Nonlocal sets of orthogonal product states in arbitrary multipartite quantum system

Recently, much attention have been paid to the constructions of nonlocal multipartite orthogonal product states. Among the existing results, some are relatively complex in structure while others have many constraint conditions. In this paper, we firstly give a simple method to construct a nonlocal set of orthogonal product states in $\otimes_{j=1}^{n}\mathbb{C}^{d}$ for $d\geq 2$. Then we give an ingenious proof for local indistinguishability of the set constructed by our method. According to the characteristics of this construction method, we get a new construction of nonlocal set with fewer states in the same quantum system. Furthermore, we generalize these two results to a more general $\otimes_{i=1}^{n}\mathbb{C}^{d_{j}}$ quantum system for $d_{j}\geq 2$. Compared with the existing results, the nonlocal set of multipartite orthogonal product states constructed by our method has fewer elements and is more simpler.