alx-ai/arxiv_papers · Datasets at Hugging Face

text stringlengths 0 44.4k
The purposeofthis paperis totryto takeourunderstandingofSI C mathematics
(as it might be called) a little further forward. The research we repo rt began with
a chance numerical discovery made while we were working on a diﬀeren t problem.
Pursuing that initial numerical hint we uncovered a rich and interest ing set of
connections between SIC-POVMs in dimension dand the Lie Algebra gl( d,C). The
existence of these connections came as a surprise to us. However , in retrospect it
is, perhaps, not so surprising. Interest in SIC-POVMs has, to dat e, focused on the
fact that an arbitrary density matrix can be expanded in terms of a SIC-POVM.
However, a SIC-POVM in dimension ddoes in fact provide a basis, not just for the
space of density matrices, but for the space of all d×dcomplex matrices— i.e.the
Liealgebragl( d,C). Boykin et al[49] haverecentlyshownthatthere isaconnection
betweentheexistenceproblemformaximalsetsofMUBs(mutuallyu nbiasedbases)
and the theory of Lie algebras. Since SIC-POVMs share with MUBs th e property
of being highly symmetrical structures in quantum state space it mig ht have been
anticipated that there are also some interesting connections betw een SIC-POVMs
and Lie algebras.2
Our main result (proved in Sections 3,4and5) is that the proposition, that a
SIC-POVM exists in dimension d, is equivalent to a proposition about the adjoint
representation of gl( d,C). Our hope is that transforming the problem in this way,
from a question about quantum state space to a question about Lie algebras, may
help to make the SIC-existence problem tractable. But even if this h ope fails to
materialize we feel that this result, along with the many other result s we obtain,
provides some additional insight into these structures.
Inddimensional Hilbert space Hda SIC-POVM is a set of d2operatorsE1,
...,Ed2of the form
Er=1
dΠr (1)
where the Π rare rank-1 projectors with the property
Tr(ΠrΠs) =/braceleftigg
1r=s
1
d+1r/ne}ationslash=s(2)
We will refer to the Π ras SIC projectors, and we will say that {Πr:r= 1,...,d2}
is a SIC set.
It follows from this deﬁnition that the Ersatisfy
d2/summationdisplay
r=1Er=I (3)
(sotheyconstitute aPOVM),andthattheyarelinearlyindependen t (sothePOVM
is informationally complete).
It is an open question whether SIC-POVMs exist for all values of d. However,
examples have been constructed analytically in dimensions 2–15 inclus ive [1,2,11,
16,19,28,39,46], and in dimensions 19, 24, 35 and 48 [ 16,46]. Moreover, high
precisionnumerical solutionshave been constructed in dimensions 2 –67inclusive [ 5,
46]. Thislendssomeplausibilitytothe speculationthat theyexistinalldime nsions.
For a comprehensive account of the current state of knowledge in this regard, and
many new results, see the recent study by Scott and Grassl [ 46].
All known SIC-POVMs have a group covariance property. In other words, there
exists
(1) a group Ghavingd2elements
(2) a projective unitary representation of GonHd:i.e.a mapg→UgfromG
to the set of unitaries such that Ug1Ug2∼Ug1g2for allg1,g2(where the
notation “ ∼” means “equals up to a phase”)
(3) a normalized vector \|ψ/an}bracketri}ht(the ﬁducial vector)
such that the SIC-projectors are given by
Πg=Ug\|ψ/an}bracketri}ht/an}bracketle{tψ\|U†
g (4)
(where we label the projector by the group element g, rather than the integer ras
above).
Most known SIC-POVMs are covariant under the action of the Weyl- Heisenberg
group (though not all—see Renes et al[5] and, for an explicit example of a non
Weyl-Heisenberg SIC-POVM, Grassl [ 19]). Here the group is Zd×Zd, and the
projective representation is p→Dp, wherep= (p1,p2)∈Zd×ZdandDpis the3
corresponding Weyl-Heisenberg displacement operator
Dp=d−1/summationdisplay
rτ(2r+p1)p2\|r+p1/an}bracketri}ht/an}bracketle{tr\| (5)
In this expression τ=eiπ(d+1)
d, the vectors \|0/an}bracketri}ht,...\|d−1/an}bracketri}htare an orthonormal basis,
and the addition in \|r+p1/an}bracketri}htis modulod. For more details see, for example, ref. [ 16].
One should not attach too much weight to the fact that all known SI C-POVMs
have a group covariance property as this may only reﬂect the fact that group co-
variant SIC-POVMs are much easier to construct. So in this paper w e will try to
prove as much as we can without assuming such a property. One pot ential beneﬁt
ofthis attitude is that, by accumulatingenough facts about SIC-P OVMsin general,
we may eventually get to the point where we can answer the question , whether all
SIC-POVMs actually do have a group covariance property.
The fact that the d2operatorsΠ rare linearly independent means that they form
a basis for the complex Lie algebra gl( d,C) (the set of all operators acting on Hd).
Since the Π rare Hermitian, then iΠrforms a basis also for the real Lie algebra
u(d) (the set of all anti-Hermitian operators acting on Hd). So for any operator
A∈gl(d,C) there is a unique set of expansion coeﬃcients arsuch that
A=d2/summationdisplay
r=1arΠr (6)
To ﬁnd the expansion coeﬃcients we can use the fact that
d2/summationdisplay
s=1Tr(ΠrΠs)/parenleftbiggd+1
dδst−1
d2/parenrightbigg
=δrt (7)
from which it follows
ar=d+1
dTr(ΠrA)−1
dTr(A) (8)
Specializing to the case A= ΠrΠswe ﬁnd
ΠrΠs=d+1
d
d2/summationdisplay
t=1TrstΠt
−dδrs+1
d+1I (9)
where

The purposeofthis paperis totryto takeourunderstandingofSI C mathematics

(as it might be called) a little further forward. The research we repo rt began with

a chance numerical discovery made while we were working on a diﬀeren t problem.

