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

Abstract

Nonlocal sets of orthogonal product states (OPSs) are widely used in quantum protocols owing to their good property. Thus a lot of attention are paid to how to construct a nonlocal set of orthogonal product states though it is a difficult problem. In this paper, we propose a novel general method to construct a nonlocal set of orthogonal product states in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ for $d\ge 3$. We give an ingenious proof for the local indistinguishability of those product states. The set of product states, which are constructed by our method, has a very good structure. Subsequently, we give a construction of nonlocal set of OPSs with smaller members in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ for $d\ge 3$. On the other hand, we present two construction methods of nonlocal sets of OPSs in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$, where $m\ge 3$ and $n\ge 3.$ Furthermore, we propose the concept of isomorphism for two nonlocal sets of OPSs. Our work is of great help to understand the structure and classification of locally indistinguishable OPSs.

arXiv commentarXiv admin note: text overlap with arXiv:2003.06852

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