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A Return to the Optimal Detection of Quantum Information

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A Return to the Optimal Detection of Quantum Information A Return to the Optimal Detection of Quantum Information Eric Chitambar 1∗ and Min-Hsiu Hsieh 2y 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 fjDii = numerical data to support both C1 and C2, but these 2 i iπ jsii ⊗ jsiigi=0, where jsii = U j0i 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 jΨ i = 1=2(j01i − j10i). 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 fΠig 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 := fΠi gi=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 fj ii; pigi=1 if for every > 0 we have P + > Pk (n) every LOCC operation belongs to SEP [4]. 1 − i=1 pih ijΠi j ii 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 = fj ii; pigi=1, the error proba- dent pure states, the global POVM attaining minimum k bility associated with some identification POVM fΠigi=1 error consists of orthonormal, rank one projectors [11] Pk is given by 1− i=1 pih ijΠij ii. 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 = fj ii; pigi=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 fjφiigi=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 fjφiigi=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 fjφ igk 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 fjΨ i;U ⊗ U jFiig , where i=0 perfectly distinguish the states jFii given by (1). How- p p i i ever, these are three entangled states which, by a result of jFii / U ⊗ U [( 2 + 1)j00i − ( 2 − 1)j11i]: (1) Walgate and Hardy, means they cannot be distinguished The corresponding error probability is perfectly by LOCC [13]. Therefore, the global minimum p 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 jFii 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 fjF~ ihF~ jg2 with jF~ ihF~ j = i i i=0 i i crimination has been recently obtained by Kleinmann et jF ihF j + 1=3jΨ−ihΨ−j. It is fairly straightforward to i i al. [14]. Here we cite their result in its strongest form, compute that F~ = 1=2(j' ih' j + j' ih' j), where 0 + + − − adapted specifically for the problem at hand. p − j'±i = jF0i± 1=3jΨ i is a product state. This suffices 2 to prove separability of the POVM. Proposition 1 ([14]). If the states fjFiigi=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 2 [1=3; 1] there is a product operator E ≥ 0 such 3 P2 that (i) i=0hFijEjFii = 1, (ii) hF0jEjF0i = χ, and tinguishing the three jsii can always be converted into 0 1 1=2 (iii) the normalized states jFi i := p E jFii are a protocol for distinguishing ρ and σ by simply coarse- hFijEjFii perfectly distinguishable by separable operations.
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