Generalized quantum state discrimination problems

0.21 MB·English

by  Kenji Nakahira

Generalized quantum state discrimination problems
Kenji Nakahira,1,4 Kentaro Kato,2 and Tsuyoshi Sasaki Usuda3,4
1Yokohama Research Laboratory, Hitachi, Ltd., Yokohama, Kanagawa 244-0817, Japan
2Quantum Communication Research Center, Quantum ICT Research Institute,
Tamagawa University, Machida, Tokyo 194-8610, Japan
3School of Information Science and Technology, Aichi Prefectural University, Nagakute, Aichi 480-1198, Japan
4Quantum Information Science Research Center, Quantum ICT Research Institute,
Tamagawa University, Machida, Tokyo 194-8610, Japan
(Dated: June23, 2015)
We address a broad class of optimization problems of finding quantum measurements, which
5 includestheproblemsoffindinganoptimalmeasurementintheBayescriterionandameasurement
1 maximizing the average success probability with a fixed rate of inconclusive results. Our approach
0 candealwith anyproblem inwhich each oftheobjective andconstraint functionsisformulated by
2 the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We
first derivedualproblems and necessary and sufficient conditions for an optimal measurement. We
n
alsoconsidertheminimaxversionoftheseproblemsandprovidenecessaryandsufficientconditions
u
for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists
J
anoptimalsolution withthesamesymmetry. Examplesareshowntoillustratehowourresultscan
0
beused.
2

] I. INTRODUCTION been studied [16–18]. Moreover, a measurement that
h
p maximizes the average success probability where a cer-
- tain fixed average error probability is allowed, which we
Discrimination between quantum states is a funda-
t
n mental topic in quantum information theory. Research callanoptimalerrormarginmeasurement,hasalsobeen
a investigated [19–21]. On the other hand, in the case in
in quantum state discrimination was pioneered by Hel-
u strom, Holevo, and Yuen et al. [1–3] in the 1970s and which prior probabilities are unknown, several types of
q
measurements basedon the minimax strategy have been
[ has attracted intensive attention. It is well known in
proposed, such as a measurement that minimizes the
quantummechanicsthatnonorthogonalstatescannotbe
2 maximumprobabilityofdetectionerrors[22]andamea-
discriminated with certainty. Thus, optimal measure-
v surement with a certain fraction of inconclusive results
ment strategies have been proposed under various crite-
4 [23]. Properties of optimal measurements in the above
4 ria. Among them, one of the most widely investigated is
criteria, such as necessary and sufficient conditions for
7 the Bayes criterion, or the criterion of minimum average
optimal solutions, have been derived for each criterion.
5 error probability [1–3]. In the Bayes criterion, necessary
0 and sufficient conditions for obtaining an optimal mea- In this paper, we investigate optimization problems of
1. surementhavebeenformulated[1–4],andclosed-forman- finding optimal quantum measurements and their mini-
0 alytical expressions for optimal measurements have also maxversionsthatareapplicabletoawiderangeofquan-
5 beenderivedinsomecases(seee.g.,[5–9]). Thiscriterion tum state discrimination problems. Our approach can
1 isbasedontheassumptionthatpriorprobabilitiesofthe deal with any problem in which each of the objective
v: states are known. In contrast, if these prior probabili- and constraintfunctions is formulated by the sum of the
i ties are unknown, then minimax criteria are often used traces of the multiplication of a Hermitian operator and
X
[10, 11]. Necessary and sufficient conditions for a mea- a detection operator, which implies that any problems
r surement minimizing the worst case of the averageerror related to finding any of the optimal measurements de-
a
probabilityintheminimaxstrategyhavebeenfound[10]. scribed above can be formulated as our problems. Thus,
This result has also been extended to the average Bayes wecansaythatourapproachcanprovideaunifiedtreat-
cost [12], mentinalargeclassofproblems. Theresultsobtainedin
this paper would be valuable from the practical point of
Other types of optimal measurements have been in-
view; for example, they not only provide a broader per-
vestigated. In the case in which prior probabilities of
spective than the results for a particular problem, but
the states are known, an example concerns a measure-
also can apply to many problems that have not been re-
ment that achieves low average error probability at the
portedpreviously,some examplesofwhicharepresented
expense of allowing for a certain fraction of inconclusive
in this paper. To obtain knowledge about an optimal
results [13–15]. In particular, an unambiguous (or error-
measurementin a new criterionhas the potential to cre-
free) measurement that maximizes the success probabil-
ate a new application of quantum state discrimination.
ity, which is called an optimal unambiguous measure-
ment, has been well studied [13–15]. A measurement InSec.II,we provideageneralizedoptimizationprob-
that maximizes the average success probability with a lem in which each of the objective and constraint func-
fixed average failure (or inconclusive) probability, which tionsis formulatedbythe sumofthe tracesofthe multi-
is called an optimal inconclusive measurement, has also plication of a Hermitian operatorand a detection opera-
2
tor. We derive its dual problem and necessary and suffi- problem:
cientconditionsforanoptimalmeasurement. InSec.III,
M−1
we discuss the minimax version of our generalized prob- maximize f(Π)= Tr(cˆ Πˆ )
lem and providenecessaryand sufficientconditions for a m m (2)
m=0
minimax solution. In Sec. IV, we demonstrate that if a subject to Π∈M◦,X
given problem has a certain symmetry, then there exists
where cˆ ∈ S holds for any m ∈ I . (Note that any
an optimal solution with the same symmetry. Finally, m M
linear combinationof positive semidefinite operatorsis a
we present some examples to illustrate the applicability
Hermitian operator.) M◦ is expressed by
of our results in Sec. V.
