A Return to the Optimal Detection of Quantum Information

Eric Chitambar 1∗ and Min-Hsiu Hsieh 2† 1 Department of Physics and Astronomy, Southern Illinois University, Carbondale, Illinois 62901, USA 2 Centre for Quantum Computation & Intelligent Systems (QCIS), Faculty of Engineering and Information Technology (FEIT), University of Technology Sydney (UTS), NSW 2007, Australia (Dated: September 5, 2018)

In 1991, Asher Peres and William Wootters wrote a seminal paper on the nonlocal processing of quantum information [Phys. Rev. Lett. 66 1119 (1991)]. We return to their classic problem and solve it in various contexts. Specifically, for discriminating the “double trine” ensemble with minimum error, we prove that global operations are more powerful than local operations with classical communication (LOCC). Even stronger, there exists a finite gap between the optimal LOCC probability and that obtainable by separable operations (SEP). Additionally we prove that a two-way, adaptive LOCC strategy can always beat a one-way protocol. Our results provide the first known instance of “nonlocality without entanglement” in two qubit pure states.

One physical restriction that naturally emerges in apply to any measure of distinguishability: quantum communication scenarios is nonlocality. Here, two or more parties share some multi-part quantum C1: LOCC is strictly sub-optimal compared to global system, but their subsystems remain localized with no operations, “global” quantum interactions occurring between them. C2: The optimal LOCC protocol involves two-way com- Instead, the system is manipulated through local quan- munication and adaptive measurements. tum operations and classical communication (LOCC) performed by the parties. The set of global POVMs will be denoted by GLOBAL, Asher Peres and William Wootters were the first to and C1 can be symbolized by GLOBAL > LOCC. A two- introduce the LOCC paradigm and study it as a re- way LOCC protocol with adaptive measurement refers to stricted class of operations in their seminal work [1]. To at least three rounds of measurement, Alice → Bob → gain insight into how the LOCC restriction affects in- Alice, where the choice of measurement in each round formation processing, they considered a seemingly sim- depends on the outcome of the other party’s measure- ple problem. Suppose that Alice and Bob each possess ment in the previous round. We symbolize C2 as LOCC

a qubit, and with equal probability, their joint system is > LOCC→. In Ref. [1] Peres and Wootters obtained

prepared in one of the states belonging to the set {|Dii = numerical data to support both C1 and C2, but these 2 i iπ |sii ⊗ |sii}i=0, where |sii = U |0i and U = exp(− 3 σy). conjectures have never been proven for the double trine. This highly symmetric ensemble is known as the “dou- Before we present our contribution to the problem, we ble trine,” and we note that lying orthogonal to all three would like to briefly highlight the legacy of the Peres- − p arXiv:1304.1555v1 [quant-ph] 4 Apr 2013 states is the singlet |Ψ i = 1/2(|01i − |10i). Wootters paper. Perhaps most notably is that it sub- Alice and Bob’s goal is to identify which double trine sequently led to the discovery of quantum teleportation element was prepared only by performing LOCC. Like [3]. Other celebrated phenomena can also directly trace any quantum operation used for state identification, Al- their roots to Ref. [1] such as so-called nonlocality with- ice and Bob’s collective action can be described by some out entanglement [4] and quantum data hiding [5]. More positive-operator valued measure (POVM). While the generally, Ref. [1] paved the way for future research into non-orthogonality of the states prohibits the duo from LOCC and its fundamental connection to quantum en- perfectly identifying their state, there are various ways to tanglement [6]. measure how well they can do. Peres and Wootters chose We finally note that in a return to Ref. [1] of his own, the notoriously difficult measure of accessible information Wootters constructed a separable POVM that obtains [2], but their paper raises the following two general con- the same information as the best known global measure-

jectures concerning the double trine ensemble, which can ment [7]. A POVM {Πi} belongs to the class of separable 2

