Construction of nonlocal multipartite quantum states

For general bipartite quantum systems, many sets of locally indistinguishable orthogonal product states have been constructed so far. Here, we first present a general method to construct multipartite orthogonal product states in $d_1\otimes d_2\otimes \cdots \otimes d_n(d_{1,2,\cdots ,n}\ge 3,n\ge 4)$ by using some locally indistinguishable bipartite orthogonal product states. And we prove that these multipartite orthogonal quantum states cannot be distinguished by local operations and classical communication. Furthermore, in $d_1\otimes d_2\otimes \cdots \otimes d_n(d_{1,2,\cdots ,n}\ge 3,n\ge 5)$, we give a general method to construct a much smaller number of locally indistinguishable multipartite orthogonal product states for even and odd $n$ separately. In addition, we also present a general method to construct complete orthogonal product bases for the multipartite quantum systems. Our results demonstrate the phenomenon of nonlocality without entanglement for the multipartite quantum systems.