  In this paper, we mainly study the local distinguishable multipartite quantum
states by local operations and classical communication (LOCC) in $m_1\otimes
m_2\otimes\ldots\otimes m_n$ , where the quantum system $m_1$ belongs to Alice,
$m_2$ belongs to Bob, \ldots 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.