(n) (n) k operations (SEP) if each POVM element can be decom- In general, a sequence of POVMs P := {Πi }i=1 posed as a tensor product Πi = Ai ⊗ Bi over the two asymptotically attains an error probability P on ensem- k systems. SEP is an important class of operations since ble {|ψii, pi}i=1 if for every  > 0 we have P +  > Pk (n) every LOCC operation belongs to SEP [4]. 1 − i=1 pihψi|Πi |ψii for sufficiently large n. If each In this paper, we prove that conjectures C1 and C2 are POVM in the sequence P(n) can be generated by LOCC, indeed true when distinguishability success is measured then P is achievable by asymptotic LOCC. by the minimum error probability, which is defined as fol- It is known that for an ensemble of linearly indepen- k lows. For an ensemble E = {|ψii, pi}i=1, the error proba- dent pure states, the global POVM attaining minimum k bility associated with some identification POVM {Πi}i=1 error consists of orthonormal, rank one projectors [11] Pk is given by 1− i=1 pihψi|Πi|ψii. Then the minimum er- (see also [12]). We strengthen this result and extend it ror probability of distinguishing E with respect to a class to the asymptotic setting. of operations S (such as LOCC, SEP, GLOBAL, etc.) is k given by the infimum of error probabilities taken over all Theorem 1. Let E = {|ψii, pi}i=1 be an ensemble of lin- POVMs that can be generated by S. Note that we can early independent states spanning some space S. Suppose replace “infimum” by “minimum” only if S is a compact that Popt is the global minimum error probability of E. k Then there exists a unique orthonormal basis {|φii}i=1 set of operations. While GLOBAL, SEP and LOCC→ all have this property, LOCC does not [8, 9]. Hence, of S such that: (a) A POVM attains an error probabil- to properly discuss the LOCC minimum error, we must ity Popt on E if and only if it can also distinguish the k consider the class of so-called asymptotic LOCC, which {|φii}i=1 with no error, and (b) A sequence of POVMs is LOCC plus all its limit operations [9]. We will prove asymptotically attains an error probability Popt on E if C1 with respect to this more general class of operations. and only if it contains a subsequence that can asymptot- ically distinguish the {|φ i}k with no error. A. Global and Separable Operations: The double trine i i=1 ensemble has a group-covariant structure which greatly The proof is given in the Appendix. Theorem 1 simplifies the analysis. In fact, Ban et al. have proven essentially reduces optimal distinguishability of non- that the so-called “Pretty Good Measurement” (PGM) orthogonal linearly independent ensembles to perfect dis- [10] is indeed an optimal global POVM for discriminating crimination of orthogonal ensembles. Applying part (a) ensembles with such symmetries [11]. For the double to the double trine ensemble, if an LOCC POVM could trine, the PGM consists of simply projecting onto the attain the error probability of Eq. (2), then it can also − i i 2 orthonormal basis {|Ψ i,U ⊗ U |Fii} , where i=0 perfectly distinguish the states |Fii given by (1). How- √ √ i i ever, these are three entangled states which, by a result of |Fii ∝ U ⊗ U [( 2 + 1)|00i − ( 2 − 1)|11i]. (1) Walgate and Hardy, means they cannot be distinguished The corresponding error probability is perfectly by LOCC [13]. Therefore, the global minimum √ error probability is unattainable by LOCC. 1/2 − 2/3 ≈ 2.86 × 10−2. (2) But is the probability attainable by asymptotic LOCC? If it is, then part (b) of Theorem 1 likewise im- To show that SEP can also obtain this probability, plies that the |Fii must be perfectly distinguishable by we explicitly construct a separable POVM. The idea asymptotic LOCC. While Ref. [13] provides simple cri- is to mix a sufficient amount of the singlet state with teria for deciding perfect LOCC distinguishability of two each of the PGM POVM elements so to obtain sepa- qubit ensembles, no analogous criteria exists for asymp- rability (a similar strategy was employed in Ref. [7]). totic LOCC. The only general result for asymptotic dis- The resulting POVM is {|F˜ ihF˜ |}2 with |F˜ ihF˜ | = i i i=0 i i crimination has been recently obtained by Kleinmann et |F ihF | + 1/3|Ψ−ihΨ−|. It is fairly straightforward to i i al. [14]. Here we cite their result in its strongest form, compute that F˜ = 1/2(|ϕ ihϕ | + |ϕ ihϕ |), where 0 + + − − adapted specifically for the problem at hand. p − |ϕ±i = |F0i± 1/3|Ψ i is a product state. This suffices 2 to prove separability of the POVM. Proposition 1 ([14]). If the states {|Fii}i=0 can be per- B. LOCC and Asymptotic LOCC : Let us begin with fectly distinguished by asymptotic LOCC, then for all a clear description of asymptotic LOCC discrimination. χ ∈ [1/3, 1] there is a product operator E ≥ 0 such 3