M−1
M◦ = Π∈M: Tr(aˆ Πˆ )≤b , ∀j ∈I ,(3)
j,m m j J
( )
m=0
X
II. GENERALIZED OPTIMAL MEASUREMENT where aˆ ∈ S and b ∈ R hold for any m ∈ I and
j,m j M
j ∈ I . J is a nonnegative integer. We should mention
J
A. Formulation thatanequalityconstraint(e.g.,Tr(aˆj,0Πˆ0)=bj)canbe
replacedbytwoinequalityconstraints(e.g.,Tr(aˆ Πˆ )≤
j,0 0
b and Tr(−aˆ Πˆ ) ≤ −b ). We call an optimal solu-
We consider a quantum measurement on a Hilbert j j,0 0 j
tion to problem (2) a generalized optimal measurement
space H. Such a quantum measurement can be mod-
or simply an optimal measurement. Problem (2) is said
eled by a positive operator-valued measure (POVM)
tobeaprimalproblem. Sincef(Π)islinearinΠandM◦
Π={Πˆ :m∈I }onH,whereM isthenumberofthe
m M is convex,problem(2)is a convexoptimizationproblem.
detection operators and I = {0,1,··· ,k−1}. An ex-
k NotethatsincetheconstraintofΠ∈M,i.e.,Eq.(1),can
ampleofaquantummeasurementisanoptimalmeasure-
be rewritten as Tr(ρˆΠˆ ) ≥ 0 and M−1Tr(ρˆΠˆ ) = 1
mentfordistinguishingRquantumstatesrepresentedby m m=0 m
for any density operator ρˆ, we can say that each of the
density operators ρˆ (r ∈ I ). The density operator ρˆ
r R r P
objective and constraint functions is formulated by the
satisfies ρˆ ≥0 and has unit trace, i.e., Tr ρˆ =1, where
r r
Aˆ≥0denotesthatAˆarepositivesemidefinite(similarly, sum of the traces of the multiplication of a Hermitian
Aˆ≥Bˆ denotes Aˆ−Bˆ ≥0). A minimum error measure- operator and a detection operator.
ment is such an optimal measurement, which can be ex-
pressedbyaPOVMwith M =Rdetectionoperators. A
B. Examples
quantum measurement that may return an inconclusive
answer can be expressed by a POVM with M = R+1
We give some examples of optimization problems of
detection operators; in this case the detection operator
Πˆ (r ∈ I ) corresponds to identification of the state finding quantum measurements that can be formulated
r R
as problem (2). Let us consider discrimination between
ρˆ , while Πˆ corresponds to the inconclusive answer.
r R R quantum states {ρˆ : r ∈ I } with prior probabilities
r R
Let M be the entire set of POVMs on H that consist
{ξ :r∈I }.
r R
of M detection operators. Π∈M satisfies
Πˆ ≥0, ∀m∈I , 1. Optimal measurement in the Bayes criterion
m M
M−1
Πˆ =ˆ1, (1) The optimization problem of finding an optimal mea-
m
surement in the Bayes criterion is formulated as [1–3]
m=0
X
R−1
where ˆ1 is the identity operator on H. In addition, let minimize Tr(WˆmΠˆm) (4)
S and S+ be the entire sets of Hermitian operators on mX=0
subject to Π∈M.
HandsemidefinitepositiveoperatorsonH,respectively.
Let R and R+ be the entire sets of real numbers and Wˆ ∈S (m∈I ) can be expressed by
nonnegative real numbers, respectively, and RN be the m + R
+
entire set of collections of N nonnegative real numbers. R−1
Wˆ = ξ B ρˆ , (5)
Here, we consider a generalizedoptimization problem. m r m,r r
The conditional probability that the measurement out- Xr=0
come is m when a quantum state ρˆ is given is repre- whereBm,r ∈R+ holdsforanym,r ∈IR. Thisproblem
sentedby Tr(ρˆΠˆ ), andthus there existmany optimiza- can be written as the form of problem (2) with
m
tionproblemsoffinding optimalquantummeasurements M =R,
such that each of the objective and constraint functions
J =0,
is expressed by a linear combination of forms Tr(ρˆ Πˆ ).
For this reason, we consider the following optimizratmion cˆm =−Wˆm. (6)
3
2. Optimal error margin measurement sive measurement. This problem is formulated as
R−1
An optimal error margin measurement is a measure- maximize ξ Tr(ρˆ Πˆ )
r r r
ment maximizing the average success probability under
r=0 (10)
the constraint that the average error probability is not subject to ΠX∈M, Tr(ρˆ Πˆ )≥q, ∀r ∈I ,
r r R
greater than a given value ε with 0 ≤ ε ≤ 1 [19–21]. Tr(GˆΠˆ )=p,
R
In particular, if ε = 0, then an optimal error margin
measurement is equivalent to an optimal unambiguous where Gˆ is defined by Eq. (9). Since the optimal value
measurement. The optimization problem of finding an
ofproblem(10)is monotonicallydecreasingwithrespect
optimal error margin measurement is formulated as
top, weobtainthe samesolutionifthe lastconstraintof
problem (10) is replaced with Tr(GˆΠˆ ) ≥ p. Thus, this
R−1 R
maximize ξ Tr(ρˆ Πˆ ) problem is equivalent to problem (2) with
r r r
Xr=0 R−1 (7) M =J =R+1,
subject to Π∈M, ξrTr[ρˆr(Πˆr +ΠˆR)]≥1−ε, cˆ = ξmρˆm, m<R,
r=0 m 0, m=R,
X (cid:26)
where we consider that the statement that the average −δm,jρˆm, j <R,
aˆ =
errorprobabilityisnotgreaterthanεisequivalenttothe j,m −δm,RGˆ, j =R,
(cid:26)
statementthatthesumoftheaveragesuccessandfailure
−q, j <R,
probabilities is not less than 1−ε. This problem can be bj = −p, j =R, (11)
written as the form of problem (2) with (cid:26)
where δk,k′ is the Kronecker delta. Note that if q >
M =R+1, q′ holds, where q′ is the average success probability of
J =1, an optimal inconclusive measurement with the average
failure probability of p, then this problem is infeasible,
ξ ρˆ , m<R,
cˆm = 0m, m m=R, i.e., M◦ is empty. We discuss this problem in detail in
(cid:26) Subsec. VA.
