Download Mixed-quantum-state detection with inconclusive results

PHYSICAL REVIEW A 67, 042309 共2003兲
Mixed-quantum-state detection with inconclusive results
Yonina C. Eldar*
Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel
共Received 19 November 2002; revised manuscript received 27 January 2003; published 15 April 2003兲
We consider the problem of designing an optimal quantum detector with a fixed rate of inconclusive results
that maximizes the probability of correct detection, when distinguishing between a collection of mixed quantum states. We show that the design of the optimal detector can be formulated as a semidefinite programming
problem, and derive a set of necessary and sufficient conditions for an optimal measurement. We then develop
a sufficient condition for the scaled inverse measurement to maximize the probability of correct detection for
the case in which the rate of inconclusive results exceeds a certain threshold. Using this condition we derive the
optimal measurement for linearly independent pure-state sets, and for mixed-state sets with a broad class of
symmetries. Specifically, we consider geometrically uniform 共GU兲 state sets and compound geometrically
uniform 共CGU兲 state sets with generators that satisfy a certain constraint. We then show that the optimal
measurements corresponding to GU and CGU state sets with arbitrary generators are also GU and CGU,
respectively, with generators that can be computed very efficiently in polynomial time within any desired
accuracy by solving a reduced size semidefinite programming problem.
DOI: 10.1103/PhysRevA.67.042309
PACS number共s兲: 03.67.Hk
I. INTRODUCTION
Quantum-information theory refers to the distinctive information processing properties of quantum systems, which
arise when information is stored in or retrieved from quantum states. A fundamental aspect of quantum information
theory is that nonorthogonal quantum states cannot be perfectly distinguished. Therefore, a central problem in quantum
mechanics is to design measurements optimized to distinguish between a collection of nonorthogonal quantum states.
We consider a quantum-state ensemble consisting of m
positive semidefinite Hermitian density operators 兵 ␳ i , 1⭐i
⭐m 其 on an n-dimensional complex Hilbert-space H, with
prior probabilities 兵 p i ⬎0,1⭐i⭐m 其 . For our measurement,
we consider general positive operator-valued measures 关1,2兴,
consisting of positive semidefinite Hermitian operators that
form a resolution of the identity on H.
Different approaches to distinguish between the density
operators ␳ i have emerged. In one approach, the measurement consists of m measurement operators which are designed to maximize the probability of correct detection. Necessary and sufficient conditions for an optimum
measurement maximizing the probability of correct detection
have been developed 关3–5兴. Closed-form analytical expressions for the optimal measurement have been derived for
several special cases 关6 –11兴. In particular, the optimal measurement for pure and mixed-state ensembles with broad
symmetry properties, referred to as geometrically uniform
共GU兲 and compound GU 共CGU兲 state sets, are considered in
Refs. 关10,11兴. Iterative procedures maximizing the probability of correct detection have also been developed for cases in
which the optimal measurement cannot be found explicitly
关12,5兴.
More recently, a different approach to the problem has
emerged, which in some cases may be more useful. This
*Electronic address: [email protected]
1050-2947/2003/67共4兲/042309共14兲/$20.00
approach, referred to as unambiguous quantum-state discrimination, combines error-free discrimination with a certain fraction of inconclusive results 关13–20兴. The basic idea,
pioneered by Ivanovic 关13兴, is to design a measurement that
with probability ␤ returns an inconclusive result, but if the
measurement returns an answer, then the answer is correct
with probability 1. In this case, the measurement consists of
m⫹1 measurement operators corresponding to m⫹1 outcomes, where m outcomes correspond to detection of each of
the states and the additional outcome corresponds to an inconclusive result. Chefles 关18兴 showed that a necessary and
sufficient condition for the existence of unambiguous measurements for distinguishing between a collection of pure
quantum states is that the states are linearly independent pure
states. Necessary and sufficient conditions on the optimal
measurement minimizing the probability ␤ of an inconclusive result were derived in Ref. 关20兴. The optimal measurement when distinguishing between GU and CGU pure-state
sets was also considered in Ref. 关20兴, and was shown under
certain conditions to be equal to the equal-probability measurement 共EPM兲.
An interesting alternative approach for distinguishing between a collection of quantum-states, first considered by
Chefles and Barnett 关21兴 and Zhang, Li, and Guo 关22兴 for
pure-state ensembles, and then later extended by Fiurášek
and Ježek 关23兴 to mixed-state ensembles, is to allow for a
certain probability of an inconclusive result, and then maximize the probability of correct detection when a conclusive
result is obtained. Thus, in this approach, the measurement
again consists of m⫹1 measurement outcomes. However,
now the outcomes do not necessarily correspond to perfect
detection of each of the states. Indeed, if, for example, the
quantum states are linearly dependent pure states, then perfect detection of each of the states is not possible 关18兴. Nonetheless, by allowing for inconclusive results, a higher probability of correct detection can be obtained for the case in
which a conclusive result is obtained in comparison with the
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PHYSICAL REVIEW A 67, 042309 共2003兲
YONINA C. ELDAR
probability of correct detection attainable without inconclusive results.
Necessary conditions as well as a set of sufficient conditions on the optimal measurement operators maximizing the
probability of correct detection subject to the constraint that
the probability of an inconclusive result is equal to a constant
␤ were derived in Ref. 关23兴, using Lagrange multiplier
theory. It was also pointed out in Ref. 关23兴 that obtaining a
closed form analytical solution to the optimal measurement
operators directly from these conditions is a difficult problem.
In this paper, we extend the results of Ref. 关23兴 in several
ways. First, using principles of duality in vector space optimization, in Sec. III we show that the conditions derived in
Ref. 关23兴 are both necessary and sufficient. We also show
that the Lagrange multipliers can be obtained by solving a
reduced size semidefinite programming problem 关24兴, which
is a convex optimization problem. By exploiting the many
well-known algorithms for solving semidefinite programs
关24 –27兴, the optimal measurement can be computed very
efficiently in polynomial time within any desired accuracy.
Furthermore, in contrast to the iterative algorithm proposed
in Ref. 关23兴, which is only guaranteed to converge to a local
optimum, algorithms based on semidefinite programming are
guaranteed to converge to the global optimum.
Second, we derive a general condition in Sec. IV under
which the scaled inverse measurement 共SIM兲 is optimal. This
measurement consists of measurement operators that are proportional to the reciprocal states associated with the given
state ensemble, and can be regarded as a generalization of
the EPM to linearly dependent pure-state sets and mixed
states.
Third, we develop the optimal measurement for state sets
with broad symmetry properties. Specifically, in Sec. V we
consider GU state sets defined over a finite group of unitary
matrices. We obtain a convenient characterization of the SIM
and show that the SIM operators have the same symmetries
as the original state set. We then show that for a pure GU
state set and for values of ␤ exceeding a certain threshold,
the SIM is optimal. For a mixed GU state set, under a certain
constraint on the generator and for values of ␤ exceeding a
threshold, the SIM is again shown to be optimal. For arbitrary values of ␤ , the optimal measurement operators corresponding to a pure or mixed GU state set are shown to be
GU with the same generating group. As we show, the generators can be computed very efficiently in polynomial time
within any desired accuracy.
In Sec. VI we consider CGU state sets 关28兴, in which the
states are generated by a group of unitary matrices using
multiple generators. We obtain a convenient characterization
of the SIM for CGU state sets, and show that the SIM vectors are themselves CGU. Under a certain condition on the
generators and for values of ␤ exceeding a threshold, the
SIM is shown to be optimal. Finally we show that for arbitrary CGU state sets and for arbitrary values of ␤ , the optimal measurement operators are also CGU, and we propose
an efficient algorithm for computing the optimal generators.
It is interesting to note that a closed form analytical expression exists for the optimal measurement when distin-
guishing between GU and CGU 共possibly mixed兲 state sets
with generators that satisfy a certain constraint, under each of
the three approaches outlined to quantum detection, where in
the last approach we assume that ␤ exceeds a given threshold. Furthermore, as shown in Refs. 关11,20兴 and in Secs. V
and VI, the optimal measurement operators corresponding to
GU and CGU state sets are also GU and CGU, respectively,
under each one of the three outlined optimality criteria.
Before proceeding to the detailed development, we provide in the following section a statement of our problem.
II. PROBLEM FORMULATION
Assume that a quantum channel is prepared in a quantumstate drawn from a collection of given states represented by
density operators 兵 ␳ i ,1⭐i⭐m 其 on an n-dimensional complex Hilbert space H. We assume without loss of generality
that the eigenvectors of ␳ i ,1⭐i⭐m, collectively span 关43兴 H
so that m⭓n. Since ␳ i is Hermitian and positive semidefinite, we can express ␳ i as ␳ i ⫽ ␾ i ␾ i* for some matrix ␾ i ,
e.g., via the Cholesky or eigendecomposition of ␳ i 关29兴. We
refer to ␾ i as a factor of ␳ i . The choice of ␾ i is not unique;
if ␾ i is a factor of ␳ i , then any matrix of the form ␾ i⬘
⫽ ␾ i Q i , where Q i is an arbitrary matrix satisfying Q i Q i*
⫽I, is also a factor of ␳ i .
To detect the state of the system a measurement is constructed comprising m⫹1 measurement operators 兵 ⌸ i ,0⭐i
⭐m 其 that satisfy
⌸ i ⭓0,
0⭐i⭐m,
m
兺 ⌸ i ⫽I.
i⫽0
共1兲
Each of the operators ⌸ i ,1⭐i⭐m correspond to detection of
the corresponding states ␳ i ,1⭐i⭐m, and ⌸ 0 corresponds to
an inconclusive result. We seek the measurement operators
⌸ i that maximize the probability P D of correct detection,
subject to the constraint that the probability P I of an inconclusive result is equal to a constant ␤ ⬍1.
Given that the transmitted state is ␳ j , the probability of
correctly detecting the state using measurement operators
兵 ⌸ i ,1⭐i⭐m 其 is Tr( ␳ j ⌸ j ) and the probability of a detection
m
error is 兺 i⫽1,i⫽
j Tr( ␳ j ⌸ i ). Therefore, the probability of correct detection is given by
m
P D⫽
兺
i⫽1
p i Tr共 ␳ i ⌸ i 兲 ,
共2兲
where p i ⬎0 is the prior probability of ␳ i with 兺 i p i ⫽1, and
the probability of a detection error is given by
m
P E⫽
m
兺 兺
i⫽1 j⫽1,j⫽i
p i Tr共 ␳ i ⌸ j 兲 .