P2 that (i) i=0hFi|E|Fii = 1, (ii) hF0|E|F0i = χ, and tinguishing the three |sii can always be converted into 0 1 1/2 (iii) the normalized states |Fi i := √ E |Fii are a protocol for distinguishing ρ and σ by simply coarse- hFi|E|Fii perfectly distinguishable by separable operations. graining over all outcomes corresponding to |s2i and |s3i. The minimum error probability in distinguishing ρ and In the appendix we prove that these three conditions can- σ is readily found to be (see Appendix): not be simultaneously satisfied; therefore, GLOBAL > LOCC for minimum error discrimination. Here, we pro- 1 1 p 2 − 2 1 − 3p1p2 − p0p1 − p0p2, (4) vide a little intuition into why Proposition 1 must be 1 1 true. For every LOCC protocol that correctly identifies which simplifies to 2 − 24 [75 + 32 cos(2θ) − 7 cos(4θ) + 2 1/2 the given state with probability 1 − , we can think of 18 cos(2φ) sin (2θ)] . In the interval −π/6 ≤ θ < π/6, the success probability as smoothly evolving from com- a minimum is obtained at θ = −π/6 and φ = 0. This corresponds to p = p = 1/2 and p = 0 with an error plete randomness (χ = 1/3) to its final average value 0 √1 2 (χ = 1 − ). Then for each χ ∈ (1/3, 1 − ), the protocol probability of 1/2 − 3/4. Now, this probability lower can be halted after some sequence of measurement out- bounds the error probability along each branch of Alice’s comes (collectively represented by the product operator measurement, and therefore it places a lower bound on E) such that given these outcomes: (1) there is one state any one-way LOCC measurement scheme. In fact, this that can be identified with probability χ (which by sym- lower bound turns out to be tight. When Alice performs 2 2 metry we can assume is |F0i), and (2) the transformed en- the POVM { 3 (I − |siihsi|)}i=0 outcome i will eliminate semble can be discriminated by a separable POVM with |sii ⊗ |sii but leave the other two states with an equal post-measurement probability. Thus, in each branch we success probability no less than 1 − . By compactness of √ −2 SEP, we let  → 0 and replace (2) by the condition that a obtain the error probability 1/2 − 3/4 ≈ 6.70 × 10 , separable POVM perfectly distinguishes the post-halted and this provides the minimum one-way error probability. ensemble. If we allow feedback from Bob, there exists better mea-

