\begin{figure}
\caption{\label{fig:decay}
\label{fig:decay}
\end{figure}
Figure~\ref{fig:g} displays the coupling functions determined from the conditions~\eqref{eq:conds} for different values of $N$. Functions with larger $N$ describe a smoother turn-on and turnoff behavior. Figure~\ref{fig:decay}(a) shows the corresponding squared Fourier transforms $\abs{G(\omega T)}^2$ of these coupling functions in the regime $\omega T \gg N$ relevant to protective measurement, with $\abs{G(\omega T)}^2$ representing the dependence of the state disturbance on the choice of $g(t)$. We have neglected the rapid oscillations of $\abs{G(\omega T)}^2$, since they are irrelevant to considerations of state disturbance in protective measurements \footnote{Targeting the zeros of $\abs{G(\omega T)}^2$ to minimize the state disturbance by tuning $T$ would require precise knowledge of $\omega$ and therefore of $\op{H}_S$. But in a protective measurement, $\op{H}_S$ is \emph{a priori} unknown \cite{Aharonov:1993:jm,Dass:1999:az}.}. Small values of $N$ already achieve a strong reduction of the state disturbance. Figures~\ref{fig:g} and \ref{fig:decay}(a) show that while increasing $N$ entails a higher rate of change of the measurement strength outside the turn-on and turnoff region and a larger peak strength at $t=T/2$, it nevertheless reduces the state disturbance. This indicates that the smoothness of the turn-on and turnoff of the interaction has a decisive influence on the state disturbance.
Increasing $N$ also makes $g(t)$ narrower (see Fig.~\ref{fig:g}), making its Fourier transform wider and the initial decay of the transition amplitude slower, as seen in Fig.~\ref{fig:decay}(b). However, Fig.~\ref{fig:decay}(a) shows that this increase in width is insignificant in the relevant regime $\omega T \gg N$. Fundamentally, if $N \rightarrow \infty$, $g(t)$ becomes infinitely narrow and the transition amplitude becomes infinitely wide. Thus, one cannot eliminate the state disturbance altogether even in the limit of infinitely many $f_n(t)$.
We now make precise the connection between smoothness and state disturbance. Mathematically, smoothness is measured by how many times a function is continuously differentiable over a given domain; we call a function that is $k$ times continuously differentiable a $C^k$-smooth function. The $j$th-order derivative of $g(t)=\frac{1}{T} \left( 1 + \sum_{n=1}^N a_n f_n(t)\right)$ [Eq.~\eqref{eq:bvdhkjbvd}] at $t=0$ and $t=T$ is proportional to $\sum_{n=1}^N a_n (2\pi n)^j$ for even $j$ and zero for odd $j$. Since all derivatives of $g(t)$ vanish for $t<0$ and $t>T$, the turn-on and turnoff points introduce a discontinuity in the derivatives. We can make all derivatives up to order $2N-1$ vanish (and thus continuous) at $t=0$ and $t=T$ by requiring that $\sum_{n=1}^N a_n (2\pi n)^{2k}=0$ for $k=1,2,\hdots,N-1$, in addition to the requirement $\sum_{n=1}^N a_n=1$ ensuring continuity of $g(t)$ itself. These, however, are precisely the conditions~\eqref{eq:conds} previously derived from the requirement of eliminating lower-order terms in the Fourier transform. Thus, increasing $N$ makes $g(t)$ arbitrarily smooth, resulting in a polynomial decay of the transition probability to arbitrary order in $1/\omega T$.
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\section{\label{sec:minim-state-dist}Minimization of state disturbance using bump coupling functions}
\begin{figure}
\caption{\label{fig:gbump}
\label{fig:gbump}
\end{figure}
The construction of coupling functions from Eq.~\eqref{eq:bvdhkjbvd} progressively increases smoothness and illuminates the relationship between smoothness and state disturbance. However, the decay of the corresponding transition probability with $T$ is only polynomial. This raises the question of whether coupling functions exist that achieve superpolynomial decay. Clearly, this will require functions with compact support $[0,T]$ that are $C^\infty$-smooth, known as bump functions \cite{Lee:2003:oo}. No such function can have a Fourier transform that follows an exponential decay in $1/\omega T$, since a function whose Fourier transform decays exponentially cannot have compact support. Thus, the state disturbance can at most exhibit subexponential decay. A suitable class of bump functions with support $[0,T]$ is given by
\begin{equation}\label{eq:bumfcts}
g_{\alpha\beta} (t) = \begin{cases}
c_{\alpha\beta}^{-1}\exp\left(-\beta \left[1-\left(\frac{2t}{T}-1\right)^2\right]^{1-\alpha}\right), \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad 0 < t <T, \\ 0 \qquad \qquad \qquad \qquad \qquad \qquad \quad \,\,\, \text{otherwise},
\end{cases}
\end{equation}
where $\alpha \ge 2$ and $\beta\ge 1$ are integers, and $c_{\alpha\beta}$ normalizes the area under $g_{\alpha\beta} (t)$. These functions are $C^\infty$-smooth with vanishing derivatives and essential singularities at $t=0$ and $t=T$. Figure~\ref{fig:gbump} shows $g_{\alpha\beta} (t)$ for several different choices of $\alpha$ and $\beta$.
\begin{figure}
\caption{\label{fig:bumpsd}
\label{fig:bumpsd}
\end{figure}
For $\alpha=2$ and $\beta=1$, the Fourier transform exhibits subexponential decay proportional to $(\omega T)^{-3/4}\text{e}^{-\sqrt{\omega T}}$ (Fig.~\ref{fig:bumpsd}). Increasing $\alpha$ and $\beta$ enhances the decay (see again Fig.~\ref{fig:bumpsd}), with Fourier transform (to leading order in $1/\omega T$) proportional to $(\omega T)^{-(\alpha+1)/2\alpha} \exp\left[-\gamma_{\alpha\beta} (\omega T)^{(\alpha-1)/\alpha}\right]$, where $\gamma_{\alpha\beta}$ is a constant. By increasing $\alpha$ we can asymptotically approach exponential decay. As seen in Fig.~\ref{fig:gbump}, this will also make $g(t)$ more narrow, rendering the initial decay less rapid, just as for $g(t)$ constructed from an increasing number of sinusoidal components. Figure~\ref{fig:bumpsd} makes clear that since $\omega T \gg 1$, bump functions are superior to coupling functions composed of the sinusoidal components defined in Eq.~\eqref{eq:bvdhkjb678678}.
\section{\label{sec:suff-first-order}Sufficiency of the first-order amplitude}
The higher-order corrections $A_m^{(\ell \ge 2)}(T)$ [Eq.~\eqref{eq:g8fbvsv1}] are of $\ell$th order in the interaction strength, but in general contain terms of first order in $1/T$ \cite{Schlosshauer:2014:pm}. This raises the question of whether the conditions~\eqref{eq:conds}, which eliminate terms up to order $(\omega T)^{-(2N+1)}$ in $A_m^{(1)}(T)$, also eliminate these orders in $A_m^{(\ell)}(T)$ for all $\ell \ge 2$. We find that this is indeed the case. Evaluating $A_m^{(\ell)}(T)$ for $g(t)$ with $N$ nonzero coefficients $a_n$ satisfying the $N$ conditions~\eqref{eq:conds} gives, to leading order in $1/\omega T$,
\begin{align}\label{eq:uuuun22243}
\widetilde{A}_m^{(\ell)}(T)&= \left(-\frac{\text{i}}{\hbar}\right)^\ell \frac{\text{i} O_{mn} }{(\ell-1)!} \left[O_{mm} ^{\ell-1}-O_{nn} ^{\ell-1}\text{e}^{\text{i} \omega T}\right] \left(2\pi\right)^{2N} \notag\\ &\quad \times \left(\sum_{n=1}^N a_n n^{2N}\right) \left(\frac{1}{\omega T}\right)^{2N+1}.
\end{align}
Since this is of the same leading order in $1/\omega T$ as the first-order transition amplitude $A_m^{(1)}(T)$ [see Eq.~\eqref{eq:uuuun2233}], the total transition amplitude $A_m(T) = \sum_{\ell=1}^\infty A_m^{(\ell)}(T)$ is also of the same leading order as $A_m^{(1)}(T)$.
We establish a stronger result still. We calculate the total transition amplitude to leading order in $1/\omega T$ by summing Eq.~\eqref{eq:uuuun22243} over all orders $\ell$. The result is
\begin{align}\label{eq:uuudfb43}
\widetilde{A}_m(T) &\approx -\frac{2\text{i}}{\hbar} O_{mn} \text{e}^{\text{i}\omega T/2}
\sin \left\{ \frac{\omega T }{2} \left[1 +\chi_{mn}(T) \right]\right\} \left(2\pi\right)^{2N} \notag\\ &\quad \times\left(\sum_{n=1}^N a_n n^{2N}\right) \left(\frac{1}{\omega T}\right)^{2N+1},
\end{align}
where $\chi_{mn}(T) = (\hbar\omega T)^{-1} \left[O_{mm} - O_{nn}\right]$ \footnote{Eq.~\eqref{eq:uuudfb43} omits an overall phase factor, which does not influence the transition probability.}. Comparison with Eq.~\eqref{eq:uuuun2233} shows that the corrections $A^{(\ell \ge 2)}_{m}(T)$ merely introduce a scaling factor $1+\chi_{nm}(T)$ into the argument of the sine function, whose oscillations, however, may be disregarded (see note~\cite{Note2}). Hence we may replace the sine function by 1, in which case Eqs.~\eqref{eq:uuuun2233} and \eqref{eq:uuudfb43} become identical. Thus, to leading order in $1/\omega T$, the first-order transition probability $\abs{A^{(1)}(T)}^2$ accurately describes the state disturbance. This offers an important calculational advantage and enables the analysis of state disturbance in terms of properties of Fourier-transform pairs.
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\section{Discussion}
A particularly intriguing application of protective measurement is the possibility of characterizing the quantum state of a single system from a set of protectively measured expectation values. While this approach is intrinsically limited by its requirement that the system initially be in an eigenstate of its Hamiltonian \cite{Aharonov:1993:qa,Aharonov:1993:jm,Dass:1999:az}, it has the distinct conceptual and practical advantage of not requiring ensembles of identically prepared systems, in contrast with conventional quantum-state tomography based on strong \cite{Vogel:1989:uu,Dunn:1995:oo,Smithey:1993:lm,Breitenbach:az:1997,White:1999:az,James:2001:uu,Haffner:2005:sc,Leibfried:2005:yy,Altepeter:2005:ll,Lvovsky:2009:zz} or weak \cite{Lundeen:2011:ii,Lundeen:2012:rr,Fischbach:2012:za,Bamber:2014:ee,Dressel:2011:au} measurements. Thus, it provides an important alternative and complementary strategy for quantum-state measurement \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp,Dass:1999:az,Vaidman:2009:po,Auletta:2014:yy,Diosi:2014:yy,Aharonov:2014:yy}.
To successfully characterize the initial state of the system with protective measurements, it is crucial that the initial state of the system is minimally disturbed during the series of protective measurements that determine the set of expectation values. We have shown how one can minimize this state disturbance, given a fixed duration $T$ and average strength ($\propto 1/T$) of each protective measurement. Specifically, we have described a systematic procedure for designing the time dependence of the system--apparatus interaction (described by the coupling function) such that the state disturbance decreases polynomially or subexponentially with $T$. The leading order in $1/T$ can be made arbitrarily large for polynomial decay, and one may also come arbitrarily close to exponential-decay behavior by using bump functions. Since strictly exponential decay cannot be attained, bump functions are the optimal choice, as they produce the least possible state disturbance in a protective measurement.
Previous discussions of protective measurement \cite{Aharonov:1993:qa,Aharonov:1993:jm,Aharonov:1996:fp} have appealed to the condition that the coupling function change slowly during the measurement such that the quantum adiabatic theorem \cite{Born:1928:yf} can be applied. But our results indicate that this condition is both too weak and too strict. It is too weak, because it concerns only the smallness of the first-order derivative of the coupling function, rather than the number of continuous derivatives. It is too strict, because our analysis shows that the state disturbance in a protective measurement is chiefly due to discontinuities in the coupling function and its derivatives during the turn-on and turnoff of the measurement interaction. Once a sufficiently smooth turn-on and turnoff is achieved, the interaction strength may be changed comparably rapidly during the remaining period without creating significant additional state disturbance. Thus, the reduction of the state disturbance through an optimization of the coupling function does not necessitate adjustment of the measurement time or average interaction strength. Furthermore, compared to the condition of smoothness, the weakness of the interaction has a small effect on the state disturbance, which depends only quadratically on the average interaction strength.
The optimization procedure described here is very general, because it solely modifies the time dependence of the coupling function and is independent of the physical details of the system and the apparatus. In particular, it is independent of the Hamiltonian and the measured observable. This raises the question of whether and how one might further improve the fidelity of the state measurement if the specifics of the physical system and measured observables are taken into account. One approach would be to make use of any available partial knowledge of the Hamiltonian of the system. Such knowledge may be used to additionally reduce the state disturbance, since then the system--apparatus interaction can be designed to target the partially known eigenspaces of the Hamiltonian \footnote{Of course, in the limiting case of a completely known Hamiltonian, a projective measurement in the energy eigenbasis permits determination of the state of the system without any state disturbance, since the system is assumed to be in one of these eigenstates.}. In this case, one may also be able to reduce particular transition amplitudes by minimizing some of the transition matrix elements $O_{mn}=\bra{m} \op{O} \ket{n}$ [see Eqs.~\eqref{eq:uuuun22243} and \eqref{eq:uuudfb43}]. However, this approach can be expected to succeed only for a subset of eigenstates and very few particular choices (if any) of observables $\op{O}$, while state determination requires the protective measurement of multiple complementary (and practically measurable) observables.
In summary, we have shown how to optimally implement protective measurements and thereby maximize the likelihood of success of protective measurements that seek to determine the quantum state of single systems. Our results dramatically improve the performance of protective measurements and may aid in their future experimental realization.
\begin{thebibliography}{42}
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\bibitem [{\citenamefont {Vogel}\ and\ \citenamefont
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{\bibinfo {volume} {70}},\ \bibinfo {pages} {1244} (\bibinfo {year}
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\bibitem [{\citenamefont {Breitenbach}\ \emph {et~al.}(1997)\citenamefont
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{Yamamoto}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {53}},\ \bibinfo {pages} {R2911}
(\bibinfo {year} {1996})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {{Hari Dass}}\ and\ \citenamefont
{Qureshi}(1999)}]{Dass:1999:az}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{{Hari Dass}}}\ and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
{Qureshi}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. A}\ }\textbf {\bibinfo {volume} {59}},\ \bibinfo {pages} {2590}
(\bibinfo {year} {1999})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Vaidman}(2009)}]{Vaidman:2009:po}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Vaidman}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Compendium of
Quantum Physics: Concepts, Experiments, History and Philosophy}}},\ \bibinfo
{editor} {edited by\ \bibinfo {editor} {\bibfnamefont {D.}~\bibnamefont
{Greenberger}}, \bibinfo {editor} {\bibfnamefont {K.}~\bibnamefont
{Hentschel}}, \ and\ \bibinfo {editor} {\bibfnamefont {F.}~\bibnamefont
{Weinert}}}\ (\bibinfo {publisher} {Springer},\ \bibinfo {address}
{Berlin/Heidelberg},\ \bibinfo {year} {2009}), pp.\ \bibinfo {pages}
{505--508}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Gao}(2014)}]{Gao:2014:cu}
\BibitemOpen
\bibinfo {editor} {\bibfnamefont {S.}~\bibnamefont {Gao}},\ ed.,\ \href@noop
{} {\emph {\bibinfo {title} {Protective Measurement and Quantum Reality:
Towards a New Understanding of Quantum Mechanics}}}\ (\bibinfo {publisher}
{Cambridge University Press},\ \bibinfo {address} {Cambridge},\ \bibinfo
{year} {2014})\BibitemShut {NoStop}
\bibitem [{\citenamefont {Auletta}(2014)}]{Auletta:2014:yy}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
{Auletta}},\ }in\ Ref.~\cite{Gao:2014:cu}, pp.\
\bibinfo {pages} {39--62}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Di{\'o}si}(2014)}]{Diosi:2014:yy}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Di{\'o}si}},\ }in\ Ref.~\cite{Gao:2014:cu}, pp.\
\bibinfo {pages} {63--67}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Aharonov}\ and\ \citenamefont
{Cohen}(2014)}]{Aharonov:2014:yy}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Cohen}},\ }in\ Ref.~\cite{Gao:2014:cu}, pp.\
\bibinfo {pages} {28--38}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Aharonov}\ and\ \citenamefont
{Vaidman}(1996)}]{Aharonov:1996:ii}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Vaidman}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Bohmian
Mechanics and Quantum Theory: An Appraisal}}},\ \bibinfo {editor} {edited by\
\bibinfo {editor} {\bibfnamefont {J.~T.}\ \bibnamefont {Cushing}}, \bibinfo
{editor} {\bibfnamefont {A.}~\bibnamefont {Fine}}, \ and\ \bibinfo {editor}
{\bibfnamefont {S.}~\bibnamefont {Goldstein}}}\ (\bibinfo {publisher}
{Kluwer},\ \bibinfo {address} {Dordrecht},\ \bibinfo {year} {1996}), pp.\
\bibinfo {pages} {141--154}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Aharonov}\ \emph {et~al.}(1999)\citenamefont
{Aharonov}, \citenamefont {Englert},\ and\ \citenamefont
{Scully}}]{Aharonov:1999:uu}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}}, \bibinfo {author} {\bibfnamefont {B.~G.}\ \bibnamefont
{Englert}}, \ and\ \bibinfo {author} {\bibfnamefont {M.~O.}\ \bibnamefont
{Scully}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys.
Lett. A}\ }\textbf {\bibinfo {volume} {263}},\ \bibinfo {pages} {137}
(\bibinfo {year} {1999})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Alter}\ and\ \citenamefont
{Yamamoto}(1997)}]{Alter:1997:oo}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
{Alter}}\ and\ \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Yamamoto}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {56}},\ \bibinfo {pages} {1057}
(\bibinfo {year} {1997})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Schlosshauer}\ and\ \citenamefont
{Claringbold}(2014)}]{Schlosshauer:2014:tp}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Schlosshauer}}\ and\ \bibinfo {author} {\bibfnamefont {T.~V.~B.}\
\bibnamefont {Claringbold}},\ }in\ Ref.~\cite{Gao:2014:cu}, pp.\
\bibinfo {pages} {180--194}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Schlosshauer}(2014)}]{Schlosshauer:2014:pm}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Schlosshauer}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {90}},\ \bibinfo {pages} {052106}
(\bibinfo {year} {2014})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Schlosshauer}(2015)}]{Schlosshauer:2015:pm}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Schlosshauer}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {92}},\ \bibinfo {pages} {062116}
(\bibinfo {year} {2015})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Born}\ and\ \citenamefont
{Fock}(1928)}]{Born:1928:yf}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Born}}\ and\ \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Fock}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Z. Phys.}\
}\textbf {\bibinfo {volume} {51}},\ \bibinfo {pages} {165} (\bibinfo {year}
{1928})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Sakurai}(1994)}]{Sakurai:1994:om}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~J.}\ \bibnamefont
{Sakurai}},\ }\href@noop {} {\emph {\bibinfo {title} {Modern Quantum
Mechanics}}},\ \bibinfo {edition} {2nd}\ ed.\ (\bibinfo {publisher}
{Addison-Wesley},\ \bibinfo {address} {Reading, Massachusetts},\ \bibinfo
{year} {1994})\BibitemShut {NoStop}
\bibitem [{Note1()}]{Note1}
\BibitemOpen
\bibinfo {note} {The full expression for $A^{(\ell )}_{m}(T)$ also contains
contributions from the apparatus subspace \cite {Schlosshauer:2014:pm}. They
are irrelevant to our present analysis.}\BibitemShut {Stop}
\bibitem [{Note2()}]{Note2}
\BibitemOpen
\bibinfo {note} {Targeting the zeros of $\left \delimiter 69640972 G(\omega
T)\right \delimiter 86418188 ^2$ to minimize the state disturbance by tuning
$T$ would require precise knowledge of $\omega $ and therefore of $\protect
\mathaccentV {hat}05E{H}_S$. But in a protective measurement, $\protect
\mathaccentV {hat}05E{H}_S$ is \protect \emph {a priori} unknown \cite
{Aharonov:1993:jm,Dass:1999:az}.}\BibitemShut {Stop}
\bibitem [{\citenamefont {Lee}(2003)}]{Lee:2003:oo}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont
{Lee}},\ }\href@noop {} {\emph {\bibinfo {title} {Introduction to Smooth
Manifolds}}}\ (\bibinfo {publisher} {Springer},\ \bibinfo {address} {New
York},\ \bibinfo {year} {2003})\BibitemShut {NoStop}
\bibitem [{Note3()}]{Note3}
\BibitemOpen
\bibinfo {note} {Eq.~\protect \textup {\hbox {\mathsurround \z@ \protect
\normalfont (\ignorespaces \ref {eq:uuudfb43}\unskip \@@italiccorr )}} omits
an overall phase factor, which does not influence the transition
probability.}\BibitemShut {Stop}
\bibitem [{Note4()}]{Note4}
\BibitemOpen
\bibinfo {note} {Of course, in the limiting case of a completely known Hamiltonian,
a projective measurement in the energy eigenbasis permits determination of the state
of the system without any state disturbance, since the system is assumed to be in
one of these eigenstates.}\BibitemShut {Stop}
\end{thebibliography}
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\begin{equation}gin{document}
\title{Spatial entanglement using a quantum walk on a many-body system}
\author{Sandeep K. \surname{Goyal}}
\email{[email protected]}
\affiliation{The Institute of Mathematical Sciences, CIT campus, Chennai 600 113, India}
\author{C. M. \surname{Chandrashekar}}
\email{[email protected]}
\affiliation{Institute for Quantum Computing, University of Waterloo,
Ontario N2L 3G1, Canada}
\affiliation{Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada}
\begin{equation}gin{abstract}
The evolution of a many-particle system on a one-dimensional lattice, subjected
to a quantum walk can cause spatial entanglement in the lattice position,
which can be exploited for quantum information/communication purposes. We
demonstrate the evolution of spatial entanglement and its dependence
on the quantum coin operation parameters, the number of particles present
in the lattice and the number of steps of the quantum walk on the
system. Thus, spatial entanglement can be controlled and optimized
using a many-particle discrete-time quantum walk.
\end{abstract}
\maketitle
\preprint{Version}
\section{Introduction}
\langlebel{intro}
Entanglement in a quantum state has been the fundamental resource in
many quantum information and computation protocols, such as
cryptography, communication, teleportation and algorithms \cite{NC00,
HHH09}. To implement these protocols, generating an entangled state
is very important. Similarly, studies on the interface between
condensed matter systems and quantum information have shown
entanglement as a signature of quantum phase transition \cite{ON02,
OAF02, OROM06}. To understand the phases and dynamics in many-body
systems an analysis of entanglement in many-body systems is very
important.
Hence, various schemes have been proposed for entanglement generation
in quantum systems \cite{BH02, LHL03, RER07, WS09} and for understanding
entanglement in many-body systems \cite{AFOV08}.
Quantum walk (QW) is one such process in which an uncorrelated
state can evolve to an entangled state and be used to analyze the
evolution of entanglement \cite{Kem03, CLX05}.
\par
The QW, which was developed as a quantum analog of the classical random
walk (CRW), evolves a particle into an entanglement between its internal and position degrees of freedom. It has played a significant role in the development of quantum algorithms \cite{Amb03}.
Furthermore, the QW has been used to demonstrate
coherent quantum control over atoms, quantum
phase transition \cite{CL08}, to explain the phenomena such as
breakdown of an electric field-driven system \cite{OKA05} and direct
experimental evidence for wavelike energy transfer within
photosynthetic systems \cite{ECR07, MRL08}. Experimental
implementation of the QW has also been
reported \cite{DLX03, RLB05, PLP08, KFC09}, and various other schemes
have
been proposed for its physical realization \cite{TM02, RKB02, EMB05, Cha06, MBD06}.
Therefore, studying entanglement during the QW process will be useful
from a quantum information theory perspective and also contribute to
further investigation of the practical applications of the QW. In this
direction, evolution of entanglement between single particle and
position with time (number of steps of the discrete-time QW) has been
reported \cite{CLX05}.
\par
In this paper, we consider a multipartite quantum walk on a one-dimensional lattice and study the evolution of {\it spatial
entanglement}, entanglement between different lattice points. All
the particles considered in the system are identical and
indistinguishable with two internal states (sides of the quantum
coin). Spatial entanglement generated using a QW can be controlled by
tuning different parameters, such as parameters in the quantum coin
operation, number of particles in the system and evolution time (number of
steps). To quantify entanglement in the system we are
using Meyer-Wallach multipartite entanglement measure.
\par
In Sec. \ref{qw}, we describe single-particle and many-particle
discrete-time QWs. In Sec. \ref{entanglement}, entanglement between a particle and
position space and spatial entanglement using single- and many-particle
QWs are discussed. In Sec. \ref{mpent}, we present the measure for
spatial entanglement of the system using the Meyer-Wallach global
entanglement measure scheme for particles in a one-dimensional lattice
and in a closed chain ($n-$cycle). We also demonstrate control
over spatial entanglement by
exploiting the dynamical properties of the QW.
We conclude with the summary in Sec. \ref{conc}.