The probability of an inconclusive result is
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MIXED-QUANTUM-STATE DETECTION WITH . . .
III. CONDITIONS FOR OPTIMALITY
m
P I⫽
兺 p i Tr共 ␳ i ⌸ 0 兲 ⫽Tr共 ⌬⌸ 0 兲 ⫽ ␤ ,
i⫽1
共4兲
where for brevity we denote
m
⌬⫽
兺
i⫽1
p i␳ i .
共5兲
Using Lagrange multipliers, it was shown in Ref. 关23兴 that
a set of measurement operators 兵 ⌸̂ i ,0⭐i⭐m 其 maximizes P D
subject to P I ⫽ ␤ for a state set 兵 ␳ i ,1⭐i⭐m 其 with prior probabilities 兵 p i ,1⭐i⭐m 其 if there exists an Hermitian X̂ and a
constant ␦ˆ satisfying
X̂⭓ p i ␳ i ,
Our problem is to find the measurement operators 兵 ⌸ i ,0⭐i
⭐m 其 that maximize P D of Eq. 共2兲 subject to the constraints
共1兲 and 共4兲.
Note that since Tr( ␳ i )⫽1 for all i,
1⭐i⭐m,
X̂⭓ ␦ˆ ⌬,
共7兲
共8兲
such that
m
P D⫹ P E⫹ P I⫽
兺
i⫽1
p i Tr共 ␳ i 兲 ⫽1.
共6兲
Each of the three approaches to quantum detection outlined
in the introduction correspond to maximizing P D subject to
different constraints on P I and P E . The standard quantum
detection problem is to maximize P D subject to P I ⫽0,
which implies that ⌸ 0 ⫽0. Therefore, this approach is
equivalent to seeking m measurement operators ⌸ i ⭓0,1⭐i
m
⌸ i ⫽I to maximize P D , or equivalently,
⭐m satisfying 兺 i⫽1
minimize P E . In unambiguous quantum-state discrimination
the problem is to choose m⫹1 measurement operators to
maximize P D , or equivalently minimize P I , subject to P E
⫽0. As shown in 关18兴, it is not always possible to choose
measurement operators such that P E ⫽0. For example, if the
state ensemble is a pure-state ensemble consisting of density
operators ␳ i of the form ␳ i ⫽ 兩 ␾ i 典具 ␾ i 兩 for a set of linearly
dependent vectors 兩 ␾ i 典 , then there is no measurement that
will result in P E ⫽0. Nonetheless, we may seek the measurement operators that minimize P E , or equivalently, maximize
P D , subject to P I ⫽ ␤ for some ␤ ⬍1. By allowing for ␤
⬎0 we can achieve a larger probability of correct detection
when a conclusive result is obtained than that which can be
achieved using the standard quantum detection approach in
which we require that ␤ ⫽0.
Equipped with the standard operations of addition and
multiplication by real numbers, the space B of all Hermitian
n⫻n matrices is an n 2 -dimensional real vector space. As
noted in Ref. 关23兴, by choosing an appropriate basis for B,
the problem of maximizing P D subject to Eqs. 共1兲 and 共4兲
can be put in the form of a standard semidefinite programming problem, which is a convex optimization problem; for
a detailed treatment of semidefinite programming problems
see, e.g., Refs. 关24 –27兴. Recently, methods based on
semidefinite programming have been employed in a variety
of different problems in quantum detection and quantum information 关5,20,30–35兴. By exploiting the many well-known
algorithms for solving semidefinite programs 关24兴, e.g., interior point methods 关44兴 关25,27兴, the optimal measurement
can be computed very efficiently in polynomial time within
any desired accuracy.
The semidefinite programming formulation can also be
used to derive necessary and sufficient conditions for optimality, which we discuss in the following section.
共 X̂⫺ p i ␳ i 兲 ⌸̂ i ⫽0,
1⭐i⭐m,
共 X̂⫺ ␦ˆ ⌬ 兲 ⌸̂ 0 ⫽0.
共9兲
共10兲
It was also shown that Eqs. 共9兲 and 共10兲 are necessary conditions for optimality.
In Appendix A we use duality arguments similar to those
used in Ref. 关5兴 to show that Eqs. 共7兲–共10兲 are necessary and
sufficient conditions for optimality, so that a set of measurement operators ⌸̂ i maximizes P D subject to P I ⫽ ␤ if and
only if there exists an Hermitian X̂ and a constant ␦ˆ satisfying Eqs. 共7兲–共10兲. Furthermore, we show that X̂ and ␦ˆ can be
determined as the solution to the following semidefinite programming problem:
min 兵 Tr共 X 兲 ⫺ ␦ ␤ 其 ,
X苸B, ␦ 苸R
共11兲
where B is the set of n⫻n Hermitian operators and R denotes the reals, subject to
X⭓ p i ␳ i ,
1⭐i⭐m,
X⭓ ␦ ⌬.
共12兲
The problem of Eqs. 共11兲 and 共12兲 is referred to as the dual
problem.
Note that the dual problem involves many fewer decision
variables than the primal maximization problem. Specifically, in the dual problem we have n 2 ⫹1 real decision variables while the primal problem has (m⫹1)n 2 real decision
variables. Therefore, it is advantageous to solve the dual
problem and then use Eqs. 共9兲 and 共10兲 to determine the
optimal measurement operators, rather than solving the primal problem directly. Once we determine X̂ and ␦ˆ , the optimal measurement operators ⌸̂ i can be computed using Eqs.
共9兲 and 共10兲, in a similar manner to that described in Ref. 关5兴.
We summarize the results of Appendix A in the following
theorem:
Theorem 1 (necessary and sufficient conditions). Let
兵 ␳ i ,1⭐i⭐m 其 denote a collection of quantum states with
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YONINA C. ELDAR
prior probabilities 兵 p i ⬎0,1⭐i⭐m 其 . Let ⌳ denote the set of
all ordered sets of Hermitian measurement operators ⌸
m
m
that satisfy ⌸ i ⭓0, 兺 i⫽0
⌸ i ⫽I and Tr(⌬⌸ 0 )⫽ ␤
⫽ 兵 ⌸ i 其 i⫽0
m
where ⌬⫽ 兺 i⫽1 p i ␳ i , and let ⌫ denote the set of Hermitian
matrices X and scalars ␦ such that X⭓p i ␳ i ,1⭐i⭐m and X
⭓ ␦ ⌬. Consider the problem max⌸苸⌳J(⌸) and the dual probm
p i Tr( ␳ i ⌸ i ) and
lem minX,␦苸⌫T(X,␦), where J(⌸)⫽ 兺 i⫽1
T(X, ␦ )⫽Tr(X)⫺ ␦ ␤ . Then
共1兲 For any X, ␦ 苸⌫ and ⌸苸⌳, T(X, ␦ )⭓J(⌸);
共2兲 there is an optimal ⌸, denoted ⌸̂, such that Ĵ
⫽J(⌸̂)⭓J(⌸) for any ⌸苸⌳;
共3兲 there are an optimal X and ␦ , denoted X̂ and ␦ˆ , such
that T̂⫽T(X̂, ␦ˆ )⭐T(X, ␦ ) for any X, ␦ 苸⌫;
共4兲 T̂⫽Ĵ;
共5兲 necessary and sufficient conditions on the optimal
measurement operators ⌸̂ i are that there exists X, ␦ 苸⌫ such
that
共 X⫺p i ␳ i 兲 ⌸̂ i ⫽0,
1⭐i⭐m,
共 X⫺ ␦ ⌬ 兲 ⌸̂ 0 ⫽0;
共6兲 given X̂ and ␦ˆ , necessary and sufficient conditions on
the optimal measurement operators ⌸̂ i are
共 X̂⫺p i ␳ i 兲 ⌸̂ i ⫽0,
1⭐i⭐m,
共 X̂⫺ ␦ˆ ⌬ 兲 ⌸̂ 0 ⫽0.
In Ref. 关5兴 the authors consider the standard quantum detection problem of choosing m measurement operators
兵 ⌸ i ,1⭐i⭐m 其 to maximize P D subject to P I ⫽0, and develop a set of necessary and sufficient conditions for optimality on the measurement operators. Specifically, using an
approach similar to that used in Appendix A they show that a
set of measurement operators 兵 ⌸̂ i ,1⭐i⭐m 其 maximizes P D
subject to P I ⫽0 for a state set 兵 ␳ i ,1⭐i⭐m 其 with prior probabilities 兵 p i ,1⭐i⭐m 其 if there exists an Hermitian X̂ satisfying
X̂⭓p i ␳ i ,
1⭐i⭐m,
共 X̂⫺p i ␳ i 兲 ⌸̂ i ⫽0,
1⭐i⭐m.
共13兲
Furthermore, they show that the conditions 共13兲 imply that
the rank of each optimal measurement operator ⌸̂ i ,1⭐i⭐m
is no larger than the rank of the corresponding density operator ␳ i ,1⭐i⭐m. In particular, if the quantum state ensemble is a pure-state ensemble consisting of 共not necessarily
independent兲 rank-one density operators ␳ i ⫽ 兩 ␾ i 典具 ␾ i 兩 , then
the optimal measurement maximizing P D subject to P I ⫽0 is
a pure-state measurement consisting of rank-one measurement operators ⌸ i ⫽ 兩 ␮ i 典具 ␮ i 兩 .
The conditions 共13兲 for a set of measurement operators to
maximize P D subject to P I ⫽0 are equivalent to conditions
共7兲 and 共9兲 on a set of measurement operators to maximize
P D subject to P I ⫽ ␤ . Therefore, using the results in Ref. 关5兴
we have the following proposition:
Proposition 2. Let 兵 ␳ i ,1⭐i⭐m 其 denote a collection of
quantum states with prior probabilities 兵 p i ,1⭐i⭐m 其 . Then
the optimal measurement that maximizes P D subject to
P I ⫽ ␤ consists of measurement operators 兵 ⌸ 0 ,1⭐i⭐m 其
with rank(⌸ i )⭐rank( ␳ i ),1⭐i⭐m. In particular if
兵 ␳ i ⫽ 兩 ␾ i 典具 ␾ i 兩 ,1⭐i⭐m 其 is a pure-state quantum ensemble,
then the optimal measurement is a pure-state measurement consisting of measurement operators of the form
m
兩 ␮ i 典具 ␮ i 兩 其 .
