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arXiv:1311.1536 · Journal of Mathematical Physics

Asymptotic state discrimination and a strict hierarchy in distinguishability norms

Eric Chitambar, Min-Hsiu Hsieh

Published Nov 1, 2014• 21 citations
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Abstract

In this paper, we consider the problem of discriminating quantum states by local operations and classical communication (LOCC) when an arbitrarily small amount of error is permitted. This paradigm is known as asymptotic state discrimination, and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic LOCC. We use this new criterion to prove a gap in the LOCC and separable distinguishability norms. We then turn to the operational advantage of using two-way classical communication over one-way communication in LOCC processing. With a simple two-qubit product state ensemble, we demonstrate a strict majorization of the two-way LOCC norm over the one-way norm.

DOI
10.1063/1.4902027

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In this paper, we consider the problem of discriminating quantum states by local operations and classical communication (LOCC) when an arbitrarily small amount of error is permitted. This paradigm is known as asymptotic state discrimination, and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic LOCC. We use this new criterion to prove a gap in the LOCC and separable distinguishability norms. We then turn to the operational advantage of using two-way classical communication over one-way communication in LOCC processing. With a simple two-qubit product state ensemble, we demonstrate a strict majorization of the two-way LOCC norm over the one-way norm.
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.
We revisit the problem of discriminating orthogonal quantum states within the local-quantum-operation-and-classical-communication (LOCC) paradigm. Our particular focus is on the asymptotic situation where the parties have infinite resources and the protocol may become arbitrarily long. Our main result is a necessary condition for perfect asymptotic LOCC discrimination. As an application, we prove that for complete product bases, unlimited resources are of no advantage. On the other hand, we identify an example for which it still remains undecided whether unlimited resources are superior.
Quantum state discrimination involves identifying a given state out of a set of possible states. When the states are mutually orthogonal, perfect state discrimination is always possible using a global measurement. In the case of multipartite systems when the parties are constrained to use multiple rounds of local operations and classical communication (LOCC), perfect state discrimination is often impossible even with the use of asymptotic LOCC, wherein an error is allowed but must vanish in the limit of an infinite number of rounds. Utilizing our recent results on asymptotic LOCC, we derive a lower bound on the error probability for LOCC discrimination of any given set of mutually orthogonal pure states. Informed by the insights gained from this lower bound, we are able to prove necessary conditions for perfect state discrimination by asymptotic LOCC. We then illustrate by example the power of these necessary conditions in significantly simplifying the determination of whether perfect discrimination of a given set of states can be accomplished arbitrarily closely using LOCC. The latter examples include a proof that perfect discrimination by asymptotic LOCC is impossible for any minimal unextendible product basis (UPB), where minimal means that for the given multipartite system no UPB with a smaller number of states can exist. We also give a simple proof that what has been called strong nonlocality without entanglement is considerably stronger than had previously been demonstrated.
Distinguishability is a fundamental and operational measure generally connected to information applications. In quantum information theory, from the postulates of quantum mechanics it often has an intrinsic limitation, which then dictates and also characterises capabilities of related information tasks. In this work, we consider discrimination between bipartite two-qubit unitary transformations by local operations and classical communication (LOCC) and its relations to entangling capabilities of given unitaries. We show that a pair of entangling unitaries which do not contain local parts, if they are perfectly distinguishable by global operations, can also be perfectly distinguishable by LOCC. There also exist non-entangling unitaries, e.g. local unitaries, that are perfectly discriminated by global operations but not by LOCC. The results show that capabilities of LOCC are strictly restricted than global operations in distinguishing bipartite unitaries for a finite number of repetitions, contrast to discrimination of a pair of bipartite states and also to asymptotic discrimination of unitaries.
We analyze the difference in the local distinguishability among the following three restrictions: (i) local operations and only one-way classical communications (one-way LOCC) are permitted; (ii) local operations and two-way classical communications (two-way LOCC) are permitted; and (iii) all separable operations are permitted. We obtain two main results concerning the discrimination between a given bipartite pure state and the completely mixed state with the condition that the given state should be detected perfectly. As the first result, we derive the optimal discrimination protocol for a bipartite pure state in cases (i) and (iii). As the second result, by constructing a concrete two-way local discrimination protocol, it is proven that case (ii) is much better than case (i), i.e. two-way classical communication remarkably improves the local distinguishability in comparison with one-way classical communication at least for a low-dimensional bipartite pure state.
We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability, thus establishing the associated Chernoff exponent. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite asymptotic separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB.