State discrimination with error margin and its locality
aa r X i v : . [ qu a n t - ph ] J u l State discrimination with error margin and its locality
A. Hayashi, T. Hashimoto, and M. Horibe
Department of Applied PhysicsUniversity of Fukui, Fukui 910-8507, Japan
There are two common settings in a quantum-state discrimination problem. One is minimum-errordiscrimination where a wrong guess (error) is allowed and the discrimination success probability ismaximized. The other is unambiguous discrimination where errors are not allowed but the incon-clusive result “I don’t know” is possible. We investigate a discrimination problem with a finitemargin imposed on the error probability. The two common settings correspond to the error margins1 and 0. For arbitrary error margin, we determine the optimal discrimination probability for twopure states with equal occurrence probabilities. We also consider the case where the states to bediscriminated are multipartite and show that the optimal discrimination probability can be achievedby local operations and classical communication.
I. INTRODUCTION
Suppose we are given an unknown quantum state ρ ,which is guaranteed to be one in a set of known states { ρ a } with some known occurrence probabilities. The taskof quantum-state discrimination [1] is to optimally iden-tify the input state ρ with one of { ρ a } by performingsome measurement on ρ . Usually, one of two conditionsis imposed on the probabilities of making a wrong guess(error). In one setting, the discrimination success prob-ability is optimized without imposing any condition onerror probabilities [2]. This is called minimum-error dis-crimination since the mean error probability is minimizedas a consequence. On the other hand, an incorrect iden-tification (error) is not allowed in unambiguous discrim-ination [3, 4, 5, 6]. Instead, the inconclusive result “I donot know” is possible when one is not certain about theidentity of the input state.Some interesting alternative approaches have been pro-posed. Croke et al. considered a maximum-confidencemeasurement [7], which optimizes the conditional prob-ability that, when a given state is identified by the mea-surement, it is indeed the correct state. In anotherscheme considered in Refs. [8, 9, 10, 11], the probabilityof correct discrimination is maximized while the rate ofinconclusive results is fixed. This scheme interpolates ina way between minimum-error discrimination and unam-biguous discrimination.In this paper we consider a discrimination problemwith a finite error margin m imposed on the probabilityof error. When the error margin m is zero, the problem isequivalent to unambiguous discrimination. In minimum-error discrimination, no condition is imposed on the errorprobability, which means the error margin m is 1. Thus,discrimination with a general error margin unifies the twocommonly adopted settings. A similar scheme was con-sidered by Touzel, Adamson, and Steinberg [12], wherethey minimized the probability of an inconclusive resultby imposing some bound on the error probability andcompared the numerical results of projective and posi-tive operator-valued measure (POVM) measurements. InSec.II, we will formulate two types of constraints on the error probability and analytically determine the maxi-mum success probability as a function of the error margin m for two pure states with equal occurrence probabilities.Let us now assume that the two states to be discrim-inated are multipartite and generally entangled. Theinteresting question here is whether the parties sharingthe input state can achieve the optimal discriminationby means of local operations and classical communica-tion (LOCC). It is known that this is possible for bothminimum-error [13, 14] and unambiguous [15, 16] dis-crimination problems. In Sec.III, we will tackle this prob-lem for discrimination with a general error margin. Byestablishing a general theorem for three-element POVMsof a two-dimensional space, we show that discrimina-tion with an error margin can be optimally performedby LOCC. II. DISCRIMINATION WITH ERROR MARGINOF TWO PURE STATES