兵 ⌸ i ⫽ 兩 ␮ i 典具 ␮ i 兩 ,1⭐i⭐m,⌸ 0 ⫽I⫺ 兺 i⫽1
As pointed out in Ref. 关23兴, obtaining a closed-form analytical expression for the optimal measurement operators directly from the necessary and sufficient conditions for optimality of Theorem 1 is a difficult problem. Since Eq. 共11兲 is
a 共convex兲 semidefinite programming 关24,25,27兴 problem,
there are very efficient methods for solving Eq. 共11兲. In particular, the optimal matrix X̂ and optimal scalar ␦ˆ minimizing Tr(X)⫺ ␦ ␤ subject to Eq. 共12兲 can be computed in Matlab using the linear matrix inequality 共LMI兲 Toolbox 共see
Refs. 关5,20兴 for further details兲.
A suboptimal measurement that has been suggested as a
detection measurement for unambiguous quantum-state discrimination between linearly independent pure quantumstates is the EPM 关18 –20兴, in which the measurement vectors are proportional to the reciprocal states associated with
the states to be distinguished. Specifically, the EPM corresponding to a set of state vectors 兵 兩 ␾ i 典 ,1⭐i⭐m 其 that collectively span H with prior probabilities 兵 p i ,1⭐i⭐m 其 consists
of the measurement vectors 兵 ␮ i ,1⭐i⭐m 其 , where 关20兴
␮ i ⫽ 冑␭ n 共 ⌿⌿ * 兲 ⫺1 ␺ i , 1⭐i⭐m.
共14兲
Here 兩 ␺ i 典 ⫽ 冑p i ␾ i , ⌿ is the matrix of columns 兩 ␺ i 典 , 兵 ␭ i ,1
⭐i⭐n 其 denote the eigenvalues of ⌿⌿ * and ␭ n ⫽min ␭i . A
general condition under which the EPM is optimal for unambiguous quantum-state discrimination between linearly independent pure quantum-states so that it maximizes P D subject
to P E ⫽0 was derived in Ref. 关20兴. It was also shown that for
GU state sets and for CGU state sets with generators satisfying a certain constraint, the EPM maximizes P D subject to
P E ⫽0.
In the following section we consider a generalization of
the EPM to linearly dependent pure states and mixed states,
which we refer to as the scaled inverse measurement 共SIM兲.
We then use the necessary and sufficient conditions for optimality of Theorem 1 to derive a general condition under
which the SIM maximizes P D subject to P I ⫽ ␤ . In analogy
to the EPM, in Secs. V and VI we show that for GU state sets
and CGU state sets with generators satisfying a certain constraint, the SIM maximizes P D subject to P I ⫽ ␤ for values
of ␤ larger than a threshold, and derive explicit formulas for
the optimal measurement operators.
IV. THE SIM AND THE OPTIMAL MEASUREMENT
A. The SIM
The SIM corresponding to a set of density operators 兵 ␳ i
⫽ ␾ i␾ *
i ,1⭐i⭐m 其 with eigenvectors that collectively span H
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and prior probabilities 兵 p i ,1⭐i⭐m 其 consists of the measurement operators 兵 ⌺ i ⫽ ␮ i ␮ *
i ,0⭐i⭐m 其 , where
␮ i ⫽ ␥ 共 ⌿⌿ * 兲
⫺1
␺ i⫽ ␥ ⌬
⫺1
␺ i , 1⭐i⭐m,
共15兲
m
and ⌺ 0 ⫽I⫺ 兺 i⫽1
␮ i␮ *
i . Here ⌿ is the matrix of 共block兲
columns ␺ i ⫽ 冑p i ␾ i and ␥ is chosen such P I ⫽ ␤ and
m
兺 i⫽0
⌸ i ⫽I. Note that since the eigenvectors of the 兵 ␳ i 其 collectively span H, the columns of the 兵 ␺ i 其 also together span
H, so ⌿⌿ * is invertible.
From Eq. 共15兲
m
兺
i⫽1
冉兺 冊
m
2 ⫺1
␮ i␮ *
i ⫽␥ ⌬
i⫽1
␺ i ␺ i* ⌬ ⫺1 ⫽ ␥ 2 ⌬ ⫺1 ,
共16兲
so that
m
⌺ 0 ⫽I⫺
兺 ␮ i ␮ *i ⫽I⫺ ␥ 2 ⌬ ⫺1 .
i⫽1
共17兲
It then follows that the probability of an inconclusive result
using the SIM is
P I ⫽Tr共 ⌬⌺ 0 兲 ⫽Tr共 ⌬ 兲 ⫺ ␥ 2 Tr共 I 兲 ⫽1⫺n ␥ 2 .
共18兲
Therefore to satisfy P I ⫽ ␤ ,
␥⫽
冑
1⫺ ␤
.
n
共19兲
In addition, from Eq. 共17兲 the SIM operators satisfy
m
兺 i⫽0
⌸ i ⫽I if and only if ␥ 2 ⭐␭ n where 兵 ␭ i ,1⭐i⭐n 其 denote
the eigenvalues of ⌬⫽⌿⌿ * and ␭ n ⫽min ␭i . From Eq. 共19兲
it follows that the SIM is defined only for values of ␤ satisfying
䉭
␤ ⭓1⫺n␭ n ⫽ ␤ min .
共20兲
Since the factors ␾ i are not unique, the SIM factors ␮ i are
also not unique. If ␮ i are the SIM factors corresponding to
␾ i , then the SIM factors corresponding to ␾ i⬘ ⫽ ␾ i Q i with
Q iQ *
i ⫽I are ␮ i⬘ ⫽ ␮ i Q i . Therefore, although the SIM factors are not unique, the SIM operators ⌺ i ⫽ ␮ i ␮ i* are unique.
In the case in which the prior probabilities are all equal to
a constant p,
␮ i⫽
␥
冑p
共 ⌽⌽ * 兲 ⫺1 ␾ i ,
1⭐i⭐m,
共21兲
where ⌽ is the matrix of 共block兲 columns ␾ i .
The SIM corresponding to a pure-state ensemble 兩 ␾ i 典
consists of the measurement vectors 兩 ␮ i 典 ⫽ ␥ ⌬ ⫺1 兩 ␺ i 典 , where
兩 ␺ i 典 ⫽ 冑p i 兩 ␾ i 典 . From Eq. 共14兲 it follows that in the special
case in which the vectors 兩 ␾ i 典 are linearly independent and
␥ ⫽ 冑␭ n , the SIM vectors are equal to the EPM vectors.
From Eqs. 共19兲 and 共20兲 we then have that ␤ min is equal to
the probability of an inconclusive result when using the
EPM.
B. Optimality of the SIM
For linearly independent pure quantum-states it was
shown in Ref. 关20兴 that the SIM with ␥ ⫽ 冑␭ n , or equivalently the EPM, maximizes P D subject to P E ⫽0 for state
sets with equal prior probabilities and strong symmetry properties. The smallest possible probability of an inconclusive
result in this case is ␤ ⫽ ␤ min given by Eq. 共20兲. Theorem 3
below asserts that for a large class of state sets, including
those discussed in Ref. 关20兴, the SIM maximizes P D subject
to P I ⫽ ␤ for ␤ ⭓ ␤ min .
From the necessary and sufficient conditions of Theorem
1 it follows that the SIM is optimal if and only if the measurement operators ⌸̂ i ⫽ ␮ i ␮ *
i ,1⭐i⭐m and ⌸̂ 0 ⫽⌺ 0 defined
by Eqs. 共15兲 and 共17兲 satisfy Eqs. 共9兲 and 共10兲 for some
Hermitian X̂ and constant ␦ˆ satisfying Eqs. 共7兲 and 共8兲. A
sufficient condition for optimality of the SIM is given in the
following theorem, the proof of which is provided in Appendix B.
Theorem 3 (optimality of the SIM). Let 兵 ␳ i ⫽ ␾ i ␾ *
i ,1⭐i
⭐m 其 denote a collection of quantum-states with prior probabilities 兵 p i ,1⭐i⭐m 其 . Let 兵 ⌺ i ⫽ ␮ i ␮ *
i ,1⭐i⭐m 其 with 兵 ␮ i
m
⫽ ␥ ⌬ ⫺1 ␺ i ,1⭐i⭐m 其 and ⌺ 0 ⫽I⫺ 兺 i⫽1
⌺ i denote the scaled
inverse measurement 共SIM兲 operators corresponding to 兵 ␺ i
⫽ 冑p i ␾ i ,1⭐i⭐m 其 , where ␥ 2 ⫽(1⫺ ␤ )/n, ⌬⫽⌿⌿ * and ⌿
is the matrix with block columns ␺ i , and let ␭ n ⫽min ␭i ,
where ␭ i are the eigenvalues of ⌬. Then the SIM maximizes
P D subject to P I ⫽ ␤ for ␤ ⭓ ␤ min with ␤ min⫽1⫺n␭n if for
each 1⭐i⭐m,
⫺1
␺ i ⫽ ␣ I,
共 1/␥ 兲 ␮ *
i ␺ i ⫽ ␺ i* ⌬
where ␣ is a constant independent of i.
It is interesting to note that the condition of Theorem 3 for
the SIM to maximize P D subject to P I ⫽ ␤ for ␤ ⭓ ␤ min is
identical to the condition given in Theorem 1 of Ref. 关11兴 for
the least-squares measurement 关10,34兴, or the square-root
measurement 关9,35–39兴, to maximize P D subject to P I ⫽0.
