Optimal quantum detectors for unambiguous detection of mixed states

Transcription

1 PHYSICAL REVIEW A 69, (004) Optial quantu detectors for unabiguous detection of ixed states Yonina C. Eldar* Departent of Electrical Engineering, Technion Israel Institute of Technology, Haifa 3000, Israel Mihailo Stojnic and Babak Hassibi Departent of Electrical Engineering, California Institute of Technology, Pasadena, California 9115, USA (Received 7 Deceber 003; published 1 June 004) We consider the proble of designing an optial quantu detector that distinguishes unabiguously between a collection of ixed quantu states. Using arguents of duality in vector space optiization, we derive necessary and sufficient conditions for an optial easureent that iniizes the probability of an inconclusive result. We show that the previous optial easureents that were derived for certain special cases satisfy these optiality conditions. We then consider state sets with strong syetry properties, and show that the optial easureent operators for distinguishing between these states share the sae syetries, and can be coputed very efficiently by solving a reduced size seidefinite progra. DOI: /PhysRevA PACS nuber(s): Hk I. INTRODUCTION The proble of detecting inforation stored in the state of a quantu syste is a fundaental proble in quantu inforation theory. Several approaches have eerged to distinguishing between a collection of nonorthogonal quantu states. In one approach, a easureent is designed to iniize the probability of a detection error [1 10]. A ore recent approach, referred to as unabiguous detection [11 19], is to design a easureent that with a certain probability returns an inconclusive result, but such that if the easureent returns an answer, then the answer is correct with probability 1. An interesting alternative approach for distinguishing between a collection of quantu states, which is a cobination of the previous two approaches, is to allow for a certain probability of an inconclusive result, and then axiize the probability of correct detection [19 1]. We consider a quantu state enseble consisting of density operators i,1i on an n-diensional coplex Hilbert space H, with prior probabilities p i 0,1i. A pure-state enseble is one in which each density operator i is a rank-one projector i i, where the vectors i, though evidently noralized to unit length, are not necessarily orthogonal. Our proble is to design a quantu detector to distinguish unabiguously between the states i. Chefles [16] showed that a necessary and sufficient condition for the existence of unabiguous easureents for distinguishing between a collection of pure quantu states is that the states are linearly independent. Necessary and sufficient conditions on the optial easureent iniizing the probability of an inconclusive result for pure states were derived in Ref. [18]. The optial easureent when distinguishing between a broad class of syetric pure-state sets was also considered in Ref. [18]. *Electronic address: yonina@ee.technion.ac.il Electronic address: ihailo@systes.caltech.edu Electronic address: hassibi@systes.caltech.edu The proble of unabiguous detection between ixed state ensebles has received considerably less attention. Rudolph, Speakkens, and Turner [] showed that unabiguous detection between ixed quantu states is possible as long as one of the density operators in the enseble has a nonzero overlap with the intersection of the kernels of the other density operators. They then consider the proble of unabiguous detection between two ixed quantu states, and derive upper and lower bounds on the