Strong quantum nonlocality without entanglement in multipartite quantum systems

In this paper, we generalize the concept of strong quantum nonlocality from two aspects. First, in a tripartite quantum system, we present a construction of strongly nonlocal quantum states containing $6{(d\ensuremath{-}1)}^{2}$ orthogonal product states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ and build $6{d}^{2}\ensuremath{-}8d+4$ product states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d+1}$, which have been proved to be strongly nonlocal. Obviously, both results turn out to be one order of magnitude smaller than the number of basis states ${d}^{3}$ for $d\ensuremath{\ge}3$. Second, we give explicit forms of strongly nonlocal orthogonal product basis in ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ and ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$ quantum systems, where four is the largest known number of subsystems in which there exists strong quantum nonlocality without entanglement up to now. All the results positively answer the open problems raised by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]; that is, there do exist a small number of quantum states that can demonstrate strong quantum nonlocality without entanglement.