A similar result for the special case in which the density
operators ␳ i are rank-one operators of the form ␳ i
⫽ 兩 ␾ i 典具 ␾ i 兩 and the vectors 兩 ␾ i 典 are linearly independent was
derived in Ref. 关37兴. The least-squares measurement is in
general a suboptimal measurement that has been employed
as a detection measurement in many applications 共see e.g.,
Refs. 关37–39兴兲 and has many desirable properties. Its construction is relatively simple; it can be determined directly
from the given collection of states; it maximizes P D subject
to P I ⫽0 for pure-state ensembles 关9,10兴 and mixed-state ensembles 关11兴 that exhibit certain symmetries; it is ‘‘pretty
good’’ when the states to be distinguished are equally likely
and almost orthogonal 关35兴; and it is asymptotically optimal
关34,36兴. Although the sufficient conditions given by Theorem
3 and Theorem 1 of Ref. 关11兴 are the same, the optimal
measurements in both cases are different: the SRM consists
of m measurement operators ⌸ i ⫽ ␮ i ␮ *
i ,1⭐i⭐m with measurement factors ␮ i ⫽(⌿⌿ * ) ⫺1/2␺ i ,1⭐i⭐m, and the SIM
consists of m⫹1 measurement operators ⌸ i ⫽ ␮ i ␮ *
i ,1⭐i
m
⌸ i with measurement factors ␮ i
⭐m and ⌸ 0 ⫽I⫺ 兺 i⫽1
⫽ ␥ (⌿⌿ * ) ⫺1 ␺ i ,1⭐i⭐m, where ␥ 2 ⫽(1⫺ ␤ )/n.
042309-5
PHYSICAL REVIEW A 67, 042309 共2003兲
YONINA C. ELDAR
We now try to gain some physical insight into the sufficient condition of Theorem 3. First we note that, as we ex⫺1
␺ i ⫽ ␣ I does not depend on the
pect, the condition ␺ *
i ⌬
choice of factors ␾ i . Indeed, if ␾ i⬘ ⫽ ␾ i Q i is another factor
of ␳ i with Q i satisfying Q i Q *
i ⫽I, and if ⌿ ⬘ is the matrix of
block columns ␺ i⬘ ⫽ 冑p i ␾ i⬘ ⫽ 冑p i ␾ i Q i , then it is easy to see
⫺1
that ( ␺ i⬘ ) * (⌿ ⬘ ⌿ ⬘ * ) ⫺1 ␺ i⬘ ⫽ ␣ I if and only if ␺ *
␺i
i ⌬
⫽ ␣ I.
Now, if the state ␳ i ⫽ ␾ i ␾ *
i is transmitted with prior probability p i , then the probability of correctly detecting the state
is
using
measurement
operators
⌺ i ⫽ ␮ i ␮ i*
*
*
*
*
p i Tr( ␮ i ␾ i ␾ i ␮ i )⫽Tr( ␮ i ␺ i ␺ i ␮ i ). It follows that if the
condition for optimality of Theorem 3 is met, then the probability of correctly detecting each of the states ␳ i using the
SIM is the same.
For a pure-state ensemble consisting of density operators
␳ i ⫽ 兩 ␾ i 典具 ␾ i 兩 with prior probabilities p i , the probability of
correct detection of the ith state is given by 兩 具 ␮ i 兩 ␺ i 典 兩 2 . Since
具 ␮ i 兩 ␺ i 典 ⫽ ␥ 具 ␺ i 兩 ⌬ ⫺1 兩 ␺ i 典 ⭓0 for any set of weighted vectors
兩 ␺ i 典 , 具 ␮ i 兩 ␺ i 典 is constant for all i if and only if 兩 具 ␮ i 兩 ␺ i 典 兩 2 is
constant for all i. Therefore, we may interpret the condition
in Theorem 3 for pure-state ensembles as follows. The SIM
is optimal for a set of states 兩 ␾ i 典 with prior probabilities p i
and for ␤ ⭓ ␤ min if the probability of correctly detecting each
one of the states using the SIM vectors is the same, regardless of the specific state chosen.
For pure-state ensembles with linearly independent state
vectors we have already seen that the SIM with ␥ ⫽ 冑␭ n is
equal to the EPM, and results in P E ⫽0. Furthermore, for
this choice of measurement vectors, P I ⫽ ␤ min , where ␤ min is
given by Eq. 共20兲. Since P I ⫹ P D ⫹ P E ⫽1, it follows that the
EPM, or equivalently the SIM with ␥ ⫽ 冑␭ n , maximizes P D
from all measurements that result in P I ⫽ ␤ min . We can now
use Theorem 3 to generalize this result to values of ␤ that are
larger than ␤ min . Specifically, we have the following Corollary to Theorem 3.
Corollary 4. Let 兵 ␳ i ⫽ 兩 ␾ i 典具 ␾ i 兩 ,1⭐i⭐m 其 denote a purestate ensemble with linearly independent vectors 兩 ␾ i 典 and
prior probabilities 兵 p i ,1⭐i⭐m 其 . Then the scaled-inverse
measurement maximizes P D subject to P I ⫽ ␤ for any ␤
⭓ ␤ min .
Proof. From Theorem 3 the SIM maximizes P D subject to
P I ⫽ ␤ for any ␤ ⭓ ␤ min if 具 ␺ i 兩 (⌿⌿ * ) ⫺1 兩 ␺ i 典 is independent
of i, where 兩 ␺ i 典 ⫽ 冑p i 兩 ␾ i 典 and ⌿ is the matrix of columns
兩 ␺ i 典 . Now, if the vectors 兩 ␾ i 典 are linearly independent, then
m⫽n so that ⌿ is invertible and ⌿ * (⌿⌿ * ) ⫺1 ⌿⫽I. Since
具 ␺ i 兩 (⌿⌿ * ) 兩 ␺ i 典 is the ith diagonal element of
⌿ * (⌿⌿ * ) ⫺1 ⌿, 具 ␺ i 兩 (⌿⌿ * ) ⫺1 兩 ␺ i 典 ⫽1 for all i, and the
SIM is optimal.
䊏
In the remainder of the paper, we use Theorem 3 to derive
the optimal measurement for mixed and 共not necessarily independent兲 pure-state sets with certain symmetry properties.
The symmetry properties we consider are quite general, and
include many cases of practical interest.
over a group of unitary matrices and are generated by a
single generating matrix. We first obtain a convenient characterization of the SIM for GU state sets, and show that
under a certain constraint on the generator the SIM is optimal when ␤ ⭓ ␤ min . In particular, for 共not necessarily independent兲 pure-state ensembles the SIM is optimal. We then
show that for arbitrary GU state sets and arbitrary values of
␤ , the optimal measurement is also GU, and we develop an
efficient computational method for finding the optimal generators.
Let G⫽ 兵 U i ,1⭐i⭐m 其 be a finite group of m unitary matrices U i . That is, G contains the identity matrix I, if G contains U i , then it also contains its inverse U ⫺1
i ⫽U *
i , and the
product U i U j of any two elements of G is in G 关41兴.
A state set generated by G using a single generating operator ␳ is a set S⫽ 兵 ␳ i ⫽U i ␳ U *
i ,U i 苸G其 . The group G is the
generating group of S. Such a state set has strong symmetry
properties and is called GU. For consistency with the symmetry of S, we will assume equiprobable prior probabilities
on S.
If the state set 兵 ␳ i ,1⭐i⭐m 其 is GU, then we can always
choose factors ␾ i of ␳ i such that 兵 ␾ i ⫽U i ␾ ,U i 苸G其 , where
␾ is a factor of ␳ , so that the factors ␾ i are also GU with
generator ␾ . In the remainder of this section we explicitly
assume that the factors are chosen to be GU.
V. GEOMETRICALLY UNIFORM STATE SETS
␾ * 共 ⌽⌽ * 兲 ⫺1 ␾ ⫽ ␣ I
In this section we consider geometrically uniform 共GU兲
关40兴 state sets in which the density operators ␳ i are defined
A. Optimality of the SIM for GU States
For a GU state set with equal prior probabilities 1/m and
generating group G, ⌽⌽ * commutes with each of the matrices U i 苸G 关28,11兴. Consequently, T⫽(⌽⌽ * ) ⫺1 also commutes with U i for all i, so that from Eq. 共21兲,
␮ i ⫽ ␨ T ␾ i ⫽ ␨ TU i ␾ ⫽ ␨ U i T ␾ ⫽U i ␮ ,
1⭐i⭐m, 共22兲
where ␨ ⫽ 冑m ␥ and
␮ ⫽ ␨ 共 ⌽⌽ * 兲 ⫺1 ␾ .
共23兲
It follows that the SIM factors ␮ i are also GU with generating group G and generator ␮ given by Eq. 共23兲. Therefore, to
compute the SIM factors for a GU state set all we need is to
compute the generator ␮ . The remaining measurement factors are then obtained by applying the group G to ␮ .
From Eq. 共22兲 we have that
共 1/␥ 兲 ␮ *
i ␺ i⫽
1
␥ 冑m
␮ * U i* U i ␾ ⫽
1
␥ 冑m
␮ *␾ ,
共24兲
where ␾ and ␮ are the generators of the state factors and the
SIM factors, respectively. Thus, the probability of correct
detection of each one of the states ␳ i using the SIM is the
same, regardless of the state transmitted. This then implies
from Theorem 3 that for a 共not necessarily independent兲
pure-state GU ensemble the SIM is optimal when ␤ ⭓ ␤ min .
For a mixed-state ensemble, if the generator ␾ satisfies
共25兲
for some ␣ , then from Theorem 3 the SIM is again optimal.
042309-6
PHYSICAL REVIEW A 67, 042309 共2003兲
MIXED-QUANTUM-STATE DETECTION WITH . . .
FIG. 1. Example of a GU state set.
B. Example of a GU state set
We now consider an example demonstrating the
ideas of the preceding section. Consider the state set
兵 ␳ i ⫽ 兩 ␾ i 典具 ␾ i 兩 ,1⭐i⭐4 其 with equal prior probabilities
兵 p i ⫽1/4,1⭐i⭐4 其 , where 兵 兩 ␾ i 典 ⫽U i 兩 ␾ 典 ,Ui 苸G其 . Here
G⫽ 兵 U i ,1⭐i⭐4 其 is an Abelian group of unitary matrices
U 1 ⫽I 2 ,
U 3⫽
冋
U 2⫽
⫺1
0
0
⫺1
册
冋
1
0
0
⫺1
U 4⫽
,
冋
册
,
⫺1
0
0
1
册
,
共26兲
FIG. 2. Symmetry property of the state vectors 兩 ␾ i 典 given by
Eq. 共28兲 and the scaled-inverse measurement 共SIM兲 vectors 兩 ␮ i 典
which are given by Eq. 共31兲. The state vectors and the SIM vectors
both have the same symmetry properties.
and the smallest eigenvalue of ⌿⌿ * ⫽(1/4)⌽⌽ * is ␭ n
⫽1/4. From Eq. 共20兲 it then follows that ␤ min⫽1/2, so that
the SIM is defined for values of ␤ satisfying ␤ ⭓1/2. Applying the group G to the generator 兩 ␮ 典 , the SIM vectors are
兩␾典⫽
冋 册
1 冑3
.
