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Optimal local discrimination of two multipartite pure states

S. Virmani, Sacchi, M F, Martin Bodo Plenio, Damian Markham

Published Jan 1, 2001• 96 citations
OpenAlex

Abstract

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.

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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.
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.
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.
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.
We consider different settings of the task to distinguish pure orthogonal quantum states under local operations and a limited amount of classical communication. In the first setting, the spatially separated parties are allowed to perform only local projective measurements without any classical communication during the measurements. Under such a restricted class of operations, if the states are indistinguishable, then within a second group of settings, the parties are allowed to use an additional resource for the distinguishing. The additional resource is either a pure entangled state or multiple identical copies of the given states. Comparisons between these two types of resources are also done in certain cases. Both probabilistic discrimination and perfect discrimination of the states are considered for the second group of settings. Within a third setting, the parties perform local projective measurements with a restriction on the availability of classical communication during the measurements. But in this setting the parties are allowed to use a maximally entangled state as a resource.
It is shown that local distinguishability of orthogonal mixed states can be completely characterized by the local distinguishability of their supports irrespective of entanglement and mixedness of the states. This leads to two kinds of upper bounds on the number of locally distinguishable orthogonal mixed states. The first one depends only on pure-state entanglement within the supports of the states and therefore may be easy to compute in many instances. The second bound is optimal in the sense that it optimizes the bounding quantities, not necessarily the function of entanglement alone, over all orthogonal mixed-state ensembles (satisfying certain conditions) admissible within the supports of the density matrices.