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\section{Quantum Walk}
\langlebel{qw}
Classical random walk (CRW) describes the dynamics of a particle in position
space with a certain probability. The QW is the quantum analog of CRW-developed exploiting
features of quantum
mechanics such as superposition and interference of quantum amplitudes
\cite{GVR58, FH65, ADZ93}. The QW, which involves superposition of
states, moves simultaneously exploring multiple possible paths with
the amplitudes corresponding to the different paths interfering. This
makes the variance of the QW on a line to grow quadratically with the
number of steps which is in sharp contrast to the linear growth for
the CRW.
\par
The study of QWs has been largely divided into two standard variants:
discrete-time QW (DTQW) \cite{ADZ93, DM96, ABN01} and a continuous-time QW (CTQW) \cite{FG98}.
In the CTQW, the walk is defined directly on the {\it position} Hilbert
space $\mathcal{H}_p$, whereas for the DTQW it is necessary to
introduce an additional {\it coin} Hilbert space $\mathcal{H}_c$, a quantum
coin operation to define the direction in which the particle amplitude
has to evolve. The connection between these two variants and the generic
version of the QW has been studied \cite{FS06,
C08}. However, the coin degree of freedom in
the DTQW is an advantage over the CTQW as it allows control of dynamics of the QW \cite{AKR05, CSL08}. Therefore, we take full advantage of the coin degree of freedom in this work and
study the DTQW on a many-particle system.
\subsection{Single-particle quantum walk}
\langlebel{spqw}
The DTQW is defined on the Hilbert space $\mathcal H= \mathcal H_{c}
\otimes \mathcal H_{p}$. In one dimension, the coin Hilbert space $\mathcal H_{c}$,
spanned by the basis state $|0\ranglengle$ and $|1\ranglengle$, represents two
sides of the quantum coin, and the position Hilbert space $\mathcal H_{p}$,
spanned by the basis states $|\psi_j\ranglengle$, $j \in \mathbb{Z}$,
represent the positions in the lattice. To implement the DTQW, we will
consider a three-parameter U(2) operator $C_{\xi, \theta, \zeta}$ of
the form
\begin{equation}
\langlebel{coin}
C_{\xi,\theta,\zeta}
\equiv \left( \begin{equation}gin{array}{clcr} e^{i\xi}\cos(\theta) & &
e^{i\zeta}\sin(\theta) \\ e^{-i\zeta} \sin(\theta) & &
-e^{-i\xi}\cos(\theta)
\end{array} \right)
\end{equation}
as the quantum coin operation \cite{CSL08}. The quantum coin operation is applied on the particle state ($C_{\xi, \theta, \zeta} \otimes {\mathbbm 1}$) when the initial state of the complete system is
\begin{equation}
\langlebel{qw:in}
|\Psi_{in}\ranglengle= \left [ \cos(\delta)|0\ranglengle +
e^{i\eta}\sin(\delta)|1\ranglengle \right ] \otimes |\psi_{0}\ranglengle.
\end{equation}
The state $\cos(\delta)|0\ranglengle + e^{i\eta}\sin(\delta)|1\ranglengle$ is the
state of the particle and $|\psi_{0}\ranglengle$ is the state of the
position at the lattice position $j=0$.
\par
The quantum coin operation on the particle is followed by the conditional unitary shift operation $S$ which acts on the complete
Hilbert space of the system:
\begin{equation}
\langlebel{eq:alter}
S =\exp(-i \sigma_{z}\otimes Pl),
\end{equation}
where $P$ is the momentum operator, $\sigma_{z}$ is the Pauli
spin operator in the $z$ direction and $l$ is the length of each step. The
eigenstates of $\sigma_{z}$ are denoted by $|0\ranglengle$ and
$|1\ranglengle$.
Therefore, $S$, which delocalizes the wave packet over the positions
$(j-1)$ and $(j+1)$, can also be written as
\begin{equation}gin{eqnarray}
\langlebel{eq:condshift} S = |0\ranglengle \langlengle 0|\otimes \sum_{j \in
\mathbb{Z}}|\psi_{j-1}\ranglengle \langlengle \psi_{j} |+|1\ranglengle \langlengle 1
|\otimes \sum_{j \in \mathbb{Z}} |\psi_{j+1}\ranglengle \langlengle \psi_{j}|.
\end{eqnarray}
\par
The process of
\begin{equation}
\langlebel{dtqwev}
W_{\xi, \theta, \zeta} =
S(C_{\xi, \theta, \zeta} \otimes {\mathbbm 1})
\end{equation}
is iterated without resorting to intermediate
measurement to help realize a large number of steps of the QW. The parameters
$\delta$ and $\eta$ in Eq. (\ref{qw:in}) can be varied to obtain
different
initial states of the particle. The three parameters $\xi$,
$\theta$ and $\zeta$ of $C_{\xi, \theta,
\zeta}$ can be varied to choose the quantum coin operation. By
varying parameter $\theta$ the variance can be
increased or decreased according to the functional form, $\sigma^{2} \approx
(1-\sin(\theta))t^{2}$, where $t$ is the number of steps of the QW, as shown in Fig. \ref{fig:qw1a}.
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\epsfig{figure=fig1.eps, width=9.0cm}
\caption{\langlebel{fig:qw1a}(color online) Spread of the probability distribution for
different values of $\theta$ using the quantum coin operator $C_{0,
\theta, 0}$. The distribution is wider for (a) $(0,
\theta, 0)= (0, \frac{\pi}{12}, 0)$ than for (b) $(0,
\theta, 0)= (0, \frac{\pi}{4}, 0)$ and (c)
$(0, \theta, 0)= (0, \frac{5 \pi}{12}, 0)$, showing the decrease in spread with
increase in $\theta$. The initial state of the particle is
$|\Psi_{ins}\ranglengle = \frac{1}{\sqrt 2}\left ( |0\ranglengle + i
|1\ranglengle \right ) \otimes |\psi_{0}\ranglengle$ and the distribution is
for 100 steps.}
\end{center}
\end{figure}
\par
{\it Biased coin operation and biased QW:} The most widely studied
form of the DTQW is the walk using the Hadamard operation
\begin{equation}
H = \frac{1}{\sqrt{2}}
\begin{equation}gin{pmatrix}
1 & \mbox{~} 1 \\
1 & -1
\end{pmatrix},
\langlebel{hadamard}
\end{equation}
corresponding to the quantum coin operation with $\xi = \zeta = 0$ and $\theta = \pi/4$
in Eq. (\ref{coin}). The Hadamard operation is an unbiased coin operation, and the resulting walk is known as the Hadamard walk. This walk implemented on a
particle initially in a symmetric superposition state,
\begin{equation}
\langlebel{qw:in1}
|\Psi_{ins}\ranglengle = \frac{1}{\sqrt 2} \left [ |0\ranglengle + i |1\ranglengle
\right ] \otimes |\psi_{0}\ranglengle,
\end{equation}
obtained by choosing $\delta = \pi/4$ and $\eta = \pi/2$ in
Eq. (\ref{qw:in}), returns a symmetric, unbiased probability
distribution of the particle in position space. However, the
Hadamard walk on any asymmetric initial state of the particle results
in an asymmetric, biased probability
distribution of the particle in position space
\cite{ABN01}. We should note that the role of the initial state on the
symmetry of the probability distribution is not vital for a QW using the
three-parameter operator given by Eq. (\ref{coin}) as a quantum coin
operation.
\par
To elaborate this further, we will consider the first-step evolution
of the DTQW using a three-parameter quantum coin operation given by
Eq. (\ref{coin}) on a particle initially in the symmetric
superposition state.
After the first step of the DTQW the state can be written as
\begin{equation}gin{eqnarray}
\langlebel{eq:condshift2}
W_{\xi, \theta, \zeta}|\Psi_{ins}\ranglengle = \frac{1}{\sqrt 2}
\left [ \left(e^{i\xi} \cos(\theta)+ i e^{i\zeta} \sin(\theta)\right )
|0\ranglengle|\psi_{-1}\ranglengle
+ \left( e^{-i\zeta}\sin (\theta) - i e^{-i\xi}
\cos(\theta)\right) |1\ranglengle|\psi_{+1}\ranglengle \right ].
\end{eqnarray}
If $\xi=\zeta$, Eq. (\ref{eq:condshift2}) has left-right symmetry in
the position probability distribution, but not otherwise. That is,
the parameters $\xi$ and $\zeta$ introduce asymmetry in the position
space probability distribution. Therefore, a coin operation with $\xi
\neq \zeta$ in Eq. (\ref{coin}) can be called as a biased quantum coin
operation which will bias the QW probability distribution of the particle
initially in a symmetric superposition state (Fig. \ref{fig:qw2})
\cite{CSL08}. However, we should note that irrespective of the
quantum coin operation used, QW can also be biased by
choosing an
asymmetric initial state of the particle (for example, the Hadamard walk
of a particle initially in the state $|0\ranglengle$ or the state $|1\ranglengle$).
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\epsfig{figure=fig2.eps, width=9.0cm}
\caption{Spread of probability distribution for different values of
$\xi$, $\theta$, $\zeta$ using the quantum coin operator $C_{\xi,
\theta, \zeta}$. The parameter $\xi$ shifts the distribution to the
left: (a)$(\xi, \theta, \zeta) = ( \frac{\pi}{6}, \frac{\pi}{6}, 0)$
and (c) $(\xi, \theta, \zeta)= (\frac{5 \pi}{12}, \frac{\pi}{3}, 0
)$. The parameter $\zeta$ shifts it to the right: (b) $(\xi, \theta,
\zeta)= (0, \frac{ \pi}{6}, \frac{\pi}{6})$ and (d) $(\xi, \theta,
\zeta) = (0, \frac{\pi}{3}, \frac{5 \pi}{12})$. The initial state
of the particle $|\Psi_{ins}\ranglengle = \frac{1}{\sqrt{2}}(|0\ranglengle +
i |1\ranglengle) \otimes |\psi_{0}\ranglengle$ and the distribution is for
100 steps.}
\langlebel{fig:qw2}
\end{center}
\end{figure}
\subsection{Many-particle quantum walk}
\langlebel{mbqw1}
To define a many-particle QW in one dimension, we will consider an $M$-particle system with one non-interacting particle at each position
(Fig. \ref{mi}). The $M$ identical particles in $M$ lattice points with
each particle having its own coin and position Hilbert space will have
a total Hilbert space $\mathcal{H}=\left(
\mathcal{H}_c \otimes \mathcal{H}_p \right)^M $. We
assume the particles to be distinguishable.
\par
The evolution of each step of the QW on the $M$-particle system is given by
the application of the operator
$W_{0,\theta, 0}^{\otimes M}$.
The initial state that we will consider for the many-particle system in one dimension will be
\begin{equation}gin{equation}
|\Psi_{ins}^{M}\ranglengle = \bigotimes_{j=-\frac{M-1}{2}}^{j=\frac{M-1}{2}}
\left( \frac{|0\ranglengle + i|1\ranglengle}{\sqrt{2}} \right) \otimes
|\psi_{j}\ranglengle.
\langlebel{initialMBQWstate}
\end{equation}
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\includegraphics[width=8.5cm]{fig3.eps}
\caption{Many-particle state with one non-interacting particle at each
position space.}
\langlebel{mi}
\end{center}
\end{figure}
\begin{equation}gin{figure}[ht]
\includegraphics[width=9.0cm]{fig4.eps}
\caption{(color online) Probability distribution of 40 particles
initially with one particle in each position space when subjected to
the QW of different number of steps. The initial state of all the
particles is $\frac{1}{\sqrt 2}(|0\rangle + i |1\rangle)$ and is evolved in
position space using the Hadamard operator, $C_{0, \pi/4, 0}$ as
the quantum coin. The distribution spreads in the position space with
an increase in number of steps.}
\langlebel{miqw}
\end{figure}
\par
For an $M$-particle system after $t$ steps of the QW, the
Hilbert space consists of the tensor product of single lattice
position Hilbert space which is $(2t+M+1)$ in number. That
is, after $t$ steps of the QW, the $M$ particles are spread
between $(j-t)$ to $(j+t)$. In principle, each lattice point is associated
with a Hilbert space spanned by two subspaces, a zero-particle
subspace and one-particle subspace spanned by two possible states of
the coin, $|0\ranglengle$ and $|1\ranglengle$.
Therefore, the dimension of each lattice point will be $3^{M}$ and the
dimension of total Hilbert space is $(3^{M})^{\otimes M}$.
Fig. (\ref{miqw}) shows the probability distribution of the many-particle
system with an increase in number of steps of the QW.
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\section{Entanglement}
\langlebel{entanglement}
To efficiently make use of entanglement as a physical resource, the
amount of entanglement in a given system has to be
quantified. Therefore, entanglement in a pure bipartite system or a
system with two Hilbert spaces is quantified using standard measures
known as entropy of entanglement or Schmidt number
\cite{NC00}. The entropy of entanglement corresponds to the von
Neumann entropy, a functional of the eigenvalues of the reduced density
matrix, and a Schmidt number is the number of non-zero Schmidt
coefficients in its Schmidt decomposition. For a multipartite state,
there are quite a few good entanglement measures that have been proposed \cite{CKW00,
BL01, EB01, MW02, VDM03, Miy08, HJ08}. However, as the number of
particles in the system increases, the complexity of finding an
appropriate entanglement measure also increases, making scalability
impractical. Among the proposed measures, to address
this scalability
problem, Mayer and Wallach proposed a {\em scalable} global entanglement
measure (polynomial measure) to quantify entanglement in many-particle
systems \cite{MW02}.
\par
In this section, we will first discuss the entanglement of a
particle with position space quantified using entropy of
entanglement. Later we will discuss spatial
entanglement quantified using the Mayer-Wallach (M-W) measure. Spatial
entanglement has been explored earlier using different methods. For
example, in an ideal bosonic gas it has been studied using
off-diagonal long-range order \cite{HAKV07}. For our investigations,
we consider a distinguishable many-particle system, implement QW and
use the M-W measure to quantify spatial entanglement. In this system the
dynamics of particles can be controlled by varying the quantum coin
parameters, the initial state of the particles, the number of particles in
the system and the number of steps of the QW. In particular, we choose
the particles in one-dimensional open and closed chains. The spatial
entanglement thus created can be used for example to create
entanglement between distant atoms in an optical lattice \cite{SGC09}
or as a channel for state transfer in spin chain systems \cite{Bos03,
CDE04, CDD05}.
\subsection{Single-particle - position entanglement}
\langlebel{cpent}
QW entangles the particle (coin) and the position degrees of freedom.
To quantify it, let us consider a DTQW on a particle
initially in a state given by
Eq. (\ref{qw:in1}) with a simple form of a coin operation
\begin{equation}
\langlebel{coin1}
C_{0,\theta, 0}
\equiv \left( \begin{equation}gin{array}{clcr} \cos(\theta) & &
\sin(\theta) \\ \sin(\theta) & &
-\cos(\theta)
\end{array} \right).
\end{equation}
After the first step, $W_{0,\theta,0} = S(C_{0, \theta, 0} \otimes
{\mathbbm 1})$, the state takes the form
\begin{equation}gin{eqnarray}
| \Psi_{1} \ranglengle &=& W_{0,\theta, 0} | \Psi_{ins}\ranglengle
= \gamma \left( |0 \ranglengle \otimes |\psi_{j-1} \ranglengle
\right) + \delta \left( |1 \ranglengle \otimes |\psi_{j+1}
\ranglengle \right)
\langlebel{evolution01}
\end{eqnarray}
where $\gamma = \left( \frac{\cos(\theta) + i\sin(\theta)}{\sqrt{2}} \right)$ and
$\delta = \left( \frac{\sin(\theta) - i\cos(\theta)}{\sqrt{2}} \right)$.
The Schmidt rank of $|\Psi_1\ranglengle$ is
$2$ which implies entanglement in the system.
The value of entanglement with an increase in the number of steps can be further quantified by
computing the von Neumann entropy of the reduced density matrix of
the position subspace.
\begin{equation}gin{figure}[ht]
\includegraphics[width=9.0cm]{fig5.eps}
\caption{(color online) Entanglement of a single particle with position
space when subjected to the QW. The initial state of a particle is
$\frac{1}{\sqrt 2}(|0\rangle + i |1\rangle)$ and is evolved in position
space using different values for $\theta$ in the quantum coin
operation $C_{0, \theta, 0}$. The entanglement initially oscillates
and approaches an asymptotic value with an increase in the number of
steps. For smaller values of $\theta$ the entanglement is higher and
decreases with an increase in $\theta$. Initial oscillation is also
larger for higher $\theta$.}
\langlebel{enta}
\end{figure}
\begin{equation}gin{figure}[ht]
\includegraphics[width=9.0cm]{fig6.eps}
\caption{(color online) Entanglement of single particle with position
space when subjected to the QW. The initial state of the particle is
given by Eq. (\ref{qw:in}) with $\delta = \frac{2\pi}{9}$ and $\eta
= \frac{\pi}{6}$ and is evolved in position space using different
values for $\theta$ in the quantum coin operation $C_{0, \theta,
0}$. The entanglement initially oscillates and approaches an
asymptotic value with an increase in the number of steps. For smaller
values of $\theta$ the entanglement is higher and decreases with an
increase in $\theta$. Initial oscillation is also larger for higher
$\theta$.}
\langlebel{enta1}
\end{figure}
\par
Fig. \ref{enta} shows a plot of the entanglement against the number of
steps of the QW on a particle initially in a symmetric
superposition state using different values for $\theta$ in the
operation $W_{\theta}$. The von Neumann entropy of the reduced density
matrix of the coin is used to quantify entanglement between the
coin and the position in Fig. \ref{enta}. That is,
\begin{equation}
E_{c}(t) = - \sum_{j} \langlembda_{j} \rm{log}_{2}(\langlembda_{j})
\end{equation}
where $\langlembda_{j}$ are eigenvalues of the reduced density matrix of
the coin after $t$ steps (time). The entanglement initially
oscillates and
reaches an asymptotic value
with increasing number of steps. In the asymptotic limit,
the entanglement value decreases with an increase in $\theta$ and this
dependence can be attributed to the spread of the amplitude distribution
in position space. That is, with an increase in $\theta$, constructive
interference of quantum amplitudes toward the origin becomes prominent
narrowing the distribution in the position space.
In Fig. \ref{enta1}, the process is repeated for a particle initially
in an asymmetric superposition state $|\Psi_{in}\ranglengle =
\left [\cos(\frac{2\pi}{9}) |0\ranglengle + e^{i \frac{\pi}{6}}
\sin(\frac{2\pi}{9})|1\ranglengle \right ] \otimes |\psi_{0}\ranglengle$.
Comparing Fig. \ref{enta1} with Fig. \ref{enta}, we can note the
increase in entanglement and decrease in the oscillation. This
observation can be explained by going back to our earlier note on
biased QW in Sec. \ref{spqw}. In Fig. \ref{fig:qw2} we note that
biasing of the coin operation leads to an asymmetry in the probability
distribution, with an increase in peak height on one side and a decrease
on the other side (increase and decrease are in reference to the symmetric
distribution). A similar biasing effect can also be reproduced by
choosing an asymmetric initial state of the particle. The biased
distribution with an increased value of probability at one side in the
distribution contributes to a reduced oscillation in the
distribution. This in turn results in the increase
of the von Neumann entropy: entanglement.
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\subsection{Spatial entanglement}
\langlebel{spentqw}
{\em Spatial entanglement} is the entanglement between the lattice
points. This entanglement takes the form of non-local particle number
correlations between spatial modes. To observe spatial entanglement
we first need to associate the lattice
with the state of a particle. Then we need to consider the evolution of
a single-particle QW followed by the evolution of a many-particle QW, in order to
understand spatial entanglement.
\subsubsection{Using a single-particle quantum walk}
\langlebel{spentqw1}
In a single-particle QW, each lattice point is associated
with a Hilbert space spanned by two subspaces. The first is the
zero-particle subspace which does not involve any coin (particle)
states. The other is the one-particle subspace spanned by the two
possible states
of the coin, $|0\ranglengle$ and $|1\ranglengle$. To obtain the spatial
entanglement we will write the state of
the particle in the form of the state of a lattice. Following from
Eq. (\ref{evolution01}), the state of the
particles after
first two steps of QW takes the form
\begin{equation}gin{eqnarray}
| \Psi_{2} \ranglengle &=& W_{0,\theta, 0} | \Psi_{1} \ranglengle = \gamma \left [
\cos (\theta) |0\ranglengle |\psi_{j-2} \ranglengle + \sin (\theta)
|1\ranglengle|\psi_{j} \ranglengle
\right ] + \delta \left [ \sin(\theta) |0\ranglengle |\psi_{j} \ranglengle -
\cos(\theta) |1\ranglengle|\psi_{j+2}
\ranglengle \right ].
\langlebel{evolution02}
\end{eqnarray}
In order to obtain the state of the lattice we can redefine the position
state in the
following way: the occupied position state $|\psi_j\ranglengle$ as
$|1_{j}\ranglengle$, which means
that the $j^{th}$ position is occupied and the rest of the lattice is
empty. Therefore, we can rewrite Eq. (\ref{evolution02}) as
\begin{equation}gin{eqnarray}
| \Psi_{2} \ranglengle & = & \gamma \left [
\cos (\theta) |0\ranglengle | 1_{j-2} \ranglengle + \sin (\theta)
|1\ranglengle| 1_{j} \ranglengle
\right ] + \delta \left [ \sin(\theta) |0\ranglengle | 1_{j} \ranglengle -
\cos(\theta) |1\ranglengle| 1_{j+2}
\ranglengle \right ].
\langlebel{evolution03}
\end{eqnarray}
Since we are interested in the spatial entanglement, we project this
state into one of the coin state so that we can ignore the
entanglement between the coin and the position state and consider only the
lattice states.
Here we will choose the coin state to be $|0\ranglengle$ and take
projection to obtain the state of the lattice in the form
\begin{equation}gin{equation}
|\Psi_{lat}\ranglengle = |0\ranglengle \left(\gamma
\cos(\theta)|1_{j-2}\ranglengle + \delta \sin(\theta)
|1_{j}\ranglengle
\right).
\langlebel{latticestate}
\end{equation}
Each lattice site $j$ can be considered as a Hilbert space with basis states
$|1_{j}\ranglengle$ (occupied state) and $|0_{j}\ranglengle$
(unoccupied state). Then, the above Eq. (\ref{latticestate}) in
the extended Hilbert space of each lattice can be rewritten in terms
of occupied and unoccupied lattice states as
\begin{equation}gin{equation}
|\Psi_{lat}^{\prime}\ranglengle = \gamma \cos(\theta) |1_{j-2}
\,0_{j}\ranglengle+\delta \sin(\theta) |0_{j-2} \,1_{j} \ranglengle.
\langlebel{latticestate1}
\end{equation}
We can see that after first two steps of the QW the lattice points $j$ and
$(j-2)$ are entangled. One
can check that the lattice points $j$ and $(j+2)$ are entangled if
we choose the coin state to be $|1\ranglengle$. With an increase in the number
of steps, the state of the particle spreads in position space and the
projection over one of the coin state reduces that state to a pure state, for which one may compute
spatial entanglement, according to the above prescription. Therefore,
with an increase in the number of steps, the spatial entanglement from a single-particle QW decreases.
\subsubsection{Using many-particle quantum walk}
\langlebel{mbqw}
We will extend the study of evolution of spatial entanglement as the QW progresses on a many-particle system.
\par
Let us first consider the analysis of first two steps of the Hadamard
walk ($\theta = \pi/4$ in Eq. (\ref{coin1})) on a three-particle
system with the initial state:
\begin{equation}gin{equation}
|\Psi_{ins}^{3p}\ranglengle = \bigotimes_{j=-1}^{+1}
\left( \frac{|0\ranglengle + i|1\ranglengle}{\sqrt{2}} \right) \otimes
|\psi_{j}\ranglengle.
\langlebel{initialMBQWstate3p}
\end{equation}
We will label the three particles at positions $-1$, $0$ and
$1$ as ${\rm A}$, ${\rm B}$ and ${\rm C}$, respectively. Since evolution of these particles is
independent, we write down the state after the first step as a tensor
product of each of the three particles:
\begin{equation}gin{align}
|\Psi^{3p}_{1}\ranglengle = W_{0, \theta, 0}^{\otimes 3}|\Psi_{ins}^{3p}\ranglengle
= \left[ \gamma |0\ranglengle |-2 \ranglengle + \delta |1\ranglengle| 0 \ranglengle \right]_{\rm A} \otimes \left[ \gamma |0\ranglengle |-1 \ranglengle + \delta |1\ranglengle| +1 \ranglengle \right]_{\rm B}
\otimes \left[ \gamma |0\ranglengle |0
\ranglengle + \delta |1\ranglengle| +2 \ranglengle \right]_{\rm C},
\langlebel{threeparticle1step}
\end{align}
where $\gamma = (1+i)/2$ and $\delta = (1-i)/2$. After two steps the
tensor product of each of the three particles is given by
\begin{equation}gin{align}
|\Psi^{3p}_{2}\ranglengle &= \left[ \gamma \left( \frac{|0\ranglengle |-3
\ranglengle+|1\ranglengle|-1 \ranglengle}{\sqrt{2}} \right) + \delta
\left( \frac{|0\ranglengle |-1 \ranglengle - |1\ranglengle|+1
\ranglengle}{\sqrt{2}} \right) \right]_{\rm A} \nonumber \\
&\otimes \left[ \gamma \left( \frac{|0\ranglengle |-2
\ranglengle+|1\ranglengle|0 \ranglengle}{\sqrt{2}} \right) + \delta \left(
\frac{|0\ranglengle |0 \ranglengle - |1\ranglengle|+2 \ranglengle}{\sqrt{2}}
\right) \right]_{\rm B} \nonumber \\
&\otimes \left[ \gamma \left( \frac{|0\ranglengle |-1
\ranglengle+|1\ranglengle|+1 \ranglengle}{\sqrt{2}} \right) + \delta \left(
\frac{|0\ranglengle |+1 \ranglengle - |1\ranglengle|+3 \ranglengle}{\sqrt{2}}
\right) \right]_{\rm C}.
