Strong quantum nonlocality without entanglement in $n$-partite system with even $n$

Abstract

In multipartite systems, great progress has been made recently on the study of strong quantum nonlocality without entanglement. However, the existence of orthogonal product sets with strong quantum nonlocality in even party systems remains unknown. Here the even number is greater than four. In this paper, we successfully construct strongly nonlocal orthogonal product sets in $n$-partite systems for all even $n$, which answers the open questions given by Halder et al. [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.040403} {Phys. Rev. Lett \textbf{122}, 040403 (2019)}] and Yuan et al. [\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.042228} {Phys. Rev. A \textbf{102}, 042228 (2020)}] for any possible even party systems. Thus, we find general construction of strongly nonlocal orthogonal product sets in space $\otimes_{i=1}^{n}\mathcal{C}^{d_{i}}$ ($n,d_{i}\geq 3$) and show that there do exist incomplete orthogonal product bases that can be strongly nonlocal in any possible $n$-partite systems for all even $n$. Our newly constructed orthogonal product sets are asymmetric. We analyze the differences and connections between these sets and the known orthogonal product sets in odd party systems. In addition, we present a local state discrimination protocol for our sets by using additional entangled resource. When at least two subsystems have dimensions greater than three, the protocol consumes less entanglement than teleportation-based protocol. Strongly nonlocal set implies that the information cannot be completely accessed as long as it does not happen that all parties are together. As an application, we connect our sets with local information hiding in multipartite system.