Local distinguishability of quantum states in multipartite System

In this paper, we mainly study the local distinguishable multipartite quantum states by local operations and classical communication (LOCC) in $m_1\otimes m_2\otimes\ldots\otimes m_n$,wherethequantumsystem$m_1$belongstoAlice,$m_2$belongstoBob,\dots and$m_n$ belongs to Susan. We first present the pure tripartite distinguishable orthogonal quantum states by LOCC in $m_1\otimes m_2\otimes m_3$. With the conclusion in $m_1\otimes m_2\otimes m_3$, we prove distinguishability or indistinguishability of some quantum states. At last, we give the $n$-party distinguishable quantum states in $m_1\otimes m_2\otimes \cdots \otimes m_n$. Our study further reveals quantum nonlocality in multipartite high-dimensional.