−ξ ρˆ , m<R,
m m
aˆ =
0,m −Gˆ, m=R,
(cid:26)
b =ε−1, (8) C. Dual problem
0
where
In this subsection, we show the dual problem of prob-
lem (2). We also show that the optimal values of primal
R−1
Gˆ = ξ ρˆ . (9) problem (2) and the dual problem are the same.
r r
r=0
X
Theorem 1 Let us consider problem (2). We also con-
Note that an optimal error margin measurement has
sider the following optimization problem
strong relationship with an optimal inconclusive mea-
surement [24, 25]. However, if one wants to obtain an J−1
optimal error margin measurement for a given ε, then minimize s(Xˆ,λ)=Tr Xˆ + λ b
j j
(12)
one needs to solve problem (7) instead of the problemof j=0
X
finding an optimal inconclusive measurement. subject to Xˆ ≥zˆ (λ), ∀m∈I
m M
withvariablesXˆ ∈S andλ={λ ∈R :j ∈I }∈RJ,
j + J +
3. Optimal inconclusive measurement with a lower bound where
on success probabilities
J−1
zˆ (λ)=cˆ − λ aˆ . (13)
m m j j,m
Anotherexampleisanextensionoftheproblemoffind-
j=0
X
ing an optimal inconclusive measurement. An optimal
inconclusive measurement is a measurement maximizing If M◦ is not empty, then the optimal values of problems
theaveragesuccessprobabilityunderthe constraintthat (2) and (12) are the same.
theaveragefailureprobabilityequalsagivenvaluepwith
Problem (12) is called the dual problem of problem
0 ≤ p ≤ 1 [16–18]. Here, we add the constraint that for
each r ∈ I the success probability of the state ρˆ , i.e., (2). Note that in general Xˆ satisfying the constraints of
R r
Tr(ρˆrΠˆr),isnotlessthanagivenvalueq with0≤q ≤1. problem (12) is not in S+; however, it is obvious that if
When q =0, an optimal solution is an optimal inconclu- m∈I exists such that zˆ ∈S , then Xˆ ∈S holds.
M m + +
4
Proof Let us define the function L as problemoffindingaminimumerrormeasurementwasde-
rivedfromgeneralprobabilistictheories. InRefs.[29,30],
L(Π,σ,Xˆ,λ)
the dual problem was derived from “ensemble steering,”
M−1 M−1 which determines what states one party can prepare on
=f(Π)+ Tr(σˆ Πˆ )+Tr Xˆ ˆ1− Πˆ
m m m the other party’s system by sharing a bipartite state. In
" !#
mX=0 mX=0 the same way, we can derive dual problem (12) without
J−1 M−1 using POVMs (see Appendix A). However, it might not
+ λj bj − Tr(aˆj,mΠˆm) , (14) be easy to prove that the optimal value of problem (12)
" #
j=0 m=0 is attained by using these approaches.
X X
where σ = {σˆ ∈ S : m ∈ I }, Xˆ ∈ S, and λ ∈ RJ.
m + M +
Note that L is called the Lagrangian for problem (2).
D. Conditions for an optimal measurement
Substituting Eqs. (2),(12), and (13) into Eq. (14) gives
L(Π,σ,Xˆ,λ) Necessaryandsufficientconditionsforanoptimalmea-
M−1 surement in several problems (such as a minimum error
=s(Xˆ,λ)+ Tr[(σˆ +zˆ (λ)−Xˆ)Πˆ ]. (15) measurementandanoptimalinconclusivemeasurement)
m m m
have been derived [1–4, 17, 18, 31, 32]. The following
m=0
X
theorem extends these results to our more general set-
Let us consider the following optimization problem:
ting.
minimize s (σ,Xˆ,λ)
σ
subject to σˆ ∈S , ∀m∈I , Theorem 2 Suppose that a POVM Π is in M◦. The
m + M (16)
Xˆ ∈S, following statements are all equivalent.
λ∈RJ,
+ (1) Π is an optimal measurement of problem (2).
where (2) Xˆ ∈S and λ∈RJ exist such that
+
s (σ,Xˆ,λ)= max L(Π,σ,Xˆ,λ) (17)
σ Π∈SM Xˆ −zˆm(λ)≥0, ∀m∈IM, (19)
+
[Xˆ −zˆ (λ)]Πˆ =0, ∀m∈I , (20)
and SM = {Πˆ ∈ S : m ∈ I }. Let X = {Xˆ : Xˆ ≥ m m M
+ m + M
M−1
σˆ +zˆ (λ), ∀m ∈ I }. The second term of the right-
m m M λ b − Tr(aˆ Πˆ ) =0, ∀j ∈I . (21)
handsideofEq.(15)isnonpositiveifXˆ ∈X,andcanbe j j j,m m J
" #
infinite if Xˆ 6∈ X. Therefore, from Eq. (17), s (σ,Xˆ,λ) mX=0
σ
can be expressed as (3) λ∈RJ exists such that
+
s (σ,Xˆ,λ)= s(Xˆ,λ), Xˆ ∈X, (18) M−1
σ ∞, otherwise. zˆ (λ)Πˆ −zˆ (λ)≥0, ∀m∈I , (22)
(cid:26) n n m M
n=0
FromEq.(18), itfollowsthat there exists anoptimalso- X
M−1
lution to problem (16) such that σˆ = 0 holds for any
m λ b − Tr(aˆ Πˆ ) =0, ∀j ∈I . (23)
m ∈ IM. Indeed, if (σ,Xˆ,λ) is an optimal solution to j" j m=0 j,m m # J
problem (16) (in this case, Xˆ ∈X holds from Eq. (18)), X
then ({σˆ′ =0: m∈I },Xˆ,λ) is also an optimal solu- Note that, from Eq. (20), the support of the detection
tion. Hemnce, problem (M16) can be rewritten by problem operator Πˆm of the optimal measurement is included in
(12). the kernel of Xˆ −zˆm(λ) for any m∈IM.
Slater’s condition is known to a sufficient condition
underwhich,iftheprimalproblemisconvex,theoptimal Proof It is sufficient to show(1) ⇒ (2), (2) ⇒ (3), and
valuesoftheprimalanddualproblemsarethesame[26]. (3) ⇒ (1).
Sinceeachconstraintofprimalproblem(2),includingthe First, we show (1) ⇒ (2). Suppose that (Xˆ,λ) is an
constraint of Π ∈ M, is expressed as a form of uj(Π) ≤ optimal solution to dual problem (12). Let σˆm = 0 for
0, where uj is an affine function of Π, from Ref. [27], any m ∈ IM. It is obvious from Eq. (12) that Eq. (19)
(therefinedformof)Slater’sconditionisthattheprimal holds. From Theorem 1, f(Π) = s(Xˆ,λ) holds. More-
problem is feasible, i.e., M◦ is not empty. Thus, since over, from Π ∈ M◦, the second and third terms of the
Slater’s condition holds, the optimal values of problems right-handside of Eq.(14) arezero, andthe fourth term
(2) and (12) are the same. (cid:4) is nonnegative, which yields
Itisworthnotingthatsomeattempts havebeenmade L(Π,σ,Xˆ,λ)≥f(Π)=s(Xˆ,λ). (24)
toobtainthemaximumaveragesuccessprobabilitywith-
out using the fact that POVMs describe quantum mea- In contrast, from Eq. (19) and the fact that the trace of
surements [28–30]. In Ref. [28], the dual problem to the the multiplication of two positive semidefinite operators
5
isnonnegative,Tr[(Xˆ−zˆ (λ))Πˆ ]≥0holdsforanym∈ distribution. WeconsiderthefollowingfunctionF(µ,Π):
m m
I , which yields L(Π,σ,Xˆ,λ) ≤ s(Xˆ,λ) from Eq. (15).
