alx-ai/arxiv_papers · Datasets at Hugging Face

text stringlengths 0 44.4k
For later reference let us note that the matrix Jr, with matrix elements
(Jr)st=Jrst (28)
is the adjoint representative of Π rin the SIC-projector basis:
adΠrΠs= [Πr,Πs] =d2/summationdisplay
t=1JrstΠt (29)
It can be seen that all the interesting features of the tensor Grs(respectively,
the tensors Trst,JrstandRrst) are contained in the order-2 angle tensor θrs(re-
spectively, the order-3 angle tensor θrst). It is also easy to see that, for any unitary
U, the transformation
Πr→UΠrU†(30)
leaves the angle tensors invariant. This suggests that we shift our focus from indi-
vidual SIC-POVMs to families of unitarily equivalent SIC-POVMs—SIC- families,
as we will call them for short.
We begin our investigation in Section 2by giving necessary and suﬃcient con-
ditions for an arbitrary tensor θrs(respectively θrst) to be the rank-2 (respectively
rank-3) angle tensor corresponding to a SIC-family. We also show t hat either angle
tensor uniquely determines the corresponding SIC-family. Finally we describe a6
method for reconstructing the SIC-family, starting from a knowle dge of either of
the two angle tensors.
In Sections 3,4and5we prove the central result of this paper: namely, that
the existence of a SIC-POVM in dimension dis equivalent to the existence of a
certain very special set of matrices in the adjoint representation of gl(d,C). In
Section3we show that, for any SIC-POVM, the adjoint matrices Jrhave the
spectral decomposition
Jr=Qr−QT
r (31)
whereQris a rankd−1 projector which has the remarkable property of being
orthogonal to its own transpose:
QrQT
r= 0 (32)
We refer to this feature of the adjoint matrices as the Q-QTproperty. In Section 3
we also show that from a knowledge of the Jmatrices it is possible to reconstruct
the corresponding SIC-family. In Section 4we characterize the general class of
projectors which have the property of being orthogonal to their own transpose.
Then, in Section 5, we prove a converse of the result established in Section 3. The
Q-QTproperty is not completely equivalent to the property of being a SIC set.
However, it turns out that it is, in a certain sense, very nearly equiv alent. To be
more speciﬁc: let Lrbe any set of d2Hermitian operators which constitute a basis
for gl(d,C) and letCrbe the adjoint representative of Lrin this basis. Then the
necessary and suﬃcient condition for the Crto have the spectral decomposition
Cr=Qr−QT
r (33)
whereQris a rankd−1 projector such that QrQT
r= 0 is that there exists a
SIC set Π rsuch thatLr=ǫr(Πr+αI) for some ﬁxed number α∈Rand signs
ǫr=±1. In particular, the existence of an Hermitian basis for gl( d,C) having the
Q-QTproperty is both necesary and suﬃcient for the existence of a SIC -POVM in
dimensiond.
In Section 6we digress brieﬂy, and consider sl( d,C) (the Lie algebra consisting
of all trace-zero d×dcomplex matrices). As we have explained, this paper is
motivated by the hope that a Lie algebraic perspective will cast light o n the SIC-
existence problem, rather than by an interest in Lie algebras as suc h. We focus on
gl(d,C) because that is the casewherethe connection with SIC-POVMsse ems most
straightforward. However a SIC-POVM also gives rise to an interes ting geometrical
structure in sl( d,C), as we show in Section 6.
In Section 7we derive a number of additional identities satisﬁed by the Jand
Qmatrices.
The complex projectors Qr,QT
rand the real projector Qr+QT
rdeﬁne three
families of subspaces. It turns out that there are some interestin g geometrical
relationships between these subspaces, which we study in Section 8.
Finally, in Section 9we show that, with the appropriate choice of gauge, the
Gram matrix corresponding to a Weyl-Heisenberg covariant SIC-fa mily has a fea-
ture analogous to the Q-QTproperty, which we call the P-PTproperty. It is an
open question whether this result generalizes to other SIC-families , not covariant
with respect to the Weyl-Heisenberg group.7
2.The Angle Tensors
The purpose of this section is to establish the necessary and suﬃcie nt conditions
for an arbitrary tensor θrs(respectively θrst) to be the order-2 (respectively order-
3) angle tensor for a SIC-family. We will also show that either one of t he angle
tensors is enough to uniquely determine the SIC-family. Moreover, we will describe
explicit procedures for reconstructing the family, starting from a knowledge of one
of the angle tensors.
We begin by considering the general class of POVMs (not just SIC-P OVMs)
which consist of d2rank-1 elements. A POVM of this type is thus deﬁned by a set
ofd2vectors\|ξ1/an}bracketri}ht,...,\|ξd2/an}bracketri}htwith the property
d2/summationdisplay
r=1\|ξr/an}bracketri}ht/an}bracketle{tξr\|=I (34)
Note that/summationtextd2
r=1/vextenddouble/vextenddouble\|ξr/an}bracketri}ht/vextenddouble/vextenddouble2=d, so the vectors \|ξr/an}bracketri}htcannot all be normalized. In the
particular case of a SIC-POVM the vectors all have the same norm/vextenddouble/vextenddouble\|ξr/an}bracketri}ht/vextenddouble/vextenddouble=1√
d.
However in the general case they may have diﬀerent norms.
Given a set of such vectors consider the Gram matrix
Prs=/an}bracketle{tξr\|ξs/an}bracketri}ht (35)
Clearly the Gram matrix cannot determine the POVM uniquely since if Uis any
unitary operator then the vectors U\|ξr/an}bracketri}htwill deﬁne another POVM having the same
Gram matrix. However, the theorem we now prove shows that this is the only free-
dom. In other words, the Gram matrix ﬁxes the POVM up to unitary e quivalence.
The theorem also provides us with a criterion for deciding whether an arbitrary
d2×d2matrixPis the Gram matrix corresponding to a POVM of the speciﬁed
type. As a corollary this will give us a criterion for deciding whether an arbitrary
tensorθrsis speciﬁcally the order-2 angle tensor for a SIC-family.
Theorem 1. LetPbe anyd2×d2Hermitian matrix. Then the following conditions
are equivalent:
(1)Pis a rankdprojector.
(2)Psatisﬁes the trace identities
Tr(P) = Tr(P2) = Tr(P3) = Tr(P4) =d (36)
(3)Pis the Gram matrix for a set of d2vectors\|ξr/an}bracketri}ht(not all normalized) such

