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arXiv:quant-ph/0104068 · Physical Review A

Optimally conclusive discrimination of nonorthogonal entangled states by local operations and classical communications

Yi-Xin Chen, Dong Yang

Published Jan 15, 2002• 69 citations
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Abstract

We consider one copy of a quantum system prepared with equal prior probability in one of two nonorthogonal entangled states of multipartite distributed among separated parties. We demonstrate that these two states can be optimally distinguished in the sense of conclusive discrimination by local operations and classical communications alone. This proves strictly the conjecture that Virmani et al. confirmed numerically and analytically. Generally the optimal protocol requires local POVM operations which are explicitly constructed. The result manifests that distinguishable information is obtained only and completely at the last operation and all prior operations give no information about that state.

DOI
10.1103/physreva.65.022320

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We consider one copy of a quantum system prepared with equal prior probability in one of two nonorthogonal entangled states of multipartite distributed among separated parties. We demonstrate that these two states can be optimally distinguished in the sense of conclusive discrimination by local operations and classical communications alone. This proves strictly the conjecture that Virmani et al. confirmed numerically and analytically. Generally the optimal protocol requires local POVM operations which are explicitly constructed. The result manifests that distinguishable information is obtained only and completely at the last operation and all prior operations give no information about that state.
We consider one copy of a quantum system prepared in one of two nonorthogonal pure product states of multipartite distributed among separated parties. We show that there exist protocols that achieve optimal conclusive discrimination by means of local operations and classical communications as good as by global measurements. Also, we show a protocol that minimizes the average number of local operations. Our result implies that two pure product multipartite states might not have the nonlocal property although more than two can have.
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 paper, Walgate et. al. demonstrated that any two orthogonal multipartite pure states can be optimally distinguished using only local operations. We utilise their result to show that this is true for any two multiparty pure states, in the sense of inconclusive discrimination. There are also certain regimes of conclusive discrimination for which the same also applies, although we can only conjecture that the result is true for all conclusive regimes. We also discuss a class of states that can be distinguished locally according to any discrimination measure, as they can be locally recreated in the hands of one party. A consequence of this is that any two maximally entangled states can always be optimally discriminated locally, according to any figure of merit.
Remote quantum-state discrimination is a critical step for the implementation of quantum communication network and distributed quantum computation. We present a protocol for remotely implementing the unambiguous discrimination between nonorthogonal states using quantum entanglements, local operations, and classical communications. This protocol consists of a remote generalized measurement described by a positive operator valued measurement (POVM). We explicitly construct the required remote POVM. The remote POVM can be realized by performing a nonlocal controlled-rotation operation on two spatially separated qubits, one is an ancillary qubit and the other is the qubit which is encoded by two nonorthogonal states to be distinguished, and a conventional local Von Neumann orthogonal measurement on the ancilla. The particular pair of states that can be remotely and unambiguously distinguished is specified by the state of the ancilla. The probability of successful discrimination is not optimal for all admissible pairs. However, for some subset it can be very close to an optimal value in an ordinary local POVM.