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arXiv:1304.1555 · Physical Review A

Revisiting the optimal detection of quantum information

Eric Chitambar, Min-Hsiu Hsieh

Published Aug 27, 2013• 57 citations
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Abstract

In 1991, Peres and 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 demonstrate ``nonlocality without entanglement'' in two-qubit pure states.

DOI
10.1103/physreva.88.020302

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In 1991, Peres and 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 demonstrate nonlocality without entanglement'' in two-qubit pure states.
In this paper, we consider the conditions under which a given ensemble of two-qubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the perfect distinguishability problem of two-qubit ensembles-both for separable operations and LOCC-by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state. Then, for the well-known task of minimum error discrimination, it is shown that almost all two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that any two pure states can be distinguished optimally by LOCC. Special attention is given to ensembles that lack entanglement, and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by LOCC, thus providing new examples of the phenomenon known as non-locality without entanglement. We next consider an example of N parties who each share the same state but who are ignorant of its identity. The state is drawn from the rotationally invariant trine ensemble, and we establish a tight connection between the N-copy ensemble and Shor's lifted single-copy ensemble. For any finite N, we prove that optimal identification of the states cannot be achieved by LOCC; however, as N→∞, LOCC can indeed discriminate the states optimally. This is the first result of its kind. Finally, we turn to the task of unambiguous discrimination and derive new lower bounds on the LOCC inconclusive probability for symmetric states. When applied to the double trine ensemble, this leads to a rather different distinguishability character than when the minimum error probability is considered.
In a recent work [Pallister, Linden, and Montanaro, Phys. Rev. Lett. 120, 170502 (2018)], Pallister et al. proposed an optimal strategy to verify nonmaximally entangled two-qubit pure states under the constraint that the accessible measurements are locally projective and nonadaptive. Their good result leads naturally to the following question: What is the optimal strategy among general local operations and classical communication (LOCC) measurements? In this paper, we answer this problem completely for two-qubit pure states. To be specific, we give the optimal strategy for each of the following available classes of measurements: (i) local operations and one-way classical communication (one-way LOCC) measurements; (ii) local operations and two-way classical communication (two-way LOCC) measurements; and (iii) separable measurements. Surprisingly, our results reveal that for the two-qubit pure state verification problem two-way LOCC measurements remarkably outperform one-way LOCC measurements and have the same power as the separable measurements.
An ensemble of product states is said to exhibit “quantum nonlocality without entanglement” if it cannot be optimally discriminated using local operations and classical communication (LOCC). We show that this property can depend on the chosen discrimination measure. Specifically, we construct a family of ensembles, each consisting of six linearly independent, equally probable bipartite product states, for which LOCC fails to achieve optimal minimum-error discrimination but succeeds in achieving optimal unambiguous discrimination. We further extend our construction to multipartite systems and provide strong numerical evidence that a similar separation between local and global optima is present for minimum-error discrimination, but not for unambiguous discrimination.
Abstract In the case of a multi-party system, through local operations and classical communication (LOCC), each party may not perform perfect discrimination of quantum states that are separable and orthogonal. This property of quantum ensemble is called “nonlocality without entanglement” since each local party has a limit to full information of given quantum states. When this property is extended to the case of minimum-error discrimination, one can see that it is revealed when a nonlocal measurement provides more information about the unentangled states than LOCC does. One may infer the fact that the property depends on quantum states composing the quantum ensemble. However, an essential but unsettled question about the property is whether an explicit dependence on prior probabilities in terms of minimum-error discrimination could be shown in nonlocality without entanglement. In a simple term, one can ask whether different quantum ensembles made of the same separable quantum states could exhibit explicitly different behavior of the nonlocality. We answer this question in the positive, and we furthermore provide the explicit functional dependence of guessing probability on prior probabilities for the mirror-symmetric ensemble.
Quantum measurements can sometimes extract more information from multiple copies of a state than from each copy separately. One of the simplest settings is the discrimination of three qubit states when two identical copies are available and one is allowed to perform an arbitrary joint measurement. Recent numerical work identified the double-trine ensemble as the apparent optimal benchmark in this setting, but an analytic proof of its optimality remained open. In this paper, we prove that the double-trine ensemble is indeed globally optimal. More precisely, among all triples of equally likely qubit states, no choice of states can give a larger optimal two-copy discrimination success probability. The result holds in full generality, whether the signal states are pure or mixed, real or complex. This exactly solved case serves as a foundation for a broader analytic understanding of optimal collective measurements.
Entangled states can help in quantum-state discrimination by local operations and classical communication (LOCC). For example, a Bell state is necessary (and sufficient) to perfectly discriminate a set of either three or four Bell states by LOCC. In this paper, we consider the task of LOCC discrimination of the states of noisy Bell ensembles, where a given ensemble consists of the states obtained by mixing the Bell states with an arbitrary two-qubit state with nonzero probabilities. It is proved that a Bell state is required for optimal discrimination by LOCC, even though the ensembles do not contain, in general, any maximally entangled state and, in specific instances, any entangled state.