M K−1
Thus, from Eq. (24), we obtain L(Π,σ,Xˆ,λ) = s(Xˆ,λ), F(µ,Π)= µ f (Π),
k k
i.e.,
k=0
X
M−1
Tr[(Xˆ −zˆm(λ))Πˆm]=0, ∀m∈IM. (25) f (Π)= Tr(cˆ Πˆ )+d , (27)
k k,m m k
m=0
Therefore, using the fact that AˆBˆ = 0 holds for any X
Aˆ,Bˆ ∈ S satisfying Tr(AˆBˆ) = 0 yields Eq. (20). From where cˆk,m ∈ S and dk ∈ R hold for any m ∈ IM and
+ k ∈ I . We want to find a POVM Π ∈ M◦ that max-
L(Π,σ,Xˆ,λ) = f(Π), the fourth term of the right-hand K
imizes the worst-case value of F(µ,Π) over µ ∈ P, i.e.,
side of Eq. (14) must be zero. Therefore, Eq. (21) holds.
min F(µ,Π), where M◦ is defined by Eq. (3). In the
µ∈P
Next, we show (2) ⇒ (3). From Eq. (20), XˆΠˆm = case of K =1, this problem is equivalent to problem (2)
zˆm(λ)Πˆm holds. Summing this equation over m = with cˆm =cˆ0,m and d0 =0. Therefore, this problem can
0,··· ,M −1 yields Xˆ = M−1zˆ (λ)Πˆ , which gives be regarded as an extension of problem (2).
m=0 m m
Eq. (22). Equation (23) obviously holds from Eq. (21). We canseethatifM◦ isnotempty,thenthe so-called
Finally,weshow(3)⇒(1P). LetXˆ = M−1zˆ (λ)Πˆ . minimaxtheoremholds,thatis,thereexists(µ⋆,Π⋆)sat-
m=0 m m isfying the following equations:
We have that for any POVM Π′ = {Πˆ′ : m ∈ I } ∈
Pm M
M◦, max minF(µ,Π)=F(µ⋆,Π⋆)
Π∈M◦µ∈P
f(Π)−f(Π′) =min max F(µ,Π). (28)
µ∈PΠ∈M◦
J−1 M−1
≥f(Π)+ λ b − Tr(aˆ Πˆ ) Indeed, M◦ and P are closed convex sets, and F(µ,Π)
j j j,m m
" # is a continuous convex function of µ for fixed Π and a
j=0 m=0
X X continuous concave function of Π for fixed µ, which are
J−1 M−1
−f(Π′)− λ b − Tr(aˆ Πˆ′ ) sufficient conditions for the minimax theorem to hold
j j j,m m [33]. We call (µ⋆,Π⋆), µ⋆, and Π⋆ respectively a min-
" #
j=0 m=0
X X imax solution, minimax probabilities, and a minimax
M−1 measurement. (µ⋆,Π⋆) is a minimax solutionif and only
=Tr Xˆ − Tr[zˆ (λ)Πˆ′ ]
m m if (µ⋆,Π⋆) is a saddle point ofF(µ,Π), i.e., the following
mX=0 inequalities hold for any µ∈P and Π∈M◦ [33]:
M−1
= Tr[(Xˆ −zˆ (λ))Πˆ′ ]≥0, (26) F(µ⋆,Π)≤F(µ⋆,Π⋆)≤F(µ,Π⋆). (29)
m m
m=0
X Let
where the first inequality follows from Eq. (23) and F⋆(µ)= max F(µ,Π) (30)
M−1Tr(aˆ Πˆ′ ) ≤ b . The last inequality follows Π∈M◦
m=0 j,m m j
from Eq. (22), i.e., Xˆ ≥ zˆ (λ). Since f(Π) ≥ f(Π′) with µ ∈ P. It follows from Eq. (29) that F⋆(µ⋆) =
P m
holdsforanyPOVMΠ′ ∈M◦,Πisanoptimalmeasure- F(µ⋆,Π⋆) holds. From Eq. (27), F(µ,Π) can be ex-
ment of problem (2). (cid:4) pressed by
K−1 M−1
F(µ,Π)= µ Tr(cˆ Πˆ )+d
k k,m m k
III. GENERALIZED MINIMAX SOLUTION k=0 "m=0 #
X X
M−1 K−1 K−1
A. Formulation = Tr µkcˆk,m Πˆm + µkdk.(31)
" ! #
m=0 k=0 k=0
X X X
In this section, we consider the quantum minimax Thus,F⋆(µ)foragivenµ∈P canbeobtainedbyfinding
strategy,whichprovidesadifferenttypeofproblemfrom Π ∈ M◦ that maximizes the first term of the second
those discussed in the previous section. The quantum line ofEq.(31), whichis formulatedas problem(2) with
minimax strategy has been investigated [10–12, 22, 23] c = K−1µ cˆ .
m k=0 k k,m
under the assumption that the collection of prior proba-
bilitiesisnotgiven. Weinvestigatetheminimaxstrategy P
for a generalized quantum state discrimination problem. B. Examples
Let K be a positive integer. Also, let P be the entire
set of collections of K nonnegative real numbers, µ = We give someexamples ofminimax problemsthat can
{µ ≥ 0 : k ∈ I }, satisfying K−1µ = 1, which beformulatedasEq.(27). Letusconsiderdiscrimination
k K k=0 k
implies that µ ∈ P can be interpreted as a probability between R quantum states {ρˆ :r ∈I }.
