Quantum nonlocality of multipartite orthogonal product states

Abstract

Local distinguishability of orthogonal quantum states is an area of active research in quantum information theory. However, most of the relevant results are about local distinguishability in bipartite Hilbert space and very little is known about the multipartite case. In this paper we present a generic method to construct a completable $n$-partite $(n\ensuremath{\ge}3)$ product basis with only $2n$ members, which exhibits nonlocality without entanglement with $n$ parties, each holding a system of any finite dimension. We give an effective proof of the nonlocality of the completable multipartite product basis. In addition, we construct another incomplete multipartite product basis with a smaller number of members that cannot be distinguished by local operations and classical communication in a ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ quantum system, where $n\ensuremath{\ge}3$ and ${d}_{i}\ensuremath{\ge}2$ for $i=1,2,...,n$. The results can lead to a better understanding of the phenomenon of nonlocality without entanglement in any multipartite quantum system.