C. LOCC > LOCC→: We will now compute the min- surement strategies. The following protocol generalizes imum one-way error probability for the double trine, the optimal one-way scheme just described. (Round I) and then describe an explicit two-way protocol with a Alice performs the measurement with Kraus operators 2 smaller error probability. In the one-way task, Alice given by {Ai}i=0 with makes a measurement and communicates her result to A = p1/3(1 − p)|s ihs | + p1/3(1 + p)|s⊥ihs⊥|. Bob. Without loss of generality, we fine-grain Alice’s i i i i i measurement so that each POVM element is rank one ⊥ Here |s i is the state orthogonal to |sii (explicitly iφ i |ηihη|, with |ηi = r cos θ|0i+re sin θ|1i. Given outcome ⊥ i |si i = U |1i). Note that this is the square-root of η, Bob’s task is to optimally discriminate the ensemble the POVM given by Peres and Wootters [1]. Without 2 2 {|sii}i=0, but now with an updated distribution {pi}i=0 loss of generality, we suppose that Alice obtains out- given by come “0” and communicates the result to Bob. Her 2 0 |hη|ski| 2 2πk iφ 2πk 2 (normalized) post-measurement states are |s i = |0i, pk = = | cos cos θ + e sin sin θ| . (3) 0 3P (η) 3 3 3 0 −1/2 √ p |s1i = [2(2 + p)] ( 1 − p|0i − 3(1 + p)|1i), and 1 P2 2 0 −1/2 √ p Here, P (η) = 3 i=0 |hη|sii| , and we’ve used the co- |s2i = [2(2 + p)] ( 1 − p|0i + 3(1 + p)|1i). (Round 1 P2 variance 3 i=0 |siihsi| = I/2. Additionally, we can as- II) From Bob’s perspective, he is still dealing with the sume that p0 ≥ p1, p2, since if |ηi fails to generate a original states |sii, but now their prior probabilities distribution with this property, by the symmetry we can have changed to Pi|A0 = PA0|i. He now proceeds as always rotate |ηi such that p0 is indeed the maximum if Alice had completely eliminated the state |s0i (i.e. post-measurement probability. This means we can only if she had chosen p = 1 as the strength of her mea- restrict attention to −π/6 ≤ θ ≤ π/6. surement). Specifically, he projects onto the eigenba-

Next, we observe that Bob’s task of distinguishing sis of |s1ihs1| − |s2ihs2| which are the states |±i = 2 p the ensemble {|sii, pi}i=0 is no easier than distinguish- 1/2(|0i ± |1i). A “+” outcome is associated with |s1i ing between the two weighted states ρ = p0|s0ihs0| and and a “−” outcome is associated with |s2i; this is the op-

σ = p1|s1ihs1| + p2|s2ihs2|. Indeed, any protocol dis- timal measurement for distinguishing between two pure 4 states [15]. By the symmetry of the states, it is suffi- this problem is to better understand the limitations of cient to only consider the “+” outcome, which he com- processing quantum information by LOCC. Our results municates to Alice. The conditional probabilities are complement a series of recent results in this direction √ P = (1 − p)/6, P = 1/24(2 + 3)(2 + p), [9, 14, 16]. In particular is Ref. [14] which provides a A0B+|0 A0√B+|1 and PA0B+|2 = 1/24(2 − 3)(2 + p). These can be in- necessary condition for perfect discrimination by asymp- verted to give Pi|A0B+ = 2PA0B+|i. (Round III) At this totic LOCC discrimination (Prop. 1 above). Theorem 0 0 point, Alice still has three distinct states |s0i, |s1i and 1 of our paper largely extends this result as we reduce 0 0 |s2i. Here, |s1i will have the greatest probability while asymptotic minimum error discrimination of linearly in- 0 |s0i will have the smallest when p is close to 1. Alice then dependent states to asymptotic perfect discrimination. 0 ignores |s2i and performs optimal discrimination between Our proofs of C1 and C2 are the first of its kind for just |s0 i and |s0 i. Letting Q = P + P , the 0 1 0|A0B+ 1|A0B+ two qubit ensembles, and we contrast it with previous minimum error probability is given by the well-known work on the subject. C1 was first shown by Massar Helstrom bound [15] with normalized probabilities: and Popescu for two qubits randomly polarized in the s same direction [17]. However, a different distinguishabil- Q P0|A B P1|A B P (A0B+) = 1 − (1 + 1 − 4 0 + 0 + |hs0 |s0 i|2). ity measure was used and the asymptotic case was not err 2 Q2 0 1 considered. Later, Koashi et al. showed an asymptotic