r R
P
6
1. Minimax solution in the Bayes strategy That is, we have
M =R+1,
TheminimaxstrategyinwhichtheaverageBayescost
K =J =R,
is used as the objective function has been investigated
ρˆ , m=k or m=R,
in Ref. [12]. We regard µ ∈ P with K = R as prior cˆ = k
k,m 0, otherwise,
probabilities of the states {ρˆr :r ∈IR}. The aim of this (cid:26)
problem is to find a POVM Π that minimizes the worst- dk =0,
case average Bayes cost B(µ,Π) over µ ∈ P. B(µ,Π) is aˆ =δ ρˆ ,
j,m m,R j
expressed by
b =p. (36)
j
R−1
B(µ,Π)= Tr[Wˆ (µ)Πˆ ], 3. Minimax solution for plural state sets
m m
m=0
X
R−1 We consider a quantum measurement that maximizes
Wˆm(µ)= µkBm,kρˆk, (32) the worst-case average success probabilities for plural
Xk=0 quantum state sets {Ψk : k ∈ IK} with K ≥ 2 as an-
other example, where, for each k ∈ I , Ψ is a set of R
K k
quantum states, Ψ = {ρˆ : r ∈ I }, with prior prob-
whereB ∈R holdsforanym,k∈I . Thisproblem k k,r R
m,k + R
abilities {ξ : r ∈ I }. This problem can be expressed
can be expressed by a form of Eq. (27) with F(µ,Π) = k,r R
by a form of Eq. (27) with
−B(µ,Π). In this case, we have
R−1
f (Π)= Tr(ρˆ′ Πˆ ), ∀k ∈I ,
R−1 k k,m m K
fk(Π)=− Tr[(Bm,kρˆk)Πˆm], ∀k ∈IR, mX=0
M◦ =M, (37)
m=0
X
M◦ =M, (33) where ρˆ′ =ξ ρˆ . That is, we have
k,r k,r k,r
M =R,
i.e.,
J =0,
cˆ =ρˆ′ ,
k,m k,m
M =K =R,
d =0. (38)
k
J =0,
We discuss this problem in detail in Subsec. VB.
cˆ =−B ρˆ ,
k,m m,k k
d =0. (34)
k
C. Properties of a minimax solution
Weshownecessaryandsufficientconditionsforamin-
2. Inconclusive minimax solution imaxsolutioninTheorem3andanoptimizationproblem
of obtaining a minimax measurement in Theorem 4.
The application to the minimax strategy to state dis- Theorem 3 Suppose that µ⋆ ∈ P and Π⋆ ∈ M◦ hold.
crimination that allows a nonzero failure probability has
The following statements are all equivalent.
been investigated in Ref. [23]. The aim of this problem
is tofinda POVMΠ, whichwe callaninconclusivemin- (1) (µ⋆,Π⋆) is a minimax solution to Eq. (27).
imax measurement, that maximizes the worst-case value (2) We have that for any k ∈IK,
ofthesumoftheaveragesuccessandfailureprobabilities
f (Π⋆)≥F⋆(µ⋆). (39)
underthe constraintthatTr(ρˆ Πˆ )is notgreaterthana k
j R
given value p with 0≤p≤1 for any j ∈IR. In particu- (3) F⋆(µ⋆)=F(µ⋆,Π⋆) holds, and we have that for any
lar,if p=0,then aninconclusive minimax measurement k,k′ ∈I such that µ⋆ >0,
K k′
isastandardminimaxmeasurementwithoutinconclusive
results [10]. Let K =R and µ∈P be prior probabilities fk(Π⋆)≥fk′(Π⋆). (40)
of the states {ρˆ : r ∈ I }; then, this problem can be
r R Proof It suffices to show (1) ⇔ (2) and (2) ⇔ (3).
expressed by a form of Eq. (27) with
First, we show (1) ⇒ (2). Let µ(k) = {µk′ = δk,k′ :
k′ ∈ I }. From Eq. (29) and F⋆(µ⋆) = F(µ⋆,Π⋆), we
K
fk(Π)=Tr[ρˆk(Πˆk+ΠˆR)], ∀k ∈IR, have that for any k ∈IK,
M◦ ={Π∈M:Tr(ρˆ Πˆ )≤p, ∀j ∈I }. (35) f (Π⋆)=F(µ(k),Π⋆)≥F(µ⋆,Π⋆)=F⋆(µ⋆). (41)
j R R k
7
Thus, Eq. (39) holds. Similarly, we call an optimal measurement and a mini-
Next, we show (2) ⇒ (1). From Eqs. (30) and (39), max solution that are invariant under the same action
We obtain, for any µ∈P and Π∈M◦, a groupcovariant(or G-covariant)optimal measurement
and a minimax solution, respectively. Optimal measure-
K−1
mentsforgroupcovariantstatesetshavebeenwellinves-
F(µ⋆,Π)≤F⋆(µ⋆)≤ µ f (Π⋆)=F(µ,Π⋆).(42)
k k tigated, and it has been derived that a G-covariantopti-
Xk=0 malmeasurementexistsforaG-covariantstatesetunder
Substituting µ = µ⋆ and Π = Π⋆ into this equation severaloptimalitycriteria[5–8,17,23,32,34,35]. These
givesF⋆(µ⋆)=F(µ⋆,Π⋆). Thus,fromEq.(42),Eq.(29) results not only help us to obtain analytical optimal so-
holds, which means that (µ⋆,Π⋆) is a minimax solution lutions (e.g., [36–38]), but also are useful for developing
to Eq. (27). computationally efficient algorithms for obtaining opti-
Then, we show (2) ⇒ (3). From Eq. (39) and the def- mal solutions [39, 40]. In this section, we will generalize
inition of F⋆(µ), F(µ⋆,Π⋆) = F⋆(µ⋆) must hold. Thus, these results to our generalized optimization problems.
we have
fk(Π⋆)=F⋆(µ⋆), ∀k ∈IK s.t. µ⋆k >0, A. Group action
f (Π⋆)≥F⋆(µ⋆), ∀k ∈I s.t. µ⋆ =0, (43)
k K k
First,letusdescribeagroupaction. Agroupactionof
from which we can easily see that Eq. (40) holds.
GonasetT isasetofmappingsfromT toT,{π (x)(x∈
Finally, we show (3) ⇒ (2). From Eq. (40), f (Π⋆) = g
k T):g ∈G} (we also denote π (x) as g◦x), such that
fk′(Π⋆) holds for any k,k′ ∈ IK satisfying µ⋆k > 0 and g
µ⋆k′ > 0. Thus, according to the definition of F⋆(µ), (1) For any g,h ∈ G and x ∈ T, (gh)◦x = g ◦(h◦x)
F⋆(µ⋆)=fk′(Π⋆) holds for any k′ ∈IK satisfying µ⋆k′ > holds.
0. Substituting this into Eq. (40) gives Eq. (39). (cid:4) (2) Foranyx∈T,e◦x=xholds,whereeistheidentity
element of G.