Pursuing that initial numerical hint we uncovered a rich and interest ing set of

connections between SIC-POVMs in dimension dand the Lie Algebra gl( d,C). The

existence of these connections came as a surprise to us. However , in retrospect it

is, perhaps, not so surprising. Interest in SIC-POVMs has, to dat e, focused on the

fact that an arbitrary density matrix can be expanded in terms of a SIC-POVM.

However, a SIC-POVM in dimension ddoes in fact provide a basis, not just for the

space of density matrices, but for the space of all d×dcomplex matrices— i.e.the

Liealgebragl( d,C). Boykin et al[49] haverecentlyshownthatthere isaconnection

betweentheexistenceproblemformaximalsetsofMUBs(mutuallyu nbiasedbases)

and the theory of Lie algebras. Since SIC-POVMs share with MUBs th e property

of being highly symmetrical structures in quantum state space it mig ht have been

anticipated that there are also some interesting connections betw een SIC-POVMs

and Lie algebras.2

Our main result (proved in Sections 3,4and5) is that the proposition, that a

SIC-POVM exists in dimension d, is equivalent to a proposition about the adjoint

representation of gl( d,C). Our hope is that transforming the problem in this way,

from a question about quantum state space to a question about Lie algebras, may

help to make the SIC-existence problem tractable. But even if this h ope fails to

materialize we feel that this result, along with the many other result s we obtain,

provides some additional insight into these structures.

Inddimensional Hilbert space Hda SIC-POVM is a set of d2operatorsE1,

...,Ed2of the form

Er=1

dΠr (1)

where the Π rare rank-1 projectors with the property

Tr(ΠrΠs) =/braceleftigg

1r=s

1

d+1r/ne}ationslash=s(2)

We will refer to the Π ras SIC projectors, and we will say that {Πr:r= 1,...,d2}

is a SIC set.

It follows from this deﬁnition that the Ersatisfy

d2/summationdisplay

r=1Er=I (3)

(sotheyconstitute aPOVM),andthattheyarelinearlyindependen t (sothePOVM

is informationally complete).

It is an open question whether SIC-POVMs exist for all values of d. However,

examples have been constructed analytically in dimensions 2–15 inclus ive [1,2,11,

16,19,28,39,46], and in dimensions 19, 24, 35 and 48 [ 16,46]. Moreover, high

precisionnumerical solutionshave been constructed in dimensions 2 –67inclusive [ 5,

46]. Thislendssomeplausibilitytothe speculationthat theyexistinalldime nsions.

For a comprehensive account of the current state of knowledge in this regard, and

many new results, see the recent study by Scott and Grassl [ 46].

All known SIC-POVMs have a group covariance property. In other words, there

exists

(1) a group Ghavingd2elements

(2) a projective unitary representation of GonHd:i.e.a mapg→UgfromG

to the set of unitaries such that Ug1Ug2∼Ug1g2for allg1,g2(where the

notation “ ∼” means “equals up to a phase”)

(3) a normalized vector |ψ/an}bracketri}ht(the ﬁducial vector)

such that the SIC-projectors are given by

Πg=Ug|ψ/an}bracketri}ht/an}bracketle{tψ|U†

g (4)

(where we label the projector by the group element g, rather than the integer ras

above).

Most known SIC-POVMs are covariant under the action of the Weyl- Heisenberg

group (though not all—see Renes et al[5] and, for an explicit example of a non

Weyl-Heisenberg SIC-POVM, Grassl [ 19]). Here the group is Zd×Zd, and the

projective representation is p→Dp, wherep= (p1,p2)∈Zd×ZdandDpis the3

corresponding Weyl-Heisenberg displacement operator

Dp=d−1/summationdisplay

rτ(2r+p1)p2|r+p1/an}bracketri}ht/an}bracketle{tr| (5)

In this expression τ=eiπ(d+1)

d, the vectors |0/an}bracketri}ht,...|d−1/an}bracketri}htare an orthonormal basis,

and the addition in |r+p1/an}bracketri}htis modulod. For more details see, for example, ref. [ 16].

One should not attach too much weight to the fact that all known SI C-POVMs

have a group covariance property as this may only reﬂect the fact that group co-

variant SIC-POVMs are much easier to construct. So in this paper w e will try to

prove as much as we can without assuming such a property. One pot ential beneﬁt

ofthis attitude is that, by accumulatingenough facts about SIC-P OVMsin general,

we may eventually get to the point where we can answer the question , whether all

SIC-POVMs actually do have a group covariance property.

The fact that the d2operatorsΠ rare linearly independent means that they form

a basis for the complex Lie algebra gl( d,C) (the set of all operators acting on Hd).

Since the Π rare Hermitian, then iΠrforms a basis also for the real Lie algebra

u(d) (the set of all anti-Hermitian operators acting on Hd). So for any operator

A∈gl(d,C) there is a unique set of expansion coeﬃcients arsuch that

A=d2/summationdisplay

r=1arΠr (6)

To ﬁnd the expansion coeﬃcients we can use the fact that

d2/summationdisplay

s=1Tr(ΠrΠs)/parenleftbiggd+1

dδst−1

d2/parenrightbigg

=δrt (7)

from which it follows

ar=d+1

dTr(ΠrA)−1

dTr(A) (8)

Specializing to the case A= ΠrΠswe ﬁnd

ΠrΠs=d+1

d

d2/summationdisplay

t=1TrstΠt

−dδrs+1

d+1I (9)

where