probability of an inconclusive result. They also develop a closed for solution for the optial easureent in the case in which both states have kernels of diension 1. In Ref. [3], the authors consider the proble of unabiguous discriination between two general density atrices. In this paper we develop a general fraework for unabiguous state discriination between a collection of ixed quantu states, which can be applied to any nuber of states with arbitrary prior probabilities. For our easureent we consider general positive operator-valued easures [,4], consisting of +1 easureent operators. We derive a set of necessary and sufficient conditions for an optial easureent that iniizes the probability of an inconclusive result, by exploiting principles of duality theory in vector space optiization. We then show that the previous optial easureents that were derived for certain special cases satisfy these optiality conditions. Next, we consider geoetrically unifor (GU) and copound GU (CGU) state sets [7,8,5], which are state sets with strong syetry properties. We show that the optial easureent operators for unabiguous discriination between such state sets are also GU and CGU, respectively, with generators that can be coputed very efficiently by solving a reduced size seidefinite progra. The paper is organized as follows. In Sec. II, we provide a stateent of our proble. In Sec. III we develop the necessary and sufficient conditions for optiality using Lagrange duality theory. Soe special cases are considered in Sec. IV. In Sec. V we consider the proble of distinguishing between a collection of states with a broad class of syetry properties /004/69(6)/06318(9)/$ The Aerican Physical Society

2 ELDAR, STOJNIC, AND HASSIBI PHYSICAL REVIEW A 69, (004) II. PROBLEM FORMULATION Assue that a quantu channel is prepared in a quantu state drawn fro a collection of ixed states, represented by density operators i,1i on an n-diensional coplex Hilbert space H. We assue without loss of generality that the eigenvectors of i,1i, collectively span [33] H. To detect the state of the syste a easureent is constructed coprising +1 easureent operators i,0i that satisfy i = I, i=0 i 0,0 i. The easureent operators are constructed so that either the state is correctly detected, or the easureent returns an inconclusive result. Thus, each of the operators i,1i correspond to detection of the corresponding states i,1i, and 0 corresponds to an inconclusive result. Given that the state of the syste is j, the probability of obtaining outcoe i is Tr j i. Therefore, to ensure that each state is either correctly detected or an inconclusive result is obtained, we ust have Tr j i = i ij,1 i, j, for soe 0 i 1. Since fro Eq. (1), 0 =I i, Eq. () iplies that Tr i 0 =1 i, so that given that the state of the syste is i, the state is correctly detected with probability i, and an inconclusive result is returned with probability 1 i. It was shown in Ref. [16] that for pure-state ensebles consisting of rank-one density operators i = i i, Eq. () can be satisfied if and only if the vectors i are linearly independent. For ixed states, it was shown in Ref. [] that Eq. () can be satisfied if and only if one of the density operators i has a nonzero overlap with the intersection of the kernels of the other density operators. Specifically, denote by K i the null space of i and let S i = j=1,ji K j denote the intersection of K j,1 j, ji. Then to satisfy Eq. () the eigenvectors of i ust be contained in S i and ust not be entirely contained in K i. This iplies that K i ust not be entirely contained in S i. Soe exaples of