We consider the problem of discriminating between twoknown pure states ρ = | φ ih φ | and ρ = | φ ih φ | with |h φ | φ i| 6 = 1. We assume that the two statesoccur with equal probabilities and there is complete clas-sical knowledge of the two states. Our measurement onthe input state can produce three outcomes, µ = 1 , , µ = 1 or 2 means that the input state isguessed to be ρ µ , and outcome 3 means that “I do notknow,” which is called the inconclusive result. Let us in-troduce the POVM, { E , E , E } , corresponding to thethree measurement outcomes.We define P E µ ,ρ a to be the joint probability that theinput state is ρ a , ( a = 1 ,
2) and the measurement out-come is µ = (1 , ,
3) produced by POVM element E µ : P E µ ,ρ a = 12 tr[ E µ ρ a ] . The task is to maximize the discrimination success prob-ability given by p ◦ = X a =1 , P E a ,ρ a = 12 (cid:16) tr[ E ρ ] + tr[ E ρ ] (cid:17) . (1)Let us define some probabilities of making errors. Sup-pose the measurement outcome is µ = 1. The probabilityof error in this case, which means that the input state was ρ , is the conditional probability defined by P ρ | E = P E ,ρ P E , (2)where P E = P E ,ρ + P E ,ρ , the probability of findingoutcome µ = 1. Similarly, when the measurement out-come is µ = 2, the conditional probability of error isgiven by P ρ | E = P E ,ρ P E . (3)In minimum-error discrimination, one maximizes thesuccess probability of Eq.(1) without imposing any condi-tions on the two conditional error probabilities of Eqs.(2)and (3). On the other hand, the conditional error prob-abilities of Eqs.(2) and (3) are required to be zero in theunambiguous discrimination problem.In this paper, we consider a discrimination problemwith the conditions that these conditional error proba-bilities should not exceed a certain error margin m (0 ≤ m ≤ P ρ | E = tr[ E ρ ]tr[ E ρ ] + tr[ E ρ ] ≤ m, (4a) P ρ | E = tr[ E ρ ]tr[ E ρ ] + tr[ E ρ ] ≤ m. (4b)It is clear that unambiguous discrimination correspondsto the case of m = 0 and minimum-error discriminationcorresponds to the case of m = 1 since the probabilitiesdo not exceed 1.We can impose a margin of error in a different way. Ifwe require that the mean error probability, denoted by p × , should not exceed the error margin m , we obtain p × ≡ P E ,ρ + P E ,ρ ≤ m. (5)It turns out that this condition is weaker than the error-margin conditions given in Eq.(4), because Eq.(5) followsfrom Eqs.(4): p × = P E ,ρ + P E ,ρ = P ρ | E P E + P ρ | E P E ≤ m ( P E + P E ) ≤ m. From now on, we call the conditions given by Eqs.(4) andEq.(5) strong and weak error-margin conditions, respec-tively. We will first consider the discrimination problemwith the strong error-margin condition. The weak error-margin condition will be discussed later.The discrimination problem with the strong error- margin condition is formulated in the following way:maximize: p ◦ = 12 (tr[ E ρ ] + tr[ E ρ ]) , (6a)subject to: E ≥ , E ≥ , (6b) E + E ≤ , (6c)tr[ E ρ ] ≤ m (tr[ E ρ ] + tr[ E ρ ]) , (6d)tr[ E ρ ] ≤ m (tr[ E ρ ] + tr[ E ρ ]) . (6e)Here, Eq.(6c) represents the positivity condition of thePOVM element E , and Eqs.(6d) and (6e) are the strongerror-margin conditions. This is a problem of semidefi-nite programming. As we will see, this problem can besolved in a closed form.We present the results first. The maximum-discrimination success probability is given as follows: p ◦ = A m (cid:16) − |h φ | φ i| (cid:17) (0 ≤ m ≤ m c ) , (cid:16) p − |h φ | φ i| (cid:17) ( m c ≤ m ≤ , (7)where m c = 12 (cid:16) − p − |h φ | φ i| (cid:17) , (8)and A m is an increasing function of the error margin anddefined to be A m = 1 − m (1 − m ) (cid:16) p m (1 − m ) (cid:17) . (9)Figure 1 displays the maximum success probability asa function of error margin. When the error margin isless than m c , the maximum success probability is givenby that of unambiguous discrimination, one minus the fi-delity of the two states, multiplied by a factor A m , whichis an increasing function of the error margin. When m = 0, the above p ◦ reproduces the success probabil-ity of unambiguous discrimination since A = 1. Notethat the maximum success probability is equal to that ofminimum-error discrimination when m c ≤ m ≤