By symmetry, each sequence of outcomes (Ai,Bµ) - with form of C1 for two qubit mixed states with respect to i ∈ {0, 1, 2}, µ ∈ {+, −} - occurs with the same prob- the different task of “unambiguous discrimination” [18] ability. Hence, the total error probability across all (the same can also be shown for the double trine ensem- (A0B+) branches is given by Perr = 6Perr . The plot is ble [19]). Finally, C2 has been observed by Owari and given in Fig. 1. It obtains a minimum of approximately Hayashi on mixed states and only for a special sort of dis- 6.47 × 10−2, which is smaller than the one-way optimal tinguishability measure [20]. Our work is distinct from √ of 1/2− 3/4 ≈ 6.70×10−2. The one-way optimal prob- all previous results in that it deals with pure states and ability is obtained at the point p = 1. minimum error probability, a highly natural measure of distinguishability. The fact that we consider pure ensem- bles with three states is significant since it is well-known that any two pure states can be distinguished optimally via LOCC (i.e. LOCC = GLOBAL) [21, 22]. Thus, with the double trine being a real, symmetric, and pure ensem- ble of two qubits, we have identified the simplest type of ensemble in which LOCC 6= GLOBAL for state discrim- ination. Even more, since the double trine ensemble consists of product states (i.e. no entanglement), we have shown that “nonlocality without entanglement” can exist in even the simplest types of ensembles with more than two FIG. 1. The error probability Perr using the above protocol as a function of Alice’s measurement strength p. The point states. This distinction is further sharpened by consid- p = 1 is the one-way minimum error probability. ering that LOCC 6= SEP for the optimal discrimination of the double trine. Separable operations are interest- Discussion and Conclusions: Our results for minimum ing since, like LOCC operations, they lack the ability to error discrimination of the double trine ensemble can be create entanglement. Nevertheless, SEP evidently pos- summarized as: sesses some nonlocal power as it can outperform LOCC in discriminating the double trine. Thus, entanglement GLOBAL = SEP > LOCC > LOCC→. and nonlocality can truly be regarded as two distinct re- We thus put substantial closure to a problem first posed sources, even when dealing with two qubit pure states. over 20 years ago. A primary motivation for studying We would like to thank Runyao Duan, Debbie Leung, 5 and Laura Manˇcinska for helpful discussions on the topic Finally, let ΠS be the compact, convex set of POVMs of LOCC distinguishability. with k = dim(S) elements, each having support on S. We have just shown that the continuous linear function k Pk f : ΠS → R given by f({Πi}i=1) = 1− i=1 pihψi|Πi|ψii Appendix can be maximized only by an extreme point of ΠS (rank one projectors). Convexity of Π implies that this ex- Proof of Theorem 1 S k treme point P0 := {|φiihφi|}i=1 must be unique. (b) For the asymptotic statement, we will need to en- (a) We first recall a few general facts about minimum k dow ΠS with a metric. For two POVMs P = {Π1, ..., Πk} error discrimination. A POVM {Πi} is optimal on E i=1 and P0 = {Π0 , ..., Π0 } in Π , we can define a dis- if and only if Λ ≥ p |ψ ihψ | for all |ψ i, in which the 1 k S j j j j 0 1 Pk 0 Pk tance measure by d(P, P ) = 2 i=1 kΠi − Πik1, where operator Λ := i=1 piΠi|ψiihψi| is hermitian [2, 11, 22]. 0 0 0 Pk kAk1 = Tr|A| [23]. Note that when Πi = |φiihφi| is pure, Since i=1 Πi = I, we have 1 0 0 0 0 we have 2 kΠi − |φiihφi|k1 ≥ 1 − hφi|Πi|φii [2]. k For any P = {Π }k , define Pˆ = {P Π P }k . Sup- X i i=1 S i S i=1 0 = tr[Λ − Λ] = tr[Πj(Λ − pjρj)]. pose there exists a sequence of POVMs P(n) such that j=1 (n) for any  > 0, f(Pˆ ) < Popt +  for sufficiently large n. (n) 1/2 As Pˆ is a sequence in the compact metric space ΠS, Then as Λ−pjρj ≥ 0 and tr[Πj(Λ−pjρj)] = tr[Πj (Λ− 1/2 by the Weierstrass Theorem from analysis, there will ex- pjρj)Πj ] ≥ 0, we must have that ist some convergent subsequence Pˆ(nj ) → P. Continuity ˆ(nj ) Πj(Λ − pj|ψjihψj|) = (Λ − pj|ψjihψj|)Πj = 0. (5) of f implies that limnj →∞ f(P ) = f(P0) = f(P) (re- call Popt = f(P0)). However, by part (a), the POVM

