Several nonlocal sets of multipartite pure orthogonal product states

TL;DR

This paper constructs small nonlocal sets of multipartite orthogonal product states, revealing new properties of nonlocality without entanglement and their implications for quantum state discrimination and bound entangled states.

Contribution

It introduces both completable and uncompletable small nonlocal sets, and explores their properties, including their use in constructing orthogonal bases and bound entangled states.

Findings

Completable sets enable perfect LOCC discrimination with a Bell state.

Uncompletable sets contain subspace UPBs leading to bound entangled states.

A Bell state suffices for LOCC discrimination regardless of system dimension.

Abstract

It is known that there exist sets of pure orthogonal product states which cannot be perfectly distinguished by local operations and classical communication (LOCC). Such sets are nonlocal sets which exhibit nonlocality without entanglement. These nonlocal sets can be completable or uncompletable. In this work both completable and uncompletable small nonlocal sets of multipartite orthogonal product states are constructed. Apart from nonlocality, these sets have other interesting properties. In particular, the completable sets lead to the construction of a class of complete orthogonal product bases with the property that if such a basis is given then no state can be eliminated from that basis by performing orthogonality-preserving measurements. On the other hand, an uncompletable set of the present kind contains several Shifts unextendible product bases (UPBs) that belong to qubit subspaces. Identifying these subspace UPBs, it is possible to obtain a class of high-dimensional multipartite bound entangled states. Finally, it is shown that a two-qubit maximally entangled Bell state shared between any two parties is sufficient as a resource to distinguish the states of any completable set (of the above kind) perfectly by LOCC. This constitutes an example where the amount of entanglement, sufficient to accomplish the aforesaid task, depends neither on the dimension of the individual subsystems nor on the number of parties.