\langlebel{threeparticlestate}
\end{align}
By projecting this state into one of the coin states (we choose state
$|0\ranglengle \otimes |0\ranglengle \otimes |0\ranglengle$)
we can obtain a state of the lattice for which spatial
entanglement may be computed. Then the
state of
the lattice after projection and normalization is
\begin{equation}gin{align}
|\Psi_{lat}\ranglengle =& \gamma^3 \,
|{\rm A}\ranglengle_{-3}|{\rm B}\ranglengle_{-2}|{\rm C}\ranglengle_{-1} \nonumber \\
& + \gamma^2 \delta \left( \,
|{\rm A}\ranglengle_{-3}|{\rm B}\ranglengle_{-2}|{\rm C}\ranglengle_{1}
+ |{\rm A}\ranglengle_{-3}|{\rm B}\ranglengle_{0}|{\rm C}\ranglengle_{-1}
+|{\rm AC}\ranglengle_{-1}|{\rm B}\ranglengle_{-2} \right) \nonumber \\
& + \gamma \delta^2 \left( \,
|{\rm A}\ranglengle_{-3}|{\rm B}\ranglengle_{0}|{\rm C}\ranglengle_{1} +
|{\rm AC}\ranglengle_{-1}|{\rm B}\ranglengle_{0}
+ |{\rm A}\ranglengle_{-1}|{\rm B}\ranglengle_{-2}|{\rm C}\ranglengle_{1} \right) \nonumber \\
& + \delta^3 \, |{\rm A}\ranglengle_{-1}|{\rm B}\ranglengle_{0}|{\rm C}\ranglengle_{1},
\langlebel{mblatticestate}
\end{align}
where $A,B$ and $C$ represent the particle labels and the subscripts represent
the position labels. In a similar manner we can obtain
$|\Psi_{lat}\ranglengle$ for a system with a large number of
particles. Then the next task is to calculate the spatial
entanglement.
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\section{Calculating spatial entanglement in a multipartite system}
\langlebel{mpent}
In a system with two particles, the state is separable if we can write
it as a tensor product of individual particle states, and
entangled if not. For a system with $M > 2$ particles, a state is
said to be fully separable if it can be written as
\begin{equation}gin{equation}
| \psi \ranglengle = | \phi_1 \ranglengle \otimes | \phi_2 \ranglengle \otimes
\cdots | \phi_k \ranglengle,
\langlebel{partialentangled}
\end{equation}
when $k=M$. $|\phi_i\ranglengle$ will then denote the state of the $i^{th}$
particle. When $k < M$ a state is said to be \emph{partially} entangled and
when $k=1$ the state will be fully entangled.
\par
Rather than using the von Neumann entropy to quantify multipartite entanglement
of a given state $\rho$, one sometimes often prefers to consider purity,
which corresponds (up to a constant) to linear entropy, that is the
first-order term in the expansion of the von Neumann entropy around its maxima,
given by
\begin{equation}gin{equation}
E = \frac{d}{d-1} \left[ 1 - {\rm Tr} \rho^2 \right]
\langlebel{linentropy}
\end{equation}
for a $d$-dimensional particle Hilbert space \cite{PWK04}.
To quantify the entanglement of multipartite pure states, one measure
commonly, used is the Meyer- Wallach (M-W) measure \cite{MW02}. It
is the entanglement measure of a single particle to the rest of the
system, averaged over the whole of the system and is given by
\begin{equation}gin{equation}
E_{MW} = \frac{d}{d-1} \left[ 1 - \frac{1}{L} \sum_{i=1}^{L} {\rm Tr} \rho_i^2 \right]
\langlebel{W-M}
\end{equation}
where $L$ is the system size and $\rho_i$ is the reduced density
matrix of the $i^{th}$ subsystem. The M-W measure does not diverge with
increasing system size and is relatively easy to calculate.
\par
In a multipartite QW the dimension at each lattice point,
after projection over
one particular state of coin, is $2^M$ where $M$ is the number of
particles. Hence, the expression for entanglement will be
\begin{equation}gin{align}
E_{MW}(|\psi_{lat}\ranglengle) & =
\frac{2^M}{2^M-1}\left(1-\frac{1}{2t+M+1}\sum_{j=-(t+\frac{M}{2})}^{t+\frac{M}{2}}{\rm
tr}\rho_j^2\right)
\langlebel{mw}
\end{align}
where $t$ is the number of steps and $\rho_j$ is the reduced
density matrix of $j^{th}$ lattice point. The reduced density matrix $\rho_{j}$ can be written as
\begin{equation}gin{align}
\rho_j &= \sum_{k} p^j_k|k\ranglengle\langlengle k|
\end{align}
where $|k\ranglengle$ is one of the $2^M$ possible states available for
a lattice point and $p^j_k$ can be calculated once we have the
probability distribution of an individual particle on the lattice.
\par
Since we have $M$ distinguishable particles, we have $2^M$
configurations depending upon whether a given particle is present in
the lattice point or not after freezing the state of the
particle. This set of configurations forms the
basis for a single-lattice point Hilbert space. Now we can calculate $p^j_k$, the probability of
$k^{th}$ configuration of a particle in the $j^{th}$ lattice point as follows.
Let us say $a_j^{(l_i)}$ is the probability of the $i^{th}$ particle to be
or not to be in the $j^{th}$ lattice point depending on
$l_i$. If $l_i$ is $1$,
then it gives us the probability of the particle to be in the lattice
point. If $l_i$ is $0$, then $a^{(l_i)}_j$ is the probability of a particle not to be in the lattice point, that is, $a_j^{(0)} =
1-a_j^{(1)}$. Hence, we can write
\begin{equation}gin{align}
p^j_k &= \prod_{i}a^{l_i}_j.
\end{align}
Once we have the probability of each particle at a given lattice position,
the spatial entanglement can be conveniently calculated.
Since the QW is a controlled evolution, one can obtain a
probability distribution of each particle over all lattice
positions. In fact, one can easily control the probability distribution
by varying quantum coin parameters during the QW process
and hence the entanglement.
\par
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\includegraphics[width=9.5cm]{fig7.eps}
\caption{(color online) Evolution of spatial entanglement with an increase in the number
of steps of the QW for different number of particles in an
open one-dimensional lattice chain. The entanglement first
increases and with further increase in the number of steps,
the number of lattice positions exceeds the number of particles in
the system resulting in the decrease of the spatial entanglement.
The distribution is obtained by implementing the QW on particles
in the initial state $\frac{1}{\sqrt 2}(|0\rangle + i |1\rangle)$ and the Hadamard
operation $C_{0, \pi/4, 0}$ as quantum coin operation.}
\langlebel{eeqw}
\end{center}
\end{figure}
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\includegraphics[width=8.5cm]{fig8.eps}
\caption{(color online) Quantity of spatial entanglement for 10
particles after 10 steps and 20 particles after 20 steps of the QW on a
one-dimensional lattice using different
values of $\theta$ in the quantum coin operation $C_{0, \theta, 0}$.
For (a) and (b), the distribution is for particles initially in the symmetric superposition state,
$\frac{1}{\sqrt{2}}(|0\ranglengle + i|1\ranglengle)$, and for (c) the particle's initial state is $|0\ranglengle$
(will be the same for state $|1\ranglengle$). Quantity of entanglement is
higher for $\theta$ closer to $0$ and $\pi/2$ compared to the
intermediate value. We note that the asymmetric probability
distribution due to an asymmetric initial state in case of (c)
contributes for an increase in the quantity of spatial
entanglement. When $\theta = \pi/2$, for every even number of steps
of the QW, the system returns to the initial state where entanglement is
$0$. Entanglement is 0 for $\theta= 0$.}
\langlebel{entqwtheta}
\end{center}
\end{figure}
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\includegraphics[width=8.5cm]{fig9.eps}
\caption{(color online) Evolution of spatial entanglement for a system with
different number of particles in a closed chain. With an increase in
the number of steps, the entanglement value remains close to
asymptotic value with some peaks in between. The peaks can be
accounted for the crossover of leftward and rightward propagating
amplitudes of the internal state of the particle during the QW.
The peaks are more for a chain with a smaller number of
particles. An increase in the number of particles in the system results
in the decrease of the entanglement value. The distribution is
obtained by using $\frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$ as the initial
states of all particles and the Hadamard operation $C_{0, \pi/4,
0}$ as quantum coin operation.}
\langlebel{Ering}
\end{center}
\end{figure}
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\includegraphics[width=8.5cm]{fig10.eps}
\caption{(color online) Quantity of spatial entanglement for 20 particles on a closed
chain after 20 steps of the QW using different values of
$\theta$ in the quantum coin operation $C_{0, \theta, 0}$. The distribution is for
particles initially in the state $\frac{1}{\sqrt{2}}(|0\ranglengle + i|1\ranglengle)$.
Since the system is a closed chain, the QW does not expand the position Hilbert space, and
therefore for all values of $\theta$ from $0$ to $\pi/2$ the
entanglement value remains roughly uniform except for
a small peak at smaller values of $\theta$. For $\theta=0$
when the number of steps equal to the number of particles, the amplitudes
goes round the chain and returns to its initial state making the
entanglement $0$ and for $\theta = \pi/2$, for every even number of
steps of the QW, the system returns to the initial state where
entanglement is again $0$.}
\langlebel{Eringtheta}
\end{center}
\end{figure}
\par
Fig. \ref{eeqw} shows the phase diagram of the spatial entanglement using a many-particle QW.
Data for the phase diagram were obtained numerically by subjecting the many-particle system
with different number of particles to the QW with increasing number of steps.
The quantity of spatial entanglement was computed using Eq. (\ref{mw}).
\par
Here, we have chosen the Hadamard operation $C_{0, \pi/4, 0}$ and
$\frac{1}{\sqrt 2}(|0\ranglengle + i |1\ranglengle)$ as the quantum coin
operation and initial state of the particles, respectively, for the evolution of the many-particle QW.
To see the variation of entanglement for a fixed number of
particles with an increase in steps, we can pick a line parallel to the $y$
axis. That is, fix the number of particles and see the variation of entanglement
with the number of steps.
\par
In Fig. \ref{eeqw}, we see that for a fixed number of particles, the entanglement at first increases to some value before gradually falling. For $M=12$ we can note that the peak value is about $0.5$ before gradually falling. With an increase in the number of steps of the QW, the
number of lattice positions to which the particles evolve increases resulting
in the decrease of the spatial entanglement (see Eq. (\ref{mw})).
The decrease in entanglement before the number of steps is equal to the number of
particles should be noted. This is because for the Hadamard walk the
spread of a probability distribution after $t$ steps is between
$\frac{-t}{\sqrt 2}$ and $\frac{t}{\sqrt 2}$ \cite{CSL08}.
\par
If we fix the number of steps and measure the entanglement by
increasing the number of particles in the system, the quantity of
spatial entanglement first decreases and then it starts increasing with
an increase in the number of particles.
\par
To show the variation of spatial entanglement with the quantum coin
parameter $\theta$, we plot the spatial entanglement by varying the
parameter $\theta$ for a system with 10 particles after 10 steps of the QW
and for a system with 20 particles after 20 steps of the QW in Fig. \ref{entqwtheta}.
In this figure, (a) and (b) are plots that use the symmetric
superposition state $\frac{1}{\sqrt 2}(|0\rangle + i|1\rangle)$ (unbiased QW)
as an initial state of all the particles, and (c) is the plot with all
the particles in one of the basis states $|0\rangle$ or $|1\rangle$ (biased QW)
as the initial state. We note that the quantity of entanglement is higher
for $\theta$ values closer to $0$ and $\pi/2$ and dips for values
in between for all the three cases.
Biasing the QW, plot (c) shows
a slight increase in the quantity of entanglement compared to the
unbiased case, plot (b). A similar effect is seen by biasing the QW
using two parameters $\xi, \zeta$ in the coin operation $C_{\xi,
\theta,\zeta}$ on particles initially in a symmetric superposition
state.
\par
{\it Closed chain: }
Since most physical systems considered for
implementation will be of a definite dimension, we extend our
calculations to one of the simplest examples of closed geometry,
an $n-$cycle. For a QW on an $n-$cycle, the shift operation,
Eq. (\ref{eq:condshift}), takes the form
\begin{equation}gin{eqnarray}
\langlebel{eq:condshift1} S = |0\ranglengle \langlengle 0|\otimes \sum_{j =
0}^{n-1} |\psi_{j-1 \mbox{~mod~}n}\ranglengle \langlengle \psi_{j} |
+|1\ranglengle \langlengle 1 |\otimes \sum_{j =0}^{n-1} |\psi_{j+1 \mbox{~mod~}n}\ranglengle \langlengle \psi_{j}|.
\end{eqnarray}
When we consider a many-particle system in a closed chain, with the number
of lattice positions equal to the number of particles $M$, the QW process
does not expand the position Hilbert space like it does on an open
chain (line). Therefore the spatial entanglement does not decrease at later times as it does for a walk on an open chain, but
remains close to the asymptotic value. Fig. \ref{Ering} shows the evolution of entanglement for
a system with different number of particles in a closed chain. The
peaks seen in the plot can be accounted for by the crossover of the leftward and
rightward propagating amplitudes of the internal state of the particle
during the QW process. The frequency of the peaks is more for a smaller
number of particles (smaller closed chain). Also, note that the
increase in the number of particles and the number of lattice points in
the closed cycle results in the decrease in spatial entanglement of
the system.
\par
In Fig. \ref{Eringtheta}, the value of spatial entanglement for 20
particles on a closed chain after 20 steps of the QW using different
values of $\theta$ in the quantum coin operation $C_{0, \theta, 0}$ is
presented. For all values of $\theta$ from $0$ to $\pi/2$ the
entanglement value remains roughly uniform except for
the extreme values of $\theta$. For $\theta=0$, the amplitude goes round the ring and
returns to its initial state making the spatial entanglement value $=
0$. For $\theta = \pi/2$, for every even number of steps of the QW,
the system returns to the initial state where spatial entanglement is
again $0$.
\par
Therefore, spatial entanglement on a large lattice space can be
created, controlled and optimized for a maximum entanglement value by
varying the quantum coin parameters and number of particles in the
multi-particle QW.
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\section{Conclusion}
\langlebel{conc}
We have presented the evolution of spatial entanglement in a
many-particles system subjected to a QW process. By considering
many particle in the one-dimensional open and closed chain we have
shown that spatial entanglement can be generated and controlled by
varying the quantum coin parameters, the initial state and the number of steps
in the dynamics of the QW process. The spatial entanglement generated
can have a potential application in quantum information theory and
other physical processes.
\begin{center}
{\bf Acknowledgement}
\end{center}
C.M.C is thankful to Mike and Ophelia Lezaridis for the financial support at IQC,
ARO, QuantumWorks and CIFAR for travel support. C.M.C also thank IMSc, Chennai,
India for the hospitality during November - December 2008. S.K.G thanks Aiswarya Cyriac for the help in programming.
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\begin{document}
\title{Characterisations of Testing Preorders for a
Finite Probabilistic $\pi$-Calculus}
\begin{abstract}
We consider two characterisations of the may and must testing preorders for a
probabilistic extension of the finite $\pi$-calculus: one based on
notions of probabilistic weak simulations, and the other on a
probabilistic extension of a fragment of Milner-Parrow-Walker modal
logic for the $\pi$-calculus. We base our notions of simulations on the
similar concepts used in previous work for probabilistic CSP.
However, unlike the case with CSP (or other non-value-passing
calculi), there are several possible definitions of simulation for the
probabilistic $\pi$-calculus, which arise from different ways of scoping
the name quantification. We show that in order to capture the
testing preorders, one needs to use the ``earliest'' simulation
relation (in analogy to the notion of early (bi)simulation in the
non-probabilistic case). The key ideas in both characterisations are
the notion of a ``characteristic formula'' of a probabilistic process,
and the notion of a ``characteristic test'' for a formula. As in an
earlier work on testing equivalence for the $\pi$-calculus by Boreale and De
Nicola, we extend the language of the $\pi$-calculus with a mismatch
operator, without which the formulation of a characteristic test will
not be possible.
\vskip12pt
Keywords:
Probabilistic $\pi$-calculus; Testing semantics; Bisimulation; Modal logic
\end{abstract}
\section{Introduction}
We consider an extension of a finite version (without replication or recursion)
of the $\pi$-calculus~\cite{Milner92IC2} with a probabilistic choice operator, alongside
the non-deterministic choice operator of the $\pi$-calculus. Such an extension has been
shown to be useful in modelling protocols and their
properties, see, e.g., \cite{Norman09,Chatzikokolakis07}.
The combination of both probabilistic and non-deterministic choice has long been a subject
of study in process theories,
see, e.g., \cite{Hansson90,Yi92,Segala94CONCUR,Deng08LMCS}.
In this paper, we consider a natural
notion of preorders for the probabilistic $\pi$-calculus, based on
the notion of {\em testing}~\cite{Nicola84,Hennessy88}.
In this testing theory, one defines a notion of test, what it means
to apply a test to a process, the outcome of a test, and how the outcomes
of tests can be compared.
In general, the outcome of a test can be any non-empty set, endowed with
a (partial) order; in the case of the original theory, this is simply a two-element lattice,
with the top element representing success and the bottom element representing failure.
In the probabilistic case, the set of outcomes is the unit interval [0,1], denoting
probabilities of success, with the standard mathematical ordering $\leq$.
In the presence of non-determinism, it is natural to consider a set of such probabilities
as the result of applying a test to a process. Two standard approaches for comparing
results of a test are the so-called Hoare preorder, written $\sqsubseteq_{Ho}$, and
the Smyth preorder, $\sqsubseteq_{Sm}$~\cite{Hennessy82}:
\begin{itemize}
\item $O_1 \sqsubseteq_{Ho} O_2$ if for every $o_1 \in O_1$ there exists $o_2 \in O_2$ such that $o_1 \leq o_2.$
\item $O_1 \sqsubseteq_{Sm} O_2$ if for every $o_2 \in O_2$ there exists $o_1 \in O_1$ such that $o_1 \leq o_2.$
\end{itemize}
Correspondingly, these give rise to two semantic preorders for processes:
\begin{itemize}
\item {\em may-testing}:
$P \sqsubseteq_{pmay} Q$ iff for every test $T$, $Apply(T,P) \sqsubseteq_{Ho} Apply(T,Q)$
\item {\em must-testing}:
$P \sqsubseteq_{pmust} Q$ iff for every test $T$, $Apply(T,P) \sqsubseteq_{Sm} Apply(T,Q)$,
\end{itemize}
where $Apply(T,P)$ refers to the result of applying the test $T$ to process $P$.
We derive two characterisations of both may-testing and must-testing: one based
on a notion of probabilistic weak (failure) simulation~\cite{Segala94CONCUR},
and the other based on a modal logic obtained by extending
Milner-Parrow-Walker (MPW) modal logic
for the (non-probabilistic) $\pi$-calculus~\cite{Milner93TCS}.
The probabilistic $\pi$-calculus that we consider here is a variant of
the probabilistic $\pi$-calculus considered in \cite{Chatzikokolakis07},
but extended with the mismatch operator.
As has already been observed in the testing semantics for the non-probabilistic
$\pi$-calculus~\cite{Boreale95IC},
the omission of mismatch would result in a strictly less discriminating test.
This is essentially due to the possibility of two kinds of output transitions
in the $\pi$-calculus,
a bound-output action, which outputs a new name, e.g., $\bar x(w).0$,
and a free-output action, e.g., $\bar x y.0.$ Without the mismatch operator, the two processes
are related via may-testing, because the test cannot
distinguish between output of a fresh
name and output of an arbitrary name (see \cite{Boreale95IC}).
The technical framework used to prove the main results in this paper
is based on previous works
on probabilistic CSP (pCSP)~\cite{Deng07ENTCS,Deng08LMCS}, an extension of Hoare's CSP~\cite{Hoare85}
with a probabilistic choice operator.
This allows us to adapt some proofs and results from
\cite{Deng07ENTCS,Deng08LMCS} that are not calculus-specific.
The name-passing feature of the $\pi$-calculus, however, gives rise to several
difficulties not found in the non-name-passing calculi such as pCSP,
and it consequently requires new techniques to deal with.
For instance, there is not a canonical notion of (weak) simulation in the $\pi$-calculus,
unlike the case with pCSP.
Different variants arise from different ways of scoping the name quantification
in the simulation clause dealing with input transitions, e.g., the ``early''
vs. the ``late'' variants of (bi)simulation~\cite{Milner92IC2}. In the case of weak simulation,
one also gets a ``delay'' variant of (bi)simulation~~\cite{Ferrari95,Sangiorgi96,vanGlabbeek96}.
As we show in Section~\ref{sec:sim}, the right notion of simulation is the early variant, as all other
weak simulation relations are strictly more discriminating than the early one.
Another difficulty is in proving congruence properties, a prerequisite
for the soundness of the (failure) simulation preorders. The possibility of performing a `close'
communication in the $\pi$-calculus requires a combination of closure
under parallel composition and name restriction (see Section \ref{sec:sound}).
We use the so-called ``up-to'' techniques~\cite{Sangiorgi98MSCS}
for non-probabilistic calculi to prove these congruences.
We show that $\sqsubseteq_{pmay}$ coincides with
a simulation preorder $\sqsubseteq_S$ and a preorder $\sqsubseteq_{{\cal L}}$
induced by a modal logic ${\cal L}$ extending the MPW logic.
Dually, the must-testing preorder is shown to coincide with
a failure simulation preorder, $\sqsubseteq_{FS}$, and
a preorder $\sqsubseteq_{{\cal F}}$ induced by a modal logic ${\cal F}$ extending ${\cal L}.$
For technical reasons in proving the completeness result of (failure) simulation,
we make use of testing preorders involving vector-based testing ($\sqsubseteq_{pmay}^\Omega$ and
$\sqsubseteq_{pmust}^\Omega$ below).
The precise relations among these preorders are as follows:
$$
\sqsubseteq_S ~ \subseteq ~ \sqsubseteq_{pmay} ~ = ~ \sqsubseteq_{pmay}^\Omega ~ \subseteq ~ \sqsubseteq_\Lcal ~ \subseteq ~ \sqsubseteq_S
$$
$$
\sqsubseteq_{FS} ~ \subseteq ~ \sqsubseteq_{pmust} ~ = ~ \sqsubseteq_{pmust}^\Omega ~ \subseteq ~ \sqsubseteq_\Fcal ~ \subseteq ~ \sqsubseteq_{FS}.
$$
The proofs of these inclusions are subjects of Section~\ref{sec:sound}, Section~\ref{sec:modal}
and Section~\ref{sec:comp}. Let us highlight the characterisations of may-testing preorder.
As with the case with pCSP \cite{Deng08LMCS}, the key idea to the proof
of the inclusion $\sqsubseteq_\Lcal\; \subseteq\; \sqsubseteq_S$ is to show that for each process $P$, there
exists a {\em characteristic formula} $\varphi_P$ such that
if $Q\models \varphi_P$ then $P\sqsubseteq_S Q$.
The inclusion $\sqsubseteq_{pmay}^\Omega \; \subseteq\; \sqsubseteq_\Lcal$ is proved by showing that
for each formula $\varphi$, there exists a {\em characteristic test}
$T_\varphi$ such that for all process $P$, $P \models \varphi$ iff
$P$ passes the test $T_\varphi$ with some threshold testing outcome.
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\section{Processes and probabilistic distributions}
\label{sec:pi}
We consider an extension of the (finite) $\pi$-calculus
with a probabilistic choice operator, $\pch p$, where $p \in (0,1].$
We shall be using the late version of the operational
semantics, formulated in the reactive style (in the sense of \cite{vanGlabbeek95})
following previous work \cite{Deng07ENTCS,Deng08LMCS}.
The use of the late semantics allows for a straightforward definition
of characteristic formulas (see Section~\ref{sec:modal}), which are used in the
completeness proof.
So our testing equivalence is essentially a ``late'' testing
equivalence. However, as has been shown in \cite{Ingolfsdottir95,Boreale95IC},
late and early testing equivalences coincide for value-passing/name-passing
calculi.
We assume a countably infinite set of {\em names}, ranged over by $a,b,x,y$ etc. Given a name $a$, its {\em co-name} is $\bar a.$
We use $\mu$ to denote a name or a co-name.
Process expressions are generated by the following two-sorted grammar:
\[\begin{array}{rcl}
P & ::= & s \mid P {\pch p} P \\
s & ::= & \nil \mid a(x).s \mid \bar a x.s \mid [x=y]s \mid [x\not = y]s \mid s + s \mid s | s \mid \nu x.s
\end{array}\]
We let $P,Q,...$ range over process terms defined by this grammar, and
$s,t$ range over the subset $S_p$ comprising only the state-based process
terms, i.e. the sub-sort $s$.
The input prefix $a(x)$ and restriction $\nu x$ are name-binding contructs; $x$ in this case
is a bound name. We denote with $fn(P)$ the set of free names in $P$
and $bn(P)$ the set of bound names. The set of names in $P$ (free or bound)
is denoted by $n(P).$
We shall assume that bound names are different from each other and
different from any free names.
