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

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 $d\geq3$. 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 $d\geq3$. On the other hand, we present two construction methods of nonlocal sets of OPSs in $\mathbb{C}^{m} \otimes \mathbb{C}^{n}$, where $m\geq3$ and $n\geq3.$ 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.