2 ⫺1
兩 ␾ 3典 ⫽
冋 册
1 ⫺ 冑3
,
2
1
1
2
冋冑 册
冋 冑册
3
1
兩 ␾ 4典 ⫽
,
1 ⫺ 3
.
2 ⫺1
⌽⌽ * ⫽
兩 ␮ 典 is given by
冋 册
3
0
0
1
冋册
,
冑3
⫺1
,
共30兲
⫺
1
冑3
,
兩 ␮ 4典 ⫽ ␥
⫺
1
冑3
.
共31兲
⫺1
1
Comparing Eq. 共31兲 with Eq. 共28兲 it is evident that the
SIM vectors have the same symmetries as the original state
vectors, as illustrated in Fig. 2.
The probability of correct detection using the SIM vectors, given that a conclusive result was obtained, is
1
m
m
兺 兩 具 ␮ i兩 ␾ i 典 兩 2 ⫽ 兩 具 ␮ 兩 ␾ 典 兩 2 ⫽1⫺ ␤ ,
i⫽1
1
␤⭓ .
2
共32兲
Since, as we have seen in Sec. V A, for GU pure-state sets
the SIM maximizes P D subject to P I ⫽ ␤ for any ␤ ⭓1/2, it
follows that the probability of correct detection P D , given
that a conclusive result was obtained, using any set of measurement operators 兵 ⌸ i ,0⭐i⭐m 其 with P I ⫽ ␤ satisfies
P D ⭐1⫺ ␤ ,
共29兲
1
兩␮典⫽␥
兩 ␮ 3典 ⫽ ␥
共28兲
The GU state set of Eq. 共28兲 is illustrated in Fig. 1.
From Eq. 共22兲 it follows that the SIM vectors corresponding to the state set of Fig. 1 are also GU with generator
兩 ␮ 典 ⫽2 ␥ (⌽⌽ * ) ⫺1 兩 ␾ 典 , where ⌽ is the matrix of columns
兩 ␾ i 典 and ␥ 2 ⫽(1⫺ ␤ )/2. Here ␤ ⭓ ␤ min is the desired rate of
inconclusive results. Since
,
1
共27兲
The matrix U 2 represents a reflection about the x axis, U 3
represents a rotation by ␲ , and U 4 represents a reflection
about the y axis. Applying the group G to the generator 兩 ␾ 典 ,
the GU state vectors are
兩 ␾ 1典 ⫽ 兩 ␾ 典 , 兩 ␾ 2典 ⫽
冋册
冋 册 冋 册
1
兩 ␮ 1 典 ⫽ 兩 ␮ 典 , 兩 ␮ 2 典 ⫽ ␥ 冑3
and the generating vector is
1
␤⭓ .
2
共33兲
Recall that the motivation for allowing for inconclusive
results is to be able to increase the probability of correct
detection with respect to the probability of correct detection
attainable when P I ⫽0. Since the state set of Fig. 1 is GU,
the measurement that maximizes P D subject to P I ⫽0 is the
least-squares measurement with measurement vectors 兩 ␹ i 典
042309-7
PHYSICAL REVIEW A 67, 042309 共2003兲
YONINA C. ELDAR
⫽(⌽⌽*)⫺1/2兩 ␾ i 典 关10,11兴. The vectors 兩 ␹ i 典 are also GU with
generating group G and generating vector
兩 ␹ 典 ⫽ 共 ⌽⌽ * 兲 ⫺1/2兩 ␾ 典 ⫽
冋 册
1
1
.
2 ⫺1
共34兲
m
兺
i⫽1
1
m
兺
⌸̂ i U *j ⭐U j U *j ⫽I.
共38兲
m
J 共 兵 ⌸̂ (i j) 其 兲 ⫽
m
兺 Tr共 ␳ U *i U j ⌸̂ r( j,i) U *j U i 兲
i⫽1
m
兩 具 ␹ i 兩 ␾ i 典 兩 2 ⫽ 兩 具 ␹ 兩 ␾ 典 兩 2 ⫽0.467.
共35兲
⫽
Thus, the largest probability of correct detection attainable
when P I ⫽0 is equal to 0.467.
If we allow for a probability ␤ ⫽1/2 of inconclusive results, then from Eq. 共32兲 the largest possible probability of
correct detection is equal to 1/2 which is larger than 0.467.
Thus, as we expect, by allowing for a nonzero probability of
an inconclusive result, we can increase the probability of
correct detection.
⫽
i⫽1
i⫽1
Using the fact that ␳ i ⫽U i ␳ U *
i for some generator ␳ for any
1⭐ j⭐m,
The probability of correct detection using the least-squares
measurement vectors 兩 ␹ i 典 is
P D⫽
冉兺 冊
m
⌸̂ (i j) ⫽U j
兺
k⫽1
Tr共 ␳ U k* ⌸̂ k U k 兲
m
⫽Ĵ.
冉兺
m
Tr
⫽Tr
共36兲
subject to
1
Tr
m
冉兺 冊
m
i, j⫽1
␳ i⌸ j ⫽ ␤ ,
共37兲
are ⌸̂ i and let Ĵ⫽J( 兵 ⌸̂ i 其 ). Let r( j,i) be the mapping from
I⫻I to I with I⫽ 兵 1, . . . ,m 其 , defined by r( j,i)⫽k if
U *j U i ⫽U k . Then the measurement operators ⌸̂ (i j)
U *j U i ␳ U i* U j ⌸̂ r( j,s)
冊
⭐ j⭐m are also optimal. Indeed, since ⌸̂ i ⭓0,1⭐i⭐m and
i,k⫽1
U i ␳ U i* ⌸̂ k
i,k⫽1
␳ i ⌸̂ k ,
共40兲
so that from Eq. 共37兲, P I ( 兵 ⌸̂ (i j) 其 )⫽ P I ( 兵 ⌸̂ i 其 ).
Since the measurement operators ⌸̂ (i j) are optimal for any
j, it follows immediately that the measurement operators
m
⌸̄ i are
兵 ⌸̄ i ⫽(1/m) 兺 mj⫽1 ⌸̂ (i j) ,1⭐i⭐m 其 and ⌸̄ 0 ⫽I⫺ 兺 i⫽1
also optimal. Now, for any 1⭐i⭐m,
⌸̄ i ⫽
1
m
⫽U i
m
兺
j⫽1
冉
1
m
U j ⌸̂ r( j,i) U *j ⫽
冊
m
兺
k⫽1
1
m
m
兺
k⫽1
U i U k* ⌸̂ k U k U *
i
U*
k ⌸̂ k U k U *
i ⫽U i ⌸̂U *
i ,
共41兲
m
where ⌸̂⫽(1/m) 兺 k⫽1
U*
k ⌸̂ k U k . We therefore conclude that
the optimal measurement operators can always be chosen to
be GU with generating group G.
䊏
From Proposition 5 it follows that the optimal measurement operators satisfy ⌸ i ⫽U i ⌸U *
i ,1⭐i⭐m for some generator ⌸. Thus, to find the optimal measurement operators
all we need is to find the generator ⌸. The remaining operators are obtained by applying the group G to ⌸.
Since ␳ i ⫽U i ␳ U *
i , Tr( ␳ i ⌸ i )⫽Tr( ␳ ⌸), and the problem
共2兲 reduces to the maximization problem
max Tr共 ␳ ⌸ 兲 ,
m
⫽U j ⌸̂ r( j,i) U *j ,1⭐i⭐m and ⌸̂ (0j) ⫽I⫺ 兺 i⫽1
⌸̂ (i j) for any 1
m
兺 i⫽1
⌸̂ i ⭐I, ⌸̂ (i j) ⭓0,1⭐i⭐m and
i,s⫽1
m
⫽Tr
m
P I 共 兵 ⌸ i 其 兲 ⫽1⫺
⫽Tr
m
If the generator ␾ does not satisfy Eq. 共25兲, or if ␤
⬍ ␤ min , then the SIM is no longer guaranteed to be optimal.
Nonetheless, Proposition 5 below asserts that for a GU state
set with generating group G, the optimal measurement operators that maximize P D subject to P I ⫽ ␤ for any ␤ are also
GU with generating group G. As we show, the optimal generator can be determined efficiently within any desired accuracy in polynomial time.
Proposition 5. Let S⫽ 兵 ␳ i ⫽U i ␳ U *
i ,U i 苸G其 be a geometrically uniform 共GU兲 state set with equal prior probabilities on an n-dimensional Hilbert space, generated by a finite
group G of unitary matrices, where ␳ is an arbitrary generator
and let 兵 ⌸̂ i ,0⭐i⭐m 其 denote the measurement operators that
maximize P D subject to P I ⫽ ␤ for any ␤ ⬍1. Then 兵 ⌸̂ i ,1
⭐i⭐m 其 are also GU with generating group G.
Proof. Suppose that the optimal measurement operators
that maximize
兺 Tr共 ␳ i ⌸ i 兲 ,
冊 冉兺
冉兺
冊
冉兺 冊
m
␳ i ⌸̂ s( j)
C. Optimal measurement for arbitrary GU states
i⫽1
共39兲
Finally for any 1⭐ j⭐m,
i,s⫽1
J 共 兵 ⌸ i其 兲⫽
兺 Tr共 ␳ i ⌸̂ i 兲
i⫽1
⌸苸B
共42兲
where B is the set of n⫻n Hermitian operators, subject to
the constraints
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PHYSICAL REVIEW A 67, 042309 共2003兲
MIXED-QUANTUM-STATE DETECTION WITH . . .
⌸⭓0,
m
兺 U i ⌸U i* ⭐I,
i⫽1
冉兺
m
1⫺Tr
i⫽1
冊
U i␳ U i⌸ ⫽ ␤ .
共43兲
The problem of Eqs. 共42兲 and 共43兲 is a 共convex兲 semidefinite programming problem, and therefore the optimal ⌸ can
be computed very efficiently in polynomial time within any
desired accuracy 关24,25,27兴, for example, using the LMI
toolbox on Matlab. Note that the problem of Eqs. 共42兲 and
共43兲 has n 2 real unknowns and 3 constraints, in contrast with
the original maximization problem 共2兲 subject to Eqs. 共1兲 and
共4兲 which has mn 2 real unknowns and m⫹2 constraints.
