Orthogonal product sets with strong quantum nonlocality on plane structure

TL;DR

This paper explores the structure of orthogonal product sets with strong quantum nonlocality, establishing a connection with plane geometry, constructing new strongly nonlocal sets, and proposing entanglement-assisted local discrimination protocols.

Contribution

It introduces a geometric approach to OPS with strong nonlocality, constructs minimal quantum state sets in multipartite systems, and develops resource-efficient local discrimination protocols.

Findings

Established a sufficient condition for triviality of orthogonality-preserving POVMs.

Constructed strongly nonlocal OPS in multipartite systems with fewer states.

Presented entanglement-assisted protocols for local discrimination using minimal resources.

Abstract

In this paper, we consider the orthogonal product set (OPS) with strong quantum nonlocality. Based on the decomposition of plane geometry, we present a sufficient condition for the triviality of orthogonality-preserving POVM on fixed subsystem and partially answer an open question given by Yuan et al. Phys. Rev. A \textbf{102}, 042228 (2020)}. The connection between the nonlocality and the plane structure of OPS is established. We successfully construct a strongly nonlocal OPS in $\mathcal{C}^{d_{A}}\otimes \mathcal{C}^{d_{B}}\otimes \mathcal{C}^{d_{C}}$ $(d_{A,B,C}\geq 4)$, which contains fewer quantum states, and generalize the structures of known OPSs to any possible three and four-partite systems. In addition, we present several entanglement-assisted protocols for perfectly local discrimination the sets. It is shown that the protocols without teleportation use less entanglement resources on average and these sets can always be discriminated locally with multiple copies of 2-qubit maximally entangled states. These results also exhibit nontrivial signification of maximally entangled states in the local discrimination of quantum states.