ixed states for which unabiguous detection is possible are given in Ref. []. If the state i is prepared with prior probability p i, then the total probability of correctly detecting the state is P D = p i Tr i i. Our proble therefore is to choose the easureent operators i,0i to axiize P D, subject to the constraints (1) and Tr j i =0, 1 i, j, i j. 5 To satisfy Eq. (5), i ust lie in S i defined by Eq. (3), so that i = P i i P i, 1 i, 6 where P i is the orthogonal projection onto S i. Denoting by i an nr atrix whose coluns for an arbitrary orthonoral basis for S i where r=dis i, we can express P i as P i = i i *. Fro Eqs. (6) and (1) we then have that i = i i i *, 1 i, 7 where i = i * i i is an rr atrix satisfying i i * i I, 8 i 0,1i. Therefore, our proble reduces to axiizing P D = p i Tr i i i * i, 9 subject to Eq. (8). To show that the proble of Eqs. (9) and (8) does not depend on the choice of orthonoral basis i, we note that any orthonoral basis for S i can be expressed as the coluns of i, where i = i U i for soe rr unitary atrix U i. Substituting i instead of i in Eqs. (9) and (8), our proble becoes that of axiizing P D = where i =U i i U i *, subject to p i Tr i i i * i = p i Tr i i i * i, i i * i = i i * i I, i 0,1i. Since i 0 if and only if i 0, the proble of Eqs. (10) and (11) is equivalent to that of Eqs. (9) and (8). Equipped with the standard operations of addition and ultiplication by real nubers, the space B of all Heritian nn atrices is an n -diensional real vector space. As noted in Ref. [], by choosing an appropriate basis for B, the proble of axiizing P D subject to Eq. (8) can be put in the for of a standard seidefinite prograing proble, which is a convex optiization proble; for a detailed treatent of seidefinite prograing probles see, e.g., Refs. [6 9]. By exploiting the any well-known algoriths for solving seidefinite progras [9], e.g., interior point ethods [6,8,34], the optial easureent can be coputed very efficiently in polynoial tie within any desired accuracy. Using eleents of duality theory in vector space optiization, in the following section we derive necessary and sufficient conditions on the easureent operators i = i i i * to axiize P D of Eq. (9) subject to Eq. (8)

3 OPTIMAL QUANTUM DETECTORS FOR UNAMBIGUOUS PHYSICAL REVIEW A 69, (004) III. CONDITIONS FOR OPTIMALITY A. Dual-proble forulation To derive necessary and sufficient conditions for optiality on the atrices i we first derive a dual proble, using Lagrange duality theory [30]. Denote by the set of all ordered sets = i = i * i satisfying Eq. (8) and define J = p i Tr i i i * i. Then our proble is axj. 1 We refer to this proble as the prial proble, and to any as a prial feasible point. The optial value of J is denoted by Ĵ. To develop the dual proble associated with Eq. (1) we first copute the Lagrange dual function, which is given by gz i 0 = in i 0 =in I p i Tr i i i * 1 +TrZ i i * i Tr i i * Z p i i i TrZ, 13 where Z0 is the Lagrange dual variable. Since i 0, 1i, we have that Tr i X0 for any X0. If X is not positive seidefinite, then we can always choose i such that Tr i X is unbounded below. Therefore, where = gz TrZ, A i 0,1 i,z 0, otherwise, 14 A i = i * Z p i i i, 1 i. 15 It follows that the dual proble associated with Eq. (1) is subject to intrz z i * Z p i i i 0, 1 i, Z Denoting by the set of all Heritian operators Z such that 1 * Z p i i i 0,1i, and Z0, and defining TZ =TrZ, the dual proble can be written as intz. Z 18 We refer to any Z as a dual feasible point. The optial value of TZ is denoted by Tˆ. B. Optiality conditions We can iediately verify that