We summarize our results regarding GU state sets in the
following theorem.
Theorem 6 (GU state sets). Let S⫽ 兵 ␳ i ⫽U i ␳ U *
i ,U i 苸G其
be a geometrically uniform 共GU兲 state set with equal prior
probabilities on an n-dimensional Hilbert space, generated
by a finite group G of m unitary matrices, where ␳ ⫽ ␾␾ * is
an arbitrary generator, and let ⌽ be the matrix of columns
␾ i ⫽U i ␾ . Then the scaled inverse measurement 共SIM兲 with
P I ⫽ ␤ is given by the measurement operators ⌺ i ⫽ ␮ i ␮ *
i ,0
⭐i⭐m with
␮ i ⫽U i ␮ ,
1⭐i⭐m,
where
␮ ⫽ 冑m ␥ 共 ⌽⌽ * 兲 ⫺1 ␾ ,
m
␥ 2 ⫽(1⫺ ␤ )/n, and ⌺ 0 ⫽I⫺ 兺 i⫽1
␮ i␮ *
i . The SIM has the
following properties.
共1兲 The measurement operators ⌺ i ,1⭐i⭐m are GU with
generating group G;
共2兲 the probability of correctly detecting each of the states
␳ i using the SIM is the same;
共3兲 if ␾ * (⌽⌽ * ) ⫺1 ␾ ⫽ ␣ I for some ␣ , then the SIM
maximizes P D subject to P I ⫽ ␤ for ␤ ⭓1⫺n␭ n where ␭ n is
m
␳ i ; in particular, if ␾
the smallest eigenvalue of (1/m) 兺 i⫽1
⫽ 兩 ␾ 典 is a vector so that the state set is a pure-state ensemble, then the SIM maximizes P D subject to P I ⫽ ␤ for
any ␤ ⭓1⫺n␭ n .
For an arbitrary generator ␾ the optimal measurement
operators ⌸̂ i ,1⭐i⭐m that maximize P D subject to P I ⫽ ␤
for any ␤ are also GU with generating group G and generator
m
U i ⌸U i*
⌸ that maximizes Tr( ␳ ⌸) subject to ⌸⭓0,兺 i⫽1
m
⭐I, and Tr( 兺 i⫽1 U i ␳ U i ⌸)⫽1⫺ ␤ .
VI. COMPOUND GEOMETRICALLY UNIFORM
STATE SETS
We now consider compound geometrically uniform
共CGU兲 关28兴 state sets which consist of subsets that are GU.
As we show, the SIM operators are also CGU so that they
can be computed using a set of generators. Under a certain
FIG. 3. A compound geometrically uniform pure-state set. The
state sets S1 ⫽ 兵 兩 ␾ 11典 , 兩 ␾ 21典 其 and S2 ⫽ 兵 兩 ␾ 12典 , 兩 ␾ 22典 其 are both geometrically uniform 共GU兲 with the same generating group; both sets
are invariant under a reflection about the dashed line. However, the
combined set S⫽ 兵 兩 ␾ 11典 , 兩 ␾ 21典 , 兩 ␾ 12典 , 兩 ␾ 22典 其 is no longer GU.
condition on the generators and for ␤ ⭓ ␤ min , we show that
the optimal measurement associated with a CGU state set is
equal to the SIM. For arbitrary CGU state sets and arbitrary
values of ␤ we show that the optimal measurement operators
are CGU, and we derive an efficient computational method
for finding the optimal generators.
A CGU state set is defined as a set of density operators
* ,1⭐i⭐l,1⭐k⭐r 其 such that ␳ ik ⫽U i ␳ k U *
S⫽ 兵 ␳ ik ⫽ ␾ ik ␾ ik
i ,
where the matrices 兵 U i ,1⭐i⭐l 其 are unitary and form a
group G, and the operators 兵 ␳ k ,1⭐k⭐r 其 are the generators.
We assume equiprobable prior probabilities on S.
If the state set 兵 ␳ ik ,1⭐i⭐l,1⭐k⭐r 其 is CGU, then we can
always choose factors ␾ ik of ␳ ik such that 兵 ␾ ik ⫽U i ␾ k ,1⭐i
⭐l 其 , where ␾ k is a factor of ␳ k , so that the factors ␾ ik are
also CGU with generators 兵 ␾ k ,1⭐k⭐r 其 . In the remainder of
this section we explicitly assume that the factors are chosen
to be CGU.
A CGU state set is in general not GU. However, for every
k, the matrices 兵 ␾ ik ,1⭐i⭐l 其 and the operators 兵 ␳ ik ,1⭐i
⭐l 其 are GU with generating group G.
An example of a CGU state set is illustrated in Fig. 3. In
this example the state set is 兵 ␳ ik ⫽ 兩 ␾ ik 典具 ␾ ik 兩 ,1⭐i,k⭐2 其 ,
where 兵 兩 ␾ ik 典 ⫽U i 兩 ␾ k 典 ,U i 苸G其 , G⫽ 兵 I 2 ,U 其 with
U⫽
1
2
冋
1
冑3
冑3
⫺1
册
共44兲
,
and the generating vectors are
兩 ␾ 1典 ⫽
冑 冋 册
1
1
2 1
,
兩 ␾ 2典 ⫽
冑 冋 册
1
1
2 ⫺1
.
共45兲
The matrix U represents a reflection about the dashed line in
Fig. 3. Thus, the vector 兩 ␾ 21典 is obtained by reflecting the
generator 兩 ␾ 11典 about this line, and similarly the vector 兩 ␾ 22典
is obtained by reflecting the generator 兩 ␾ 12典 about this line.
As can be seen from the figure, the state set is not GU. In
particular, there is no isometry that transforms 兩 ␾ 11典 into
042309-9
PHYSICAL REVIEW A 67, 042309 共2003兲
YONINA C. ELDAR
兩 ␾ 12典 while leaving the set invariant. However, the sets S1
⫽ 兵 兩 ␾ 11典 , 兩 ␾ 21典 其 and S2 ⫽ 兵 兩 ␾ 12典 , 兩 ␾ 22典 其 are both GU with
generating group G.
From Eq. 共51兲 it follows that the generators ␮ k are GU with
¯ . Then
generating group Q⫽ 兵 V k ,1⭐k⭐r 其 and generator ␮
for all k,
A. Optimality of the SIM for CGU state sets
¯ k V k␾ ⫽ ␮
¯ *␾ .
␮*
k ␾ k⫽ ␮ *V *
共52兲
¯ * ␾ ⫽ ␥␾ * T ␾ ⫽ ␣ I
␮
共53兲
With ⌽ denoting the matrix of 共block兲 columns ␾ ik , it
was shown in Refs. 关11,28兴 that ⌽⌽ * , and consequently T
⫽(⌽⌽ * ) ⫺1 , commutes with each of the matrices U i 苸G.
* ,1⭐i⭐l,1⭐k⭐r
Thus, the SIM operators are ⌺ ik ⫽ ␮ ik ␮ ik
with
␮ ik ⫽ ␨ T ␾ ik ⫽ ␨ TU i ␾ k ⫽U i ␮ k ,
共46兲
where ␨ ⫽ ␥ 冑lr and
␮ k ⫽ ␨ T ␾ k ⫽ ␥ 冑lr 共 ⌽⌽ * 兲 ⫺1 ␾ k .
共47兲
Therefore, the SIM factors are also CGU with generating
group G and generators ␮ k given by Eq. 共47兲. To compute the
SIM factors all we need is to compute the generators ␮ k .
The remaining measurement factors are then obtained by applying the group G to each of the generators.
From Eq. 共46兲,
* ␾ ik ⫽ ␮ *k U *i U i ␾ k ⫽ ␮ *k ␾ k ,
␮ ik
共48兲
so that from Theorem 3 the SIM is optimal if
␮ k* ␾ k ⫽ ␥ 冑lr ␾ k* T ␾ k ⫽ ␣ I,
1⭐k⭐r
共49兲
for some constant ␣ .
B. CGU state sets with GU generators
A special class of CGU state sets is CGU state sets with
GU generators in which the generators 兵 ␳ k ⫽ ␾ k ␾ k* ,1⭐k
⭐r 其 and the factors ␾ k are themselves GU. Specifically,
兵 ␾ k ⫽V k ␾ 其 for some generator ␾ , where the matrices
兵 V k ,1⭐k⭐r 其 are unitary, and form a group Q.
Suppose that U i and V k commute up to a phase factor for
all i and k so that U i V k ⫽V k U i e j ␪ (i,k) , where ␪ (i,k) is an
arbitrary phase function that may depend on the indices i and
k. In this case we say that G and Q commute up to a phase
factor and that the corresponding state set is CGU with commuting GU generators. 共In the special case in which ␪ ⫽0 so
that U i V k ⫽V k U i for all i,k, the resulting state set is GU
关28兴兲. Then for all i,k, ⌽⌽ * commutes with U i V k 关11兴, and
the SIM factors ␮ ik are given by
¯,
␮ ik ⫽ ␨ T ␾ ik ⫽ ␨ TU i V k ␾ ⫽U i V k ␮
共50兲
If in addition,
for some ␣ , then combining Eqs. 共48兲, 共52兲, and 共53兲 with
Theorem 3 we conclude that the SIM is optimal. In particu¯ * ␾ is a scalar so that Eq.
lar, for a pure-state ensemble, ␮
共53兲 is always satisfied. Therefore, for a pure CGU state set
with commuting GU generators, the SIM maximizes P D subject to P I ⫽ ␤ for ␤ ⭓ ␤ min .
C. Optimal measurement for arbitrary CGU states
If the generators ␾ k do not satisfy Eq. 共49兲, or if ␤
⬍ ␤ min , then the SIM is no longer guaranteed to be optimal.
Nonetheless, in a manner similar to that of Sec. V C, we
show that the optimal measurement operators that maximize
P D subject to P I ⫽ ␤ are CGU with generating group G. The
corresponding generators can be computed very efficiently in
polynomial time within any desired accuracy.
