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arXiv:1112.3437 · Physical Review A

Entanglement, mixedness, and perfect local discrimination of orthogonal quantum states

Somshubhro Bandyopadhyay

Published Apr 17, 2012• 31 citations
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Abstract

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.

DOI
10.1103/physreva.85.042319

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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.
It is shown that the number of orthogonal density matrices that can be perfectly distinguished locally is bounded above by the total dimension over the average of maximal pure state entanglement in the supports of the density matrices. Thus we establish a direct connection between pure state entanglement and local distinguishability for any se t of orthogonal quantum states, pure or mixed. The result holds for any finite dimensional composite quantum system irrespective of the multipartite structure and dimensions of the individual su bsystems.
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.
The (im)possibility of local distinguishability of orthogonal multipartite quantum states still remains an intriguing question. Beyond ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, the problem remains unsolved even for maximally entangled states (MESs). So far, the only known condition for the local distinguishability of states is the well-known orthogonality preservation (OP). Using an upper bound on the locally accessible information for bipartite states, we derive a very simple necessary condition for any set of pairwise orthogonal MESs in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ to be perfectly locally distinguishable. It is seen that particularly when the number of pairwise orthogonal MES states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ is equal to ddd, then this necessary condition, along with the OP condition, imposes more constraints (for said states to be perfectly locally distinguishable) than the OP condition does. When testing this condition for the local distinguishability of all sets of four generalized Bell states in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$, we find that it is not only necessary but also sufficient to determine their local distinguishability. This demonstrates that the aforementioned upper bound may play a significant role in the general scenario of local distinguishability of bipartite states.
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.