both the prial and the dual proble are strictly feasible. Therefore, their optial values are attainable and the duality gap is zero [9], i.e., Ĵ = Tˆ. In addition, for any = i = i i * i, and Z, 0, TZ J =Tr i i * i Z p i i + 0 Z 19 0 where 0 =I i i * 1 0. Note that Eq. (0) can be used to develop an upper bound on the optial probability of correct detection Ĵ. Indeed, since for any Z,TZ J, we have that ĴTZ for any dual feasible Z. Now, let ˆ i= i ˆ i * i,1i and ˆ 0=I ˆ i denote the optial easureent operators that axiize Eq. (9) subject to Eq. (8), and let Ẑ denote the optial Z that iniizes Eq. (16) subject to Eq. (17). Fro Eqs. (19) and (0) we conclude that Tr ˆ i * i Ẑ p i i i + ˆ 0Ẑ =0. 1 Since ˆ i0,ẑ0, and i * Ẑ p i i i 0, 1i, Eq. (1) is satisfied if and only if Ẑˆ 0 =0, i * Ẑ p i i i ˆ i =0, 1 i. 3 Once we find the optial Ẑ that iniizes the dual proble (16), the constraints () and (3) are necessary and sufficient conditions on the optial easureent operators ˆ i. We have already seen that these conditions are necessary. To show that they are sufficient, we note that if a set of feasible easureent operators ˆ i satisfies Eqs. () and (3), then Tr ˆ i * i Ẑ p i i i +ˆ 0Ẑ=0 so that fro Eq. (0), Jˆ =TẐ=Ĵ. We suarize our results in the following theore. Theore 1. Let i, 1i denote a set of density operators with prior probabilities p i 0,1i, and let i,1i denote a set of atrices such that the coluns of i for on orthonoral basis for S i = j=1,ji K j, where K i the null space of i. Let denote the set of all ordered sets of Heritian easureent operators = i i=0 that satisfy i 0, i=0 i =I, and Tr j i =0, 1i, i j and let denote the set of Heritian atrices Z such that Z0, * 1 Z p i i i,1i. Consider the proble ax J and the dual proble in Z TZ, where J= p i Tr i i and TZ=TrZ. Then we have the following. (1) For any Z and, TZJ

4 ELDAR, STOJNIC, AND HASSIBI PHYSICAL REVIEW A 69, (004) () There is an optial, denoted ˆ, such that Ĵ =Jˆ J for any. (3) There is an optial Z, denoted Ẑ and such that Tˆ =TẐTZ for any Z. (4) Tˆ =Ĵ. (5) Necessary and sufficient conditions on the optial easureent operators ˆ i are that there exists a Z such that Zˆ 0 =0, 4 i * Z p i i ˆ i =0, 1 i, 5 where ˆ i= i ˆ i *,1i, and ˆ i0. (6) Given Ẑ, necessary and sufficient conditions on the optial easureent operators ˆ i are Ẑˆ 0 =0, 6 * i Ẑ p i i i ˆ i =0, 1 i. 7 Although the necessary and sufficient conditions of Theore 1 are hard to solve, they can be used to verify a solution and to gain soe insight into the optial easureent operators. In the following section we show that the previous optial easureents that were derived in the literature for certain special cases satisfy these optiality conditions. IV. SPECIAL CASES We now consider two special cases that were addressed in Ref. [], for which a closed for solution exists. In Sec. IV A we consider the case in which the spaces S i defined by Eq. (3) are orthogonal, and in Sec. IV B we consider the proble of distinguishing unabiguously between two density operators with dis i =1,1i. A. Orthogonal null spaces S i The first case we consider is the case in which the null spaces S i are orthogonal, so that P i P j = ij, 1 i, j, 8 where P i is an orthogonal projection onto S i. It was shown in Ref. [] that in this case the optial easureent operators are ˆ i = P i = i * i, 1 i. 9 In Appendix A we show that the optial solution of the dual proble can be expressed as Ẑ = p i P i i P i. 30 It can easily be shown that Ẑ and ˆ i of Eqs. (30) and (9) satisfy the optiality conditions of Theore 1. B. Null spaces of diension 1 We now consider the case of distinguishing between two density operators 1 and, where S 1 and S both have diension equal to 1. In this case, as we show in Appendix B, the optial dual solution is Ẑ = d 1P 1, d d 1 f 0 d P, d 1 d f 0 d + s + s *, otherwise, 31 where P i is an orthogonal projection onto S i, is a unit nor vector in the span of 1 and such that * =0, and d i = p i i * p i i, 1 i, s = f* e * d i d f 1, f = * 1, e = * 1. 