Strong quantum nonlocality for unextendible product bases in heterogeneous systems

A set of multipartite orthogonal product states is strongly nonlocal if it locally irreducible in every bipartition, which shows the phenomenon strong quantum nonlocality without entanglement. It known that unextendible bases (UPBs) can show Thus interesting to investigate for UPBs. Most UPBs with minimum size cannot demonstrate nonlocality. In this paper, we construct a series different large sizes $d_A\otimes d_B\otimes d_C$ and d_C\otimes d_D$ $d_A, d_B, d_C, d_D\geq 3$, also these have nonlocality, answers an open question given by Halder \emph{et al.} [Phys. Rev. Lett. \textbf{122}, 040403 (2019)] Yuan \textbf{102}, 042228 (2020)] any possible three four-partite systems. Furthermore, propose entanglement-assisted protocol discriminate UPB $3\otimes 3\otimes 4$, consumes less entanglement resource than teleportation-based protocol. Our results build connection between