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

• Published in: Quantum information processing Vol. 20; no. 4
• Main Authors: Xu, Guang-Bao, Jiang, Dong-Huan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2021; Springer Nature B.V
• Subjects: Data Structures and Information Theory; Mathematical Physics; Physics; Physics and Astronomy; Quantum Computing; Quantum Information Technology; Quantum Physics; Quantum theory; Spintronics; Quantum nonlocality without entanglement; Orthogonal product state; Local operations and classical communication
• ISSN: 1570-0755; 1573-1332
• DOI: 10.1007/s11128-021-03062-8

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.