The construction of sets with strong nonlocality using fewer states

Reducing the number of states is significant for exploring which states are closely related to nonlocality. In this paper, we investigate how to construct the set exhibiting strong nonlocality using fewer states. Firstly, we focus on the tripartite system and establish a general set of orthogonal product states with the assistance of Rubik’s cube. By removing some states which differ by relative phases, the size of the set decreases from Od2 to Od while the strength of nonlocality is not affected. Furthermore, similar to the tripartite system, two irreducible sets with different nonlocality strength in four-party quantum system are demonstrated, both of which contain fewer states. Our research significantly reduces the number of states and can be used in a variety of quantum cryptographic protocols.