Theorem 4 Let us consider the following optimization
problem The actionofG onT is calledfaithful if, for any distinct
g,h∈G,there existsx∈T suchthatg◦x6=h◦x. Here,
maximize fmin(Π)= min fk(Π) we assume that the number of elements in G, which is
k∈IK (44)
subject to Π∈M◦ denoted as |G|, is greater than one.
Letus consideranactionofG onthe setI withN ≥
N
withaPOVMΠ. APOVMΠ+ is anoptimalsolutionto 1, that is, {g◦n ∈I (n ∈I ): g ∈G}. This action is
N N
problem(44)ifandonlyifΠ+ isaminimaxmeasurement not faithful in general. We also consider the action of G
of Eq. (27). on S, expressed by
Proof Suppose that Π+ is anoptimalsolutionto prob- g◦Aˆ=UˆgAˆUˆg† (46)
lem (44), and that (µ⋆,Π⋆) is a minimax solution to
with g ∈ G and Aˆ ∈ S, where Uˆ is a unitary or anti-
Eq. (27). Equations (30) and (39) give g
unitary operator and Uˆ† is conjugate transpose of Uˆ .
g g
fmin(Π⋆)≥F⋆(µ⋆)=Πm∈Max◦F(µ⋆,Π)≥Πm∈Max◦fmin(Π), (Note that if Uˆg is an anti-unitary operator, then Uˆg† is
(45) an anti-unitary operator such that Uˆg†Uˆg = UˆgUˆg† = ˆ1.)
Uˆ = ˆ1 and Uˆ = Uˆ† obviously hold, where g¯ is the
which indicates that Π⋆ is an optimal solution to prob- e g¯ g
inverse element of g. We assume that the action of G
lem (44). Since Π+ is also an optimal solution to prob- on S is faithful, which is equivalent to Uˆ 6= Uˆ for any
lem (44), f (Π+) = f (Π⋆) ≥ F⋆(µ⋆) holds from g h
min min distinct g,h ∈ G. From Eq. (46), we can easily verify
Eq. (45), and thus Statement (2) of Theorem 3 holds.
Therefore,Π+ isa minimax measurementofEq.(27). (cid:4) that for any g ∈G, c∈R and Aˆ,Bˆ ∈S, we have
g◦(Aˆ±Bˆ)=g◦Aˆ±g◦Bˆ,
g◦(cAˆ)=c(g◦Aˆ),
IV. GROUP COVARIANT OPTIMIZATION
PROBLEM g◦ˆ1=ˆ1,
Tr(g◦Aˆ)=Tr Aˆ,
In this section, we show that if an optimization prob-
Tr[(g◦Aˆ)(g◦Bˆ)]=Tr(AˆBˆ),
lem of obtaining an optimal measurement or a minimax
solution has a certain symmetry, the optimal solution g◦Aˆ ∈ S , ∀Aˆ∈S ,
+ +
also has the same symmetry. A quantum state set that g◦Aˆ≥g◦Bˆ, ∀Aˆ≥Bˆ. (47)
is invariant under the action of a group G in which each
element corresponds to a unitary or anti-unitary opera- In this section, we use these facts without mentioning
tor is called a groupcovariant(or G-covariant)state set. them.
8
B. Group covariant optimal measurement WenowshowthataG-covariantoptimalmeasurement
existsifoptimizationproblem(2)hasacertainsymmetry
As a preparation, we first prove the following lemma. with respect to G.
Lemma 5 SupposethatM◦isnotempty. Also,suppose Theorem 6 Let us consider optimization problem (2).
that there exist actions of G on S, IM, and IJ such that SupposethatM◦ is notempty. Also,supposethatthere
exist actions of G on S, I , and I satisfying Eq. (48)
g◦aˆ =aˆ , ∀g ∈G,j ∈I ,m∈I , M J
j,m g◦j,g◦m J M
and
b =b , ∀g ∈G,j ∈I . (48)
j g◦j J
Let κg(Φ) and κ(Φ) be mappings of Φ ∈ M◦ expressed g◦cˆm =cˆg◦m, ∀g ∈G,m∈IM. (54)
by
Then, for any Φ ∈ M◦ there exists Π ∈ M◦ such that
κg(Φ)={g¯◦Φˆg◦m :m∈IM}, f(Π)=f(Φ) and Eq.(50) hold, where f is the objective
function of problem (2). In particular, an optimal mea-
1
κ(Φ)= g¯◦Φˆ :m∈I . (49) surement Π exists satisfying Eq. (50). Moreover, there
g◦m M
|G|  exists an optimal solution (Xˆ,λ) to dual problem (12)
 gX∈G 
such that
Then, κ is a bijective mapping onto M◦ for any g ∈G,
g  
andκisamappingontoM◦. Moreover,foranyΦ∈M◦, g◦Xˆ =Xˆ, ∀g ∈G,
we have
λ =λ , ∀g ∈G,j ∈I . (55)
j g◦j J
g◦Πˆ =Πˆ , ∀g ∈G,m∈I , (50)
m g◦m M
As examples of Theorem 6, we can derive that there
where Π=κ(Φ).
exist a minimum error measurement, an optimal unam-
Proof First,we showthatκ is bijective ontoM◦. Let biguous measurement, andan optimal inconclusive mea-
g surement that are G-covariant if a given state set is G-
Φ∈M◦ and Φ(g) =κ (Φ). Since Φ(g) =g¯◦Φˆ ∈S+
g m g◦m covariant, which is shown in Ref. [32].
and Mm=−01Φm(g) = g¯◦ˆ1 = ˆ1 hold, Φ(g) ∈ M holds. We Note that let M◦G be the entire set of Π ∈ M◦ satis-
also Pobtain for any j ∈IJ, fying Eq. (50); then, we can easily see that, since M◦G
is convex, problem (2) remains in convex programming
M−1 M−1
Tr(aˆj,mΦˆ(mg))= Tr[aˆj,m(g¯◦Φˆg◦m)] even if we restrict the feasible set from M◦ to M◦G.
