Genuinely Nonlocal Sets in General Multipartite System and their Entanglement-assisted Discrimination

A set of multipartite orthogonal states is called genuinely nonlocal if it is locally indistinguishable across every possible bipartition of the subsystems. In this paper, we first construct nonlocal sets in \( \mathbb {C}^{d_1}\otimes \mathbb {C}^{d_2}(4\le d_1\le d_2) \). Then we construct genuinely nonlocal sets in \( \mathbb {C}^{d_1}\otimes \mathbb {C}^{d_2}\otimes \mathbb {C}^{d_3}(4\le d_1\le d_2\le d_3) \). Next, we generalize our construction to general multipartite systems \( \bigotimes _{i=1}^{n}\mathbb {C}^{d_i}(4\le d_1\le d_2\le \cdots \le d_n, n\ge 4) \), constructing genuinely nonlocal sets containing \( 2\sum _{i=2}^{n}d_i-3n+5 \) states. More importantly, we use entanglement as a resource to distinguish the constructed nonlocal and genuinely nonlocal orthogonal product states. All quantum state discrimination protocols we design use only one or multiple ancillary \( \mathbb {C}^3\otimes \mathbb {C}^3 \) maximally entangled state and are more efficient than protocols based on teleportation. These findings provide a broad understanding of how to utilize entanglement resources more effectively, and they also reveal the phenomenon of less nonlocality with more entanglement.