Proposition 7. Let S⫽ 兵 ␳ ik ⫽U i ␳ k U i* ,1⭐i⭐l,1⭐k⭐r 其
be a compound geometrically uniform 共CGU兲 state set with
equal prior probabilities on an n-dimensional Hilbert space,
generated by a finite group G of unitary matrices and generators ␳ k and let 兵 ⌸̂ ik ,1⭐i⭐,1⭐k⭐r 其 and ⌸̂ 0 denote the measurement operators that maximize P D subject to P I ⫽ ␤ for
any ␤ ⬍1. Then 兵 ⌸̂ ik ,1⭐i⭐l,1⭐k⭐r 其 are also CGU with
generating group G.
Proof. The proof is analogous to the proof of Proposition
5 and is given in Appendix C.
䊏
From Proposition 7 it follows that the optimal measurement operators satisfy ⌸ ik ⫽U i ⌸ k U *
i ,1⭐i⭐l,1⭐k⭐r for
some generators ⌸ k ,1⭐k⭐r. Thus, to find the optimal measurement operators all we need is to find the optimal generators ⌸ k ,1⭐k⭐r. The remaining operators are obtained by
applying the group G to each of the generators.
Since ␳ ik ⫽U i ␳ k U i* , Tr( ␳ ik ⌸ ik )⫽Tr( ␳ k ⌸ k ), and the
problem 共2兲 reduces to the maximization problem
r
⌸ k 苸B k⫽1
Tr共 ␳ k ⌸ k 兲 ,
共54兲
subject to the constraints
¯ ⫽ ␨ T ␾ and ␨ ⫽ ␥ 冑lr. Thus even though the state set
where ␮
is not in general GU, the SIM factors can be computed using
a single generator.
Alternatively, we can express ␮ ik as ␮ ik ⫽U i ␮ k , where
the generators ␮ k are given by
¯.
␮ k ⫽V k ␮
兺
max
共51兲
042309-10
⌸ k ⭓0,
l
r
兺兺
i⫽1 k⫽1
1
1⫺ Tr
r
1⭐k⭐r,
U i⌸ kU *
i ⭐I,
冉兺 兺
l
r
i⫽1 k,l⫽1
冊
U i␳ kU i⌸ l ⫽ ␤ .
共55兲
MIXED-QUANTUM-STATE DETECTION WITH . . .
PHYSICAL REVIEW A 67, 042309 共2003兲
The problem of Eqs. 共54兲 and 共55兲 is a 共convex兲 semidefinite programming problem, and therefore the optimal generators ⌸ k can be computed very efficiently in polynomial
time within any desired accuracy 关24,25,27兴, for example,
using the LMI toolbox on Matlab. Note that the problem of
Eqs. 共54兲 and 共55兲 has rn 2 real unknowns and r⫹2 constraints, in contrast with the original maximization 共2兲 subject to Eqs. 共1兲 and 共4兲 which has lrn 2 real unknowns and
lr⫹2 constraints.
We summarize our results regarding CGU state sets in the
following theorem.
Theorem 8 (CGU state sets). Let S⫽ 兵 ␳ ik ⫽U i ␳ k U i* ,1
⭐i⭐l,1⭐k⭐r 其 be a compound geometrically uniform
共CGU兲 state set with equal prior probabilities on an
n-dimensional Hilbert space generated by a finite group G of
unitary matrices and generators 兵 ␳ k ⫽ ␾ k ␾ k* ,1⭐k⭐r 其 , and
let ⌽ be the matrix of columns ␾ ik ⫽U i ␾ k . Then the scaled
inverse measurement 共SIM兲 with P I ⫽ ␤ is given by the mea* ,1⭐i⭐l,1⭐k⭐r and ⌺ 0 ⫽I
surement operators ⌺ ik ⫽ ␮ ik ␮ ik
*
⫺ 兺 i,k ␮ ik ␮ ik with
VII. CONCLUSION
␮ ik ⫽U i ␮ k ,
In this paper, we considered the optimal measurement operators that maximize the probability of correct detection
given a fixed probability ␤ of an inconclusive result, when
distinguishing between a collection of mixed quantum states.
We first derived a set of necessary and sufficient conditions
for optimality by exploiting principles of duality theory in
vector space optimization. Using these conditions, we derived a general condition under which the SIM is optimal.
We then considered state sets with a broad class of symmetry
properties for which the SIM is optimal. Specifically, we
showed that for GU state sets and for CGU state sets with
generators that satisfy certain constraints and for values of ␤
exceeding a threshold, the SIM is optimal. We also showed
that for arbitrary GU and CGU state sets and for arbitrary
values of ␤ , the optimal measurement operators have the
same symmetries as the original state sets. Therefore, to
compute the optimal measurement operators, we need only
to compute the corresponding generators. As we showed, the
generators can be computed very efficiently in polynomial
time within any desired accuracy by solving a semidefinite
programming problem.
where
ACKNOWLEDGMENTS
␮ k ⫽ 冑rl ␥ 共 ⌽⌽ * 兲 ⫺1 ␾ k ,
and ␥ 2 ⫽(1⫺ ␤ )/n. The SIM has the following properties.
共1兲 The measurement operators ⌺ ik ,1⭐i⭐l,1⭐k⭐r are
CGU with generating group G;
共2兲 the probability of correctly detecting each of the states
␾ ik for fixed k using the SIM is the same;
⫺1
␾ k ⫽ ␣ I for some ␣ and for 1⭐k⭐r,
共3兲 if ␾ *
k (⌽⌽ * )
then the SIM maximizes P D subject to P I ⫽ ␤ with ␤ ⭓1
⫺n␭ n where ␭ n is the smallest eigenvalue of (1/lr) 兺 i,k ␳ ik .
If in addition the generators 兵 ␾ k ⫽V k ␾ ,1⭐k⭐r 其 are geometrically uniform with U i V k ⫽V k U i e j ␪ (i,k) for all i,k, then
¯ where ␮
¯ ⫽ 冑rl ␥ (⌽⌽ * ) ⫺1 ␾ so that the
共1兲 ␮ ik ⫽U i V k ␮
SIM operators are CGU with geometrically uniform generators;
共2兲 the probability of correctly detecting each of the states
␾ ik using the SIM is the same;
共3兲 if ␾ * (⌽⌽ * ) ⫺1 ␾ ⫽ ␣ I for some ␣ , then the SIM
maximizes P D subject to P I ⫽ ␤ with ␤ ⭓1⫺n␭ n . In particular, if ␾ ⫽ 兩 ␾ 典 is a vector so that the state set is a purestate ensemble, then the SIM maximizes P D subject to P I
⫽ ␤ with ␤ ⭓1⫺n␭ n .
For arbitrary CGU state sets the optimal measurement operators ⌸̂ ik ,1⭐i⭐l,1⭐k⭐r that maximize P D subject to
P I ⫽ ␤ for any ␤ are CGU with generating group G and
r
Tr( ␳ k ⌸ k ) subject to
generators ⌸ k that maximize 兺 k⫽1
⌸ k ⭓0,1⭐k⭐r,
兺 i,k U i ⌸ k U i* ⭐I,
and
l
r
兺 k,l⫽1
U i ␳ k U i ⌸ l 兲 ⫽r 共 1⫺ ␤ 兲 .
Tr共 兺 i⫽1
The author wishes to thank Professor A. Megretski and
Professor G. C. Verghese for many valuable discussions and
Professor J. Fiurášek for drawing her attention to Ref. 关23兴.
The author is financially supported by the Taub foundation.
APPENDIX A: NECESSARY AND SUFFICIENT
CONDITIONS FOR OPTIMALITY
Denote by ⌳ the set of all ordered sets
m
,⌸ i 苸B
⌸⫽ 兵 ⌸ i 其 i⫽0
satisfying Eq. 共1兲 and 共4兲 with ␤ ⬍1, and define J(⌸)
m
p i Tr( ␳ i ⌸ i ). Then our problem is
⫽ 兺 i⫽1
max J 共 ⌸ 兲 .
⌸苸⌳
共A1兲
We refer to this problem as the primal problem, and to any
⌸苸⌳ as a primal feasible point. The optimal value of J(⌸)
is denoted by Ĵ.
To derive necessary and sufficient conditions for optimality, we now formulate a dual problem whose optimal value
serves as a certificate for Ĵ. As described in Ref. 关5兴, a general method for deriving a dual problem is to invoke the
separating hyperplane theorem 关42兴, which states that two
disjoint convex sets 关45兴 can always be separated by a hyperplane. We will take one convex set to be the point 0, and
then carefully construct another convex set that does not contain 0, and that captures the equality constraints in the primal
problem and the fact that for any primal feasible point, the
value of the primal function is no larger than the optimal
value. The dual variables will then emerge from the parameters of the separating hyperplane.
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YONINA C. ELDAR
In our problem we have two equality constraints,
m
兺 i⫽0
⌸ i ⫽I and Tr(⌬⌸ 0 )⫽ ␤ and we know that Ĵ⭓J(⌸).
Our constructed convex set will accordingly consist of mam
⌸ i , where ⌸ i 苸B and ⌸ i ⭓0,
trices of the form ⫺I⫹ 兺 i⫽0
scalars of the form ␤ ⫺Tr(⌬⌸ 0 ), and scalars of the form r
⫺J(⌸), where r⬎Ĵ. We thus consider the real vector space
where T(X, ␦ )⫽Tr(X)⫺ ␦ ␤ , Eq. 共A6兲 implies that X̂⭓p i ␳ i
for 1⭐i⭐m, and Eq. 共A7兲 implies that X̂⭓ ␦ˆ ⌬.
Let ⌫ be the set of X苸B, ␦ 苸R satisfying X⭓p i ␳ i ,1⭐i
⭐m and X⭓ ␦ ⌬. Then for any X, ␦ 苸⌫, ⌸苸⌳, we have
L⫽B⫻R⫻R⫽ 兵 共 S,x,y 兲 : S苸B, x,y苸R其 ,
共A2兲
We then define the subset ⍀ of L as points of the form
冉
m
m
⍀⫽ ⫺I⫹
兺 ⌸ i , ␤ ⫺Tr共 ⌬⌸ 0 兲 ,r⫺ i⫽1
兺 p i Tr共 ⌸ i ␳ i 兲
i⫽0
冊
冠冉
冉
兺 ⌸i
i⫽0
冊冡
⫹b„␤ ⫺Tr共 ⌬⌸ 0 兲 …
冊
m
⫹a r⫺
兺
i⫽1
p i Tr共 ⌸ i ␳ i 兲 ⭓0
1⭐i⭐m.
