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 C m ⊗ C n quantum system for n ≥ m ≥ 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 C m ⊗ C n quantum system for m ≥ 3 and n ≥ 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 C m ⊗ 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. [ABSTRACT FROM AUTHOR]