Nonlocal sets of orthogonal product states with the less amount of elements in tripartite quantum systems

Many achievements have been made since quantum nonlocality without entanglement was found. A natural question is how many orthogonal product states (OPSs) are needed at least to form a nonlocal set in a given quantum system. In this paper, we achieve some interesting results about nonlocal sets with the less amount of OPSs. Firstly, we construct new nonlocal sets of OPSs with only 6 and 9 members in the ℂ^3⊗ℂ^3⊗ℂ^3 and ℂ^4⊗ℂ^4⊗ℂ^4 quantum systems, respectively. Secondly, we give a general construction of nonlocal set of OPSs with only 3(d-1) members in ℂ^d⊗ℂ^d⊗ℂ^d quantum system for d≥ 3 . Finally, we compare our work with existing results and make a series of generalizations. The comparison result shows that the nonlocal sets of OPSs constructed by us have the least number of elements in their corresponding quantum systems.