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arXiv:0708.3154 · New Journal of Physics

Two-way classical communication remarkably improves local distinguishability

Masaki Owari, Masahito Hayashi

Published Jan 14, 2008• 33 citations
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Abstract

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.

DOI
10.1088/1367-2630/10/1/013006

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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.
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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.
The problem studied in this paper is, if the number of queries to unitary operations is fixed, say kkk, then when do local operations and classical communication (LOCC) suffice for optimally distinguishing bipartite unitary operations? We consider the above problem for two-qubit unitary operations in the case of k=1k=1k=1, showing that for two two-qubit entangling unitary operations without local parties, LOCC achieves the same distinguishability as the global operations. Specifically, we obtain the following: (i) if such two unitary operations are perfectly distinguishable by global operations, then they are perfectly distinguishable by LOCC also, and (ii) if they are not perfectly distinguishable by global operations, then LOCC can achieve the same optimal discrimination probability as the global operations.
Unambiguous state discrimination of two mixed bipartite states via local operations and classical communications (LOCC) is studied and compared with the result of a scheme realized via global measurement. We show that the success probability of a global scheme for mixed-state discrimination can be achieved perfectly by the local scheme. In addition, we simulate this discrimination via a pair of pure entangled bipartite states. This simulation is perfect for local rather than global schemes due to the existence of entanglement and global coherence in the pure states. We also prove that LOCC protocol and the sequential state discrimination (SSD) can be interpreted in a unified view. We then hybridize the LOCC protocol with three protocols (SSD, reproducing, and broadcasting) relying on classical communications. Such hybridizations extend the gaps between the optimal success probability of global and local schemes, which can be eliminated only for the SSD rather than the other two protocols.
In this paper, we treat a local discrimination problem in the framework of asymmetric hypothesis testing. We choose a known bipartite pure state |ψ) as an alternative hypothesis and the completely mixed state as a null hypothesis. As a result, we analytically derive an optimal type-2 error and an optimal positive operator valued measure (POVM) for one-way local operations and classical communication (LOCC) POVM and separable POVM. For two-way LOCC POVM, we study a family of simple three-step LOCC protocols, and show that the best protocol in this family has strictly better performance than any one-way LOCC protocol in low-dimensional systems when there may exist differences between two-way LOCC POVM and one-way LOCC POVM.
In this work, we construct small sets of bipartite orthogonal pure states that cannot be perfectly distinguished by local operations and classical communication (LOCC). We mention that not all the states within the constructed sets are necessarily entangled. However, such a set contains at least one entangled state which cannot be conclusively identified by LOCC (with nonzero probability). Then, we show that the states of any such set can be perfectly distinguished by LOCC using a minimal-dimensional entangled resource state. Clearly, here the entangled resource state provides an advantage irrespective of the dimension of the given set. Using this result, we also prove that any pure entangled state is useful as a resource to distinguish the states of any present set unambiguously with nonzero probability under LOCC. These sets exist in all two-qudit Hilbert spaces. Furthermore, it is possible to decrease the average entanglement content of such a set or to increase the cardinality of the set without changing certain properties of the same.