Our argument now proceeds analogously to the one given in ΠS obtaining Popt is unique and so P = P0. Thus, (nj ) in Ref. [12]. Let PS be the projector onto S, and d(Pˆ , P0) → 0, and so the error on E of each subse- k (n ) for some POVM {Πi}i=1 that obtains Popt on E, de- quence P j satisfies ˆ fine Πi = PSΠiPS. As the |ψii are linearly independent, k ⊥ ⊥ 1 X (nj ) 1 (n ) there exists a set of dual states |ψ i such that hψ |ψji = 1 − hφ |Π |φ i ≤ d(Pˆ j , P ) → 0. i i k i i i k 0 i=1 δij. We first note that Λ − pi|ψiihψi| ≥ 0 implies † Pk k (n ) Λˆ − pi|ψiihψi| ≥ 0, where Λˆ = Λˆ = pjΠˆ j|ψjihψj|. 1 P j j=1 Conversely, if 1 − k i=1hφi|Πi |φii → 0, then 1 − k (n ) (n ) Thus, the POVM { − PS, Πˆ i} also obtains Popt on E. j 1 j 2 I i=1 hφi|Πi |φii ≥ 4 kΠi − |φiihφi|k → 0 [2], which ˆ (nj ) We next note that Πj|ψji 6= 0 for all j. For if this were means d(Pˆ , P0) → 0. By continuity of f, we have ⊥ Pk (nj ) (nj ) not true for some |ψji, then we could contract with |ψj i 1 − pihψi|Π |ψii = f(Pˆ ) → Popt.   i=1 i ⊥ ˆ ⊥ to obtain 0 ≤ hψj | Λ − pj|ψjihψj| |ψj i = −pj. ˆ k Next, since {I − PS, Πi}i=1 is an optimal POVM, All Conditions of Proposition 1 Cannot be the corresponding equality of Eq. (5) is 0 = Πˆ j(Λˆ − Simultaneously Satisfied ⊥ pj|ψjihψj|). Applying |ψi i to the RHS yields

Condition (iii) requires orthogonality hFi|E|Fji = 0 for Πˆ j(piΠˆ i|ψii) = δijpiΠˆ i|ψii. (6) − 2 i 6= j, and so in the basis {|Ψ i, |Fii}i=0, E must take the form Thus, |φ i := √ 1 Πˆ |ψ i (which is nonzero) lies in i ˆ 2 i i hψi|Πi |ψii 2 − − X − the kernel of Πˆ j for i 6= j, while |φii is an eigenvector s|Ψ ihΨ | + (ai|FiihFi| + [bi|Ψ ihFi| + C.C.]) (7) i=0 of Πˆ j with eigenvalue +1 when i = j. Hence, Πˆ i = Pk P2 |φiihφi| and hφi|φji = δij, with i=1 |φiihφi| = PS. We where s, ai ≥ 0, i=0 ai = 1, and C.C. denotes the com- obviously have hφi|Πj|φii = δij, which means the original plex conjugate. If E is a product operator across Alice

POVM can perfectly distinguish the |φii. Conversely, and Bob’s system, then γ01 = Ah0|E|1iA must commute k any POVM {Πi}i=1 that perfectly distinguishes the |φii with γ10 = Ah1|E|0iA. Here we are taking partial con- will satisfy PSΠiPS = |φiihφi|, and will therefore obtain tractions on Alice’s system so that γ01 and γ10 are op-

Popt on E. erators acting on Bob’s system. By directly computing 6 the commutator using Eqs. (1) and (7), the condition σk1, where k · k1 is the trace norm. Since ρ − σ is her- P h0|[γ01, γ10]|0i = 0 becomes mitian with eigenvalues λi, we have kρ − σk1 = i |λi|. Thus, it is just a matter of computing the eigenvalues 2 2 1 0 = 6[Im(b2 − b3)] + (s + a0 − 3 )(s − 3 ). (8) of ∆ := ρ − σ = p0|ψ0ihψ0| − p1|ψ1ihψ1| − p2|ψ2ihψ2|. 1 1 Taking |ψii = ci0|0i + ci1|1i, we write ∆ in coordinates: With a0 = χ ≥ 3 (condition (ii)), it is clear that s ≤ 3 . 1 However, if s < 3 , then this equation cannot hold for any 2 ∗ ! 2 2 ∗ ! 1 2 |c00| c00c01 X |ci0| ci0ci1 a ∈ [ , − s). Thus, the product form constraint on E ∆ =p0 − pi . 0 3 3 c∗ c |c |2 c∗ c |c |2 1 00 01 01 i=1 i0 i1 i1 requires a0 = 3 . Next, we focus on the range a ∈ [ 1 , 1 ), which because a b
0 3 2 For a 2 × 2 matrix, M = c d , its eigenvalues are given p 2 of (iii), guarantees that E is full rank. It is known that by the expression λ± = 1/2(a+d± (a + d) − 4 det M). 0 the |Fi i can be perfectly distinguished by separable op- Thus, have that P2 0 0 erations if and only if i=0 C(Fi ) = C(Ψ ), where C(·)  −1/2 − is the concurrence of the state and |Ψ0i = √ E |Ψ i |a + d| if det M ≥ 0 hΨ−|E−1|Ψ−i |λ+| + |λ−| = p 2 (see Thm. 2 of [24]). We combine this with the fact  (a + d) − 4 det M if det M ≤ 0. that for a general two qubit state √ M⊗N|ϕi , its hϕ|M †M⊗N †N|ϕi Letting M = ∆, we can compute that a+d = p0 −p1 −p2 |det(M)||det(N)| and concurrence is given by is C(ϕ) × hϕ|M †M⊗N †N|ϕi [25]. Therefore after noting that C(F ) = 1/3 and writing i det ∆ =p p (1 − |hψ |ψ i|2) E = A ⊗ B, condition (iii) of Proposition 1 can be satis- 1 2 1 2 2 2 fied if and only if − p0p1(1 − |hψ0|ψ1i| ) − p0p2(1 − |hψ0|ψ2i| ).

1 X hΨ−|A−1 ⊗ B−1|Ψ−i Therefore, we arrive at the following 1 = pdet(A ⊗ B) . (9) 3 hF |A ⊗ B|F i i i i Lemma 1. For the weighted states ρ and σ, the mini- To compute hΨ−|A−1 ⊗ B−1|Ψ−i, we use Cramer’s Rule mum error probability is which says (A ⊗ B)−1 = 1 Adj(A ⊗ B), where v det(A⊗B) u 2 Adj(·) denotes the adjugate matrix. From (7), we have 1 1 u 2 X 2 2 − 2 t1 − 4p1p2(1 − |hψ1|ψ2i| ) − 4p0 pi|hψ0|ψii| − − Q3 that hΨ |Adj(A ⊗ B)|Ψ i = i=1hFi|A ⊗ B|Fii. Sub- i=1 stituting this into the above equation gives 1 1 if det(ρ−σ) ≤ 0, and 2 − 2 |p0 −p1 −p2| if det(ρ−σ) ≥ 0. Q3 1 X 1 j=1hFj|A ⊗ B|Fji 1 = Now, we use this result with Eq. (3) to prove Eq. (4). 3 p hF |A ⊗ B|F i i det(A ⊗ B) i i 2 1 Specifically, since |hsi|sji| = 4 for the trine states, we 1 a a + a a + a a ≥ 0 1 0 2 1 2 , (10) have that det ∆ is given by 3 p1/3a a a 0 1 2 −1 (3(3+cos(2φ))+32 cos(2θ)−(13+3 cos(2φ)) cos(4θ)). where we have used (7) and Hadamard’s inequality: 192 det(A ⊗ B) ≤ sa0a1a2 = 1/3a0a1a2. It is a straight- It is straightforward to verify that this is not positive for forward optimization calculation to see that under the φ ∈ [0, 2π) and θ ∈ [−π/6, π/6). Therefore, by Lemma P2 constraint i=0 ai = 1, the RHS of (10) obtains a min- 1, we obtain Eq. (4). 1 imum of 1 if and only if a0 = a1 = a2 = 3 . This proves 1 that condition (iii) is impossible whenever χ > 3 .

