Strong quantum nonlocality and unextendibility without entanglement in N-partite systems with odd N

A set of orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. Although such a phenomenon has been shown to any three-, four-, and five-partite systems, the existence of strongly nonlocal orthogonal product sets in multipartite systems remains unknown. In this paper, by using a general decomposition of the N-dimensional hypercubes, we present strongly nonlocal orthogonal product sets in N-partite systems for all odd N≥3. Based on this decomposition, we give explicit constructions of unextendible product bases in N-partite systems for odd N≥3. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.