1. Thiscan be understood in the following way. In minimum-error discrimination, the success probability is optimizedwith no explicit conditions on errors. The resultant con-ditional error probabilities, P ρ | E and P ρ | E , turn outto be m c given in Eq.(8). Thus, an error margin greaterthan m c has no effect on the optimization of success prob-ability. For m c ≤ m ≤
1, the optimal POVM is given bythat of the minimum error discrimination problem.In what follows, we derive the maximum success prob-ability of Eq.(7). We work in the two-dimensional sub-space spanned by states | φ i and | φ i so that the Blochvector representation can be used: ρ a = 1 + n a · σ , ( a = 1 , . p : s u ccess p r ob a b ili t y ο m : error margin p m p u m c StrongWeak
FIG. 1: Discrimination success probability p ◦ . The solid lineis the discrimination success probability with the strong error-margin condition. The dotted line is that with the weak error-margin condition. p u represents the success probability ofunambiguous discrimination and p m is that of minimum-errordiscrimination. The fidelity |h φ | φ i| is taken to be 0 . We also parametrize POVM elements by Pauli matrices: E µ = α µ + β µ · σ , ( µ = 1 , , . In terms of these parameters, the optimization problemtakes the following form:maximize: p ◦ = 12 ( α + β · n + α + β · n ) , (10a)subject to: α ≥ | β | , α ≥ | β | , (10b) α + α + | β + β | ≤ , (10c) α + β · n ≤ m (2 α + β · ( n + n )) , (10d) α + β · n ≤ m (2 α + β · ( n + n )) . (10e)The variables in this optimization problem are the pa-rameters { α , β , α , β } . Note that the conditions(10b), (10c), (10d), and (10e) define a convex set in theparameter space; if each parameter set of { α , β , α , β } and { α ′ , β ′ , α ′ , β ′ } satisfies the conditions, so does theset { pα + qα ′ , p β + q β ′ , pα + qα ′ , p β + q β ′ } , for all p, q ≥ p + q = 1. This convexity enables usto impose two useful symmetries on optimal parameterswithout loss of generality.We consider the symmetry with respect to exchange ofthe two Bloch vectors n and n . Let O be the reflectionmatrix with respect to the plane which is perpendicularto n − n and contains the origin of the Bloch sphere,so that O n = n and O n = n . Suppose the parame-ter set { α , β , α , β } is optimal. It is easy to see thatthe parameter set { α , O β , α , O β } is optimal. Fur-thermore, by convexity, the arithmetic average of these two sets of parameters is also optimal since the successprobability p ◦ is linear in the parameters. The parame-ter set obtained in this way clearly satisfies the followingsymmetry: α = α , β = O β . (11)We can repeat a similar argument for the reflection withrespect to the plane that contains the vectors n and n .This consideration enables us to safely assume that thevectors β and β lie on this plane.Let us now show that equality should actually hold inthe inequality conditions (10b) and (10c) for optimal pa-rameters. This implies that the rank of each element ofthe POVM { E , E , E } does not exceed 1; this propertywill be important in the next section. We begin with thecondition (10c) and assume that equality does not holdfor an optimal set of parameters. Then we can multi-ply all parameters by a common positive number greaterthan one so that all the conditions (10b), (10c), (10d),and (10e) are still satisfied while the success probability p ◦ increases. Since this is a contradiction, we can con-clude that equality holds in condition (10c).A similar, but rather involved, argument can also beapplied to condition (10b). Assume that a strict inequal-ity holds in Eq.(10b) for an optimal set of parameterswith the symmetry given by Eq.