Nonlocal sets of orthogonal product states with less members in multipartite quantum systems

Abstract

The structures of nonlocal sets of orthogonal product states are of great significance to understanding the essence of quantum nonlocality. Recently, Zhen et al. [Phys. Rev. A 106:062432, 2022] propose general constructions of nonlocal sets with smaller size in multipartite systems. In this paper, we first give a novel method to construct a nonlocal set of orthogonal product states in \((\mathbb {C}^{d})^{\otimes n}\) quantum systems for \(d\ge 3\) and \(n\ge 3\). The new set has the same number of elements with zhen et al.’s set in a same quantum system while its structure is different from that of zhen et al.’s set. Then, we generalize this construction method to \(\otimes _{i=1}^{n}\mathbb {C}^{d_{i}}\) quantum system and construct a nonlocal set of states with \(\sum _{j=1}^{n-2} d_{j}+2d_{n}-n+1\) members which is lower than that of zhen et al.’s set, where \(3\le d_{1}\le d_{2}\le \cdots \le d_{n}\). Comparing with the previous works, the nonlocal sets we constructed have fewer elements and good symmetric properties. This contributes to further research on the structures of nonlocal sets in multipartite systems.