Novel methods to construct nonlocal sets of orthogonal product states in an arbitrary bipartite high-dimensional system

Nonlocal sets of orthogonal product states (OPSs) are widely used in quantum protocols owing to their good property. In [Phys. Rev. A 101, 062329 (2020)], the authors constructed some unextendible product bases in \({\mathbb {C}}^{m} \otimes {\mathbb {C}}^{n}\) quantum system for \(n\ge m\ge 3\). We find that a subset of their unextendible product basis (UPB) cannot be perfectly distinguished by local operations and classical communication (LOCC). We give a proof for the nonlocality of the subset with Vandermonde determinant and Kramer’s rule. Meanwhile, we give a novel method to construct a nonlocal set with only \(2(m+n)-4\) OPSs in \({\mathbb {C}}^{m} \otimes {\mathbb {C}}^{n}\) quantum system for \(m\ge 3\) and \(n\ge 3\). By comparing the number of OPSs in our nonlocal set with that of the existing results, we know that \(2(m+n)-4\) is the minimum number of OPSs to construct a nonlocal and completable set in \({\mathbb {C}}^{m} \otimes {\mathbb {C}}^{n}\) quantum system so far. This means that we give the minimum number of elements to construct a completable and nonlocal set in an arbitrary given space. Our work is of great help to understand the structure and classification of locally indistinguishable OPSs in an arbitrary bipartite high-dimensional system.