(11). Let us increasethe component of β in the direction of ( n − n ) whileleaving the component along vector ( n + n ) unchanged.Note that β also changes in accordance with the symme-try of (11). The left-hand side of (10c) and the right-handsides of (10d) and (10e) remain the same. The left-handsides of (10d) and (10e) decrease, while the success proba-bility p ◦ increases. It is clear that there exists a positiveincrement of the component of β along the direction( n − n ) such that p ◦ increases while all the conditionsare still satisfied. We thus conclude that equality holdsin condition (10b).From these considerations, we can rewrite the problemin terms of variables α ≡ α and β ≡ β :maximize: p ◦ = α + β · n , subject to: α = | β | , α + | (1 + O ) β | = 1 ,α + β · n ≤ m (2 α + β · ( n + n )) . Since the vector β lies on the plane spanned by n and n , we can expand β as β = x n + n y n − n . The problem is further simplified in terms of variables α , x , y and takes the following form:maximize: p ◦ = α + Sx + T y, (12a)subject to: α = p Sx + T y , (12b)2 α + 2 √ S | x | = 1 , (12c)(1 − m )( α + Sx ) − T y ≤ . (12d)Here, we introduced positive constants S and T : S ≡ n · n |h φ | φ i| ,T ≡ − n · n − |h φ | φ i| . By using Eqs.(12b) and (12c), we can express α and x interms of y : α = T y + 14 , √ S | x | = 14 (1 − T y ) . Condition (12d) is then a quadratic inequality of variable y . Optimization can now be explicitly performed sincethe success probability p ◦ becomes a quadratic functionof y . After a long tedious calculation, which is outlined inthe appendix, the maximum success probability is foundto be given by Eq.(7). When the error margin is in therange of 0 ≤ m ≤ m c , the optimal POVM are given bythe parameters α max , x max , and y max : y max ≡ p m (1 − m )2(1 + √ S )(1 − m ) , (13) x max ≡ − √ S (1 − T y ) , (14) α max ≡ T y + 14 . (15)Before concluding this section, we briefly summarizethe results of the weak error-margin condition given inEq.(5). For the weak error-margin condition, conditions(10d) and (10e) should be replaced by α + β · n + α + β · n ≤ m. We can proceed in a similar way to the strong error-margin case. Omitting the details of derivation, wepresent the maximum success probability with the weakerror-margin condition: p ◦ = (cid:16) √ m + p − |h φ | φ i| (cid:17) (0 ≤ m ≤ m c ) , (cid:16) p − |h φ | φ i| (cid:17) ( m c ≤ m ≤ . (16)Unlike the strong error-margin case, the success probabil-ity for 0 ≤ m ≤ m c is not simply proportional to that of unambiguous discrimination. For purpose of comparison,the success probabilities of the two error-margin condi-tions are plotted in Fig. 1. We note that each elementof the optimal POVM in this case is again of rank 0 or1, though it is more involved to show than in the strongerror-margin case. The optimal POVM for 0 ≤ m ≤ m c is given by the parameters: y max ≡ √ S ) (cid:18) r m − √ S (cid:19) ,x max ≡ − √ S (1 − T y ) ,α max ≡ T y + 14 . III. LOCAL DISCRIMINATION WITH ERRORMARGIN
Suppose that the two pure states to be discriminatedare multipartite and generally entangled. Can the partiessharing the input state perform optimal discriminationby LOCC? It has been shown that this is possible inboth the minimum-error [13, 14] and unambiguous [15,16] discrimination problems. As shown in the precedingsection, those correspond to the cases of m c ≤ m ≤ m = 0. In this section we will show that this is truefor any value of error margin.In the preceding section we determined the optimalPOVM in the two-dimensional subspace spanned by | φ i and | φ i and showed that the rank of each POVM el-ement does not exceed 1 for any error margin. We willshow that the following general theorem holds. Theorem 1 : Let V be a two-dimensional subspace ofa multipartite tensor-product space H , and P be theprojector onto the subspace V . Then, for any three-element POVM { E , E , E } of V with every element be-ing of rank 0 or 1, there exists a one-way LOCC