3 The optial easureent operators for this case were developed in Ref. [], and can be written as where 1 = P 1,ˆ =0, d d 1 ˆ i =ˆ f 0 ˆ 1 =0,ˆ = P, d 1 d f 0 ˆ 1 = 1 P 1,ˆ = P, 1 d f d 1 1 = 1 f, otherwise, 33 1 d 1f d = 1 f. 34 We now show that Ẑ and ˆ of Eqs. (31) and (33) satisfy the optiality conditions of Theore 1. To this end we note that fro Eq. (33), 1 =1,ˆ =0, d d 1 ˆ i =ˆ f 0 ˆ 1 =0,ˆ =1, d 1 d f 0 ˆ 1 = 1,ˆ =, otherwise. Fro Eqs. (31) (35) we have that if d d 1 f 0, then 1 * Ẑ p ˆ 1 = d 1 1 * p =0, * Ẑ p ˆ =0,

5 OPTIMAL QUANTUM DETECTORS FOR UNAMBIGUOUS PHYSICAL REVIEW A 69, (004) Ẑˆ 0 = ẐI ˆ 1 = d 1 1 * 1 d 1 1 * 1 =0. Siilarly, if d 1 d f 0, then * 1 Ẑ p ˆ 1 =0, * Ẑ p ˆ = d * p =0, 36 Under a variety of different optiality criteria the optial easureent for distinguishing between GU and CGU state sets was shown to be GU and CGU, respectively [7,8,18,19]. In particular, it was shown in Ref. [18] that the optial easureent for unabiguous detection between linearly independent GU and CGU pure states is GU and CGU, respectively. We now generalize this result to unabiguous detection of ixed GU and CGU state sets. Ẑˆ 0 = ẐI ˆ = d * d * =0. 37 Finally, if neither of the conditions d 1 d f 0, d d 1 f 0 hold, then 1 d f 1 * Ẑ p ˆ 1=d f * + e * sf * + e * s * d 1 and =d f d 1 d f d 1 1 f 1 1 df d d 1 1 f =0, 38 1 d 1f * Ẑ p ˆ = * d Ẑ d 1 f =0, 39 Ẑˆ 0 = Ẑ Ẑˆ 1 Ẑˆ = Ẑ ˆ 1Ẑ 1 * 1 ˆ Ẑ *. 40 To show that Ẑˆ 0=0, we note that Ẑ 1 1 * = d f + s * ef * * + d sf + ss * ef * * and + d e * f + s * e * + d se * f + ss * e *, 41 Ẑ * = d * + d s *. 4 Substituting Eqs. (41) and (4) into Eq. (40), and after soe algebraic anipulations, we have that Ẑˆ 0 = Ẑ ˆ 1Ẑ 1 1 * ˆ Ẑ * =0. 43 Cobining Eqs. (36) (43) we conclude that the optial easureent operators of Eq. () satisfy the optiality conditions of Theore 1. V. OPTIMAL DETECTION OF SYMMETRIC STATES We now consider the case in which the quantu state enseble has syetry properties referred to as GU and GCU. These syetry properties are quite general, and include any cases of practical interest. VI. GU STATE SETS A GU state set is defined as a set of density operators i,1i such that i =U i U 1 * where is an arbitrary generating operator and the atrices U i,1i are unitary and for an Abelian group G [8,31]. For concreteness, we assue that U 1 =I. The group G is the generating group of S. For consistency with the syetry of S, we will assue equiprobable prior probabilities on S. As we now show, the optial easureent operators that axiize the probability of correct detection when distinguishing unabiguously between the density operators of a GU state set are also GU with the sae generating group. The corresponding generator can be coputed very efficiently in polynoial tie. Suppose that the optial easureent operators that axiize J i = Tr i i 44 subject to Eqs. (8) and (5) are ˆ i, and let Ĵ=Jˆ i = Tr i ˆ i. Let rj,i be the apping fro II to I with I=1,...,, defined by rj,i=k if U * j U i =U k. Then the easureent operators ˆ j * i =U j ˆ rj,iu j and ˆ j 0 =I ˆ j i for any 1 j are also optial. Indeed, since ˆ i0,1i and ˆ ii,ˆ j i 0, 1i and ˆ i j = U j ˆ iu * j U j U * j = I. Using the fact that i =U i U i * for soe generator, Finally, for li, Jˆ i j = TrU * j U j ˆ rj,iu * j U i = k=1 = =Ĵ. TrU k * ˆ ku k Tr i ˆ i