Processes are considered equivalent modulo renaming of bound names.
Processes are ranged over by $P$,$Q$,$R$, etc.
We shall refer to our probablistic extension of the $\pi$-calculus
as $\pi_p.$
We shall sometimes use an $n$-ary version of the binary operators.
For example, we use $\bigoplus_{i \in I} p_iP_i$, where $\sum_{i\in I} p_i = 1$,
to denote a process obtained by several applications of the probabilistic choice
operator. Simiarly, $\sum_{i \in I} P_i$ denotes several applications of
the non-deterministic choice operator $+.$
We shall use the $\tau$-prefix, as in $\tau.P$, as an abbreviation of
$\nu x(x(y).\nil \mid \bar x x. P),$ where $x,y \not \in fn(P).$
In this paper, we take the viewpoint that a probabilistic process
represents an unstable state that may probabilistically evolve into
some stable states. Formally, we describe
unstable states as distributions and stable states as state-based
processes.
Note that in a state-based process, probablistic choice can only
appear under input/output prefixes.
The operational semantics of $\pi_p$ will be defined only
for state-based processes.
Probabilistic distributions are ranged over by $\Delta.$
A {\em discrete probabilistic distribution} over a set $S$ is a mapping
$\Delta : S \rightarrow [0,1]$ with $\sum_{s \in S} \Delta(s) = 1.$
The {\em support} of a distribution $\Delta$, denoted
by $\supp \Delta$, is the set $\{s \mid \Delta(s) > 0 \}.$
From now on, we shall restrict to only probabilistic distributions
with finite support, and we let ${\cal D}(S)$ denote the collection
of such distributions over $S.$
If $s$ is a state-based process, then $\pdist s$ denote
the point distribution that maps $s$ to $1.$
For a finite index set $I$, given $p_i$ and distribution $\Delta_i$, for each $i\in I$,
such that $\sum_{i\in I} p_i = 1$, we define another probability distribution
$\sum_{i\in I} p_i \cdot \Delta_i$ as
$(\sum_{i\in I} p_i \cdot \Delta_i)(s) = \sum_{i\in I} p_i \cdot \Delta_i(s),$
where $\cdot$ here denotes multiplication.
We shall sometimes write this distribution as a summation
$p_1 \cdot \Delta_1 + p_2 \cdot \Delta_2 + \ldots + p_n \cdot \Delta_n$
when the index set $I$ is $\{1,\ldots,n\}.$
Probabilistic processes are interpreted
as distributions over state-based processes as follows.
\[\begin{array}{rcl}
\interp s & ::= & \pdist s \ \mbox{ for $s\in S_p$}\\
\interp {P \pch p Q} & ::= & p \cdot \interp P + (1-p) \cdot \interp Q
\end{array}\]
Note that for each process term $P$ the distribution $\interp P$ is
finite, that is it has finite support.
A transition judgment can take one of the following forms:
$$
\one{s}{a(x)}{\Delta}
\qquad
\one{s}{\tau}{\Delta}
\qquad
\one{s}{\bar a x}{\Delta}
\qquad
\one{s}{\bar a(x)}{\Delta}
$$
The action $a(x)$ is called a {\em bound-input action};
$\tau$ is the silent action; $\bar ax$ is a {\em free-output action}
and $\bar a(x)$ is a {\em bound-output action}.
In actions $a(x)$ and $\bar a(x)$, $x$ is a bound name.
Given an action $\alpha$, we denote with $fn(\alpha)$ the set
of free names in $\alpha$, i.e., those names in $\alpha$ which
are not bound names. The set of bound names in $\alpha$
is denoted by $bn(\alpha)$, and the set of all names (free and bound)
in $\alpha$ is denoted by $n(\alpha).$
The free names of a distribution is the union of free names
of its support, i.e.,
$
fn(\Delta) = \bigcup \{fn(s) \mid s \in \supp \Delta \}.
$
A substitution is a mapping from names to names; substitutions are
ranged over by $\rho, \sigma$ and $\theta.$
A substitution $\theta$ is a {\em renaming substitution}
if $\theta$ is an injective map, i.e., $\theta(x) = \theta(y)$
implies $x = y$.
A substitution is extended to a mapping between processes
in the standard way, avoiding capture of free variables.
We use the notation $s[y/x]$ to denote the result of substituting free
occurrences of $x$ in $s$ with $y.$
Substitution is lifted to a mapping between distributions as follows:
$$
\Delta[y/x] (s) =
\sum \{\Delta(s') \mid s'[y/x] = s \}.
$$
It can be verified that $\interp {P[y/x]} = \interp P [y/x]$
for every process $P.$
The operational semantics is given in Figure~\ref{fig:pi}.
The rules for parallel composition and restriction use an obvious notation for
distributing an operator over distributions, for example:
\[\begin{array}{rcl}
(\Delta_1 ~|~ \Delta_2)(s) & = &
\left\{
\begin{array}{ll}
\Delta_1(s_1) \cdot \Delta_2(s_2) & \hbox{ if $s = s_1 | s_2$ } \\
0 & \hbox{ otherwise}
\end{array}
\right .
\\
(\nu x.\Delta)(s) & = &
\left\{
\begin{array}{ll}
\Delta(s') & \hbox{ if $s = \nu x.s'$ } \\
0 & \hbox{ otherwise.}
\end{array}
\right.
\end{array}\]
The symmetric counterparts of \textbf{Sum}, \textbf{Par}, \textbf{Com} and \textbf{Close}
are omitted.
The semantics of $\pi_p$ processes
is presented
in terms of simple probabilistic automata \cite{Segala94CONCUR}.
\begin{figure}
\caption{The operational semantics of $\pi_p$.}
\label{fig:pi}
\end{figure}
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\section{Testing probabilistic processes}
\label{sec:test}
As standard in testing theories~\cite{Nicola84,Hennessy88,Boreale95IC},
to define a test, we introduce
a distinguished name $\omega$ which can only be used in tests and
is not part of the processes being tested.
A {\em test} is just a probabilistic process with possible free
occurrences of the name $\omega$ as channel name in output prefixes,
i.e., a test is a process which may have subterms of the form
$\bar \omega a.P$. Note that the object of the action prefix
(i.e., the name $a$) is irrelevant for the purpose of testing.
Note also that it makes no differences whether the
name $\omega$ appears in input prefixes instead of output prefixes;
the notion of testing preorder will remain the same.
Therefore we shall often simply write $\omega.P$ to denote $\bar \omega a.P$,
and $\one {P} {\omega} {\Delta}$ to denote
$\one {P}{\bar \omega a}{\Delta}.$
The definitions of may-testing preorder, $\sqsubseteq_{pmay}$, and must-testing preorder, $\sqsubseteq_{pmust}$,
have already been given in the introduction, but we left out the definition of
the $Apply$ function. This will be given below.
Following \cite{Deng07ENTCS}, to define the $Apply$ function,
we first define a {\em results-gathering function} ${\mathbb V}: S_p \rightarrow {\cal P}([0,1])$
as follows:
$$
{\mathbb V}(s) =
\left\{
\begin{array}{ll}
\{1\} & \qquad \hbox{if $\one s \omega {}$} \\
\bigcup \{{\mathbb V}(\Delta) \mid \one s \tau \Delta \} & \qquad
\hbox{if $s \not \stackrel{\omega}{\longrightarrow} {}$ but $\one s {\tau} {}$} \\
\{0\} & \qquad \hbox{ otherwise.}
\end{array}
\right.
$$
Here the notation ${\cal P}([0,1])$ stands for the powerset of $[0,1]$, and
we use ${\mathbb V}(\Delta)$ to denote the set of probabilities
$\{\sum_{s\in \supp \Delta} \Delta(s) \cdot p_s \mid p_s\in{\mathbb V}(s)\}$.
The $Apply$ function is then defined as follows:
given a test $T$ and a process $P$,
$$
Apply(T,P) = {\mathbb V}(\interp {\nu \vec x.(T~|~P)})
$$
where $\{\vec x\}$ is the set of free names in $T$ and $P$, excluding $\omega.$
So the process (or rather, the distribution) $\nu \vec x.(T ~|~P)$
can only perform an observable action on $\omega.$
\paragraph{Vector-based testing.}
Following \cite{Deng08LMCS}, we introdude another approach of testing called {\em vector-based testing}, which will play an important role in Section~\ref{sec:comp}.
Let $\Omega$ be a set of fresh success actions different from any
normal channel names. An $\Omega$-test is a $\pi_p$-process, but
allowing subterms $\omega.P$ for any $\omega\in\Omega$. Applying such
a test $T$ to a process $P$ yields a non-empty set of test
outcome-tuples $Apply^\Omega(T,P)\subseteq [0,1]^\Omega$.
For each such tuple, its $\omega$-component gives the probability of successfully performing action $\omega$.
To define a results-gathering function for vector-based testing, we need some auxiliary notations.
For any action $\alpha$ define $\alpha!:[0,1]^\Omega\rightarrow[0,1]^\Omega$ by
\[\alpha!o(\omega)=\left\{\begin{array}{ll}
1 & \mbox{if $\omega=\alpha$}\\
o(\omega) & \mbox{otherwise}
\end{array}\right.\]
so that if $\alpha$ is a success action in $\Omega$ then $\alpha!$ updates the tuple $1$ at that point, leaving it unchanged otherwise, and when $\alpha\not\in\Omega$ the function $\alpha!$ is the identity. For any set $O\subseteq [0,1]^\Omega$, we write $\alpha!O$ for the set $\{\alpha!o \mid o\in O\}$.
For any set $X$ define its \emph{convex closure} $\updownarrow X$ by
\[\updownarrow X ~:=~ \{\sum_{i\in I}p_i\cdot o_i \mid o_i\in X \mbox{ for each $i\in I$ and $\sum_{i\in I}p_i=1$}\}.\]
Here, $I$ is assumed to be a finite index set. Finally, zero vector $\vec{0}$ is given by $\vec{0}(\omega)=0$ for all $\omega\in\Omega$. Let $S_p^\Omega$ be the set of state-based $\Omega$-tests.
\begin{definition}
\label{def:vector-based-results}
The vector-based results-gathering function ${\mathbb V}^\Omega:S_p^\Omega\rightarrow {\cal P}([0,1]^\Omega)$ is given by
\[{\mathbb V}^\Omega(s)~:=~ \left\{\begin{array}{ll}
\updownarrow\bigcup\{\alpha!({\mathbb V}^\Omega(\Delta)) \mid s\ar{\alpha}\Delta\} & \mbox{if $s\rightarrow$}\\
\{\vec{0}\} & \mbox{otherwise}
\end{array}\right.\]
The notation $s\rightarrow$ means that $s$ is not a deadlock state, i.e. there is some $\alpha$ and $\Delta$ such that $s\ar{\alpha}\Delta$.
For any process $P$ and $\Omega$-test $T$, we define
$Apply^\Omega(T,P)$ as ${\mathbb V}^\Omega(\interp{\nu\vec{x}.(T|P)})$, where
$\{\vec x\} = fn(T,P) - \Omega.$
The vector-based may and must preorders are given by
\[\begin{array}{rcl}
P \sqsubseteq_{pmay}^\Omega Q & \mbox{ iff } & \mbox{for all $\Omega$-test $T: Apply^\Omega(T,P) \sqsubseteq_{Ho} Apply^\Omega(T,Q)$}\\
P \sqsubseteq_{pmust}^\Omega Q & \mbox{ iff } & \mbox{for all $\Omega$-test $T: Apply^\Omega(T,P) \sqsubseteq_{Sm} Apply^\Omega(T,Q)$}\\
\end{array}\]
where $\sqsubseteq_{Ho}$ and $\sqsubseteq_{Sm}$ are the Hoare and Smyth preorders on ${\cal P}([0,1]^\Omega)$
generated from $\leq$ index-wise on $[0,1]^\Omega$.
\end{definition}
Notice a subtle difference between the definition of ${\mathbb V}^\Omega$ above and the definition
of ${\mathbb V}$ given earlier. In ${\mathbb V}^\Omega$, we use {\em action-based testing}, i.e.,
the actual execution of $\omega$ constitutes a success.
This is in contrast to the {\em state-based testing} in ${\mathbb V}$, where a success is defined
for a state where a success action $\omega$ is possible, without having to
actually perform the action $\omega.$ In the case where there is no divergence,
as in our case, these two notions of testing coincide; see \cite{Deng08LMCS} for more
details.
The following theorem can be shown by adapting the proof of Theorem 6.6 in \cite{Deng08LMCS}, which states a general property about probabilistic automata \cite{DGMZ07}.
\begin{theorem}\label{thm:multi-uni}
Let $P$ and $Q$ be any $\pi_p$-processes.
\begin{enumerate}
\item $P\sqsubseteq_{pmay}^\Omega Q$ iff $P\sqsubseteq_{pmay} Q$
\item $P\sqsubseteq_{pmust}^\Omega Q$ iff $P\sqsubseteq_{pmust} Q$.
\end{enumerate}
\end{theorem}
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0.143.3
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\section{Simulation and Failure Simulation}
\label{sec:sim}
To define simulation and failure simulation, we need to generalise the transition relations between
states and distributions to those between distributions and distributions.
This is defined via a notion of lifting of a relation.
\begin{definition}[Lifting \cite{Deng09CONCUR}]
\label{def:lifting}
Given a relation ${\cal R} \subseteq S_p \times {\cal D}(S_p)$,
define a {\em lifted relation} $\overline {\cal R} \subseteq {\cal D}(S_p) \times {\cal D}(S_p)$ as the smallest relation that satisfies
\begin{enumerate}
\item $s {\cal R} \Theta$ implies $\pdist{s} \lift{\cal R} \Theta$
\item (Linearity) $\Delta_i \lift{\cal R} \Theta_i$ for all $i\in I$ implies $(\sum_{i\in I}p_i\cdot\Delta_i) \lift{\cal R} (\sum_{i\in I}p_i\cdot\Theta_i)$ for any $p_i\in [0,1]$ with $\sum_{i\in I}p_i = 1$.
\end{enumerate}
\end{definition}
The following is a useful properties of the lifting operation.
\begin{proposition}[\cite{Deng07ENTCS}]
\label{prop:lifting}
Suppose ${\cal R} \subseteq S \times {\cal D}(S)$ and $\sum_{i \in I} p_i = 1.$ If $(\sum_{i\in I} p_i \cdot \Delta_i) \lift {\cal R} \Theta$ then
$\Theta = \sum_{i \in I} p_i \cdot \Theta_i$ for some set of distributions $\Theta_i$ such that
$\Delta_i \lift {\cal R} \Theta_i$ for all $i\in I$.
\end{proposition}
For simplicity of presentation, the lifted version of the
transition relation $\sstep{\alpha}$
will be denoted by the same notation as the unlifted version.
So we shall write $\one \Delta \alpha \Theta$ when $\Delta$
and $\Theta$ are related by the lifted relation from $\sstep{\alpha}.$
Note that in the lifted transition $\one \Delta \alpha \Theta$,
{\em all} processes in $\supp{\Delta}$ must be able to simultaneously
make the transition $\alpha$. For example,
$$
\one{
\frac{1}{2} \cdot \pdist{\bar a x.s} +
\frac{1}{2} \cdot \pdist{\bar a x.t}
}
{\bar a x}
{\frac{1}{2} \cdot \pdist s + \frac{1}{2} \cdot \pdist t}
$$
but the distribution $\frac{1}{2} \cdot \pdist{\bar a x.s} + \frac{1}{2} \cdot \pdist{\bar b x.t}$
will not be able to make that transition.
We need a few more relations to define (failure) simulation:
\begin{itemize}
\item We write $\one s {\hat \tau} \Delta$
to denote either $\one s \tau \Delta$ or $\Delta = \pdist s.$
Its lifted version will be denoted by the same notation, e.g.,
$\one {\Delta_1}{\hat\tau} {\Delta_2}.$
The reflexive-transitive closure of the latter is denoted by
$\stackrel{\hat \tau}{\Longrightarrow}.$
\item$\bigstep {\Delta_1} {\hat \alpha} {\Delta_2}$, for $\alpha \not = \tau$,
iff $\Delta_1 \bstep{\hat\tau} \Delta' \sstep{\alpha} \Delta'' \bstep{\hat\tau} \Delta_2$
for some $\Delta'$ and $\Delta''.$
\item We write $s\barb{a}$ to denote $s\ar{a(x)}$,
and $s\barb{\bar a}$ to denote either $s\ar{\bar{a}(x)}$ or $s\ar{\bar{a}x}$;
$s\not \barb{\mu}$ stands for the negation.
We write $s\not\barb{X}$ when $s \not \! \ar{\tau}$ and $\forall\mu\in X: s\not\barb{\mu}$,
and $\Delta\not\barb{X}$ when $\forall s\in\supp{\Delta}:s\not\barb{X}$.
\end{itemize}
\begin{definition}
\label{def:sim}
A relation ${\cal R} \subseteq S_p \times {\cal D}(S_p)$ is said to be a {\em failure simulation}
if $s {\cal R} \Theta$ implies:
\begin{enumerate}
\item If $\one s {a(x)} {\Delta}$ and $x \not \in fn(s,\Theta)$,
then for every name $w$, there exists $\Theta_1$, $\Theta_2$ and
$\Theta'$ such that
$$\Theta \bstep{\hat \tau} \Theta_1 \sstep {a(x)} {\Theta_2},
\qquad \Theta_2[w/x] \bstep{\hat \tau} \Theta',
\qquad \hbox{ and } \qquad (\Delta[w/x]) ~ \overline {\cal R} ~ \Theta'.
$$
\item If $\one s \alpha \Delta$ and $\alpha$ is
not an input action, then
there exists $\Theta'$ such that $\bigstep \Theta {\hat \alpha} {\Theta'}$
and $\Delta ~ \overline {\cal R} ~ \Theta'$
\item If $s\not\barb{X}$ then there exists $\Theta'$ such that $\Theta\dar{\hat \tau} \Theta'\not\barb{X}$.
\end{enumerate}
We denote with $\triangleleft_{FS}$ the largest failure simulation relation. Similarly, we define \emph{simulation} and $\triangleleft_S$ by dropping the third clause above.
The {\em simulation preorder} $\sqsubseteq_S$ and
\emph{failure simulation preorder} $\sqsubseteq_{FS}$
on process terms are defined by letting
\[\begin{array}{rll}
P \sqsubseteq_S Q \mbox{ iff } \mbox{there is a distribution
$\Theta$ with $\bigstep {\interp Q}{\hat \tau} \Theta$
and $\interp P ~ \lift{\triangleleft_S} ~ \Theta.$}\\
P \sqsubseteq_{FS} Q \mbox{ iff } \mbox{there is a distribution
$\Theta$ with $\bigstep {\interp P}{\hat \tau} \Theta$
and $\interp Q ~ \lift{\triangleleft_{FS}} ~ \Theta.$}
\end{array}\]
\end{definition}
Notice the rather unusual clause for input action, where no silent
action from $\Theta_2$ is permitted after the input transition. This is reminiscent of
the notion of {\em delay (bi)simulation}~\cite{Ferrari95,Sangiorgi96,vanGlabbeek96}.
If instead of that clause, we simply require $\Theta \bstep {\widehat{a(x)}} {\Theta''}$
and $\Delta[w/x] ~ \overline {\cal R} ~ \Theta''[w/x]$ then,
in the presence of mismatch, simulation is not sound w.r.t.
the may-testing preorder, even in the non-probabilistic case.
Consider, for example, the following
processes:
$$
P = a(x).\bar a b
\qquad
Q = a(x).[x \not = c] \tau.\bar a b
$$
where we recall that $\tau.R$ abbreviates $\nu z.(z(u) ~|~ \bar z z.R)$ for
some $z \not \in fn(R).$
The process $P$ can make an input transition, and regardless of the
value of the input, it can then output $b$ on channel $a.$
Notice that for $Q$, we have
$$
Q \sstep{a(x)} [x \not = c] \tau. \bar a b \sstep{\tau} \nu z(0 ~|~ \bar a b) = Q'.
$$
$Q'$ can also outputs $b$ on channel $a$, so under this alternative definition,
$Q$ can simulate $P.$ But $P \not \sqsubseteq_{pmay} Q$, as the test
$\bar a c.a(y).\omega$ will distinguish them.
This issue has also appeared in the theory of weak (late) bisimulation
for the non-probabilistic $\pi$-calculus; see, e.g., \cite{Sangiorgi01book}.
Note that the above definition of $\triangleleft_S$ is what is usually called the
``early'' simulation.
One can obtain different variants of ``late'' simulation using different
alternations of the universal quantification on names and the existential quantifications on distributions in
clause 1 of Definition~\ref{def:sim}.
Any of these variants leads to a strictly more discriminating simulation. To see why,
consider the weaker of such late variants, i.e., one in which the universal quantifier
on $w$ comes after the existential quantifier on $\Theta_1$:
\begin{quote}
If $\one s {a(x)} {\Delta}$ and $x \not \in fn(s,\Theta)$,
then there exists $\Theta_1$ such that for every name $w$, there exist $\Theta_2$ and
$\Theta'$ such that
$$\Theta \bstep{\hat \tau} \Theta_1 \sstep {a(x)} {\Theta_2},
\qquad \Theta_2[w/x] \bstep{\hat \tau} \Theta',
\qquad \hbox{ and } \qquad (\Delta[w/x]) ~ \overline {\cal R} ~ \Theta'.
$$
\end{quote}
Let us denote this variant with $\sqsubseteq_{S'}.$ Consider the following processes:
$$
P = a(x).\bar b x.\nil + a(x).\nil + a(x).[x=z]\bar b x.\nil
\qquad
Q = \tau.a(x).\bar b x.\nil + \tau.a(x).\nil
$$
It is easy to see that $P \sqsubseteq_S Q$ but $P {\not \sqsubseteq}_{S'} Q.$
If we drop the silent transitions $\Theta_2[w/x] \bstep{\hat \tau} \Theta'$
in clause (1) of Definition~\ref{def:sim}, i.e., we let $\Theta' = \Theta_2[w/x]$
(hence, we get a delay simulation), then again we get a strictly stronger relation than $\sqsubseteq_S$.
Let us refer to this stronger relation as $\sqsubseteq_{D}$.
Let $P$ be $a(x).(c {\pch {\frac{1}{2}}} d)$ and let $Q$ be $a(x).\tau.(c {\pch {\frac{1}{2}}} d).$
Here we remove the parameters in the input prefixes $c$ and $d$ to simplify presentation.
Again, it can be shown that $P \sqsubseteq_S Q$ but $P ~ {\not \sqsubseteq}_{D} ~ Q.$ For the latter to hold,
we would have to prove
$
\frac{1}{2} \cdot \pdist c + \frac{1}{2} \cdot \pdist d ~ \overline{\triangleleft_S} ~ \pdist{\tau.(c {\pch {\frac{1}{2}}} d)},
$
which is impossible.
Note that (failure) simulation is a relation between processes and distributions,
rather than between processes, so it is not immediately obvious that it is a preorder.
This is established in Corollary~\ref{cor:sim.fail.preorder} below, whose proof requires a series of lemmas.
In the following, when we apply a substitution to an action,
we assume that the substitution affects both the free and the bound names
in the action. For example, if $\alpha = a(x)$ and
$\theta = [b/a, y/x]$ then $\alpha\theta = b(y).$
However, application of a substitution to processes or distributions
must still avoid capture.
\begin{lemma}\label{lm:rename}
Suppose $\sigma$ is a renaming substitution.
\begin{enumerate}
\item If $\one s \alpha \Delta$ then $\one {s\sigma}{\alpha \sigma}{\Delta\sigma}.$
\item If $\bigstep {\Delta} {\hat \alpha} {\Delta'}$
then $\bigstep {\Delta\sigma}{\hat \alpha\sigma} {\Delta'\sigma}.$
\end{enumerate}
\end{lemma}
\begin{lemma}
\label{lm:lifted-trans-rename}
Let $I$ be a finite index set, and let $\sum_{i \in I} p_i = 1.$
Suppose $\one {s_i} {a(x_i)} {\Delta_i}$ for each $i \in I$.
Let $x$ be a fresh name not occuring in any of $s_i$, $a(x_i)$ or $\Delta_i.$
Then
$$
\one {\sum_{i \in I} p_i \cdot \pdist {s_i}} {a(x)}{\sum_{i \in I} p_i \cdot \Delta_i[x/x_i]}.
$$
\end{lemma}
Given the above lemma, given transitions $\one {s_i}{a(x_i)}{\Delta_i}$, we can always assume that,
all the $x_i$'s are the same fresh name, so that when lifting those transitions to
distributions, we shall omit the explicit renaming of individual $x_i.$ This will simplify the
presentation of the proofs in the following. The same remark applies to bound output transitions.
\begin{lemma}
\label{lm:bigstep-dist}
Suppose $\sum_{i \in I} p_i = 1$ and $\Delta_i \bstep {\hat \alpha} \Phi_i$ for each $i \in I,$ where
$I$ is a finite index set. Then
$$
\sum_{i \in I} p_i \cdot \Delta_i \bstep {\hat \alpha} \sum_{i \in I} p_i \cdot \Phi_i.
$$
\end{lemma}
\begin{proof}
Same as in the proof of Lemma 6.6. in \cite{Deng07ENTCS}. \qed
\end{proof}
\begin{lemma}
\label{lm:sim-refl}
For every state-based process $s$, we have $s \triangleleft_S \pdist s$ and $s \triangleleft_{FS} \pdist s.$
\end{lemma}
\begin{proof}
Let ${\cal R} \subseteq S_p \times {\cal D}(S_p)$ be the relation
defined as follows: $s ~ {\cal R} ~ \Theta$ iff $\Theta = \pdist s.$
It is easy to see that ${\cal R}$ is a simulation and also a failure simulation.
