The construction of sets with strong quantum nonlocality using fewer states

In this paper, we investigate the construction of orthogonal product states with strong nonlocality in multiparty quantum systems. Firstly, we focus on the tripartite system and propose a general set of orthogonal product states exhibiting strong nonlocality in $d\otimes d\otimes d$ quantum system, which contains $6{{\left( d-1 \right)}^{2}}$ states. Secondly, we find that the number of the sets constructed in this way could be further reduced. Then using $4\otimes 4\otimes 4$ and $5\otimes 5\otimes 5$ quantum systems as examples, it can be seen that when d increases, the reduced quantum state is considerable. Thirdly, by imitating the construction method of the tripartite system, two 3-divisible four-party quantum systems are proposed, $3\otimes 3\otimes 3\otimes 3$ and $4\otimes 4\otimes 4\otimes 4$, both of which contains fewer states than the existing ones. Our research gives a positive answer to an open question raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist fewer quantum states that can exhibit strong quantum nonlocality without entanglement.
Comment: Hello, I am sorry to find that this version of the paper is not rigorous in some places and needs to be revised