Genuinely nonlocal sets without entanglement in multipartite systems

A set of multipartite orthogonal states is genuinely nonlocal if it is locally indistinguishable in every bipartition of the subsystems. If the set is locally reducible, we say it has genuine nonlocality of type I. Otherwise, we say it has genuine nonlocality of type II. Due to the complexity of the problem, the construction of genuinely nonlocal sets in general multipartite systems has not been completely solved so far. In this paper, we first provide a nonlocal set of product states in bipartite systems. We obtain a genuinely nonlocal set of type I without entanglement in general $n$-partite systems ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $[3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3]$. Then we present two constructions with genuine nonlocality of type II in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3})$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}4)$. Our results further positively answer the open problem that there does exist a genuinely nonlocal set of type II in multipartite systems [M. S. Li, Y. L. Wang, F. Shi, and M. H. Yung, J. Phys. A: Math. Theor. 54, 445301 (2021)] and highlight its related applications in quantum information processing.