6 ELDAR, STOJNIC, AND HASSIBI PHYSICAL REVIEW A 69, (004) Tr l ˆ i j =TrU l U l * U j ˆ rj,iu j * =TrU s U s * ˆ rj,i =Tr s ˆ k =0, 47 where U s =U * j U l and U k =U * j U i and the last equality follows fro the fact that since li, sk. It was shown in Refs. [8,19] that if the easureent operators ˆ i j are optial for any j, then i =1/ j=1 ˆ j i,1i and 0=I i are also optial. Furtherore, i=u i ˆ U * i where ˆ =1/ k=1 U * k ˆ ku k. We therefore conclude that the optial easureent operators can always be chosen to be GU with the sae generating group G as the original state set. Thus, to find the optial easureent operators all we need is to find the optial generator ˆ. The reaining operators are obtained by applying the group G to ˆ. Since the optial easureent operators satisfy i =U i U * i,1i and i =U i U * i,tr i i =Tr, so that the proble (9) reduces to the axiization proble axtr, 48 B where B is the set of nn Heritian operators, subject to the constraints 0, U i U * i I, TrU i U * i =0, i. 49 The proble of Eqs. (48) and (49) is a (convex) seidefinite prograing proble, and therefore the optial can be coputed very efficiently in polynoial tie within any desired accuracy [6,8,9], for exaple, using the LMI toolbox on Matlab. Note that the proble of Eqs. (48) and (49) has n real unknowns and +1 constraints, in contrast with the original axiization proble (9) subject to Eqs. (8) and (5) which has n real unknowns and +1 constraints. VII. CGU STATE SETS A CGU state set is defined as a set of density operators ik, 1il 1kr such that ik =U i k U i * for soe generating density operators k, 1,kr, where the atrices U i,1il are unitary and for an Abelian group G [8,5]. A CGU state set is, in general, not GU. However, for every k, the operators ik,1il are GU with generating group G. Using arguents siilar to those of Sec. VI and Ref. [19] we can show that the optial easureent operators corresponding to a CGU state set can always be chosen to be GU with the sae generating group G as the original state set. Thus, to find the optial easureent operators all we need is to find the optial generators ˆ k. The reaining operators are obtained by applying the group G to each of the generators ˆ k. Since the optial easureent operators satisfy ik =U i k U i *,1il,1kr and ik =U i k U i *,Tr ik ik =Tr k k, so that the proble (9) reduces to the axiization proble ax subject to the constraints l r k=1 r k B k=1 Tr k k, U i k U i * I, k 0,1 k r, Tr k U i j U i * =0, 1 k, j r,1 i l, 50 if i = 1 then k j. 51 Since this proble is a (convex) seidefinite prograing proble, the optial generators k can be coputed very efficiently in polynoial tie within any desired accuracy [6,8,9]. Note that the proble of Eqs. (50) and (51) has rn real unknowns and lr+1 constraints, in contrast with the original axiization which has lrn real unknowns and lr +1 constraints. VIII. CONCLUSION We considered the proble of distinguishing unabiguously between a collection of ixed quantu states. Using eleents of duality theory in vector space optiization, we derived a set of necessary and sufficient conditions on the optial easureent operators. We then considered soe special cases for which closed for solutions are known, and showed that they satisfy our optiality conditions. We also showed that in the case in which the states to be distinguished have strong syetry properties, the optial easureent operators have the sae syetries, and can be deterined efficiently by solving a seidefinite prograing proble. An interesting future direction to pursue is to use the optiality conditions we developed in this paper to derive closed for solutions for other special cases. APPENDIX A: PROOF OF Eq. (30) To develop the optial dual solution in the case of orthogonal null spaces, let = 1, and define a atrix such that is a square, unitary atrix, i.e., * =I. Denoting Z= Y *, the dual proble can be expressed as subject to intr Y * Y A