∗ Eq. (4) and the Minimum Error for one Mixed and [email protected] † one Pure State [email protected] [1] A. Peres and W. K. Wootters, Phys. Rev. Lett. 66, 1119 (1991). We compute an analytic formula for the minimum error [2] A. Holevo, J. Multivar. Anal. 3, 337 (1973); M. A. probability in distinguishing weighted qubit states ρ = Nielsen and I. L. Chuang, Quantum Computation and p0|ψ0ihψ0| and σ = p1|ψ1ihψ1| + p2|ψ2ihψ2|. The mini- Quantum Information (Cambridge University Press, mum error probability is given by Perr = 1/2 − 1/2kρ − 2000). 7

[3] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, ory (Academic Press, New York, 1976). A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, [16] A. M. Childs, D. Leung, L. Manˇcinska, and M. Ozols, 1895 (1993). (2012), arXiv:1206.5822. [4] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, [17] S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, (1995). Phys. Rev. A 59, 1070 (1999), quant-ph/9804053. [18] M. Koashi, F. Takenaga, T. Yamamoto, and N. Imoto, [5] B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Phys. (2007), 0709.3196. Rev. Lett. 86, 5807 (2001); D. P. DiVincenzo, D. W. [19] E. Chitambar and M.-H. Hsieh, (2013), manuscript in Leung, and B. M. Terhal, IEEE Trans. Inf. Theory 48, Preparation. 580 (2002), quant-ph/0103098. [20] M. Owari and M. Hayashi, New J. Phys. 10, 013006 [6] R. Horodecki, P. Horodecki, M. Horodecki, and (2008), 0708.3154. K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). [21] J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Phys. [7] W. Wootters, (2005), quant-ph/0506149. Rev. Lett. 85, 4972 (2000). [8] E. Chitambar, W. Cui, and H.-K. Lo, Phys. Rev. Lett. [22] S. Virmani, M. F. Sacchi, M. B. Plenio, and 108, 240504 (2012). D. Markham, Phys. Lett. A 288, 62 (2001); S. M. Bar- [9] E. Chitambar, D. Leung, L. Manˇcinska, M. Ozols, and nett and S. Croke, J. Phys. A 42, 062001 (2009). A. Winter, (2012), arXiv:1210.4583. [23] A perhaps more natural distance measure between two [10] Recall that the “Pretty Good Measurement” for an POVMs is the difference in measurement probabili- k ensemble {|φii, pi}i=1 is the POVM with elements ties, maximized over all trace one, non-negative inputs: −1/2 −1/2 Pk 1 Pk 0 piρ |φiihφi|ρ , where ρ = i=1 pi|φiihφi| [26]. maxρ 2 i=1 |tr(ρ[Πi − Πi])|. As we are only concerned [11] M. Ban, K. Kurokawa, R. Momose, and O. Hirota, Int. with issues of convergence, it suffices to consider the J. Theor. Phys. 36, 1269 (1997); H. Yuen, R. Kennedy, equivalent metric d introduced above. and M. Lax, IEEE Trans. Inf. Theory 21, 125 (1975). [24] R. Duan, Y. Feng, Y. Xin, and M. Ying, IEEE Trans. [12] C. Mochon, Phys. Rev. A 73, 032328 (2006). Inf. Theory 55, 1320 (2009), 0705.0795. [13] J. Walgate and L. Hardy, Phys. Rev. Lett. 89, 147901 [25] F. Verstraete, J. Dehaene, and B. DeMoor, Phys. Rev. (2002). A 64, 010101 (2001). [14] M. Kleinmann, H. Kampermann, and D. Bruß, Phys. [26] P. Hausladen and W. K. Wootters, J. Mod. Opt. 41, 2385 Rev. A 84, 042326 (2011), 1105.5132. (1994). [15] C. W. Helstrom, Quantum detection and estimation the-