For later reference let us note that the matrix Jr, with matrix elements

(Jr)st=Jrst (28)

is the adjoint representative of Π rin the SIC-projector basis:

adΠrΠs= [Πr,Πs] =d2/summationdisplay

t=1JrstΠt (29)

It can be seen that all the interesting features of the tensor Grs(respectively,

the tensors Trst,JrstandRrst) are contained in the order-2 angle tensor θrs(re-

spectively, the order-3 angle tensor θrst). It is also easy to see that, for any unitary

U, the transformation

Πr→UΠrU†(30)

leaves the angle tensors invariant. This suggests that we shift our focus from indi-

vidual SIC-POVMs to families of unitarily equivalent SIC-POVMs—SIC- families,

as we will call them for short.

We begin our investigation in Section 2by giving necessary and suﬃcient con-

ditions for an arbitrary tensor θrs(respectively θrst) to be the rank-2 (respectively

rank-3) angle tensor corresponding to a SIC-family. We also show t hat either angle

tensor uniquely determines the corresponding SIC-family. Finally we describe a6

method for reconstructing the SIC-family, starting from a knowle dge of either of

the two angle tensors.

In Sections 3,4and5we prove the central result of this paper: namely, that

the existence of a SIC-POVM in dimension dis equivalent to the existence of a

certain very special set of matrices in the adjoint representation of gl(d,C). In

Section3we show that, for any SIC-POVM, the adjoint matrices Jrhave the

spectral decomposition

Jr=Qr−QT

r (31)

whereQris a rankd−1 projector which has the remarkable property of being

orthogonal to its own transpose:

QrQT

r= 0 (32)

We refer to this feature of the adjoint matrices as the Q-QTproperty. In Section 3

we also show that from a knowledge of the Jmatrices it is possible to reconstruct

the corresponding SIC-family. In Section 4we characterize the general class of

projectors which have the property of being orthogonal to their own transpose.

Then, in Section 5, we prove a converse of the result established in Section 3. The

Q-QTproperty is not completely equivalent to the property of being a SIC set.