\qed
\end{proof}
\begin{lemma}
\label{lm:sim-like1}
Suppose $\Delta ~ \triangleleft_Sl_S ~ \Phi$ and $\one {\Delta} {\alpha} {\Delta'}$, where $\alpha$
is either $\tau$, a free action or a bound output action. Then
$\one \Phi {\hat \alpha} {\Phi'}$ for some $\Phi'$ such that $\Delta' ~ \triangleleft_Sl_S ~ \Phi'.$
\end{lemma}
\begin{proof}
Similar to the proof of Lemma 6.7 in \cite{Deng07ENTCS}. \qed
\end{proof}
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\begin{lemma}
\label{lm:sim-refl}
For every state-based process $s$, we have $s \triangleleft_S \pdist s$ and $s \triangleleft_{FS} \pdist s.$
\end{lemma}
\begin{proof}
Let ${\cal R} \subseteq S_p \times {\cal D}(S_p)$ be the relation
defined as follows: $s ~ {\cal R} ~ \Theta$ iff $\Theta = \pdist s.$
It is easy to see that ${\cal R}$ is a simulation and also a failure simulation.
\qed
\end{proof}
\begin{lemma}
\label{lm:sim-like1}
Suppose $\Delta ~ \triangleleft_Sl_S ~ \Phi$ and $\one {\Delta} {\alpha} {\Delta'}$, where $\alpha$
is either $\tau$, a free action or a bound output action. Then
$\one \Phi {\hat \alpha} {\Phi'}$ for some $\Phi'$ such that $\Delta' ~ \triangleleft_Sl_S ~ \Phi'.$
\end{lemma}
\begin{proof}
Similar to the proof of Lemma 6.7 in \cite{Deng07ENTCS}. \qed
\end{proof}
\begin{lemma}
\label{lm:sim-like2}
Suppose $\Delta ~ \triangleleft_Sl_S ~ \Phi$ and $\one{\Delta}{a(x)}{\Delta'}$.
Then for all name $w$, there exist $\Psi_1$, $\Psi_2$ and $\Psi$ such that
$$
\Phi \bstep {\hat \tau} \Psi_1 \sstep {a(x)} \Psi_2,
\qquad
\Psi_2[w/x] \bstep {\hat \tau} \Psi,
\qquad \mbox{ and }
\qquad
(\Delta'[w/x]) ~ \triangleleft_Sl_S ~ \Psi.
$$
\end{lemma}
\begin{proof}
From $\Delta ~ \triangleleft_Sl_S ~ \Phi$ we have that
\begin{equation}
\label{eq:sim-like2-1}
\Delta = \sum_{i \in I} p_i \cdot \pdist{s_i}, \qquad s_i \triangleleft_S \Phi_i, \qquad
\Phi = \sum_{i \in I} p_i \cdot \Phi_i.
\end{equation}
and from $\Delta \sstep {a(x)} \Delta'$ we have:
\begin{equation}
\label{eq:sim-like2-2}
\Delta = \sum_{j \in J} q_j \cdot \pdist{t_j}, \qquad t_j \sstep {a(x)} \Theta_j,
\qquad
\Delta' = \sum_{j \in J} q_j \cdot \Theta_j.
\end{equation}
We assume w.l.o.g. that all $p_i$ and $q_j$ are non-zero.
Following \cite{Deng07ENTCS}, we define two index sets:
$I_j = \{ i \in I \mid s_i = t_j \}$ and $J_i = \{ j \in J \mid t_j = s_i \}.$
Obviously, we have
\begin{equation}
\label{eq:sim-like2-3}
\{(i,j) \mid i \in I, j \in J_i\} = \{(i,j) \mid j \in J, i \in J_i \}, \quad \mbox{and}
\end{equation}
\begin{equation}
\label{eq:sim-like2-4}
\Delta(s_i) = \sum_{j \in J_i} q_j
\qquad
\Delta(t_j) = \sum_{i \in I_j} p_i.
\end{equation}
It follows from (\ref{eq:sim-like2-4}) that we can rewrite $\Phi$ as
$$
\Phi = \sum_{i \in I} \sum_{j \in J_i} \frac{p_i \cdot q_j}{\Delta(s_i)} \cdot \Phi_i.
$$
Note that $s_i = t_j$ when $j \in I_i.$
Since $s_i \triangleleft_S \Phi_i$, and $s_i = t_j \sstep {a(x)}{\Theta_j}$, we have,
given any name $w$, some $\Phi_{ij}^1$, $\Phi_{ij}^2$ and $\Phi_{ij}$ such that:
\begin{equation}
\label{eq:sim-like2-5}
\Phi_i \bstep {\hat \tau} \Phi_{ij}^1 \sstep {a(x)} \Phi_{ij}^2, \qquad
\Phi_{ij}^2[w/x] \bstep{\hat \tau}{\Phi_{ij}}, \qquad
\Theta_j[w/x] ~ \triangleleft_Sl_S ~ \Phi_{ij}.
\end{equation}
Let
$$
\Psi_1 = \sum_{i \in I} \sum_{j \in J_i} \frac{p_i \cdot q_j}{\Delta(s_i)} \cdot \Phi_{ij}^1
\qquad
\Psi_2 = \sum_{i \in I} \sum_{j \in J_i} \frac{p_i \cdot q_j}{\Delta(s_i)} \cdot \Phi_{ij}^2
\qquad
\Psi = \sum_{i \in I} \sum_{j \in J_i} \frac{p_i \cdot q_j}{\Delta(s_i)} \cdot \Phi_{ij}.
$$
Lemma~\ref{lm:bigstep-dist} and (\ref{eq:sim-like2-5}) above give us:
$$
\Phi = \sum_{i \in I} \sum_{j \in J_i} \frac{p_i \cdot q_j}{\Delta(s_i)} \cdot \Phi_{i}
\bstep {\hat \tau}
\Psi_1
\sstep {a(x)}
\Psi_2
\qquad
\Psi_2[w/x]
\bstep {\hat \tau}
\Psi
$$
It remains to show that $\Delta'[w/x] ~ \triangleleft_Sl_S ~ \Psi.$
\begin{align*}
\Delta'[w/x]
& = \sum_{j \in J} q_j \cdot \Theta_j[w/x] & \\
& = \sum_{j \in J} q_j \cdot \sum_{i \in I_j} \frac{p_i}{\Delta(t_j)} \cdot \Theta_j[w/x] & \mbox{ using (\ref{eq:sim-like2-4})} \\
& = \sum_{j \in J} \sum_{i \in I_j} \frac{p_i \cdot q_j}{\Delta(t_j)} \cdot \Theta_j[w/x] \\
& = \sum_{i \in I} \sum_{j \in J_i} \frac{p_i \cdot q_j}{\Delta(s_i)} \cdot \Theta_j[w/x] & \mbox{ using (\ref{eq:sim-like2-3}) } \\
& \triangleleft_Sl_S ~ \sum_{i \in I} \sum_{j \in J_i} \frac{p_i \cdot q_j}{\Delta(t_j)} \cdot \Phi_{ij} = \Psi &
\mbox{ using (\ref{eq:sim-like2-5}) and linearity of $\triangleleft_Sl_S$}
\end{align*}
\qed
\end{proof}
\begin{lemma}
\label{lm:sim-like3}
Suppose $\Delta ~ \triangleleft_Sl_S ~ \Phi$ and $\Delta ~ \bstep {\hat \alpha} \Delta'$,
where $\alpha$ is either $\tau$, a free action or a bound output.
Then $\Phi \bstep {\hat \alpha} \Phi'$ for some $\Phi'$ such that
$\Delta' ~ \triangleleft_Sl_S ~ \Phi'$.
\end{lemma}
\begin{proof}
Similar to the proof of Lemma 6.8 in \cite{Deng07ENTCS}.
\qed
\end{proof}
\begin{proposition}\label{prop:sim-refl-trans}
The relation $\triangleleft_Sl_S$ is reflexive and transitive.
\end{proposition}
\begin{proof}
Reflexivity of $\triangleleft_Sl_S$ follows from Lemma~\ref{lm:sim-refl}.
To show transitivity, let us define a relation ${\cal R} \subseteq S_p \times {\cal D}(S_p)$
as follows:
$
s ~ {\cal R} ~ \Theta
$
iff there exists $\Delta$ such that
$s ~ \triangleleft_S ~ \Delta$ and $\Delta ~ \triangleleft_Sl_S ~ \Theta.$
We show that ${\cal R}$ is a simulation.
But first, we claim that
$\Theta ~ \triangleleft_Sl_S ~ \Delta ~ \triangleleft_Sl_S ~ \Phi$ implies $\Theta ~ \overline {\cal R} ~ \Phi.$
This can be proved similarly as in the case of CSP (see the proof of Proposition 6.9
in \cite{Deng07ENTCS}).
Now to show that ${\cal R}$ is a simulation, there are two cases to consider.
Suppose $s ~ {\cal R} ~ \Phi$, i.e., $s ~ \triangleleft_S ~ \Delta ~ \triangleleft_Sl_S ~ \Phi.$
\begin{itemize}
\item Suppose $s \sstep {\alpha} \Theta$, where $\alpha$ is either $\tau$,
a free action or a bound output action.
From $s ~ \triangleleft_S ~ \Delta$, we have
\begin{equation}
\label{eq:sim-trans-1}
\Delta \bstep {\hat \alpha} \Delta' \qquad \mbox{ and } \qquad
\Theta ~ \triangleleft_Sl_S ~ \Delta'.
\end{equation}
By Lemma~\ref{lm:sim-like3} and (\ref{eq:sim-trans-1}), we have
$\Phi \bstep{\hat \alpha} \Phi'$ and $\Delta' ~ \triangleleft_Sl_S ~ \Phi'$,
and by the above claim and (\ref{eq:sim-trans-1}), $\Theta ~ \overline {\cal R} ~ \Phi'$.
\item Suppose $s \sstep {a(x)} \Theta,$ so we have: for all $w$, there exist
$\Delta_1$, $\Delta_2$, and $\Delta'$ such that
\begin{equation}
\Delta \bstep{\hat \tau} \Delta_1 \sstep{a(x)} \Delta_2, \qquad
\Delta_2[w/x] \bstep{\hat \tau} \Delta', \qquad \mbox{ and }
\Theta[w/x] ~ \triangleleft_Sl_S ~ \Delta'.
\end{equation}
Since $\Delta ~ \triangleleft_Sl_S ~ \Phi$, by Lemma~\ref{lm:sim-like3} we have
$\Phi \bstep {\hat \tau} \Phi_1$ and $\Delta_1 ~ \triangleleft_Sl_S ~ \Phi_1.$
And since $\Delta_1 \sstep{a(x)} \Delta_2$, by Lemma~\ref{lm:sim-like2},
for all $w$, there exist $\Phi_2$, $\Phi_3$ and $\Phi_4$ such that:
$$
\Phi_1 \bstep{\hat\tau} \Phi_2 \sstep{a(x)} \Phi_3,
\qquad
\Phi_3[w/x] \bstep{\hat\tau} \Phi_4,
\qquad
\Delta_2[w/x] ~ \triangleleft_Sl_S ~ \Phi_4.
$$
Lemma~\ref{lm:sim-like3}, together with $\Delta_2[w/x] ~ \triangleleft_Sl_S ~ \Phi_4$
and $\Delta_2[w/x] \bstep{\hat\tau} \Delta'$, implies that
$\Phi_4 \bstep{\hat\tau} \Phi_5$ and $\Delta' ~ \triangleleft_Sl_S ~ \Phi_5$ for some
$\Phi_5.$
From $\Theta[w/x] ~ \triangleleft_Sl_S ~ \Delta'$ and $\Delta' ~ \triangleleft_Sl_S ~ \Phi_5$,
we have $\Theta[w/x] ~ \overline {\cal R} ~ \Phi_5.$
Putting it all together, we have:
$$
\Phi \bstep{\hat\tau} \Phi_2 \sstep{a(x)} \Phi_3,
\qquad
\Phi_3[w/x] \bstep{\hat\tau} \Phi_5,
\qquad
\Theta[w/x] ~ \overline {\cal R} ~ \Phi_5.
$$
\end{itemize}
Thus ${\cal R}$ is indeed a simulation.
\qed
\end{proof}
\begin{proposition}
\label{prop:failsim-refl-trans}
The relation $\triangleleft_Sl_{FS}$ is reflexive and transitive.
\end{proposition}
\begin{proof}
Reflexivity of $\triangleleft_Sl_{FS}$ follows from Lemma~\ref{lm:sim-refl}.
To show transivity, we use a similar argument as in
the proof of Proposition~\ref{prop:sim-refl-trans}:
define ${\cal R}$ such that
$s ~ {\cal R} ~ \Theta$ iff there exists $\Delta$ such that
$s ~ \triangleleft_{FS} ~ \Delta$ and $\Delta ~ \triangleleft_Sl_{FS} ~ \Theta.$ We show that ${\cal R}$ is a failure simulation.
Suppose $s ~ {\cal R} ~ \Theta$.
The matching up of transitions between $s$ and $\Theta$
is proved similarly to the case with simulation, by
proving the analog of Lemmas~\ref{lm:sim-like1} - \ref{lm:sim-like3} for failure simulation.
It then remains to show that
when $s \not \barb{X}$ then there exists
$\Theta'$ such that $\Theta \dar{\hat \tau} \Theta' \not \barb{X}.$
Since $s ~{\cal R} ~ \Theta$, by the definition of ${\cal R}$,
we have a $\Delta$ s.t. $s ~ \triangleleft_{FS} ~ \Delta$
and $\Delta ~ \triangleleft_Sl_{FS} ~ \Theta.$ The former
implies that $\Delta \dar{\hat \tau} \Delta' \not \barb{X}$,
for some $\Delta'$.
It can be shown that, using arguments similar to
the proof of Lemma~\ref{lm:sim-like3} that
$\Theta \dar{\hat\tau} \Theta'$ for some $\Theta'$
such that $\Delta' ~ \triangleleft_Sl_{FS} \Theta'.$
Suppose $\supp {\Delta'} = \{s_i\}_{i \in I},$ i.e.,
$\Delta' = \sum_{i\in I} p_i \cdot \pdist{s_i}$
with $\sum_{i\in I} p_i = 1.$
Obviously, $s_i \not \barb{X}$ for each $i\in I.$
By Proposition~\ref{prop:lifting},
$\Theta = \sum_{i\in I} p_i \cdot \Theta_i$
for some distributions $\Theta_i$ such that
$\pdist{s_i} ~ \triangleleft_Sl_{FS} ~ \Theta_i.$
The latter implies, by Definition~\ref{def:lifting},
that $s_i ~ \triangleleft_{FS} ~ \Theta_i.$
Since $s_i \not \barb{X}$, it follows that
$\Theta_i \dar{\hat \tau} \Theta_i' \not \barb{X}$,
for some $\Theta_i'.$
Thus $\Theta \dar{\hat \tau} (\sum_{i \in I} p_i \cdot \Theta_i) \not \barb{X}.$
\qed
\end{proof}
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\begin{proposition}
\label{prop:failsim-refl-trans}
The relation $\triangleleft_Sl_{FS}$ is reflexive and transitive.
\end{proposition}
\begin{proof}
Reflexivity of $\triangleleft_Sl_{FS}$ follows from Lemma~\ref{lm:sim-refl}.
To show transivity, we use a similar argument as in
the proof of Proposition~\ref{prop:sim-refl-trans}:
define ${\cal R}$ such that
$s ~ {\cal R} ~ \Theta$ iff there exists $\Delta$ such that
$s ~ \triangleleft_{FS} ~ \Delta$ and $\Delta ~ \triangleleft_Sl_{FS} ~ \Theta.$ We show that ${\cal R}$ is a failure simulation.
Suppose $s ~ {\cal R} ~ \Theta$.
The matching up of transitions between $s$ and $\Theta$
is proved similarly to the case with simulation, by
proving the analog of Lemmas~\ref{lm:sim-like1} - \ref{lm:sim-like3} for failure simulation.
It then remains to show that
when $s \not \barb{X}$ then there exists
$\Theta'$ such that $\Theta \dar{\hat \tau} \Theta' \not \barb{X}.$
Since $s ~{\cal R} ~ \Theta$, by the definition of ${\cal R}$,
we have a $\Delta$ s.t. $s ~ \triangleleft_{FS} ~ \Delta$
and $\Delta ~ \triangleleft_Sl_{FS} ~ \Theta.$ The former
implies that $\Delta \dar{\hat \tau} \Delta' \not \barb{X}$,
for some $\Delta'$.
It can be shown that, using arguments similar to
the proof of Lemma~\ref{lm:sim-like3} that
$\Theta \dar{\hat\tau} \Theta'$ for some $\Theta'$
such that $\Delta' ~ \triangleleft_Sl_{FS} \Theta'.$
Suppose $\supp {\Delta'} = \{s_i\}_{i \in I},$ i.e.,
$\Delta' = \sum_{i\in I} p_i \cdot \pdist{s_i}$
with $\sum_{i\in I} p_i = 1.$
Obviously, $s_i \not \barb{X}$ for each $i\in I.$
By Proposition~\ref{prop:lifting},
$\Theta = \sum_{i\in I} p_i \cdot \Theta_i$
for some distributions $\Theta_i$ such that
$\pdist{s_i} ~ \triangleleft_Sl_{FS} ~ \Theta_i.$
The latter implies, by Definition~\ref{def:lifting},
that $s_i ~ \triangleleft_{FS} ~ \Theta_i.$
Since $s_i \not \barb{X}$, it follows that
$\Theta_i \dar{\hat \tau} \Theta_i' \not \barb{X}$,
for some $\Theta_i'.$
Thus $\Theta \dar{\hat \tau} (\sum_{i \in I} p_i \cdot \Theta_i) \not \barb{X}.$
\qed
\end{proof}
\begin{corollary}\label{cor:sim.fail.preorder}
The relations $\sqsubseteq_S$ and $\sqsubseteq_{FS}$ are preorders.
\end{corollary}
\begin{proof}
The fact that $\sqsubseteq_S$ is a preorder
follows from Lemma~\ref{lm:sim-like3} and Proposition~\ref{prop:sim-refl-trans}.
Similar arguments hold for $\sqsubseteq_{FS}$, using an analog of
Lemma~\ref{lm:sim-like3} and Proposition~\ref{prop:failsim-refl-trans}.
\qed
\end{proof}
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\section{Soundness of the simulation preorders}
\label{sec:sound}
In proving soundness of the simulation preorders with respect to
testing preorders, we first need to prove
certain congruence properties, i.e., closure
under restriction and parallel composition. For this, it is helpful
to consider a slightly more general definition of simulation,
which incorporates another relation. This technique, called
the {\em up-to} technique, has been used in the literature
to prove congruence properties of various (pre-)order for
the $\pi$-calculus~\cite{Sangiorgi98MSCS}.
\begin{definition}[Up-to rules]
Let ${\cal R} \subseteq S_p \times {\cal D}(S_p).$
Define the relation ${\cal R}^t$ where $t \in \{r, \nu, p\}$
as the smallest relation which satisfies
the closure rule for $t$, given below (where $\sigma$ is a renaming substitution):
$$
\infer[r]
{s \sigma ~{\cal R}^r ~ \Delta \sigma}
{s ~ {\cal R} ~ \Delta}
\qquad
\infer[\nu]
{(\nu \vec x.s) ~ {\cal R}^\nu ~ (\nu \vec x.\Delta)}
{s ~ {\cal R} ~ \Delta}
\qquad
\infer[p]
{(s_1 ~|~ s_2) ~ {\cal R}^p ~ (\Delta_1 ~|~ \Delta_2)}
{
s_1 ~ {\cal R} ~ \Delta_1
&
s_2 ~ {\cal R} ~ \Delta_2
}
$$
\end{definition}
\begin{definition}[(Failure) Simulation up-to]
A relation ${\cal R} \subseteq S_p \times {\cal D}(S_p)$ is said to be a {\em (failure) simulation up to
renaming} (likewise, restriction and parallel composition) if
it satisfies the clauses 1, and 2, (and 3 for failure simulation)
in Definition~\ref{def:sim}, but with
$\overline {\cal R}$ in the clauses replaced by
$\overline {{\cal R}^r}$ (respectively, $\overline{{\cal R}^\nu}$ and
$\overline{{\cal R}^p}$).
\end{definition}
It is easy to see that ${\cal R} \subseteq {\cal R}^t$ for any $t \in \{r,\nu\}$
(i.e., via the identity relation as renaming substitution in the former, and via the empty restriction
in the latter). The following lemma is then an easy consequence.
\begin{lemma}
\label{lm:sim-upto}
If ${\cal R}$ is a (failure) simulation then it is a (failure) simulation up-to renaming,
and also a (failure) simulation up to restriction.
\end{lemma}
Our objective is really to show that simulation up-to parallel composition
is itself a simulation. This would then entail that (the lifted) simulation
is closed under parallel composition, from which soundness w.r.t. may-testing
follows. We prove this indirectly in three stages:
\begin{itemize}
\item simulation up-to renaming is a simulation;
\item simulation up-to restriction is a simulation up-to renaming (hence also
a simulation by the previous item);
\item and, finally, simulation up-to parallel composition is a simulation up-to
restriction.
\end{itemize}
\subsection{Up to renaming}
Note that as a consequence of Lemma~\ref{lm:rename} (1),
given an injective renaming substitution
$\sigma$, we have:
if $\one {s\sigma}{\alpha'}{\Delta'}$
then there exists $\alpha$ and $\Delta$ such that $\alpha' = \alpha \sigma$,
$\Delta' = \Delta\sigma$ and $\one s \alpha \Delta.$ This is proved by
simply applying Lemma~\ref{lm:rename} (1) to $\one {s\sigma}{\alpha'}{\Delta'}$
using the inverse of $\sigma$.
In the following, we shall write ${\cal R}^{tt}$ to denote
$({\cal R}^t)^t$, i.e., the result of applying the up-to closure
rule $t$ twice to ${\cal R}.$
\begin{lemma}
${\cal R}^{rr} = {\cal R}^r.$
\end{lemma}
\begin{lemma}
\label{lm:lift-renaming}
If $\Delta_1 ~ \overline{{\cal R}^r} ~ \Delta_2$ then
$(\Delta_1\sigma) ~ \overline {{\cal R}^r} ~ (\Delta_2 \sigma)$
for any renaming substitution $\sigma.$
\end{lemma}
\begin{proof}
This follows from the fact that $\Delta_1 ~ \overline{{\cal R}^r} ~ \Delta_2$
implies $\Delta_1\sigma ~ \overline{{\cal R}^{rr}} ~ \Delta_2\sigma$
and that ${\cal R}^{rr} = {\cal R}^r.$
\qed
\end{proof}
\begin{lemma}
\label{lm:upto-renaming}
If ${\cal R}$ is a (failure) simulation up to renaming, then
${\cal R}^r \subseteq \triangleleft_S$ (respectively, ${\cal R}^r \subseteq \triangleleft_{FS}$).
\end{lemma}
\begin{proof}
Suppose ${\cal R}$ is a simulation. It is enough to show that ${\cal R}^r$ is a simulation.
So suppose $s ~{\cal R}^r ~ \Delta$ and $\one {s} \alpha \Theta.$
By the definition of ${\cal R}^r$, $s = s'\sigma$ and $\Delta = \Delta'\sigma$
for some renaming substitution $\sigma$ and some $s'$
and $\Delta'$ such that $s' ~ {\cal R} ~ \Delta'.$
There are several cases to consider depending on the type of $\alpha$.
\begin{itemize}
\item $\alpha$ is $\tau$ or a free action:
By Lemma~\ref{lm:rename} (1) we have $\one {s'} {\alpha'} {\Theta'}$
for some $\alpha'$ and $\Theta'$ such that $\alpha = \alpha'\sigma$
and $\Theta = \Theta'\sigma.$ Since ${\cal R}$ is a simulation up to renaming,
$s' {\cal R} \Delta'$ implies that
$\bigstep {\Delta'}{\hat{\alpha'}}{\Delta_1}$
and $\Theta' ~ \overline{{\cal R}^r} ~ \Delta_1.$
The former implies, by Lemma~\ref{lm:rename} (2), that
$\bigstep{\Delta}{\hat \alpha}{\Delta_2}$
for some $\Delta_2$ such that $\Delta_2 = \Delta_1\sigma,$
while the latter implies, by Lemma~\ref{lm:lift-renaming},
that $\Theta = (\Theta'\sigma) ~ \overline {{\cal R}^r} ~ (\Delta_1\sigma) = \Delta_2.$
\item $\alpha = a(x)$ for some $a$ and $x$:
In this case, $x \not \in fn(s,\Delta),$ so we can assume, without loss of generality,
that $x$ does not occur in $\sigma.$
Using a similar argument as in the previous case, we have that
$\one {s'} {b(x)} {\Theta'}$
for some $b$ and $\Theta'$ such that $\sigma(b) = a$
and $\Theta = \Theta'\sigma.$
Since ${\cal R}$ is a simulation up to renaming,
$s' {\cal R} \Delta'$ implies that for every name $w$, there exist
$\Delta_w^1$, $\Delta_w^2$ and $\Delta_w$ such that:
\begin{equation}
\label{eq:ren1}
\Delta' \bstep {\hat \tau} \Delta_w^1 \sstep {b(x)} \Delta_w^2,
\qquad
\Delta_w^2[w/x] \bstep {\hat \tau} \Delta_w, \quad \mbox{ and }
\end{equation}
\begin{equation}
\label{eq:ren2}
\Theta'[w/x] ~ \overline{{\cal R}^r} ~ \Delta_w.