7 OPTIMAL QUANTUM DETECTORS FOR UNAMBIGUOUS PHYSICAL REVIEW A 69, (004) i * Y * i i * p i i i, 1 i ; * Y d, B3 Y 0. A Using the orthogonality properties of i and, the proble of Eqs. (A1) and (A) is equivalent to subject to where intry Y Y i i * p i i i, 1 i ; Y =Y1 Y 0, Y A3 A4 Y 0. A5 Since TrY= TrY i, a solution to Eq. (A3) subject to Eq. (A4) is where Ŷ =Ŷ1 Ŷ Ŷ 0, A6 Ŷ = * i p i i i, 1 i. A7 Then, Ẑ = Ŷ * = p i P i i P i, A8 as in Eq. (30). APPENDIX B: PROOF OF Eq. (31) To develop the optial dual solution Ẑ for onediensional null spaces, we note that Ẑ lies in the space spanned by 1 and. Denoting by a atrix whose coluns represent an orthonoral basis for this space, Ẑ can be written as Ẑ=Ŷ *, where the atrix Ŷ is the solution to subject to intry Y 1 * Y 1 d 1, B1 B Y 0. B4 Here i = * i and d i = p i i * i i for 1i. To develop a solution to Eq. (B1) subject to Eqs. (B) (B4), we for the Lagrangian L =TrY i * i Y i d i TrXY, B5 where fro the Karush-Kuhn-Tucker conditions [3] we ust have that i 0,X0, and i i * Y i d i =0, i = 1,, B6 TrXY =0. Differentiating L with respect to Y and equating to zero, I i i * i X =0. B7 B8 If X=0, then we ust have that I= i i * i, which is possible only if 1 and are orthogonal. Therefore, X 0, which iplies fro Eq. (B7) that Eq. (B4) is active. Now, suppose that only Eq. (B4) is active. In this case our proble reduces to iniizing Try * y whose optial solution is y=0, which does not satisfy Eqs. (B) and (B3). We conclude that at the optial solution (B4) at least one of the constraints (B) and (B3) is active. Thus, to deterine the optial solution we need to deterine the solutions under each of the three possibilities: only Eq. (B) is active, only Eq. (B3) is active, both Eqs. (B) and (B3) are active, and then choose the solution with the sallest objective. Consider first the case in which Eqs. (B) and (B4) are active. In this case, Ŷ =ŷŷ * for soe vector ŷ, and without loss of generality we can assue that 1 * ŷ = d 1. To satisfy Eq. (B9), ŷ ust have the for ŷ = d1 1 + ŝ 1, B9 B10 where 1 is a unit nor vector orthogonal to 1, so that 1 * 1 =0, and ŝ is chosen to iniize TrŶ. Since TrŶ = ŷ * ŷ = d 1 + ŝ, B11 ŝ=0. Thus, Ŷ =d *, and TrŶ=d 1. This solution is valid only if Eq. (B3) is satisfied, i.e., only if Here we used the fact that * Ŷ = d 1 f d. * 1 = * * 1 = * 1 = f, B1 B13 since * is an orthogonal projection onto the space spanned by 1 and

8 ELDAR, STOJNIC, AND HASSIBI PHYSICAL REVIEW A 69, (004) Next, suppose that Eqs. (B3) and (B4) are active. In this case, Ŷ =ŷŷ * where without loss of generality we can choose ŷ such that and * ŷ = d ŷ = d + ŝ, B14 B15 where is a unit nor vector orthogonal to, and ŝ is chosen to iniize TrŶ. Since TrŶ=d +ŝ, ŝ=0, and TrŶ=d. This solution is valid only if Eq. (B) is satisfied, i.e., 1 * Y 1 = d f d 1. B16 Finally, consider the case in which Eqs. (B) (B4) are active. In this case, we can assue without loss of generality that * ŷ= d. Then, where ŝ is chosen such that ŷ = d + ŝ, 1 * Ŷ 1 = d 1 B17 B18 and TrŶ=ŷ * ŷ is iniized. Now, for ŷ given by Eq. (B17), Ŷ = d * + ŝ * + ŝ d * + ŝ * d *, so that 1 * Ŷ 1 = d f + ŝ e + d ŝe * f + d ŝ * f * e B19 = d f + ŝ * e, B0 where we defined =, and e and f are given by