Locally distinguishing genuinely nonlocal sets with one GHZ state

Abstract

A set of multipartite orthogonal product states is deemed genuinely nonlocal if it is locally indistinguishable under any bipartition of the subsystems. The proposal of genuine nonlocality makes many people interested in the construction of genuinely nonlocal sets. However, less attention has been paid to the entanglement-assisted discrimination of genuinely nonlocal sets in multipartite systems. In this paper, we first construct genuinely nonlocal product states in \(\mathbb {C}^4\otimes \mathbb {C}^4\otimes \mathbb {C}^4\) and \(\mathbb {C}^{m+2}\otimes (\mathbb {C}^4)^{\otimes {m}}\) with a set of nonlocal product states in \(\mathbb {C}^3\otimes \mathbb {C}^4\). Second, we generalize the dimension of the system to arbitrary and construct genuinely nonlocal product states in \(\mathbb {C}^{n+1}\otimes \mathbb {C}^l\otimes \mathbb {C}^l\) and \(\mathbb {C}^{m+n-1}\otimes (\mathbb {C}^l)^{\otimes {m}}\) using a set of nonlocal product states in \(\mathbb {C}^n\otimes \mathbb {C}^l,3\le n\le l\). More importantly, we achieve a perfect discrimination for the constructed genuinely nonlocal set with only one GHZ state as a resource. From the perspective of the amount of entangled resources, our discrimination protocol is highly efficient. And the Hilbert space in which the entanglement resource we use lies has the minimum dimension, so the set of product states we construct have the minimum genuine nonlocality.