\end{equation}
Let $\Phi_1 = \Delta_w^1\sigma$, $\Phi_2 = \Delta_w^2 \sigma$
and $\Phi = \Delta_w\sigma.$
From (\ref{eq:ren1}) and Lemma~\ref{lm:rename} (2) we get:
$$
\Delta = \Delta' \sigma \bstep {\hat \tau}
\Delta_w^1\sigma = \Phi_1 \sstep{a(x)} \Delta_w^2 \sigma = \Phi_2.
$$
By (\ref{eq:ren1}), the freshness assumption of $x$ w.r.t. $\sigma$,
and Lemma~\ref{lm:rename} (2), we get
$$
\Phi_2[w/x] = \Delta_w^2\sigma [w/x] = \Delta_w^2 [w/x] \sigma
\bstep{\hat\tau} \Delta_w\sigma = \Phi.
$$
Finally, by (\ref{eq:ren2}) and Lemma~\ref{lm:lift-renaming},
$
\Theta[w/x] = \Theta'\sigma [w/x] =
\Theta'[w/x] \sigma ~ \overline{{\cal R}^r} ~ \Delta_w\sigma = \Phi.
$
\item $\alpha = \bar a(x)$: This case can be proved similarly to
the previous cases.
\end{itemize}
For the case where ${\cal R}$ is a failure simulation, we additionally
need to show that whenever $s ~ {\cal R}^r ~ \Delta$ and $s \not \barb{X}$,
we have $\Delta \dar{\hat \tau} \Theta \not \barb{X}$ for some $\Theta$.
Since $s {\cal R} \Delta$, we have $s = s'\sigma$ and $\Delta = \Delta'\sigma$
for some $s'$, $\Delta$ and renaming substitution $\sigma.$
Let $X' = X\sigma^{-1}$, i.e., $X'$ is the inverse image of $X$ under $\sigma.$
Then we have that $s' \not \barb{X'}$, and $\Delta' \dar{\hat \tau} \Theta' \not \barb{X'}.$
Applying $\sigma^{-1}$ to the latter,
we obtain
$\Delta \dar{\hat \tau} \Theta \not \barb{X}.$
\qed
\end{proof}
\begin{lemma}
\label{lm:clo-renaming}
Suppose $P \sqsubseteq_S Q$ ($P \sqsubseteq_{FS} Q$)
and $\sigma$ is a
renaming substitution. Then $P \sigma \sqsubseteq_S Q\sigma$
(respectively, $P \sigma \sqsubseteq_{FS} Q\sigma$).
\end{lemma}
\begin{proof}
Immediate from Lemma~\ref{lm:upto-renaming}. \qed
\end{proof}
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\subsection{Up to name restriction}
The following lemma says that transitions are closed under name
restriction, if certain conditions are satisfied.
\begin{lemma}\label{lm:res}
\begin{enumerate}
\item For every state-based process $s$, every action $\alpha$
and every list of names $\vec x$ such that
$\{\vec x\}\cap n(\alpha) = \emptyset$,
$\one{s}{\alpha}{\Delta}$
implies $\one{\nu \vec x.s}{\alpha}{\nu \vec x.\Delta}.$
\item For every $\Delta$ and $\Phi$, every action $\alpha$
and every list of names $\vec x$ such that
$\{\vec x\}\cap n(\alpha) = \emptyset$,
$\Delta \sstep{\alpha} \Phi$
implies $\nu \vec x.\Delta \sstep{\alpha} \nu \vec x.\Phi.$
\item Suppose $\one s {\bar a b} \Delta$ and suppose $\vec x$ and $\vec y$
are names such that $\{\vec x,\vec y\} \cap \{a,b\} = \emptyset.$
Then $\one {\nu \vec x \nu b\nu \vec y. s} {\bar a(b)}{\nu \vec x\nu \vec y.\Delta}$.
\end{enumerate}
\end{lemma}
\begin{lemma}
\label{lm:nu}
If $\Delta ~ \overline{{\cal R}^\nu} ~ \Theta$ then
$(\nu \vec x.\Delta) ~ \overline{{\cal R}^\nu} ~ (\nu \vec x.\Theta) $
\end{lemma}
\begin{lemma}
\label{lm:upto-restriction}
If ${\cal R}$ is a (failure) simulation up to restriction, then
${\cal R}^\nu \subseteq \triangleleft_S$ (respectively, ${\cal R}^\nu \subseteq \triangleleft_{FS}$).
\end{lemma}
\begin{proof}
Suppose ${\cal R}$ is a simulation up to restriction.
We show that ${\cal R}^\nu$ is a simulation up to renaming,
hence by Lemma~\ref{lm:upto-renaming} we have
${\cal R}^\nu \subseteq {\cal R}^{\nu r} \subseteq \triangleleft_S.$
Suppose $s ~ {\cal R}^\nu \Delta$
and $\one {s} {\alpha}{\Theta}.$
By the definition of ${\cal R}^\nu$, we have that
$s = \nu \vec x.s'$, $\Delta = \nu \vec x.\Delta'$,
and $s'[\vec y/\vec x] ~ {\cal R} ~ \Delta'[\vec y/\vec x]$
for some $\vec y$ such that
$\{\vec y\} \cap fn(s,\Delta) = \emptyset.$
There are several cases depending on how the transition
$\one s \alpha \Theta$ is derived.
Note that there may be implicit $\alpha$-renaming involved
in the derivations of a transition judgment.
We assume that the names $\vec x$ are chosen such that
no $\alpha$-renaming is needed in deriving the transition
relation $\one{\nu \vec x.s'}{\alpha}{\Theta}$,
e.g., one such choice would be one that avoids clashes with the free
names in $\vec y$, $s$, and $\Delta$.
\begin{itemize}
\item $\alpha$ is either $\tau$
or a free action. In this case, the transition must have been
derived as follows:
$$
\infer=[res]
{\one {\nu \vec x.s'}{\alpha}{\nu \vec x.\Theta'}}
{
\one {s'}{\alpha}{\Theta'}
}
$$
where $\Theta = \nu \vec x.\Theta'$
and $n(\alpha) \cap \{\vec x\} = \emptyset.$
Here a double-line in the inference rule indicates zero or more applications
of the rule.
An inspection on the operational semantics will reveal that
in this case, $n(\alpha) \subseteq fn(s)$ and $fn(\Theta) \subseteq fn(s)$.
So in particular, $\{ \vec y\} \cap n(\alpha) = \emptyset.$
We thus can apply the renaming substitution $[\vec y/\vec x, \vec x/\vec y]$
to get
$
\one {s'[\vec y/\vec x]} {\alpha} {\Theta'[\vec y/\vec x]}.
$
Since $s'[\vec y/\vec x] ~ {\cal R} ~ \Delta'[\vec y/\vec x]$, we have
that
$
\bigstep{\Delta'[\vec y/\vec x]}{\alpha}{\Delta''[\vec y/\vec x]}
$
and
$
\Theta'[\vec y/\vec x] ~ \overline{{\cal R}^\nu} ~ \Delta''[\vec y/\vec x].
$
The former implies, via Lemma~\ref{lm:res} (1), that
$
\bigstep{\nu \vec x. \Delta'}{\alpha}{\nu \vec x.\Delta''}
$
and the latter implies, via Lemma~\ref{lm:nu},
that
$
(\nu \vec x. \Theta') ~ \overline{{\cal R}^\nu} ~ (\nu \vec x. \Delta'')
$.
Since ${\cal R}^\nu \subseteq ({\cal R}^{\nu})^r$, we also have
$
(\nu \vec x. \Theta') ~ \overline{{\cal R}^{\nu r}} ~ (\nu \vec x. \Delta'').
$
\item $\alpha = a(z)$:
With a similar argument as in the previous case, we can show that
in this case we must have $\one {s} {a(z)}{\Theta'}$
where $\Theta = \nu \vec x.\Theta'.$ We need to show that
for every name $w$, there exist $\Gamma_w^1$, $\Gamma_w^2$ and $\Gamma_w$ such that
$\Delta \bstep {\hat \tau} \Gamma_w^1 \sstep {a(z)} \Gamma_w^2$,
$\Gamma_w^2[w/z] \bstep{\hat \tau} \Gamma_w$,
and $\Theta[w/z] ~ \overline{{\cal R}^{\nu r}} ~ \Gamma_w.$
Note that $z \not \in \{\vec x\}$,
but it may be the case that $z \in \{\vec y\}.$
So we first apply a renaming $[u/z,z/u, \vec y/\vec x,\vec x/\vec y]$, for some fresh name $u$,
to the transition $\one {s'}{a(z)}{\Theta'}$ to get:
$$
\one{s'[\vec y/\vec x]}{a(u)}{\Theta'[u/z,\vec y/\vec x]}.
$$
Since $s'[\vec y/\vec x] ~ {\cal R} ~ \Delta'[\vec y/\vec x]$,
we have, for every name $w$, some $\Delta_w^1$, $\Delta_w^2$ and $\Delta_w$
such that
\begin{equation}
\label{eq:clo-nu2a}
\Delta'[\vec y/\vec x] \bstep{\hat\tau}
\Delta_w^1 \sstep{a(u)} \Delta_w^2,
\qquad
\Delta_w^2[w/u] \bstep{\hat \tau} \Delta_w, \qquad \mbox{and }
\end{equation}
\begin{equation}
\label{eq:clo-nu3a}
\Theta'[u/z, \vec y/\vec x][w/u] = \Theta'[w/z,\vec y/\vec x] ~ \overline{{\cal R}^\nu} ~ \Delta_w[w/u].
\end{equation}
Let $\Phi_w^1$, $\Phi_w^2$ and $\Phi_w$ be distributions
such that $\Delta_w^1 = \Phi_w^1[\vec y/\vec x]$,
$\Delta_w^2 = \Phi_w^2[u/z, \vec y/\vec x]$,
and $\Delta_w = \Phi_w[\vec y/\vec x].$
So in particular, $\Delta_w^2[w/u] = \Phi_w^2[w/z, \vec y/\vec x]$
and $\Delta_w[w/u] = \Phi_w[w/z, \vec y/\vec x].$
Then (\ref{eq:clo-nu2a}) can be rewritten as:
\begin{equation}
\label{eq:clo-nu2b}
\Delta'[\vec y/\vec x] \bstep{\hat\tau}
\Phi_w^1[\vec y/\vec x] \sstep{a(u)} \Phi_w^2[u/z, \vec y/\vec x]
\qquad
\Phi_w^2[w/z, \vec y/\vec x] \bstep{\hat \tau} \Phi_w[\vec y/\vec x],
\end{equation}
and (\ref{eq:clo-nu3a}) can be rewritten as:
\begin{equation}
\label{eq:clo-nu3b}
\Theta'[w/z,\vec y/\vec x] ~ \overline{{\cal R}^\nu} ~ \Phi_w[w/z, \vec y/\vec x].
\end{equation}
Now, to define $\Gamma_w^1$, $\Gamma_w^2$ and $\Gamma_w$, we need to consider
two cases, based on the value of $w$. The reason is that in the construction
of $\Gamma_w$ we need to bound the free names in $\Phi_w$, so if $z$ is substituted
with a name in $\vec y$, it could get captured.
\begin{itemize}
\item $w \not \in \{\vec x, \vec y\}$.
In this case, define:
$$
\Gamma_w^1 = \nu \vec x. \Phi_w^1, \qquad
\Gamma_w^2 = \nu \vec x. \Phi_w^2, \qquad
\Gamma_w = \nu \vec x. \Phi_w.
$$
By Lemma~\ref{lm:res} (1) and (\ref{eq:clo-nu2b}), we have:
$$
\nu \vec x. \Delta' \bstep {\hat \tau} \Gamma_w^1
\sstep {a(z)} \Gamma_w^2,
\qquad
\Gamma_w^2[w/z] \bstep {\hat \tau} \Gamma_w
$$
and by Lemma~\ref{lm:nu} and (\ref{eq:clo-nu3b}), we have
$$
(\Theta[w/z]) = (\nu \vec x. \Theta')[w/z] ~ \overline{{\cal R}^\nu} ~
\Gamma_w,
$$
hence also,
$
(\Theta[w/z]) = (\nu \vec x. \Theta')[w/z] ~ \overline{{\cal R}^{\nu r}} ~
\Gamma_w.
$
\item $w \in \{\vec x, \vec y\}.$ Let $v$ be a new name (distinct from
all other names considered so far). From the previous case, we
know how to construct $\Gamma_v^1$, $\Gamma_v^2$ and $\Gamma_v$ such that
\begin{equation}
\label{eq:clo-nu3c}
\nu \vec x. \Delta' \bstep {\hat \tau} \Gamma_v^1
\sstep {a(z)} \Gamma_v^2,
\qquad
\Gamma_v^2[v/z] \bstep {\hat \tau} \Gamma_v
\qquad
(\Theta[v/z]) ~ \overline{{\cal R}^{\nu r}} ~
\Gamma_v.
\end{equation}
In this case, let $\Gamma_w^1 = \Gamma_v^1$, $\Gamma_w^2 = \Gamma_v^2$
and $\Gamma_w = \Gamma_v[w/v].$ (Note that because subsitution is capture-avoiding,
the bound names in $\Gamma_v$ will be renamed via $\alpha$-conversion).
Then by Lemma~\ref{lm:rename} (2) and Lemma~\ref{lm:lift-renaming}
and (\ref{eq:clo-nu3c}):
$$
\nu \vec x. \Delta' \bstep {\hat \tau} \Gamma_w^1
\sstep {a(z)} \Gamma_w^2,
\qquad
\Gamma_v^2[w/z] \bstep {\hat \tau} \Gamma_w
\qquad
(\Theta[w/z]) ~ \overline{{\cal R}^{\nu r}} ~
\Gamma_w.
$$
\end{itemize}
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Since $s'[\vec y/\vec x] ~ {\cal R} ~ \Delta'[\vec y/\vec x]$,
we have, for every name $w$, some $\Delta_w^1$, $\Delta_w^2$ and $\Delta_w$
such that
\begin{equation}
\label{eq:clo-nu2a}
\Delta'[\vec y/\vec x] \bstep{\hat\tau}
\Delta_w^1 \sstep{a(u)} \Delta_w^2,
\qquad
\Delta_w^2[w/u] \bstep{\hat \tau} \Delta_w, \qquad \mbox{and }
\end{equation}
\begin{equation}
\label{eq:clo-nu3a}
\Theta'[u/z, \vec y/\vec x][w/u] = \Theta'[w/z,\vec y/\vec x] ~ \overline{{\cal R}^\nu} ~ \Delta_w[w/u].
\end{equation}
Let $\Phi_w^1$, $\Phi_w^2$ and $\Phi_w$ be distributions
such that $\Delta_w^1 = \Phi_w^1[\vec y/\vec x]$,
$\Delta_w^2 = \Phi_w^2[u/z, \vec y/\vec x]$,
and $\Delta_w = \Phi_w[\vec y/\vec x].$
So in particular, $\Delta_w^2[w/u] = \Phi_w^2[w/z, \vec y/\vec x]$
and $\Delta_w[w/u] = \Phi_w[w/z, \vec y/\vec x].$
Then (\ref{eq:clo-nu2a}) can be rewritten as:
\begin{equation}
\label{eq:clo-nu2b}
\Delta'[\vec y/\vec x] \bstep{\hat\tau}
\Phi_w^1[\vec y/\vec x] \sstep{a(u)} \Phi_w^2[u/z, \vec y/\vec x]
\qquad
\Phi_w^2[w/z, \vec y/\vec x] \bstep{\hat \tau} \Phi_w[\vec y/\vec x],
\end{equation}
and (\ref{eq:clo-nu3a}) can be rewritten as:
\begin{equation}
\label{eq:clo-nu3b}
\Theta'[w/z,\vec y/\vec x] ~ \overline{{\cal R}^\nu} ~ \Phi_w[w/z, \vec y/\vec x].
\end{equation}
Now, to define $\Gamma_w^1$, $\Gamma_w^2$ and $\Gamma_w$, we need to consider
two cases, based on the value of $w$. The reason is that in the construction
of $\Gamma_w$ we need to bound the free names in $\Phi_w$, so if $z$ is substituted
with a name in $\vec y$, it could get captured.
\begin{itemize}
\item $w \not \in \{\vec x, \vec y\}$.
In this case, define:
$$
\Gamma_w^1 = \nu \vec x. \Phi_w^1, \qquad
\Gamma_w^2 = \nu \vec x. \Phi_w^2, \qquad
\Gamma_w = \nu \vec x. \Phi_w.
$$
By Lemma~\ref{lm:res} (1) and (\ref{eq:clo-nu2b}), we have:
$$
\nu \vec x. \Delta' \bstep {\hat \tau} \Gamma_w^1
\sstep {a(z)} \Gamma_w^2,
\qquad
\Gamma_w^2[w/z] \bstep {\hat \tau} \Gamma_w
$$
and by Lemma~\ref{lm:nu} and (\ref{eq:clo-nu3b}), we have
$$
(\Theta[w/z]) = (\nu \vec x. \Theta')[w/z] ~ \overline{{\cal R}^\nu} ~
\Gamma_w,
$$
hence also,
$
(\Theta[w/z]) = (\nu \vec x. \Theta')[w/z] ~ \overline{{\cal R}^{\nu r}} ~
\Gamma_w.
$
\item $w \in \{\vec x, \vec y\}.$ Let $v$ be a new name (distinct from
all other names considered so far). From the previous case, we
know how to construct $\Gamma_v^1$, $\Gamma_v^2$ and $\Gamma_v$ such that
\begin{equation}
\label{eq:clo-nu3c}
\nu \vec x. \Delta' \bstep {\hat \tau} \Gamma_v^1
\sstep {a(z)} \Gamma_v^2,
\qquad
\Gamma_v^2[v/z] \bstep {\hat \tau} \Gamma_v
\qquad
(\Theta[v/z]) ~ \overline{{\cal R}^{\nu r}} ~
\Gamma_v.
\end{equation}
In this case, let $\Gamma_w^1 = \Gamma_v^1$, $\Gamma_w^2 = \Gamma_v^2$
and $\Gamma_w = \Gamma_v[w/v].$ (Note that because subsitution is capture-avoiding,
the bound names in $\Gamma_v$ will be renamed via $\alpha$-conversion).
Then by Lemma~\ref{lm:rename} (2) and Lemma~\ref{lm:lift-renaming}
and (\ref{eq:clo-nu3c}):
$$
\nu \vec x. \Delta' \bstep {\hat \tau} \Gamma_w^1
\sstep {a(z)} \Gamma_w^2,
\qquad
\Gamma_v^2[w/z] \bstep {\hat \tau} \Gamma_w
\qquad
(\Theta[w/z]) ~ \overline{{\cal R}^{\nu r}} ~
\Gamma_w.
$$
\end{itemize}
\item If $\alpha$ is a bound output action, i.e., $\alpha = \bar a(b)$ for some $a$ and $b.$
There are two subcases to consider, depending on whether $b \in \{\vec x\}$ (i.e., one of
the restriction names $\vec x$ is extruded) or not. The latter can be proved similarly to
the previous case. We show here a proof of the former case.
So suppose $b \in \vec x$, i.e., $\nu \vec x = \nu \vec x_1 \nu b \nu \vec x_2$ and
suppose that $[\vec y/\vec x]$ maps $b$ to $c$, i.e., $\nu \vec y = \nu \vec y_1 \nu c \nu \vec y_2.$ Suppose
the transition relation is derived as follows:
$$
\infer=[res]
{\one{\nu\vec x_1\nu b \nu \vec x_2.s'}{\bar a(b)}{\nu \vec x_1 \nu \vec x_2. \Theta'}}
{
\infer[open]
{\one {\nu b \nu \vec x_2.s} {\bar a(b)}{\nu \vec x_2.\Theta'}}
{
\infer=[res]
{\one {\nu \vec x_2.s}{\bar a b}{\nu \vec x_2.\Theta'} }
{\one {s}{\bar a b}{\Theta'}}
}
}
$$
Applying the renaming $[\vec y/\vec x, \vec x/\vec y]$ we have:
$
\one{s[\vec y/\vec x]}{\bar a c}{\Theta'[\vec y/\vec x]}.
$
Since $s'[\vec y/\vec x] ~ {\cal R} ~ \Delta'[\vec y/\vec x]$, we have that
\begin{equation}
\label{eq:clo-nu4}
\bigstep {\Delta'[\vec y/\vec x]}{\bar a c}{\Phi}, \qquad \mbox{ and } \qquad
\Theta'[\vec y/\vec x] ~ \overline{{\cal R}^\nu} ~ \Phi.
\end{equation}
Let $\Psi[\vec y/\vec x] = \Phi.$
Lemma~\ref{lm:res} (3) and (\ref{eq:clo-nu4}) imply that
$$
\bigstep{\nu \vec x.\Delta' = \nu \vec y_1\nu c\vec y_2.\Delta'[\vec y/\vec x]}
{\bar a(c)}{\nu\vec y_1\nu \vec y_2.\Psi[\vec y/\vec x] = \nu \vec x_1\vec x_2. \Psi[c/b]}
$$
and by an application of a renaming (Lemma~\ref{lm:rename} (1))
we get
$$
\bigstep {\nu \vec x.\Delta'}{\bar a (b)}{\nu\vec x_1\nu \vec x_2.\Psi}.
$$
Lemma~\ref{lm:nu} and (\ref{eq:clo-nu4}) imply
$$
(\nu \vec x_1\nu\vec x_2.\Theta'[c/b]) ~ \overline{{\cal R}^\nu} ~
(\nu\vec x_1\nu \vec x_2.\Psi[c/b])
$$
hence, via the renaming $[c/b,b/c]$,
$
(\nu \vec x_1\nu\vec x_2.\Theta') ~ \overline{{\cal R}^{\nu r}} ~
(\nu\vec x_1\nu \vec x_2.\Psi).
$
\end{itemize}
If ${\cal R}$ is a failure simulation up to restriction, we need to additionally
show that ${\cal R}^\nu$ satisfies clause 3 of Definition~\ref{def:sim}.
Suppose $s ~ {\cal R}^\nu ~ \Theta$. Then $s = \nu \vec x. s'$ and $\Theta = \nu \vec x.\Theta'$
for some $\vec x$, $s'$ and $\Theta'$ such that $s' ~ {\cal R} ~ \Theta'.$
Suppose $s \not \barb{X}.$ We need to show that $\Theta \dar{\hat \tau} \Delta$
such that $\Delta \not \barb{X}$ for some $\Delta.$
Since name restriction hides visible actions, it can be shown that
$s' \not \barb{X \setminus \{\vec x\}}$
iff $\nu \vec x. s' \not \barb{X}.$
So from $s' ~ {\cal R} ~ \Theta'$ we have that $\Theta' \dar{\hat \tau} \Delta' \not \barb{X \setminus \{\vec x\}}.$
Let $\Delta = \nu \vec x.\Delta'.$
Then by Lemma~\ref{lm:res} (2), we have
$\Theta = \nu \vec x.\Theta' \dar{\hat \tau} \nu \vec x.\Delta' = \Delta \not \barb{X}.$
\qed
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\end{proof}
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\begin{lemma}
\label{lm:clo-res}
If $P \sqsubseteq_S Q$ ($P \sqsubseteq_{FS} Q$) then
$(\nu \vec x. P) ~ \sqsubseteq_S (\nu \vec x.Q)$ (respectively, $(\nu \vec x.P) \sqsubseteq_{FS} (\nu \vec x.Q)$).
\end{lemma}
\begin{proof}
This is a simple corollary of Lemma~\ref{lm:sim-upto} and Lemma~\ref{lm:upto-restriction}. \qed
\end{proof}
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\subsection{Up to parallel composition}
The following lemma will be useful in proving the closure of simulation
under parallel composition. It is independent of the underlying calculus,
and is originally proved in \cite{Deng07ENTCS}.
\begin{lemma}
\label{lm:par0}
\begin{enumerate}
\item $
(\sum_{j\in J} p_j \cdot \Phi_j) ~|~ (\sum_{k\in K} q_k \cdot \Delta_k)
= \sum_{j\in J} \sum_{k \in K} (p_j \cdot q_k) \cdot (\Phi_j ~|~ \Delta_k).
$
\item Suppose ${\cal R}, {\cal R}' \subseteq S_p \times {\cal D}(S_p)$ are two relations
such that $s {\cal R}' \Delta$ whenever $s = s_1 ~|~ s_2$ and
$\Delta = \Delta_1 ~|~ \Delta_2$ with $s_1 {\cal R} \Delta_1$
and $s_2 {\cal R} \Delta_2.$
Then $\Phi_1 \overline {\cal R} \Delta_1$
and $\Phi_2 \overline {\cal R} \Delta_2$ imply
$(\Phi_1 ~|~ \Phi_2) \overline {{\cal R}'} (\Delta_1 ~|~ \Delta_2)$.
\end{enumerate}
\end{lemma}
We also need a slightly more general substitution lemma for
transitions than the one given in Lemma~\ref{lm:rename} (1).