However, it turns out that it is, in a certain sense, very nearly equiv alent. To be

more speciﬁc: let Lrbe any set of d2Hermitian operators which constitute a basis

for gl(d,C) and letCrbe the adjoint representative of Lrin this basis. Then the

necessary and suﬃcient condition for the Crto have the spectral decomposition

Cr=Qr−QT

r (33)

whereQris a rankd−1 projector such that QrQT

r= 0 is that there exists a

SIC set Π rsuch thatLr=ǫr(Πr+αI) for some ﬁxed number α∈Rand signs

ǫr=±1. In particular, the existence of an Hermitian basis for gl( d,C) having the

Q-QTproperty is both necesary and suﬃcient for the existence of a SIC -POVM in

dimensiond.

In Section 6we digress brieﬂy, and consider sl( d,C) (the Lie algebra consisting

of all trace-zero d×dcomplex matrices). As we have explained, this paper is

motivated by the hope that a Lie algebraic perspective will cast light o n the SIC-

existence problem, rather than by an interest in Lie algebras as suc h. We focus on

gl(d,C) because that is the casewherethe connection with SIC-POVMsse ems most

straightforward. However a SIC-POVM also gives rise to an interes ting geometrical

structure in sl( d,C), as we show in Section 6.

In Section 7we derive a number of additional identities satisﬁed by the Jand

Qmatrices.

The complex projectors Qr,QT

rand the real projector Qr+QT

rdeﬁne three

families of subspaces. It turns out that there are some interestin g geometrical

relationships between these subspaces, which we study in Section 8.

Finally, in Section 9we show that, with the appropriate choice of gauge, the

Gram matrix corresponding to a Weyl-Heisenberg covariant SIC-fa mily has a fea-

ture analogous to the Q-QTproperty, which we call the P-PTproperty. It is an

open question whether this result generalizes to other SIC-families , not covariant

with respect to the Weyl-Heisenberg group.7

2.The Angle Tensors

The purpose of this section is to establish the necessary and suﬃcie nt conditions

for an arbitrary tensor θrs(respectively θrst) to be the order-2 (respectively order-

3) angle tensor for a SIC-family. We will also show that either one of t he angle

tensors is enough to uniquely determine the SIC-family. Moreover, we will describe

explicit procedures for reconstructing the family, starting from a knowledge of one

of the angle tensors.

We begin by considering the general class of POVMs (not just SIC-P OVMs)

which consist of d2rank-1 elements. A POVM of this type is thus deﬁned by a set

ofd2vectors|ξ1/an}bracketri}ht,...,|ξd2/an}bracketri}htwith the property

d2/summationdisplay

r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=I (34)

Note that/summationtextd2

r=1/vextenddouble/vextenddouble|ξr/an}bracketri}ht/vextenddouble/vextenddouble2=d, so the vectors |ξr/an}bracketri}htcannot all be normalized. In the

particular case of a SIC-POVM the vectors all have the same norm/vextenddouble/vextenddouble|ξr/an}bracketri}ht/vextenddouble/vextenddouble=1√

d.

However in the general case they may have diﬀerent norms.

Given a set of such vectors consider the Gram matrix

Prs=/an}bracketle{tξr|ξs/an}bracketri}ht (35)

Clearly the Gram matrix cannot determine the POVM uniquely since if Uis any

unitary operator then the vectors U|ξr/an}bracketri}htwill deﬁne another POVM having the same

Gram matrix. However, the theorem we now prove shows that this is the only free-

dom. In other words, the Gram matrix ﬁxes the POVM up to unitary e quivalence.

The theorem also provides us with a criterion for deciding whether an arbitrary

d2×d2matrixPis the Gram matrix corresponding to a POVM of the speciﬁed

type. As a corollary this will give us a criterion for deciding whether an arbitrary

tensorθrsis speciﬁcally the order-2 angle tensor for a SIC-family.

Theorem 1. LetPbe anyd2×d2Hermitian matrix. Then the following conditions

are equivalent:

(1)Pis a rankdprojector.

(2)Psatisﬁes the trace identities

Tr(P) = Tr(P2) = Tr(P3) = Tr(P4) =d (36)

(3)Pis the Gram matrix for a set of d2vectors|ξr/an}bracketri}ht(not all normalized) such