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

Quantum 8, 1349 (2024).

https://doi.org/10.22331/q-2024-05-16-1349

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 $Ngeq 3$. Based on this decomposition, we give explicit constructions of unextendible product bases in $N$-partite systems for odd $Ngeq 3$. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.