Locally distinguishing tripartite strongly nonlocal quantum states with entanglement resource

Recently, Yuan et al. (Phys Rev A 102:042228, 2020) construct a set of strongly nonlocal orthogonal product states in \(\mathbb {C}^{d} \otimes \mathbb {C}^{d} \otimes \mathbb {C}^{d}\), where \(d \ge 3\). Then, it is interesting how much entanglement resource is sufficient to locally distinguish these states, and very little is known about entanglement-assisted distinguishing strongly nonlocal quantum states. In this paper, we first design one entanglement-assisted state discrimination protocols for strongly nonlocal orthogonal product states in \(\mathbb {C}^{4} \otimes \mathbb {C}^{4} \otimes \mathbb {C}^{4}\) quantum system. Then, in the same way, we give how much entanglement resource is sufficient to locally distinguish these strongly nonlocal orthogonal product states in \(\mathbb {C}^{5} \otimes \mathbb {C}^{5} \otimes \mathbb {C}^{5}\), \(\mathbb {C}^{6} \otimes \mathbb {C}^{6} \otimes \mathbb {C}^{6}\) quantum systems. Finally, we propose a conjecture that less entanglement resource than quantum teleportation can be used to locally distinguish these strongly nonlocal quantum states in \(\mathbb {C}^{d} \otimes \mathbb {C}^{d} \otimes \mathbb {C}^{d}\), where \(d \ge 3\). The above results can illustrate the important role of entanglement resource in locally distinguishing orthogonal quantum states and also reveal the phenomenon of less nonlocality with more entanglement.