m=0 m=0
X X
M−1 Proof First, we show that Π ∈ M◦ exists such that
= Tr[(g◦aˆj,m)Φˆg◦m] f(Π)=f(Φ)andEq.(50)holdforanyΦ∈M◦. LetΠ=
m=0 κ(Φ), where κ is defined by Eq. (49). From Lemma 5, Π
X
M−1 satisfies Π∈M◦ and Eq. (50). Moreover,we obtain
= Tr(aˆ Φˆ )
g◦j,g◦m g◦m
M−1
m=0
X f(Π)= Tr(cˆ Πˆ )
≤b =b , (51) m m
g◦j j
m=0
X
where the inequality follows fromthe groupactionbeing
M−1
bijective and Φ ∈ M◦. Thus, Φ(g) ∈ M◦ holds. More- = 1 Tr[cˆ (g¯◦Φˆ )]
m g◦m
over, since κ [κ (Φ)] = κ [κ (Φ)] = Φ, κ is the inverse |G|
g¯ g g g¯ g¯ m=0g∈G
mapping of κ . Therefore, κ is bijective onto M◦. X X
g g M−1
Next, we show that κ is a mapping onto M◦ and that = 1 Tr[(g◦cˆ )Φˆ ]
m g◦m
Eq. (50) holds. From Eq. (51), we have that for any |G|
m=0g∈G
j ∈I , X X
J
M−1
1
M−1Tr(aˆ Πˆ )= 1 M−1Tr(aˆ Φˆ(g))≤b ,(52) = |G| Tr(cˆg◦mΦˆg◦m)
j,m m |G| j,m m j gX∈GmX=0
mX=0 gX∈GmX=0 1
= f(Φ)=f(Φ). (56)
which means that Π ∈ M◦ holds for any Φ ∈ M◦, that |G|
is, κ is a mapping onto M◦. We also have that for any gX∈G
g ∈G and m∈I ,
M In particular, if Φ is an optimal measurement, then so is
1 1 Π.
g◦Πˆm = |G| g◦Φˆ(mh) = |G| h¯′◦Φˆh′◦g◦m Next, we show that there exists an optimal solution
hX∈G hX′∈G (Xˆ,λ) to dual problem (12) satisfying Eq. (55). Let ν =
=Πˆg◦m, (53) {ν : j ∈ I } ∈ RJ. Suppose that (Yˆ,ν) is an optimal
j J +
where h′ =h◦g¯. Thus, Eq. (50) holds. (cid:4) solution to problem (12). Also, let Yˆ(g) = g ◦ Yˆ and
9
ν(g) = {ν(g) = ν : j ∈ I }. Yˆ(g) ∈ S and ν(g) ∈ RJ Theorem 7 Let us consider a minimax solution to
j g¯◦j J +
obviously hold. We obtain for any g ∈G and m∈I , Eq. (27). Suppose that M◦ is not empty. Also, sup-
M
pose that there exist actions of G on S, I , I , and I
J−1 M J K
Yˆ(g) ≥g◦zˆ (ν)=cˆ − ν aˆ satisfying Eq. (48) and
m g◦m j g◦j,g◦m
Xj=0 g◦cˆk,m =cˆg◦k,g◦m, ∀g ∈G,k ∈IK,m∈IM,
J−1
d =d , ∀g ∈G,k ∈I . (63)
=cˆ − ν(g)aˆ =zˆ (ν(g)). (57) k g◦k K
g◦m g◦j g◦j,g◦m g◦m
j=0 Then, a minimax solution (µ,Π) exists such that
X
We also obtain
µ =µ , ∀g ∈G,k ∈I ,
k g◦k K
s(Yˆ(g),ν(g))=Tr Yˆ(g)+J−1ν(g)b g◦Πˆm =Πˆg◦m, ∀g ∈G,m∈IM. (64)
j j
Xj=0 Proof Let (η⋆,Π⋆) be a minimax solution to Eq. (27).
J−1 Also, let µ = {µ = |G|−1 η⋆ : k ∈ I } and
=Tr Yˆ + ν b =s(Yˆ,ν). (58) k g∈G g◦k K
g¯◦j g¯◦j Π = κ(Π⋆), where κ is defined by Eq. (49). Then, it
Xj=0 follows that µ ∈ P, Π ∈ MP◦, and Eq. (64) hold (also
From Eqs. (57) and (58), (Yˆ(g),ν(g)) is also an optimal see Lemma 5). Here, we show that (µ,Π) is a minimax
solution to problem (12). Let Xˆ ∈ S and λ = {λ : j ∈ solutiontoEq.(27). FromStatement(2)ofTheorem3,it
j
IJ}∈RJ+ be expressed by sWueffiwceislltsohsohwowft(hΠa)tf≥k(FΠ⋆)(η≥⋆)Fa⋆n(µd)Fh⋆o(ldη⋆s)fo≥raFn⋆y(µk)∈. IK.
k
Xˆ = 1 Yˆ(g), λ = 1 ν(g). (59) First, we show fk(Π) ≥ F⋆(η⋆) for any k ∈ IK. Let
|G| j |G| j Π(g) =κ (Π⋆); then for any k ∈I , we have
g∈G g∈G g K
X X
We can easily see that Eq. (55) holds. For any m∈IM, 1 M−1
we have fk(Π)= |G| Tr(cˆk,mΠˆ(mg))+dk
m=0g∈G
J−1 X X
1
zˆm(λ)=cˆm− |G| νj(g)aˆj,m = 1 M−1Tr(cˆ Πˆ(g))+d
gX∈GXj=0 |G| " k,m m k#
g∈G m=0
J−1 X X
= 1 cˆ − ν(g)aˆ 1 M−1
|G|gX∈G m Xj=0 j j,m = |G|g∈G"m=0Tr[(g◦cˆk,m)Πˆ⋆g◦m]+dk#
1   X X
= zˆ (ν(g)). (60) M−1
m 1
|G|gX∈G = |G|g∈G"m′=0Tr(cˆg◦k,m′Πˆ⋆m′)+dk#
From Eqs. (57), (59), and (60), we obtain for any m ∈ X X
1
IM, = fg◦k(Π⋆)≥F⋆(η⋆), (65)
|G|
Xˆ −zˆ (λ)= 1 [Yˆ(g)−zˆ (ν(g))]≥0. (61) gX∈G
m m
|G| where m′ =g◦m. The inequality in the last line follows
g∈G
X fromf (Π⋆)≥F⋆(η⋆) for anyk ∈I , whichis obtained
k K
Moreover,from Eqs. (58) and (59), we have from Theorem 3.