共A5兲
共A6兲
With r⫽Ĵ⫹1, ⌸ j ⫽0 for j⫽0, ⌸ 0 ⫽t 兩 x 典具 x 兩 , where 兩 x 典
苸Cn is fixed and t→⫹⬁, 共A4兲 yields 具 x 兩 Z⫺b⌬ 兩 x 典 ⭓0,
which implies
Z⭓b⌬.
X苸B, ␦ 苸R
X⭓ p i ␳ i ,
1⭐i⭐m,
X⭓ ␦ ⌬.
共A11兲
Furthermore, we have shown that there exists an optimal
X̂, ␦ˆ 苸⌫ and an optimal value T̂⫽T(X̂, ␦ˆ ) such that T̂⫽Ĵ.
共A7兲
共 X̂⫺ p i ␳ i 兲 ⌸̂ i ⫽0, 1⭐i⭐m,
共 X̂⫺ ␦ˆ ⌬ 兲 ⌸̂ 0 ⫽0.
共A12兲
Once we find the optimal X̂ and ␦ˆ that minimize the dual
problem Eq. 共A10兲, the constraints Eq. 共A12兲 are necessary
and sufficient conditions on the optimal measurement operators ⌸̂ i . We have already seen that these conditions are necessary. To show that they are sufficient, we note that if a set
of feasible measurement operators ⌸ i satisfies 共A12兲, then
m
兺 i⫽1
Tr„⌸ i (X̂⫺ p i ␳ i )…⫽0 and Tr„(X̂⫺ ␦ˆ ⌬)⌸̂ 0 …⫽0 so that
from 共A9兲, J(⌸)⫽T(X̂, ␦ˆ )⫽Ĵ.
APPENDIX B: PROOF OF THEOREM 3
In this appendix we prove Theorem 3. Specifically, we
show that for a set of states ␳ i ⫽ ␾ i ␾ *
i with prior probabilities p i ,
if (1/␥ ) ␮ i* ␺ i ⫽ ␣ I,1⭐i⭐m,
where ␮ i
⫽ ␥ (⌿⌿ * ) ⫺1 ␺ i ⫽ ␥ ⌬ ⫺1 ␺ i are the SIM factors and ␺ i
⫽ 冑p i ␾ i , then there exists an Hermitian X and a constant ␦
such that
With ⌸ i ⫽0,0⭐i⭐m, r→⫹⬁, Eq. 共A4兲 implies a⭓0. If a
⫽0, then Eq. 共A5兲 yields Tr(Z)⭐b ␤ ⬍b and 共A7兲 yields
Tr(Z)⭓b. Therefore we conclude that a⬎0, and define X̂
⫽Z/a, ␦ˆ ⫽b/a. Then Eq. 共A5兲 implies that
T 共 X̂, ␦ˆ 兲 ⭐Ĵ,
共A10兲
subject to
共A4兲
Similarly, Eq. 共A4兲 with r⫽Ĵ⫹1, ⌸ j ⫽0 for j⫽i, ⌸ i
⫽t 兩 x 典具 x 兩 for one value 1⭐i⭐m, where 兩 x 典 苸Cn is fixed and
t→⫹⬁ yields 具 x 兩 Z⫺ap i ␳ i 兩 x 典 ⭓0. Since 兩 x 典 and i are arbitrary, this implies
Z⭓ap i ␳ i ,
共A9兲
Let ⌸̂ i denote the optimal measurement operators. Then
combining Eq. 共A9兲 with T̂⫽Ĵ, we conclude that
for all ⌸ i 苸B and r苸R such that ⌸ i ⭓0, r⬎Ĵ.
As we now show, the hyperplane parameters (Z,a,b)
have to satisfy certain constraints, which lead to the formulation of the dual problem. Specifically, Eq. 共A4兲 with ⌸ i
⫽0, r→Ĵ implies
aĴ⭓Tr共 Z 兲 ⫺b ␤ .
冊
min 兵 Tr共 X 兲 ⫺ ␦ ␤ 其 ,
where ⌸ i 苸B,⌸ i ⭓0,r苸R and r⬎Ĵ.
It is easily verified that ⍀ is convex, and 0苸⍀. Therefore, by the separating hyperplane theorem, there exists a
nonzero vector (Z,a,b)苸L such that 具 (Z,a,b),(Q,c,d) 典
⭓0 for all (Q,c,d)苸⍀, i.e.,
Tr Z ⫺I⫹
⌸ i 共 X⫺ p i ␳ i 兲
Since X̂苸⌫, from Eqs. 共A8兲 and 共A9兲 we conclude that
T(X̂, ␦ˆ )⫽Ĵ.
Thus we have proven that the dual problem associated
with Eq. 共A1兲 is
,
共A3兲
m
i⫽1
⫹Tr„⌸ 0 共 X⫺ ␦ ⌬ 兲 …⭓0.
where R denotes the reals, with inner product defined by
具 共 W,z,t 兲 , 共 S,x,y 兲 典 ⫽Tr共 WS 兲 ⫹zx⫹ty.
冉兺
m
T 共 X, ␦ 兲 ⫺J 共 ⌸ 兲 ⫽Tr
共A8兲
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X⭓ ␺ i ␺ *
i ,
1⭐i⭐m,
X⭓ ␦ ⌬,
共 X⫺ ␺ i ␺ i* 兲 ␮ i ␮ i* ⫽0,
共B1兲
共B2兲
1⭐i⭐m,
共 X⫺ ␦ ⌬ 兲共 I⫺ ␥ 2 ⌬ ⫺1 兲 ⫽0.
共B3兲
共B4兲
PHYSICAL REVIEW A 67, 042309 共2003兲
MIXED-QUANTUM-STATE DETECTION WITH . . .
Let X⫽ ␣ ⌬ and ␦ ⫽ ␣ . Then Eqs. 共B2兲 and 共B4兲 are im␣ I⫽ ␺ i* ⌬ ⫺1 ␺ i
mediately
satisfied.
Next,
since
⫺1/2
⫺1/2
⫽ ␺ i* ⌬
⌬
␺ i , it follows that
l
( j)
J 共 兵 ⌸̂ ik
其 兲⫽
兺兺
i⫽1 k⫽1
l
␣ I⭓⌬ ⫺1/2␺ i ␺ i* ⌬ ⫺1/2.
共B5兲
⫽
␣ ⌬⭓ ␺ i ␺ i* ,
兺兺
l
⫽
共B6兲
which verifies that the conditions 共B1兲 are satisfied.
Finally,
⫺1
␺ i ⫺ ␣␥␺ i ⫽0,
共 X⫺ ␺ i ␺ *
i 兲 ␮ i ⫽ ␣␥ ⌬⌬
兺兺
冉兺
l
共B7兲
Tr
r
兺
i,s⫽1 k,t⫽1
兺 兺 Tr共 ␳ ik ⌸ ik 兲 ,
i⫽1 k⫽1
兺
i, j⫽1 k,s⫽1
共C1兲
冊
␳ ik ⌸ js ⫽ ␤ ,
r
兺兺
i⫽1 k⫽1
冉兺 兺 冊
l
⌸̂ (ikj) ⫽U j
r
i⫽1 k⫽1
共C4兲
⫽Tr
⌸̂ ik U *j ⭐U j U *j ⫽I. 共C3兲
Using the fact that ␳ ik ⫽U i ␳ k U *
i for some generators ␳ k ,
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r
兺
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兵 ⌸̄ ik ⫽(1/l) 兺 lj⫽1 ⌸̂ (ikj) ,1⭐i⭐l,1⭐k⭐r 其 and ⌸̄ 0 ⫽I⫺ 兺 i,k ⌸̄ ik
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共C2兲
are ⌸̂ ik , and let Ĵ⫽J( 兵 ⌸̂ ik 其 ). Let r( j,i) be the mapping
from I⫻I to I with I⫽ 兵 1, . . . ,l 其 , defined by r( j,i)⫽s if
U *j U i ⫽U s . Then the measurement operators ⌸̂ (ikj)
⫽U j ⌸̂ r( j,i)k U *j for any 1⭐ j⭐l are also optimal. Indeed,
l
r
兺 k⫽1
⌸̂ ik ⭐I, ⌸̂ (ikj)
since ⌸̂ ik ⭓0,1⭐i⭐l,1⭐k⭐r and 兺 i⫽1
⭓0,1⭐i⭐l,1⭐k⭐r and
l
⫽Tr
⫽Tr
subject to
冉兺
冊 冉兺
冉兺
冉兺
l
r
r
Tr共 ␳ ik ⌸̂ ik 兲 ⫽Ĵ.
l
( j)
␳ ik ⌸̂ st
Suppose that the optimal measurement operators that
maximize
l
r
i⫽1 k⫽1
APPENDIX C: PROOF OF PROPOSITION 7
1
P I 共 兵 ⌸ ik 其 兲 ⫽1⫺ Tr
lr
Tr共 ␳ k U s* ⌸̂ sk U s 兲
Finally,
so that the conditions 共B3兲 are also satisfied.
J 共 兵 ⌸ ik 其 兲 ⫽
Tr共 ␳ k U i* U j ⌸̂ r( j,i)k U *j U i 兲
r
s⫽1 k⫽1
Multiplying both sides of Eq. 共B5兲 by ⌬ 1/2 we have
l
r
⌸̄ ik ⫽
1
l
l
1
l
兺 U j ⌸̂ r( j,i)k U *j ⫽ l s⫽1
兺 U i U s* ⌸̂ sk U s U *i
j⫽1
⫽U i
冉
1
l
l
兺
s⫽1
冊
U s* ⌸̂ sk U s U *
i ⫽U i ⌸̂ k U i* ,
共C6兲
l
U s* ⌸̂ sk U s .
where ⌸̂ k ⫽(1/l) 兺 s⫽1
We therefore conclude that the optimal measurement operators can always be chosen to be CGU with the same generating group G as the original state set. Thus, to find the
optimal measurement operators all we need is to find the
optimal generators 兵 ⌸̂ k ,1⭐k⭐r 其 . The remaining operators
are obtained by applying the group G to each of the generators.
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YONINA C. ELDAR
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关43兴 Otherwise we can transform the problem to a problem equivalent to the one considered in this paper by reformulating the
problem on the subspace spanned by the eigenvectors of
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grows no faster than a polynomial of the problem size. In
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