Nonlocality without entanglement in general multipartite quantum systems

The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two lemmas related to the triviality of orthogonality-preserving local measurements. Then, we propose a general construction of nonlocal set of $n(d\ensuremath{-}1)+1$ orthogonal product states in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$. The sets of nonlocal orthogonal product states are also put forward for multipartite quantum systems with arbitrary dimensions. Our construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.