Next,we showF⋆(η⋆)≥F⋆(µ). Letη(g) ={η⋆ :k ∈
J−1 g◦k
s(Xˆ,λ)=Tr Xˆ + λjbj IK}. We have that for any g ∈G,
j=0
X K−1 M−1
= |G1| (Tr Yˆ(b)+νj(g)bj)=s(Yˆ,ν). (62) F⋆(η(g))=Φm∈Max◦ ηg⋆◦k" Tr(cˆk,mΦˆm)+dk#
gX∈G Xk=0 mX=0
K−1 M−1
T(1h2e)r.efore, (Xˆ,ν) is also an optimal solution to problem(cid:4) =Φm∈Max◦k′=0ηk⋆′"m=0Tr[cˆk′,m′(g◦Φˆm)]+dk′#
X X
K−1 M−1
C. Group covariant minimax solution =Φm′∈aMx◦k′=0ηk⋆′"m′=0Tr(cˆk′,m′Φˆ′m′)+dk′#
X X
=F⋆(η⋆), (66)
SimilartoTheorem6,wecanshowthatifEq.(27)has
a certain symmetry with respect to G, then there exists where k′ =g◦k, m′ =g◦m, and Φ′ =κ (Φ). The third
g¯
a G-covariant minimax solution. linefollowsfromthemappingκ beingbijectiveontoM◦
g¯
10
(see Lemma 5). From Eq. (66), we obtain where
K−1 R−1
F⋆(µ)=Φm∈Max◦ |G1|gX∈G kX=0 ηk(g)fk(Φ) Xˆ(λ)= Xr=0(ξr +λr)ρˆrΠˆr+λRGˆΠˆR. (71)
≤ 1 F⋆(η(g))=F⋆(η⋆). (67) In the case in which problem (10) has a certain sym-
|G| metry, we can apply Theorem 6. Suppose that a given
g∈G
X state set is G-covariant, that is, there exist actions of G
Therefore, (µ,Π) is a minimax solution. (cid:4) on S and IR satisfying g ◦(ξrρˆr) = ξg◦rρˆg◦r, which is
equivalent to g◦ρˆ = ρˆ and ξ = ξ , for any g ∈ G
r g◦r r g◦r
and r ∈ I . Let the action of G on I = I , g◦m
R M R+1
V. EXAMPLES OF OPTIMAL MEASUREMENT (m ∈ I ), be g ◦m = π(R)(m) for any m ∈ I and
M g R
AND MINIMAX SOLUTION (R)
g◦R=R,where {π :g ∈G} is the actionofG onI .
g R
Also, let the action of G on I be the same as the action
J
As an example of a generalized optimal measurement,
of G on I . Then, since Eqs. (48) and (54) hold, there
M
we discuss the problem of finding an optimal inconclu-
exists an optimal measurement satisfying Eq. (50).
sive measurement with a lower bound on success proba-
bilities,whichisintroducedinSubsubsec.IIB3. Also,as
anexampleofageneralizedminimaxsolution,wediscuss B. Minimax solution for plural state sets
theproblemoffindingaminimaxsolutionforpluralstate
sets,whichisintroducedinSubsubsec.IIIB3. Moreover,
In this example, we can apply Theorem 3, that is,
TablesI andII summarizethe problemformulationsand
(µ⋆,Π⋆) is a minimax solution if and only if Eq. (39)
their examples shown in Subsecs. IIB and IIIB, respec-
holds, or F⋆(µ⋆)=F(µ⋆,Π⋆) and Eq. (40) hold. Substi-
tively.
tuting Eq. (37) into Eq. (40) gives
R−1 R−1
A. Optimal inconclusive measurement with a lower Tr(ρˆ′k,mΠˆ⋆m)≥ Tr(ρˆ′k′,mΠˆ⋆m),
bound on success probabilities
m=0 m=0
X X
∀k,k′ ∈IK s.t. µ⋆k′ >0. (72)
Inthisexample,wecanapplyTheorems1and2. Sub-
stituting Eq. (11) into Eq. (13) gives From Eq. (38), F(µ,Π) is expressed by
(ξ +λ )ρˆ , m<R, K−1 R−1
zˆm(λ)= λmGˆ, m m m=R. (68) F(µ,Π)= µk Tr(ρˆ′k,mΠˆm)
R
(cid:26) k=0 m=0
X X
Thus,fromTheorem1,dualproblem(12)canbe rewrit- R−1 K−1
ten as = Tr µkρˆ′k,m Πˆm . (73)
" ! #
R−1 mX=0 Xk=0
minimize s(Xˆ,λ)=Tr Xˆ −q λr−pλR Thus, F⋆(µ) is equivalent to the optimal value of f(Π)
r=0 (69) of optimization problem (2) with
subject to Xˆ ≥(ξ +λ )ρˆ , ∀Xr ∈I ,
r r r R
Xˆ ≥λ Gˆ. M =R,
R
J =0,
λ Gˆ ∈ S yields Xˆ ∈ S . In particular, in the case
R + + K−1
of q = 0, Eq. (69) is equivalent to the dual problem of cˆ = µ ρˆ′ . (74)
m k k,m
finding an optimal inconclusive measurement, which is
k=0
shown in Theorem 1 of Ref. [17]. X
We can also obtain necessary and sufficient conditions This indicates that F⋆(µ) is also equivalent to the aver-
for an optimal measurement from Theorem 2. For ex- agesuccessprobabilityofaminimumerrormeasurement
ample, from Statement (3) of this theorem, Π ∈ M◦ is for the state set {cˆm/Tr cˆm :m∈IR} with prior proba-
an optimal measurement of problem (10) if and only if bilities {Tr cˆm :m∈IR} (note that cˆm ∈S+ holds from
λ∈RJ exists such that Eq. (74)).
+
WecanalsoapplyTheorem7inthecaseinwhichgiven
Xˆ(λ)−(ξ +λ )ρˆ ≥0, ∀r ∈I , state sets have a certain symmetry. Assume that there
r r r R
Xˆ(λ)−λ Gˆ ≥0, existactions of G on S, IR, andIK satisfying g◦ρˆ′k,m =
R ρˆ′ for any g ∈ G, m ∈ I , and k ∈ I . For
λr Tr(ρˆrΠˆr)−q =0, ∀r ∈IR, exg◦akm,gp◦mle, this assumption holds iRf each state seKt Ψk =
{ρˆ :m∈I }isG-covariantunderthesameactionsof
h i k,m R
λR Tr(GˆΠˆR)−p =0, (70) G on S and IR, i.e., g◦ρˆ′k,m =ρˆ′k,g◦m for any g ∈G and
h i