POVM { E L1 , E L2 , E L3 } of H such that E µ = P E L µ P ( µ = 1 , , . This implies that a POVM satisfying the conditionsof Theorem 1 can be implemented by a one-way LOCCprotocol as far as measurement for states in subspace V is concerned. The optimal POVM of discriminationwith error margin satisfies the conditions of Theorem 1;therefore, it is achievable by a one-way LOCC protocol.In the rest of the section, we prove Theorem 1. To doso, it suffices to develop the proof of Ji et al., which showsthat unambiguous discrimination of two pure states witharbitrary occurrence probabilities can be optimally real-ized by means of a one-way LOCC [16]. Our strategy forthe proof is the following. First, we show that the opti-mal POVMs of the global and the LOCC schemes sharethe same matrix elements for all states in V , which meansthat Theorem 1 holds for the optimal POVM of any un-ambiguous discrimination problem. Then, we show thatany POVM satisfying the conditions of Theorem 1 can beregarded as the optimal POVM of a certain unambiguousdiscrimination problem.Let us consider unambiguous discrimination betweenpure states | Φ i and | Φ i with occurrence probabilities s and t , respectively. We assume |h Φ | Φ i| 6 = 1 anddenote by V the two-dimensional subspace spanned by | Φ i and | Φ i . We begin with the global discriminationscheme. For our purpose, it suffices to consider the casewhere the following conditions are satisfied: r st , r ts ≥ |h Φ | Φ i| . In this case the optimal POVM is given by E = a | Φ ⊥ ih Φ ⊥ | ,E = a | Φ ⊥ ih Φ ⊥ | ,E = P − E − E . (17)Here, | Φ ⊥ a i ( a = 1 ,
2) is a normalized state in V which isorthogonal to | Φ a i . The | Φ ⊥ a i is unique up to a phasefactor. The coefficients a and a are given by a = 1 − q ts |h Φ | Φ i| − |h Φ | Φ i| , (18a) a = 1 − p st |h Φ | Φ i| − |h Φ | Φ i| , (18b)which implies that E is also of rank 0 or 1.Now suppose that the states | Φ i AB and | Φ i AB arebipartite, shared by Alice and Bob. We can choose anappropriate phase for the states so that h Φ | Φ i has anonnegative real value. Ji et al. showed that, by ap-pending an ancillary system R to, say, Alice’s system A ,we can choose an orthonormal basis {| I i RA } for Alice’scombined system RA so that the following relations hold: | i R ⊗ | Φ i AB = X I √ s I | I i RA ⊗ | η I i B , (19) | i R ⊗ | Φ i AB = X I √ t I | I i RA ⊗ | γ I i B , (20)where | η I i and | γ I i are normalized states of Bob’s sys-tem B , and h η I | γ I i is a nonnegative real number satis-fying r ss I tt I , r tt I ss I ≥ h η I | γ I i ≥ . (21)This decomposition of the states defines a one-way LOCCprotocol: Alice first performs measurement in the basis {| I i} and informs Bob of the outcome I ; he then dis-criminates between states | η I i and | γ I i . Ji et al. showedthat this one-way LOCC protocol achieves the maximumsuccess probability given by the global optimal POVM ofEq.(17).We can show that the POVM { E L µ } µ =1 , , correspond-ing to the protocol of Ji et al. actually satisfies stronger conditions; the global POVM and the LOCC POVMshare the same matrix element between any states in thesubspace V . To see this, let us write the LOCC POVM { E L µ } µ =1 , , as E L µ = X I e AI ⊗ e Bµ ( I ) ( µ = 1 , , , (22)where { e AI } I is a POVM of Alice’s system A defined by e AI = R h | I i ( RA ) h I | i R , and { e Bµ ( I ) } µ =1 , , is Bob’s POVM: e B ( I ) = 1 − q tt I ss I h η I | γ I i − h η I | γ I i | γ ⊥ I ih γ ⊥ I | , (23a) e B ( I ) = 1 − q ss I tt I h η I | γ I i − h η I | γ I i | η ⊥ I ih η ⊥ I | , (23b) e B ( I ) = P B ( I ) − e B ( I ) − e B ( I ) . (23c)Here, operator P B ( I ) is the projector onto the two-dimensional subspace V B ( I ) spanned by | η I i and | γ I i ,and the state | η ⊥ I i is orthogonal to | η I i in this subspace. | γ ⊥ I i is defined similarly.The global E has the following vanishing matrix ele-ments: h Φ | E | Φ i = h Φ | E | Φ i = h Φ | E | Φ i = 0 . It is clear that the corresponding matrix elements of E L1 are