Eq. (3). Therefore, to satisfy Eq. (B18), ŝ ust be of the for ŝ = 1 e *ej d1 f * d for soe. The proble of Eq. (B1) then becoes 1 in e ej d1 f * d, which is equivalent to axree j f. Since Re e j f e j f = f, the optial choice of is e j = f * /f, and For this choice of ŝ, ŝ = f* d e * d 1 d f 1. TrŶ = d + ŝ =d 1+ f e d 1 d f 1. B1 B B3 B4 B5 B6 Clearly, d. Therefore, to coplete the proof of Eq. (31) we need to show that d 1. Now, e d 1 = e d d 1 + f d 1 f d = 1 e d 1 + e + f d d1 d f = f d1 d 0, B7 where we used the fact that e + f = * 1 * 1 + * 1 * 1 = * 1 1 =1, B8 since * + * is an orthogonal projection onto the space spanned by 1 and. [1] A. S. Holevo, J. Multivariate Anal. 3, 337 (1973). [] C. W. Helstro, Quantu Detection and Estiation Theory (Acadeic, New York, 1976). [3] Y. C. Eldar, A. Megretski, and G. C. Verghese, IEEE Trans. Inf. Theory 49, 101 (003). [4] M. Charbit, C. Bendjaballah, and C. W. Helstro, IEEE Trans. Inf. Theory 35, 1131 (1989). [5] M. Osaki, M. Ban, and O. Hirota, Phys. Rev. A 54, 1691 (1996). [6] M. Ban, K. Kurokawa, R. Moose, and O. Hirota, Int. J. Theor. Phys. 36, 169 (1997). [7] Y. C. Eldar and C. D. Forney, Jr., IEEE Trans. Inf. Theory 47, 858 (001). [8] Y. C. Eldar, A. Megretski, and G. C. Verghese, e-print quantph/ [9] Y. C. Eldar, Phys. Rev. A 68, (003). [10] H. P. Yuen, R. S. Kennedy, and M. Lax, IEEE Trans. Inf. Theory IT-1, 15 (1975). [11] I. D. Ivanovic, Phys. Lett. A 13, 57 (1987). [1] D. Dieks, Phys. Lett. A 16, 303 (1988). [13] A. Peres, Phys. Lett. A 18, 19(1988). [14] G. Jaeger and A. Shiony, Phys. Lett. A 197, 83(1995). [15] A. Peres and D. R. Terno, J. Phys. A 31, 7105 (1998). [16] A. Chefles, Phys. Lett. A 39, 339 (1998). [17] A. Chefles and S. M. Barnett, Phys. Lett. A 50, 3 (1998). [18] Y. C. Eldar, IEEE Trans. Inf. Theory 49, 446 (003). [19] Y. C. Eldar, Phys. Rev. A 67, (003). [0] C. W. Zhang, C. F. Li, and G. C. Quo, Phys. Lett. A 61, 5 (1999). [1] J. Fiurášek and M. Ježek, Phys. Rev. A 67, 0131 (003). [] T. Rudolph, R. W. Spekkens, and P. S. Turner, Phys. Rev. A 68, (003)

9 OPTIMAL QUANTUM DETECTORS FOR UNAMBIGUOUS PHYSICAL REVIEW A 69, (004) [3] P. Raynal, N. Lutkenhaus, and S. J. van Enk, Phys. Rev. A 68, 0308 (003). [4] A. Peres, Found. Phys. 0, 1441 (1990). [5] Y. C. Eldar and H. Bölcskei, IEEE Trans. Inf. Theory 49, 993 (003). [6] F. Alizadeh, Ph.D. thesis, University of Minnesota, Minneapolis, MN 1991 (unpublished). [7] F. Alizadeh, in Advances in Optiization and Parallel Coputing, edited by P. Pardalos (North-Holland, Netherlands, 199). [8] Y. Nesterov and A. Neirovski, Interior-Point Polynoial Algoriths in Convex Prograing (SIAM, Philadelphia, 1994). [9] L. Vandenberghe and S. Boyd, SIAM Rev. 38, 40(1996). [30] D. P. Bertsekas, Nonlinear Prograing, nd ed. (Athena Scientific, Belont, MA, 1999). [31] G. D. Forney, Jr., IEEE Trans. Inf. Theory 37, 141 (1991). [3] A. Ben-Tal and A. Neirovski, Lectures on Modern Convex Optiization (SIAM, Philadelphia, 001). [33] Otherwise we can transfor the proble to a proble equivalent to the one considered in this paper by reforulating the proble on the subspace spanned by the eigenvectors of i,1i. [34] Interior point ethods are iterative algoriths that terinate once a prespecified accuracy has been reached. A worst-case analysis of interior point ethods shows that the effort required to solve a seidefinite progra to a given accuracy grows no faster than a polynoial of the proble size. In practice, the algoriths behave uch better than predicted by the worst-case analysis, and in fact in any cases the nuber of iterations is alost constant in the size of the proble