Entanglement of minimal dimension and a class of local state discrimination problems

TL;DR

This paper constructs small sets of bipartite states that are hard to distinguish locally but can be distinguished with minimal entanglement resources, highlighting the utility of entanglement in local state discrimination.

Contribution

It introduces a method to distinguish certain bipartite states using minimal entangled resources and proves the universal usefulness of any pure entangled state for this task.

Findings

Sets exist in all two-qudit spaces that are locally indistinguishable without entanglement.

A minimal-dimensional entangled state enables perfect local discrimination of these sets.

Any pure entangled state can serve as a resource for unambiguous local discrimination.

Abstract

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.