also zero. As for the matrix element between | Φ i and | Φ i , E gives h Φ | E | Φ i = a |h Φ | Φ ⊥ i| = 1 − r ts h Φ | Φ i . We find that E L1 has the same matrix element: h Φ | E L1 | Φ i = X I s I − q tt I ss I h η I | γ I i − h η I | γ I i |h η I | γ ⊥ I i| = 1 − r ts h Φ | Φ i . The same thing can be readily verified for E L2 and E L3 .We thus have shown that E µ = P E L µ P, ( µ = 1 , , { E µ } µ =1 , , thatsatisfies the conditions of Theorem 1, and write it interms of normalized states | ψ µ i and nonnegative coef-ficients b µ : E µ = b µ | ψ µ ih ψ µ | ( µ = 1 , , . (24)There are two linearly independent states among {| ψ µ i} µ =1 , , , which can be assumed to be | ψ i and | ψ i . The two-dimensional subspace V is spanned bythose states. Operator E = P − E − E is of rank 0 or1 by definition; therefore, the greater eigenvalue of thetwo eigenvalues of operator E + E is equal to 1. Thisimplies that the coefficients b and b can be expressedas b = 1 − r |h ψ | ψ i| − |h ψ | ψ i| , (25a) b = 1 − r |h ψ | ψ i| − |h ψ | ψ i| , (25b)where r is a positive number that satisfies the condition r, /r ≥ |h ψ | ψ i| . This can be seen in the followingway. It is evident that there exists a positive number r such that the coefficient b can be written in the form ofEq.(25a) since 0 ≤ b ≤
1. It is also easy to see that ifthe greater eigenvalue of E + E is equal to 1, then thecoefficients b and b should satisfy the relation: b + b = 1 + b b (cid:0) − |h ψ | ψ i| (cid:1) . From this relation we find that b is expressed byEq.(25b).For any positive r , there is a set of positive numbers s and t such that r = p t/s and s + t = 1. We alsonote that the relation |h ψ | ψ i| = |h ψ ⊥ | ψ ⊥ i| holds.Therefore, a general POVM (24) satisfying the conditionsof Theorem 1 can be identified with the optimal POVMof unambiguous discrimination between | ψ ⊥ i and | ψ ⊥ i with the occurrence probabilities s and t , respectively.Thus, we conclude that Theorem 1 holds for the bipartitecase.The general multipartite case of Theorem 1 followsby induction. Remember that Bob’s POVM is given byEq.(23), which satisfies the conditions of Theorem 1: thisis a POVM of the two-dimensional subspace V B ( I ) andeach element is of rank 0 or 1. If Bob’s system is compos-ite and actually shared by Bob and Charles, we can con-struct a one-way LOCC POVM for Bob and Charles inthe same way. The whole one-way LOCC POVM wouldtake the form E L µ = X I A I B e AI A ⊗ e BI B ( I A ) ⊗ e Cµ ( I A , I B ) ( µ = 1 , , . It is clear that we can repeat the same procedure if nec-essary. This completes the proof of Theorem 1.
IV. CONCLUDING REMARKS
We considered a quantum-state discrimination prob-lem with an error margin imposed, which unifies theminimum-error and the unambiguous discriminationproblems. We determined the optimal discriminationsuccess probability for two pure states with equal occur-rence probabilities. When the states are multipartite, wehave shown that a LOCC scheme achieves the globally at-tainable optimal success probability. This was shown byestablishing a general theorem for three-element POVMsof a two-dimensional space. In this paper, we assumed that complete classicalknowledge is given for the states to be discriminated.Instead, we can consider a situation where no classicalknowledge of the states is given, but a certain number oftheir copies are available as reference states. One’s taskis to correctly identify a given input state with one ofthe reference states by some measurement on the wholestate [17, 18, 19]. This problem is called the quantum-state identification problem to distinguish it from dis-crimination problems with classical knowledge assumed.We studied the identification problem for two bipartitepure states [20] and found that the optimal unambiguousidentification cannot be achieved by LOCC while this ispossible in minimum-error identification. It would be ofgreat interest to consider identification problems with ageneral error margin imposed.
