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Optimal Discrimination of Quantum States on a Two-Dimensional Hilbert Optimal Discrimination of Quantum States on a Two-Dimensional Hilbert Space by Local Operations and Classical Communication Kenji Nakahira1, 3 and Tsuyoshi Sasaki Usuda2, 3 1Yokohama Research Laboratory, Hitachi, Ltd., Yokohama, Kanagawa 244-0817, Japan 2School of Information Science and Technology, Aichi Prefectural University, Nagakute, Aichi 480-1198, Japan 3Quantum Information Science Research Center, Quantum ICT Research Institute, Tamagawa University, Machida, Tokyo 194-8610, Japan (Dated: August 15, 2018) We study the discrimination of multipartite quantum states by local operations and classical communication. We derive that any optimal discrimination of quantum states spanning a two- dimensional Hilbert space in which each party’s space is finite dimensional is possible by local operations and one-way classical communication, regardless of the optimality criterion used and how entangled the states are. I. INTRODUCTION mation theory [11]. We denote a measurement on a 2D Hilbert space as a 2D measurement. Some important examples of 2D non-projective measurements are a mea- If two or more physically separated parties cannot surement maximizing the success rate for more than two communicate quantum information, their possibilities of states on a 2D Hilbert space and a measurement giving measuring quantum states are severely restricted. Intu- the result “don’t know” with non-zero probability, such itively, product states seem to be able to be optimally as an inconclusive measurement [12–14]. distinguished using only local operations and classical communication (LOCC), while entangled states seem to Let ex be a composite Hilbert space and sub be H H be indistinguishable. However, Bennett et al. found a subspace of ex. For simplicity, we say that a mea- H that orthogonal pure product states exist that cannot be surement described by the POVM Πm on sub can { } H perfectly distinguished by LOCC [1]. Later, Walgate et be realized by LOCC (or one-way LOCC) if there exists al. proved that any two pure orthogonal states in finite- an LOCC measurement (or a one-way LOCC measure- dimensional systems can be distinguished with certainty ment) described by the POVM Em on ex such that { } H using local operations and one-way classical communica- Πm = PsubEmPsub for any index m, where Psub is the or- tion (one-way LOCC) no matter how entangled they are thogonal projection operator onto sub. If any measure- H [2]. These results encourage further investigations on the ment on sub can be realized by LOCC, then any quan- H distinguishability of quantum states by LOCC, and sev- tum states on sub can be optimally distinguished using H eral important results have been reported in the case of only LOCC. Walgate et al.’s question can be rephrased as orthogonal states [3–8]. In this paper, we consider only “Can any measurement on a 2D Hilbert space be realized finite-dimensional systems. by LOCC?” The problem of LOCC discrimination for non- We emphasize that this question would be quite dif- orthogonal states is much more complicated. One of the ficult to answer. Instead of a 2D non-projective mea- main reasons is that perfect discrimination between them surement, one might consider realizing a corresponding is impossible, even without LOCC restriction. Instead, projective measurement, which is obtained by Naimark’s optimal discrimination can be sought. Walgate et al. [2] theorem [15], by LOCC. According to Naimark’s theo- posed the question: “Can any non-orthogonal states on a rem, any non-projective measurement can be realized by two-dimensional (2D) Hilbert space be optimally distin- a projective measurement on an extended Hilbert space. arXiv:1501.06222v1 [quant-ph] 26 Jan 2015 guished by LOCC?” To definitively answer this question, However, if a 2D non-projective measurement has more we must consider all optimality criteria. Various opti- than two POVM operators, then so does the correspond- mality criteria have been suggested, such as the Bayesian ing projective measurement, and such a measurement criterion, the Neyman-Pearson criterion, and the mutual often cannot be realized by LOCC [16–18]. Thus, this information criterion, but the above question is not an- approach cannot directly answer the question. Alter- swered except for very special cases, such as an opti- natively, one might try to decompose a given 2D non- mal error-free measurement for two non-orthogonal pure projective measurement into several 2D projective mea- states [9, 10]. Another reason is that optimal discrim- surements. It is known that there exist “decompos- ination for non-orthogonal states often requires a non- able” measurements, which statistically give the same projective measurement on the space spanned by the results as randomly choosing among measurements each given states, while any orthogonal states can be per- of which has fewer POVM operators than the original fectly distinguished by projective measurement. A posi- one [19]. If a 2D measurement can be decomposed into tive operator-valued measure (POVM) is the most gen- 2D projective measurements, then from [2], it can ob- eral formulation of a measurement permitted by quantum viously be realized by LOCC. However, only a few 2D mechanics and is commonly adopted in quantum infor- non-projective measurements are decomposable [19]. 2 In this paper, we show that any 2D measurement dim B. We consider finite-dimensional systems; NA H can be realized by one-way LOCC no matter how many and NB are finite. Assume that Alice and Bob share one POVM operators it has. Our result answers the above of a known collection of L quantum states represented L question: A global measurement is not needed for a 2D by density operators ρl l=1 on and want to optimally measurement in finite-dimensional systems, regardless of discriminate between{ them} in aH certain optimality crite- the optimality criterion used. rion. Our main result is that any 2D measurement (in It is worth noting that the problem of realizing a mea- finite-dimensional systems) can be realized by one-way surement by one-way LOCC is closely related to real- LOCC (see Corollary 3), which indicates that any op- izing a quantum receiver. Realization of an optimal or timal discrimination of ρl can be realized by one-way suboptimal receiver for optical states using linear opti- LOCC. { } cal feedback (or feedforward) and photon counting has We can easily extend our result to multipartite systems been widely studied both theoretically and experimen- in a way similar to [2]. Here, let us imagine a tripar- tally [21–30]. This type of receiver performs an individ- tite system: Alice, Bob, and Charlie share two linearly ual measurement on each temporal or spatial slot. A independent quantum states ψ and φ , which can be measurement can be decomposed into such individual represented by | i | i measurements if it can be realized by one-way LOCC; ′ ′ thus, our result indicates that any 2D measurement can ψ = pn A qn BC , be decomposed into individual measurements, at least | i n | i | i X in finite-dimensional systems. It is often important to ′ ′ φ = pn A rn BC , (2) investigate whether a measurement can be realized by | i n | i | i one-way LOCC to check whether it can be implemented X using only feasible resources when the whole system is instead of (1). In (2), Bob and Charlie are first grouped spatially or temporally separated. as one party. Asuume that our main result, i.e., Corol- In Section II, we present some necessary preliminaries, lary 3, holds in a bipartite system; then, we can show that where we show that any 2D measurement can be real- any measurement on any tripartite 2D Hilbert space can ized by one-way LOCC if any measurement with finite also be realized by one-way LOCC. Indeed, Alice per- rank-one POVM operators on any 2D bipartite Hilbert forms a measurement on her system according to the space in which Alice’s subspace is two-dimensional can bipartite one-way LOCC protocol that we will propose be realized by one-way LOCC. In Section III, we recall in this paper and tells the result to Bob and Charlie. the idea of Walgate et al. [2], which provides a method Then, Bob and Charlie can again use the same protocol. for realizing a 2D projective measurement by one-way This argument can easily be extended to any multipar- LOCC. In Section IV, we consider realizing a 2D non- tite system, and thus, in the rest of paper, we consider projective measurement by one-way LOCC. We show only bipartite systems. that, by extending Walgate et al.’s idea, any measure- First, we show that our main reulst can be reduced ment with finite rank-one POVM operators on any 2D to a simpler one. For example, from [20], any quantum bipartite Hilbert space in which Alice’s subspace is two- measurement with a continuous set of outcomes (includ- dimensional can be realized by one-way LOCC (Proposi- ing the discrete outcomes) on a finite-dimensional Hilbert tions 6 and 8; also Theorem 2). We conclude the paper space is equivalent to a continuous random choice of mea- in Section V. surements with finite outcomes. Thus, it suffices to show that any 2D measurement with finite outcomes can be re- alized by one-way LOCC. We show the following lemma: II. PRELIMINARIES We first consider a bipartite system. Let ψ and φ Lemma 1 If any measurement with finite rank-one | i | i be two linearly independent quantum states shared by POVM operators on a 2D (2,N)-space (N is finite in- Alice and Bob. We can write, in general form, teger) can be realized by one-way LOCC, then any 2D measurement (in finite-dimensional systems) can be re- ψ = pn qn , A B alized by one-way LOCC.
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