In the following, we denote with $n(\theta)$ the set of
all names appearing in
the domain and range of $\theta$.
\begin{lemma}
\label{lm:trans-subst}
For any substitution $\sigma$, the following hold:
\begin{enumerate}
\item If $\one s \alpha \Delta$ and $bn(\alpha) \cap n(\sigma) = \emptyset$
then $\one{s\sigma}{\alpha\sigma}{\Delta\sigma}.$
\item If $\bigstep {\Delta}{\hat \alpha}{\Phi}$ and
$bn(\alpha) \cap n(\sigma) = \emptyset$
then $\bigstep{\Delta\sigma}{\hat \alpha\sigma}{\Phi\sigma}.$
\end{enumerate}
\end{lemma}
The following lemma shows that transitions are closed under
parallel composition, under suitable conditions.
\begin{lemma}\label{lm:trans-par}
\begin{enumerate}
\item If $\one {s}{\alpha}{\Delta}$ and $fn(s') \cap bn(\alpha) = \emptyset$
then $\one{s~|~s'} {\alpha}{\Delta~|~\pdist{s'}}$
and $\one{s'~|~s}{\alpha}{\pdist{s'}~|~\Delta}.$
\item If $\bigstep {\Phi}{\hat \alpha}{\Delta}$, where $\alpha$ is either $\tau$,
a free action or a bound output, and $fn(\Phi') \cap bn(\alpha) = \emptyset$
then $\bigstep {\Phi~|~\Phi'} {\hat \alpha}{\Delta~|~\Phi'}$
and $\bigstep{\Phi'~|~\Phi}{\hat \alpha}{\Phi'~|~\Delta}.$
\item If $\one{\Phi}{a(y)}{\Phi'}$ and $\one{\Delta}{\bar a w}{\Delta'}$
then $\one{\Phi~|~\Delta}{\tau}{\Phi'[w/y]~|~\Delta'}.$
\item If $\one{\Phi}{a(y)}{\Phi'}$ and $\one{\Delta}{\bar a(y)}{\Delta'}$
then $\one{\Phi~|~\Delta}{\tau}{\nu y.(\Phi'~|~\Delta')}.$
\end{enumerate}
\end{lemma}
\begin{lemma}
\label{lm:upto-par}
If ${\cal R}$ is a simulation, then ${\cal R}^p \subseteq \triangleleft_S$.
\end{lemma}
\begin{proof}
We show that ${\cal R}^p$ is a simulation up to restriction, and
therefore, by Lemma~\ref{lm:upto-restriction}, it is included in $\triangleleft_S$.
So suppose $s ~ {\cal R}^p ~ \Delta$ and $\one{s}{\alpha}{\Theta}.$
By definition, we have $s = s_1 ~|~ s_2$ and $\Delta = \Delta_1 ~|~ \Delta_2$
such that $s_1 ~ {\cal R} ~ \Delta_1$ and $s_2 ~{\cal R} ~ \Delta_2.$
There are several cases to consider depending on the type of $\alpha$:
\begin{itemize}
\item $\alpha$ is a free output action.
There can be two ways in which the transition $\one{s}{\alpha}{\Theta}$ is derived.
We show here one case; the other case is symmetric.
So suppose the transition is derived as follows:
$$
\infer[par]
{\one{s_1~|~s_2}{\alpha}{\Theta'~|~\pdist{s_2}}}
{
\one{s_1}{\alpha}{\Theta'}
}
$$
where $\Theta = \Theta'~|~ \pdist{s_2}.$
Since $s_1 ~ {\cal R} ~ \Delta_1$, we have
$$
\bigstep{\Delta_1}{\hat \alpha}{\Delta_1'}
$$
and $\Theta' ~ \overline{{\cal R}} ~ \Delta_1'$.
The former implies, via Lemma~\ref{lm:trans-par} (2),
that $\bigstep{\Delta_1 ~|~ \Delta_2}{\hat\alpha}{\Delta_1' ~|~ \Delta_2}.$
Since $s_2 ~ {\cal R} ~ \Delta_2$ by assumption, and therefore
$\pdist{s_2} ~ \overline {{\cal R}} ~\Delta_2$,
by Lemma~\ref{lm:par0} (2) we have
$$\Theta = (\Theta'~|~\pdist{s_2})~\overline {{\cal R}^p} ~ (\Delta_1' ~|~ \Delta_2)$$
and therefore, also
$$\Theta = (\Theta'~|~\pdist{s_2})~\overline {{\cal R}^{p \nu}} ~ (\Delta_1' ~|~ \Delta_2).$$
\item $\alpha = a(y)$ and $y \not \in fn(s,\Delta).$ That is, in this case, the transition
is derived as follows:
$$
\infer[par]
{\one{s_1 ~|~ s_2}{a(y)}{\Theta' ~|~ \pdist{s_2}}}
{
\one{s_1}{a(y)}{\Theta'}
}
$$
and $y \not \in fn(s_2).$
(There is another symmetric case which we omit here.)
Since $s_1 ~ {\cal R} ~\Delta_1$, we have, for every name $w$,
some $\Delta_w^1$, $\Delta_w^2$ and $\Delta_w$ such that:
\begin{equation}
\label{eq:clo-par1}
\Delta_1 \bstep{\hat \tau} \Delta_w^1 \sstep{a(y)} \Delta_w^2, \qquad
\Delta_w^2[w/y] \bstep{\hat \tau} \Delta_w, \quad \mbox{ and }
\end{equation}
\begin{equation}
\label{eq:clo-par2}
\Theta'[w/y]~ \overline{{\cal R}} ~ \Delta_w.
\end{equation}
From (\ref{eq:clo-par1}) above and Lemma~\ref{lm:trans-par} (2), and the assumption that $y \not \in fn(s,\Delta)$,
we have
$$
\Delta_1~|~\Delta_2 \bstep{\hat \tau} \Delta_w^1 ~|~ \Delta_2
\sstep{a(y)} \Delta_w^2 ~|~ \Delta_2, \qquad
\Delta_w^2[w/y] ~|~ \Delta_2 \bstep{\hat\tau} \Delta_w ~|~ \Delta_2.
$$
Since $s_2 ~ {\cal R} ~ \Delta_2$, and therefore
$\pdist{s_2} ~ \overline{{\cal R}} ~ \Delta_2$,
it then follows from (\ref{eq:clo-par2}) and Lemma~\ref{lm:par0} (2) that
$$
\Theta[w/y] = (\Theta'[w/y]~|~ \pdist{s_2})
~ \overline {{\cal R}^p} ~ (\Delta_w ~|~ \Delta_2)
$$
and therefore
$$
\Theta[w/y] = (\Theta'[w/y]~|~ \pdist{s_2})
~ \overline {{\cal R}^{p \nu}} ~ (\Delta_w ~|~ \Delta_2).
$$
\item $\alpha = \bar a(y)$ and $y \not \in fn(s,\Delta)$. This case is similar to the
previous cases, except that we only need to consider an instantiation of $y$ with a fresh
name. This is left as an exercise for the reader.
\item $\alpha = \tau$ and the transition $\one{s}{\tau}{\Theta}$ is derived via
a \textbf{Com}-rule. We show here one case; the other case can be dealt with symmetrically.
So suppose the transition is derived as follows:
$$
\infer[com]
{\one{s_1 ~|~ s_2}{\tau}{\Theta_1[w/y]~|~\Theta_2}}
{
\one{s_1}{a(y)}{\Theta_1}
&
\one{s_2}{\bar a w}{\Theta_2}
}
$$
Without loss of generality, we can assume that $y \not \in fn(s,\Delta).$
Since $s_1 ~ {\cal R} ~ \Delta_1$ and $s_2 ~ {\cal R} ~ \Delta_2$,
we have:
\begin{itemize}
\item For every name $w$, there are $\Lambda_1$, $\Lambda_2$ and
$\Delta_1^w$ such that
\begin{equation}
\label{eq:clo-par3}
\Delta_1 \bstep{\hat \tau} \Lambda_1 \sstep{a(y)} \Lambda_2, \qquad
\Lambda_2[w/y] \bstep{\hat \tau} \Delta_1^w \qquad \mbox{ and }
\end{equation}
\begin{equation}
\label{eq:clo-par4}
\Theta_1[w/y] ~ \overline{{\cal R}} ~ \Delta_1^w
\end{equation}
\item There exists $\Delta_2'$ such that
\begin{equation}
\label{eq:clo-par5}
\Delta_2 \bstep {\hat \tau} \Phi_1 \sstep{\bar a w} \Phi_2 \bstep{\hat \tau} \Delta_2'
\qquad \mbox{ and }
\end{equation}
\begin{equation}
\label{eq:clo-par6}
\Theta_2 ~ \overline{{\cal R}} ~ \Delta_2'
\end{equation}
\end{itemize}
From (\ref{eq:clo-par3}), (\ref{eq:clo-par5}), and
Lemma~\ref{lm:trans-par} (2)-(3), we have:
$$
\Delta_1 ~|~ \Delta_2 \bstep{\hat \tau} {\Lambda_1 ~|~ \Phi_1}
\sstep {\tau} {\Lambda_2[w/y] ~|~ \Phi_2}
\bstep {\hat \tau} {\Delta_1^w ~|~ \Delta_2'},
$$
and Lemma~\ref{lm:par0} (2), together with (\ref{eq:clo-par4}) and (\ref{eq:clo-par6}),
implies
$$
(\Theta_1[w/y] ~|~ \Theta_2) ~ \overline{{\cal R}^p} ~ (\Delta_1^w ~|~ \Delta_2')
$$
and therefore
$$
(\Theta_1[w/y] ~|~ \Theta_2) ~ \overline{{\cal R}^{p \nu}} ~ (\Delta_1^w ~|~ \Delta_2').
$$
\item $\alpha = \tau$ and the transition $\one s \tau \Theta$ is derived
via the \textbf{Close}-rule:
$$
\infer[close .]
{\one{s_1 ~|~ s_2}{\tau}{\nu y.(\Theta_1 ~|~ \Theta_2)}}
{
\one{s_1}{a(y)}{\Theta_1}
&
\one{s_2}{\bar a(y)}{\Theta_2}
}
$$
Again, we only show one of the two symmetric cases.
Without loss of generality, assume that $y$ is chosen to be
fresh w.r.t. $s$ and $\Delta.$
Since $s_1 ~ {\cal R} \Delta_1$ and $s_2 ~ {\cal R} \Delta_2$, we have:
\begin{itemize}
\item For every name $w$, there are $\Lambda_1$, $\Lambda_2$ and $\Delta_1^w$ such that
$$
\Delta_1 \bstep{\hat \tau} \Lambda_1 \sstep{a(y)} \Lambda_2, \qquad
\Lambda_2[w/y] \bstep{\hat \tau} \Delta_1^w
\qquad \mbox{and} \qquad
\Theta_1[w/y] ~ \overline{{\cal R}} ~ \Delta_1^w.
$$
Note that letting $w = y$, we have
\begin{equation}
\label{eq:clo-par7}
\Delta_1 \bstep{\hat \tau} \Lambda_1 \sstep{a(y)} \Lambda_2, \qquad
\Lambda_2 \bstep{\hat \tau} \Delta_1^y \qquad \mbox{and}
\end{equation}
\begin{equation}
\label{eq:clo-par8}
\Theta_1 ~ \overline{{\cal R}} ~ \Delta_1^y
\end{equation}
\item There exist $\Phi_1$, $\Phi_2$ and $\Delta_2'$ such that
\begin{equation}
\label{eq:clo-par9}
\Delta_2 \bstep {\hat \tau} \Phi_1 \sstep{\bar a(y)} \Phi_2 \bstep{\hat \tau} \Delta_2' \qquad \mbox{and}
\end{equation}
\begin{equation}
\label{eq:clo-par10}
\Theta_2 ~ \overline{{\cal R}} ~ \Delta_2'
\end{equation}
\end{itemize}
Then, by (\ref{eq:clo-par7}), (\ref{eq:clo-par9}),
Lemma~\ref{lm:trans-par} (2) and (4),
and Lemma~\ref{lm:res} (1), we have:
$$
\Delta_1 ~|~ \Delta_2 \bstep{\hat \tau} {\Lambda_1 ~|~ \Phi_1}
\sstep {\tau} {\nu y.(\Lambda_2 ~|~ \Phi_2)}
\bstep {\hat \tau} {\nu y.(\Delta_1^y ~|~ \Delta_2')}.
$$
Lemma~\ref{lm:par0} (2), together with (\ref{eq:clo-par8}) and (\ref{eq:clo-par10}),
implies
$$
(\Theta_1 ~|~ \Theta_2) ~ \overline{{\cal R}^p} ~ (\Delta_1^y ~|~ \Delta_2'),
$$
which also means:
$$
(\Theta_1 ~|~ \Theta_2) ~ \overline{{\cal R}^{p \nu}} ~ (\Delta_1^y ~|~ \Delta_2').
$$
Now by Lemma~\ref{lm:nu}, the latter implies that
$$
\nu y.(\Theta_1 ~|~ \Theta_2) ~ \overline{{\cal R}^{p \nu}} ~ \nu y. (\Delta_1^y ~|~ \Delta_2').
$$
\end{itemize}
\qed
\end{proof}
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\begin{lemma}
\label{lm:upto-par-failsim}
If ${\cal R}$ is a failure simulation, then ${\cal R}^p \subseteq \triangleleft_{FS}$.
\end{lemma}
\begin{proof}
Suppose $s {\cal R}^p \Delta$ and $s\not\barb{X}$. By definition, we have
$s = s_1 ~|~ s_2$ and $\Delta = \Delta_1 ~|~ \Delta_2$ such that
$s_1 ~ {\cal R} ~ \Delta_1$ and $s_2 ~{\cal R} ~ \Delta_2.$ Then we have
$s_i\not\barb{X}$ for $i=1,2$.
Define a set $A$ as follows:
$$
A = \{ a, \bar a \mid a \in fn(s_1,s_2,\Delta_1,\Delta_2) \} \cup X.
$$
That is, $A$ contains the set of free (co-)names in $s_i$ and $\Delta_i$ and $X.$
Let $X_i$ be the largest set such that $X \subseteq X_i \subseteq A$ and $s_i \not\barb{X_i}.$
Since ${\cal R}$ is a failure simulation, it follows that there exist $\Delta_i'$ such that
$\Delta_i \dar{\tau} \Delta_i' \not \barb{X_i}.$
By Lemma~\ref{lm:trans-par} (2), we have $\Delta_1~|~\Delta_2
\dar{\tau} \Delta'_1~|~\Delta'_2.$
We claim that $(\Delta_1' ~|~ \Delta_2') \not \barb{X}.$
Suppose otherwise, that is, there exist $t_1 \in \supp {\Delta_1'}$
and $t_2 \in \supp {\Delta_2'}$ such that either
$(t_1 ~|~ t_2) \barb \mu$, for some $\mu \in X$, or
$(t_1 ~|~ t_2) \ar{\tau}$. If $(t_1 ~|~ t_2) \barb \mu$ then our operational semantics entails
that either $t_1 \barb \mu$ or $t_2 \barb \mu$, which contradicts the
fact that $\Delta_i' \not\barb {X_i}.$ So let's assume that
$(t_1 ~|~ t_2) \ar{\tau}.$ Again, from the assumption $\Delta_i' \not\barb {X_i}$, we can immediately
rule out the cases where $t_i \ar{\tau}$ or $t_i \barb \mu$, for some $\mu \in X.$
This leaves us only with the cases where $t_1 \ar{\mu}$ and $t_2 \ar{\bar \mu}$
where $\mu \not \in X$ and $\bar\mu \not \in X.$ But since $\Delta_i' \not \barb{X_i}$, this
can only be the case if $\mu \not \in X_1$ and $\bar\mu \not \in X_2.$
From the operational semantics, it is easy to see that $fn(\Delta_1',\Delta_2') \subseteq fn(\Delta_1,\Delta_2)$,
so it must be the case that $\mu \in A$ and $\bar\mu \in A.$
It also must be the case that $s_1 \barb \mu$, for otherwise, it would contradict the ``largest'' property
of $X_1$. Similarly, we can argue that $s_2 \barb {\bar \mu}$. But then this would imply that
$(s_1 ~|~ s_2) \ar{\tau}$, contradicting the fact that $(s_1 ~|~ s_2) \not\barb{X}.$
The matching up of transitions and the using of ${\cal R}$ to prove the
preservation property of $\triangleleft_{FS}$ under parallel composition are
similar to those in the corresponding proof in Lemma~\ref{lm:upto-par}
for simulations, so we omit them.
\qed
\end{proof}
\begin{lemma}\label{lm:clo-par}
\begin{enumerate}
\item
If $P_1 \sqsubseteq_S Q_1$ and $P_2 \sqsubseteq_S Q_2$
then
$P_1 ~|~ P_2 ~ \sqsubseteq_S Q_1 ~|~ Q_2.$
\item
If $P_1 \sqsubseteq_{FS} Q_1$ and $P_2 \sqsubseteq_{FS} Q_2$
then
$P_1 ~|~ P_2 ~ \sqsubseteq_{FS} Q_1 ~|~ Q_2.$
\end{enumerate}
\end{lemma}
\begin{proof}
It is enough to show that $(\triangleleft_S)^p \subseteq \triangleleft_S$ and $(\triangleleft_{FS})^p \subseteq \triangleleft_{FS}$,
which follow directly from Lemmas~\ref{lm:upto-par} and \ref{lm:upto-par-failsim} respectively. \qed
\end{proof}
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\subsection{Soundness}
We now proceed to proving the main result, which is that
$P \sqsubseteq_S Q$ implies $P \sqsubseteq_{pmay} Q$, and $P \sqsubseteq_{FS} Q$ implies $P \sqsubseteq_{pmust} Q$.
The structure of the proof follows closely that of \cite{Deng08LMCS}.
Most of the intermediate lemmas in this section are not specific
to the $\pi$-calculus; rather, they utilise the underlying probabilistic
automata semantics.
Let $\pi^\omega$ be the set of all $\pi$ processes that may use action $\omega$.
We write $s\ar{\alpha}_\omega\Delta$ if either $\alpha=\omega$ or
$\alpha\not=\omega$ but both $s\nar{\omega}$ and $s\ar{\alpha}\Delta$ hold.
We define $\ar{\hat{\tau}}_w$ as we did for $\ar{\hat{\tau}}$, using $\ar{\tau}_\omega$ in place
of $\ar{\tau}$. Similarly, we define $\dar{}_\omega$ and $\dar{\hat{\alpha}}_\omega$.
Simulation and failure simulation are adapted to $\pi^\omega$ as follows.
\begin{definition}
Let $\triangleleft_{FS}^e \subseteq \pi^\omega\times
{\cal D}(\pi^\omega)$ be the largest relation such that $s \triangleleft_{FS}^e \Theta$ implies
\begin{itemize}
\item If $ s \ar{a(x)}_\omega {\Delta}$ and $x \not \in fn(s,\Theta)$,
then for every name $w$, there exists $\Theta_1$, $\Theta_2$ and
$\Theta'$ such that
$$\Theta \dar{\hat \tau}_\omega \Theta_1 \ar{a(x)}_\omega {\Theta_2},
\qquad \Theta_2[w/x] \dar{\hat \tau}_\omega \Theta',
\qquad \hbox{ and } \qquad (\Delta[w/x]) ~ \overline {\cal R} ~ \Theta'.
$$
\item if $s\ar{\alpha}_\omega \Delta$ and $\alpha$ is
not an input action, then there is some $\Theta'$
with $\Theta\dar{\hat{\alpha}}_\omega\Theta'$ and
$\Delta\lift{\triangleleft_{FS}^e}\Theta'$
\item if $s\not \barb{X}$ with $\omega\in X$ then there is some
$\Theta'$ with $\Theta\dar{\hat{\tau}}_\omega\Theta'$ and $\Theta'\not \barb{X}$.
\end{itemize}
Similarly we can define $\triangleleft_S^e$ by dropping the third clause.
Let $P \sqsubseteq_{FS}^e Q$ if $\interp{P}\dar{\hat{\tau}}_\omega\Theta$ for some $\Theta$ with $\interp{Q}\lift{\triangleleft_{FS}^e}\Theta$. Similarly, $P\sqsubseteq_S^e Q$ if $\interp{Q}\dar{\hat{\tau}}_\omega\Theta$ for some $\Theta$ with $\interp{P}\lift{\triangleleft_S^e}\Theta$.
\end{definition}
Note that for $\pi$-processes $P,Q$, there is no action $\omega$, therefore we have $P\sqsubseteq_{FS} Q$ iff $P\sqsubseteq_{FS}^e Q$, and $P\sqsubseteq_S Q$ iff $P\sqsubseteq_S^e Q$.
\begin{lemma}\label{lem:preserve.par}
Let $P,Q$ be processes in $\pi$ and $T$ be a process in $\pi^\omega$.
\begin{enumerate}
\item If $P\sqsubseteq_S Q$ then $T~|~ P \sqsubseteq_S^e T ~|~Q$.
\item If $P\sqsubseteq_{FS} Q$ then $T~|~ P \sqsubseteq_{FS}^e T ~|~Q$.
\end{enumerate}
\end{lemma}
\begin{proof}
Similar to the proof of Lemma~\ref{lm:clo-par}.
\qed
\end{proof}
\begin{lemma}
\label{lem:pmay-max}
\begin{enumerate}
\item
$P \sqsubseteq_{pmay} Q$ if and only if for every test $T$ we have
$$
max({\mathbb V}(\interp {\nu \vec x.(T ~|~ P)})) \leq
max({\mathbb V}(\interp {\nu \vec x.(T ~|~ Q)}))
$$
where $\vec x$ contain the free names of $T$, $P$ and $Q$, excluding $\omega.$
\item
$P \sqsubseteq_{pmust} Q$ if and only if for every test $T$ we have
$$
min({\mathbb V}(\interp {\nu \vec x.(T ~|~ P)})) \leq
min({\mathbb V}(\interp {\nu \vec x.(T ~|~ Q)}))
$$
where $\vec x$ contain the free names of $T$, $P$ and $Q$, excluding $\omega.$
\end{enumerate}
\end{lemma}
\begin{proof}
The results follow from the simple fact that, for non-empty finite outcome sets $O_1,O_2$,
\begin{itemize}
\item $O_1\sqsubseteq_{Ho}O_2$ iff $max(O_1)\leq max(O_2)$
\item $O_1\sqsubseteq_{Sm}O_2$ iff $min(O_1)\leq min(O_2)$
\end{itemize}
which is established as Proposition 2.1 in \cite{Deng07ENTCS}.
\qed
\end{proof}
\begin{lemma}
\label{lm:trans-max}
$\Delta_1 \dar{\hat \tau} \Delta_2$ implies
$max({\mathbb V}(\Delta_1)) \geq max({\mathbb V}(\Delta_2))$ and $min({\mathbb V}(\Delta_1)) \leq min({\mathbb V}(\Delta_2))$.
\end{lemma}
\begin{proof}
Similar properties are proven in \cite[Lemma 6.15]{Deng07ENTCS} using a function $maxlive$ instead of $max\circ{\mathbb V}$. Essentially the same arguments apply here.
\qed
\end{proof}
\begin{proposition}
\label{prop:sim-max}
\begin{enumerate}
\item $\Delta_1\lift{\triangleleft_S^e}\Delta_2$ implies $max({\mathbb V}(\Delta_1)) \leq max({\mathbb V}(\Delta_2))$.
\item $\Delta_1\lift{\triangleleft_{FS}^e}\Delta_2$ implies $min({\mathbb V}(\Delta_1)) \geq min({\mathbb V}(\Delta_2))$.
\end{enumerate}
\end{proposition}
\begin{proof}
The first clause is proven in \cite[Proposition 6.16]{Deng07ENTCS} using a function $maxlive$ instead of $max\circ{\mathbb V}$. The second clause is proven in \cite[Proposition 4.10]{Deng08LMCS}
\qed
\end{proof}
\begin{theorem}\label{thm:sim-sound}
\begin{enumerate}
\item
$P \sqsubseteq_S Q$ implies $P \sqsubseteq_{pmay} Q$
\item
$P \sqsubseteq_{FS} Q$ implies $P \sqsubseteq_{pmust} Q.$
\end{enumerate}
\end{theorem}
\begin{proof}
We prove the second statement; similar is the first one.
Suppose $P \sqsubseteq_{FS} Q$. Given Proposition~\ref{lem:pmay-max},
it is sufficient to show that for every test $T$,
$$
min({\mathbb V}(\interp{\nu \vec x (T ~|~ P) })) \leq min({\mathbb V}(\interp{\nu \vec x(T ~|~ Q)}))
$$
where $\vec x$ contain the free names of $T$, $P$ and $Q$, but excluding $\omega.$
Since $\sqsubseteq_{FS}$ is preserved by parallel composition (cf. Lemma~\ref{lem:preserve.par}) and name restriction, we have
that
$$
\nu \vec x(T ~|~ P) \sqsubseteq_{FS}^e \nu \vec x(T~|~Q),
$$
which means there is a $\Theta$ such that
$\interp {\nu \vec x(T~|~P)} \bstep{\hat \tau} \Theta$ and
$\interp {\nu \vec x(T~|~Q)} ~ \lift{\triangleleft_{FS}^e} \Theta.$
The result then follows from Proposition~\ref{prop:sim-max} and
Lemma~\ref{lm:trans-max}.
\qed
\end{proof}
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\section{A modal logic for $\pi_p$}
\label{sec:modal}
We consider a modal logic based on a fragment of
Milner-Parrow-Walker's (MPW) modal logic for the (non-probabilistic)
$\pi$-calculus~\cite{Milner93TCS}, but extended with a probabilistic
disjunction operator $\oplus$, similar to that used in \cite{Deng08LMCS}.