APPENDIX
Here we outline the last part of the derivation for theoptimal success probability given by Eq.(7). We beginwith the optimization problem:maximize: p ◦ = α + Sx + T y, (A.1a)subject to:(1 − m )( α + Sx ) − T y ≤ , (A.1b)where α = T y + 14 , (A.1c) √ S | x | = 14 (1 − T y ) . (A.1d)We consider only the case of 0 ≤ m < m c ≡ (1 − √ T ) / m c ≤ m ≤
1. By our assumption |h φ | φ i| 6 = 1, we have S = 1, T = 0, and m c < / | y | ≤ / (2 √ T ) and alsoshows that it is convenient to treat separately the twocases of different signs of x .First we will show that there is no feasible solution if x ≥
0. In this case, the condition of Eq.(A.1b) meansthat the quadratic function of y defined by f ( y ) ≡ T (1 − √ S ) y − T − m y + 1 + √ S , should be nonpositive. The function f ( y ) has the vertexat y = 12(1 − m )(1 − √ S ) ≥ √ T .
On the other hand, we find f (cid:18) √ T (cid:19) = 12 − √ T − m ! > , which implies f ( y ) > y ≤ / (2 √ T ). Thus the caseof x ≥ x <
0. The success probability is thengiven by p ◦ ( y ) ≡ T (1 + √ S ) y + T y + 1 − √ S . The condition of Eq.(A.1b) implies g ( y ) ≤
0, where thequadratic function g ( y ) is defined to be g ( y ) ≡ T (1 + √ S ) y − T − m y + 1 − √ S . The function g ( y ) has the vertex at y = y ≡ − m )(1 + √ S ) < √ T , and we also find g ( y ) = T √ S ) (cid:18) − − m ) (cid:19) ≤ . On the other hand, we have g (cid:18) √ T (cid:19) = 12 − √ T − m ! > . Since p ◦ ( y ) is an increasing function for positive y , theoptimal p ◦ is given by the greater root of the quadraticequation g ( y ) = 0, which is given by y max = 1 + 2 p m (1 − m )2(1 + √ S )(1 − m ) . Substituting y max in p ◦ ( y ), we obtain the optimal successprobability p ◦ ( y max ) = g ( y max ) + T − m y max + T y max = A m (1 − √ S ) , where A m is defined by Eq.(9). It is clear that the otheroptimal parameters x and α are given by Eq.(14) andEq.(15), respectively. [1] A. Chefles, Contemp. Phys. , 401 (2000).[2] C. W. Helstrom, Quantum Detection and EstimationTheory (Academic Press, New York, 1976).[3] I. D. Ivanovic, Phys. Lett. A , 257 (1987).[4] D. Dieks, Phys. Lett. A , 303 (1988).[5] A. Peres, Phys. Lett. A , 19 (1988).[6] G. Jaeger and A. Shimony, Phys. Lett. A , 83 (1995).[7] Sarah Croke, Erika Andersson, Stephen M. Barnett,Claire R. Gilson, and John Jeffers, Phys. Rev. Lett. ,070401 (2006).[8] A. Chefles and S. M. Barnett, J. Mod. Opt. , 1295(1998).[9] Chuan-Wei Zhang, Chuan-Feng Li, and Guang-Can Guo,Phys. Lett. A , 25 (1999).[10] J. Fiurasek and M. Jezek, Phys. Rev. A , 012321(2003).[11] Y. C. Eldar, Phys. Rev. A , 042309 (2003).[12] M. A. P. Touzel, R. B. A. Adamson, and A. M. Steinberg Phys. Rev. A , 062314 (2007).[13] Jonathan Walgate, Anthony J. Short, Lucien Hardy, andVlatko Vedral, Phys. Rev. Lett. , 4972 (2000).[14] S. Virmani, M. F. Sacchi, M. B. Plenio, and D. Markham,Phys. Lett. A , 62 (2001).[15] Y.-X. Chen and D. Yang, Phys. Rev. A , 022320(2002).[16] Zhengfeng Ji, Hongen Cao, and Mingsheng Ying, Phys.Rev. A , 032323 (2005).[17] A. Hayashi, M. Horibe, and T. Hashimoto, Phys. Rev. A , 052306 (2005).[18] Janos A. Bergou and Mark Hillery, Phys. Rev. Lett. ,160501 (2005).[19] A. Hayashi, M. Horibe, and T. Hashimoto, Phys. Rev. A , 012328 (2006).[20] Y. Ishida, T. Hashimoto, and M. Horibe, and A. Hayashi,Phys. Rev. A78