The language of formulas is given by the following grammar:
$$
\varphi ::= \top ~ \mid ~ \Ref{X} ~ \mid ~ \ldia{a(x)} \varphi ~ \mid ~
\ldia{\bar a x} \varphi ~ \mid ~
\ldia{\bar a(x)} \varphi ~ \mid ~
\varphi_1 \wedge \varphi_2 ~ \mid ~
\varphi_1 {\pch p} \varphi_2
$$
The $x$'s in $\ldia{a(x)}\varphi$ and $\ldia{\bar a(x)}\varphi$ are
binders, whose scope is over $\varphi.$
The diamond operator $\ldia{a(x)}$ is called a bound input modal operator,
$\ldia{\bar a x}$ a free output modal operator and $\ldia{\bar a(x)}$ a bound output
modal operator.
Instead of binary conjunction and probabilistic disjunction, we sometimes write $\bigwedge_{i\in I}\varphi_i$ and $\varphi_1 {\pch p} \varphi_2$ for finite index set $I$; they can be expressed by nested use of their binary forms.
We refer to this modal logic as ${\cal F}$.
Let ${\cal L}$ be the sub-logic of ${\cal F}$ by skipping the $\Ref{X}$ clause.
The semantics of each operator is defined as follows.
\begin{definition}
\label{def:sat}
The {\em satisfaction relation} $\models$ between a distribution and
a modal formula is defined inductively as follows:
\begin{itemize}
\item $\Delta \models \top$ always.
\item $\Delta \models \Ref{X}$ iff there is a $\Delta'$ with $\Delta\dar{\hat{\tau}} \Delta'$ and $\Delta'\not\barb{X}$.
\item $\Delta \models \ldia{a(x)}\varphi$ iff
for all $z$ there are $\Delta_1,$ $\Delta_2,$ $\Delta'$ and $w$ such that
$\Delta \bstep{\hat \tau} \Delta_1 \sstep{a(w)} \Delta_2$,
$\Delta_2[z/w] \bstep{\hat \tau} \Delta'$
and
$\Delta' \models \varphi[z/x].$
\item $\Delta \models \ldia{\bar a x}\varphi$ iff for some $\Delta'$,
$\Delta \bstep{\widehat{\bar a x}} \Delta'$
and $\Delta' \models \varphi.$
\item $\Delta \models \ldia{\bar a(x)}\varphi$ iff for some
$\Delta'$ and $w \not \in fn(\varphi, \Delta)$,
$\Delta \bstep{\widehat{\bar a(w)}} \Delta'$
and $\Delta' \models \varphi[w/x].$
\item $\Delta \models \varphi_1 \wedge \varphi_2$ iff
$\Delta \models \varphi_1$ and $\Delta \models \varphi_2$.
\item $\Delta \models \varphi_1 {\pch p} \varphi_2$ iff there are $\Delta_1,\Delta_2 \in {\cal D}(S_p)$ with $\Delta_1\models\varphi_1$ and $\Delta_2\models\varphi_2$, such that
$\bigstep{\Delta}{\hat \tau}{p \cdot \Delta_1 + (1-p)\cdot\Delta_2.}$
\end{itemize}
We write $\Delta \sqsubseteq_\Lcal \Theta$ just when
$\Delta \models \psi$ implies $\Theta \models \psi$ for
all $\psi \in {\cal L}$, and $\Delta \sqsubseteq_\Fcal \Theta$ just when $\Theta \models \varphi$ implies $\Delta \models \varphi$ for all $\varphi\in{\cal F}$.
We write $P \sqsubseteq_\Lcal Q$
when $\interp P \sqsubseteq_\Lcal \interp Q$, and $P \sqsubseteq_\Fcal Q$ when $\interp P \sqsubseteq_\Fcal \interp Q$.
\end{definition}
Following \cite{Deng08LMCS}, in order to show soundness
of the logical preorders w.r.t. the simulation pre-orders, we need to define a notion
of characteristic formulas.
\begin{definition}[Characteristic formula]
\label{def:char-form}
The {\em ${\cal F}$-characteristic formulas} $\varphi_s$ and $\varphi_\Delta$
of, respectively, a state-based process $s$ and a distribution $\Delta$
are defined inductively as follows:
$$
\begin{array}{rcl}
\varphi_s & := & \bigwedge\{\ldia{\alpha}\varphi_\Delta \mid s\ar{\alpha}\Delta\} \wedge \Ref{\{\mu \mid s\not\barb{\mu}\}} \qquad\mbox{ if $s\not\ar{\tau}$},\\
\varphi_s & := & \bigwedge\{\ldia{\alpha}\varphi_\Delta \mid s\ar{\alpha}\Delta, ~ \alpha \not = \tau\} ~ \wedge ~
\bigwedge\{\varphi_\Delta \mid s\ar{\tau}\Delta\} \qquad
\mbox{ otherwise}.\\
\varphi_\Delta & := & \bigoplus_{s\in\supp{\Delta}}\Delta(s) \cdot \varphi_s
\end{array}
$$
where $\bigoplus$ is a generalised probabilistic choice as in Section~\ref{sec:pi}.
The \emph{${\cal L}$-characteristic formulas} $\psi_s$ and $\psi_\Delta$ are defined likewise, but omitting the conjuncts $\Ref{\{\mu \mid s\not\barb{\mu}\}}$.
\end{definition}
Note that because we use the late semantics (cf. Figure~\ref{fig:pi}), the conjunction in $\varphi_s$ is finite even though there can
be infinitely many (input) transitions from $s.$
Given a state based process $s$, we define its {\em size}, $|s|$, as the
number of process constructors and names in $s.$
The following lemma is straightforward from the definition of the operational
semantics of $\pi_p$.
\begin{lemma}
\label{lm:trans-size}
If $s \sstep{\alpha} \Delta$ then $|s| > |t|$ for every $t \in \supp \Delta.$
\end{lemma}
\begin{lemma}
\label{lm:char-form}
For every $\Delta \in {\cal D}(S_p)$, $\Delta \models \varphi_\Delta$, as well as $\Delta\models\psi_\Delta$.
\end{lemma}
\begin{proof}
It is enough to show that $\bar s \models \varphi_s.$
This is proved by by induction on $|s|.$
So suppose $s\not\ar{\tau}$. Then we have
$$
\begin{array}{ll}
\varphi_s = & \Ref{\{\mu \mid s\not\barb{\mu}\}}\wedge \\
& \bigwedge \{\ldia {a(x)} \varphi_\Delta \mid \one s {a(x)}{\Delta} \} \wedge
\bigwedge \{\varphi_\Delta \mid \one s \tau \Delta \} \wedge \\
& \bigwedge \{\ldia {\bar a x } \varphi_\Delta \mid \one s {\bar a x }{\Delta} \} \wedge
\bigwedge \{\ldia {\bar a(x) } \varphi_\Delta \mid \one s {\bar a(x) }{\Delta} \}.
\end{array}
$$
where $
\varphi_\Delta = \bigoplus_{s \in \supp \Delta} \Delta(s).\varphi_s.
$
For each of the conjunct $\phi$, we prove that $\pdist s \models \phi.$
We show here two cases; the other cases are similar.
\begin{itemize}
\item $\phi= \Ref{X}$, where $X=\{\mu \mid s\not\barb{\mu}\}$.
For each $\mu \in X$ we have $s\not\barb{\mu}$. Moreover, since $s\not\ar{\tau}$,
we see that $s\not\barb{X}$.
\item $\phi = \ldia {a(x)} \varphi_\Delta$.
So suppose $s \sstep{a(x)} \Delta$ and
$\supp \Delta = \{s_i \mid i \in I\}$ and $\Delta = \sum_{i \in I} p_i \cdot \pdist{s_i}.$
Since $|s_i| < |s|$, by the induction hypothesis,
for every name $w$, we have
$$
\pdist{s_i[w/x]} \models \varphi_{s_i[w/x]}
$$
and therefore:
$$
\Delta[w/x] = \sum_{i\in I} p_i \cdot \pdist{s_i[w/x]} \models
\bigoplus_{i \in I} p_i \cdot \varphi_{s_i[w/x]} = \varphi_\Delta[w/x].
$$
Let $\Phi_1 = \Phi_2 = \pdist s.$
Obviously we have, for every $w$,
$$
\Phi_1 \bstep {\hat \tau} \Phi_2 \sstep {a(x)} \Delta, \qquad
\Delta[w/x] \models \varphi_\Delta[w/x].
$$
So by Definition~\ref{def:sat}, $\pdist s \models \phi.$
\end{itemize}
\qed
\end{proof}
\begin{lemma}
\label{lm:modal-sim}
For any processes $P$ and $Q$,
$\interp{P}\models\varphi_{\interp{Q}}$ implies $P\sqsubseteq_{FS} Q$, and likewise
$\interp Q \models \psi_{\interp P}$
implies $P \sqsubseteq_S Q.$
\end{lemma}
\begin{proof}
Let ${\cal R}$ be the relation defined as follows:
$s ~ {\cal R} ~ \Theta$ iff $\Theta \models \varphi_s.$
We first prove the following claim:
\begin{equation}
\label{claim}
\mbox{
$\Theta \models \varphi_\Delta$ implies
there exists $\Theta'$ such that
$\bigstep{\Theta}{\hat \tau}{\Theta'}$
and $\Delta ~ \overline {\cal R} ~ \Theta'.$
}
\end{equation}
To prove this claim (following \cite{Deng08LMCS}), suppose that $\Theta \models \Delta$.
By definition, $\varphi_\Delta = \bigoplus_{i \in I} p_i \cdot \varphi_{s_i}$
and $\Delta = \sum_{i \in I} p_i \cdot \pdist {s_i}$. For every $i \in I$,
we have $\Theta_i \in {\cal D}(S_p)$ with $\Theta_i \models \varphi_{s_i}$
such that $\bigstep {\Theta} {\hat \tau}{\Theta'}$ with
$\Theta' = \sum_{i\in I} p_i \cdot \Theta_i.$ Since $s_i ~ {\cal R} ~ \Theta_i$
for all $i \in I$, we have $\Delta ~ \overline {\cal R} ~ \Theta'.$
We now proceed to show that ${\cal R}$ is a failure simulation, hence proving
the first statement of the lemma. So suppose $s ~ {\cal R} ~ \Theta$.
\begin{enumerate}
\item Suppose $\one s \tau \Delta$. By the definition of ${\cal R}$,
we have $\Theta \models \varphi_s.$
By Definition~\ref{def:char-form}, we also have $\Theta \models \varphi_\Delta.$
By (\ref{claim}) above, there exists $\Theta'$
such that $\bigstep{\Theta}{\hat \tau}{\Theta'}$ and $\Delta ~ \overline {\cal R} ~ \Theta'.$
\item Suppose $\one s {\bar a x} \Delta$.
Then by Definition~\ref{def:char-form}, $\Theta \models \ldia{\bar a x}\varphi_\Delta.$
So $\Theta \bstep {\bar a x } \Theta'$ and $\Theta' \models \varphi_\Delta$,
for some $\Theta'.$ By (\ref{claim}), there exists $\Theta''$ such that
$\Theta' \bstep{\hat {\tau}}{\Theta''}$ and $\Delta ~ \overline {\cal R} ~ \Theta''.$
This means that $\Theta \bstep{\bar a x} \Theta''$ and
$\Delta ~ \overline {\cal R} ~ \Theta''.$
\item Suppose $\one s {a(x)} \Delta$ for some $x \not \in fn(s,\Theta).$
By Definition~\ref{def:char-form}, $\Theta \models \ldia{a(x)}\varphi_\Delta.$
This means for every name $z$, there exists $\Theta_z^1$, $\Theta_z^2$ and $\Theta_z$ such that
$\Theta \bstep {\hat \tau} \Theta_z^1 \sstep {a(x)} {\Theta_z^2}$,
$\Theta_z^2[z/x] \bstep{\hat \tau} \Theta_z$ and $\Theta_z \models \varphi_\Delta[z/x].$\footnote{Strictly
speaking, we should also consider the case where $\Theta_z^1 \sstep{a(w)} \Theta_z^2$,
but it is easy to see that since $x \not \in fn(s,\Theta)$ we can always apply a renaming to rename
$w$ to $x.$}
Then by (\ref{claim}) we have
$\bigstep {\Theta_z }{\hat \tau}{\Theta_z'}$
and $\Delta[z/x] ~ \overline {\cal R} ~ \Theta_z'.$
So we indeed have, for every name $z$, $\Theta_z^1$, $\Theta_z^2$ and $\Theta_z'$ such that
$$
\Theta \bstep{\hat \tau} \Theta_z^1 \sstep{a(x)} \Theta_z^2,
\qquad
\Theta_z^2[z/x] \bstep{\hat \tau} \Theta_z'
\qquad
\hbox{ and }
\qquad
\Delta[z/x] ~ \overline {\cal R} ~ \Theta_z'.
$$
\item Suppose $\one s {\bar a(x)}{\Delta}.$ This case is similar
to the previous one, except that we need only to consider one
instance of $x$ with a fresh name.
\item Suppose $s\not\barb{X}$ for a set of channel names $X$. By Definition~\ref{def:char-form}, we have $\Theta\models\Ref{X}$. Hence, there is some $\Theta'$ with $\Theta\dar{\hat{\tau}}\Theta'$ and $\Theta'\not\barb{X}$.
\end{enumerate}
To establish the second statement, define ${\cal R}$ by $s{\cal R}\Theta$ iff $\Theta\models\psi_s$. Just as above it can be shown that ${\cal R}$ is a simulation. Then the second statement of the lemma easily follows.
\qed
\end{proof}
\begin{theorem}\label{thm:modal-sim}
\begin{enumerate}
\item
If $P \sqsubseteq_\Lcal Q$ then $P \sqsubseteq_S Q.$
\item
If $P \sqsubseteq_\Fcal Q$ then $P \sqsubseteq_{FS} Q.$
\end{enumerate}
\end{theorem}
\begin{proof}
Suppose $P \sqsubseteq_{\cal L} Q$. By Lemma~\ref{lm:char-form}, we have $\interp P \models \psi_{\interp P}$,
hence $\interp Q \models \psi_{\interp P}$. Then by Lemma~\ref{lm:modal-sim}, we have $P \sqsubseteq_S Q.$
For the second statement, assume $P \sqsubseteq_{FS} Q$, we have $\interp{Q}\models\varphi_{\interp{Q}}$ and hence $\interp{P}\models\varphi_{\interp{Q}}$, and thus $P\sqsubseteq_{FS} Q$.
\qed
\end{proof}
\newcommand\vsim[1]{{\widehat \sqsubseteq}^{#1}_{pmay}}
\newcommand\vapply[2]{{\widehat {\cal A}}^\Omega_{\updownarrow}(#1,#2) }
\newcommand\pr[2]{\langle #1, #2\rangle}
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\section{Completeness of the simulation preorders}
\label{sec:comp}
In the following, we assume a function $new$ that takes as an argument
a finite set of names and outputs a fresh name, i.e.,
if $new(N) = x$ then $x\not \in N.$
If $N = \{x_1,\ldots,x_n\}$, we write $[x \not = N]P$ to abbreviate
$[x \not = x_1][x \not = x_2] \cdots [x \not = x_n]P.$
For convenience of presentation, we write $\vec{\omega}$ for the vector in $[0,1]^\Omega$
defined by $\vec{\omega}(\omega)=1$ and $\vec{\omega}(\omega')=0$ for
any $\omega'\not=\omega$.
We also extend the $Apply^\Omega$ function to
allow applying a test to a distribution, defined as
$Apply^\Omega(T, \Delta) = {\mathbb V}({\nu \vec x(\interp T~|~\Delta)})$
where $\vec x = fn(T,\Delta) - \Omega.$
\begin{lemma}
\label{lm:sat-renaming}
If $\Delta \models \varphi$ then $\Delta \sigma \models \varphi \sigma$
for any renaming substitution $\sigma.$
\end{lemma}
In the following, given a name $a$, we write $a.P$ to denote $a(y).P$ for
some $y \not \in fn(P).$ Similarly, we write $\bar a.P$ to denote
$\bar a a.P.$
Recall that the size of a state-based process, $|s|$, is the number of
symbols in $s.$ The {\em size} of a distribution $\Delta$, written $|\Delta|$, is
the {\em multiset} $\{|s| \mid s \in \supp \Delta \}.$
There is a well-founded ordering on $|\Delta|$, i.e., the multiset (of natural
numbers) ordering, which we shall denote with $\prec$.
\begin{lemma}
\label{lm:test1}
Let $P$ be a process and $T, T_i$ be tests.
\begin{enumerate}
\item $o\in Apply^\Omega(\omega,P)$ iff $o=\vec{\omega}$.
\item Let $X=\{\mu_1,...,\mu_n\}$ and $T=\mu_1.\omega+...+\mu_n.\omega$. Then
$\vec{0} \in Apply^\Omega(T,P)$ iff
$\interp{P}\dar{\hat{\tau}}\Delta$ for some $\Delta$ with $\Delta\not\barb{X}$.
\item Suppose the action $\omega$ does not occur in the test $T$. Then
$o\in Apply^\Omega(\omega+a(x).([x=y]\tau.T+\omega), P) $ with $o(\omega)=0$ iff
there is $\Delta$ such that
$\interp P \bstep {\widehat{\bar a y}} \Delta$
and $o\in Apply^\Omega(T[y/x], \Delta).$
\item Suppose the action $\omega$ does not occur in the test $T$ and $fn(P)\subseteq N$. Then
$o\in Apply^\Omega(\omega+a(x).([x\not=N]\tau.T+\omega), P) $ with $o(\omega)=0$ iff
there is $\Delta$ such that
$\interp P \bstep {\widehat{\bar a (y)}} \Delta$
and $o\in Apply^\Omega(T[y/x], \Delta).$
\item Suppose the action $\omega$ does not occur in the test $T$. Then
$o\in Apply^\Omega(\omega+\bar a x.T, P)$ with $o(\omega)=0$
iff there are $\Delta$, $\Delta_1$ and $\Delta_2$
such that $\interp P \bstep{\hat \tau} \Delta_1 \sstep{a(y)} \Delta_2,$
$\Delta_2[x/y] \bstep{\hat \tau} \Delta$
and
$o\in Apply^\Omega(T,\Delta).$
\item $o\in Apply^\Omega(\bigoplus_{i\in I} p_i \cdot T_i, P) $
iff $o=\sum_{i\in I}p_i\cdot o_i$ for some
$o_i\in Apply^\Omega(T_i,P)$ for all $i\in I.$
\item $o\in Apply^\Omega(\sum_{i\in I} \tau.T_i, P)$
if for all $i \in I$ there are $q_i \in [0,1]$ and
$\Delta_i$ such that $\sum_{i\in I} q_i = 1$,
$\interp P \bstep{\hat \tau} \sum_{i \in I} q_i \cdot \Delta_i$
and $o=\sum_{i\in I}q_i\cdot o_i$ for some
$o_i\in Apply^\Omega(T_i,\Delta_i).$
\end{enumerate}
\end{lemma}
\begin{proof}
The proofs of items 1 and 2 are similar to the proofs of Lemma 6.7(1) and 6.7(2)
in \cite{Deng08LMCS} for pCSP; items 6 and 7 correspond to
Lemma 6.7(4) and Lemma 6.7(5) in \cite{Deng08LMCS}, respectively.
Items 3, 4 and 5 have a counterpart in Lemma 6.7(3) of \cite{Deng08LMCS}, but they
are quite different, due to the name-passing feature of the $\pi$-calculus, and the
possibility of checking the identity of the input value via the match and the mismatch
operators.
We show here a proof of item 3; the proofs of items 4 and 5 are similar.
We first generalize item 3 to distributions: given $\omega$ and $T$ as above, we have,
for every distribution $\Theta$,
\begin{quote}
$o\in Apply^\Omega(\omega+a(x).([x=y]\tau.T+\omega), \Theta) $ with $o(\omega)=0$ iff
there is $\Delta$ such that
$\Theta \bstep {\widehat{\bar a y}} \Delta$
and $o\in Apply^\Omega(T[y/x], \Delta).$
\end{quote}
The `if' part is straightforward from Definition~\ref{def:vector-based-results}.
We show the `only if' part here.
The proof will make use of the following claim (easily proved by induction on $|\Theta|$):
\begin{equation}
\label{eq:claim}
\begin{array}{l}
\mbox{\bf Claim:} ~ o\in Apply^\Omega([y=y]\tau.T[y/x]+\omega, \Theta) \mbox{ with } o(\omega)=0 \mbox{ iff } \\
\mbox{there is $\Delta$ such that $\Theta \bstep {\hat \tau} \Delta$ and $o\in Apply^\Omega(T[y/x], \Delta).$ }
\end{array}
\end{equation}
So, suppose we have $o\in Apply^\Omega(\omega+a(x).([x=y]\tau.T+\omega), \Theta) $ with $o(\omega)=0$.
We show, by induction on $|\Theta|$, that there exists $\Delta$ such that
$\Theta \bstep {\bar a y} \Delta$
and $o\in Apply^\Omega(T[y/x], \Delta).$
Let $T' = \omega+a(x).([x=y]\tau.T+\omega)$, and suppose
$\Theta = p_1 \cdot \pdist{s_1} + \ldots + p_n \cdot \pdist{s_n}$,
for pairwise distincts state-based processes $s_1,\ldots,s_n$,
and suppose that
$\vec z$ is an enumeration of the set $fn(T',\Theta) - \Omega.$
Then
$$
Apply^\Omega(T',\Theta) = {\mathbb V}^\Omega(p_1 \cdot \pdist{\nu \vec z(T'|s_1)} + \ldots + p_n \cdot \pdist{\nu \vec z(T'|s_n)}).
$$
From Definition~\ref{def:vector-based-results}, in order to have $o(\omega) = 0$,
it must be the case that $\nu \vec z(T' | s_j) \ar{\tau} $ for
every $j \in \{1,\dots,n\}.$
From the definition of the operational semantics, there are exactly two cases where this might happen:
\begin{itemize}
\item For some $i$, $s_i \ar{\tau} \Lambda$ for some distribution $\Lambda.$
Let $\Theta' = p_1 \cdot \pdist{s_1} + \ldots + p_i \cdot \Lambda + \ldots + p_n \cdot \pdist{s_n}.$
Then we have $\Theta \ar{\hat \tau} \Theta'$ and
$\nu\vec z(T' | \Theta) \ar{\hat \tau} \nu \vec z(T'|\Theta').$
The latter means that $o \in {\mathbb V}^\Omega(\nu \vec z(T'|\Theta'))$ as well.
By Lemma~\ref{lm:trans-size}, we know that $|\Lambda| \prec \{|s_i|\}$,
and therefore $|\Theta'| \prec |\Theta|.$ By the induction hypothesis,
$$
\Theta \ar{\hat\tau} \Theta' \bstep{\widehat{\bar a y}} \Delta
$$
and $o \in Apply^\Omega(T[y/x], \Delta).$
\item For every $i \in \{1,\dots,n\}$, we have $s_i \not \ar{\tau}.$
This can only mean that the $\tau$ transition from $\nu \vec z(T'|s_i)$
derives from a communiation between $T'$ and $s_i.$
This means that $s_i\barb{\bar a}$, for every $i \in \{1,\dots,n\}.$
We claim that, in fact, for every $i$, we have $s_i \ar{\bar a y} \Theta_i$,
for some $\Theta_i.$ For otherwise, we would have that for some
$j$, $\nu \vec z(T' | s_j) \ar{\tau} \nu \vec z(([u = y]\tau.T[y/x] + \omega) ~|~ \Theta_j)$,
for some $u$ distinct from $y.$ But this means that only the $\omega$ action is enabled
in the test, so all results of ${\mathbb V}^\Omega(\nu \vec z(([u = y]\tau.T[y/x] + \omega) ~|~ \Theta_i))$
in this case would have a non-zero $\omega$ component, which would mean that
$o(\omega)$ would be non-zero as well, contradicting the assumption that $o(\omega) = 0$.
So, we have $s_i \ar{\bar a y} \Theta_i$ for every $i \in \{1,\dots,n\}.$
Let $\Theta'=p_1 \cdot \Theta_1 + \ldots + p_n \cdot \Theta_n.$
Then we have $\Theta \ar{\bar a y} \Theta'$ and
$\nu \vec z(T' ~|~ \Theta) \ar{\tau} \nu \vec z(T'' ~|~ \Theta')$
where $T'' = [y=y]\tau.T[y/x] + \omega$.
The latter transition means that $o \in {\mathbb V}^\Omega(\nu \vec z(T'' ~|~ \Theta')) = Apply^\Omega(T'',\Theta').$
We can therefore apply Claim~\ref{eq:claim} to get:
$$
\Theta \ar{\bar a y} \Theta' \bstep{\hat \tau} \Delta
$$
and $o \in Apply^\Omega(T[y/x], \Delta).$
\end{itemize}
\qed
\end{proof}
\begin{lemma}
\label{lm:test2}
If $o\in Apply^\Omega(\sum_{i\in I} \tau.T_i, P)$ then for all
$i \in I$ there are $q_i \in [0,1]$ and $\Delta_i$ with $\sum_{i\in I} q_i = 1$
such that $\interp P \bstep {\hat \tau} \sum_{i\in I} q_i \cdot \Delta_i$
and $o=\sum_{i\in I}q_i\cdot o_i$ for some $o_i\in Apply^\Omega(T_i,\Delta_i).$
\end{lemma}
\begin{proof}
The proof is similar to the proof of Lemma 6.8 in \cite{Deng08LMCS}.
\qed
\end{proof}
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