Strongest nonlocal sets with small sizes
Jicun Li, Fei Shi, Xiande Zhang · Physical Review A · 2023
A set of orthogonal states is the strongest nonlocal set if it is locally stable in every bipartition, which shows the strong quantum nonlocality proposed by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. The existence of the strongest nonlocal sets with the minimum size is an open question. In this work, we partially solve this question by constructing the strongest nonlocal sets with the minimum size in $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}2$ and $2\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$, where $2\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3}$. Moreover, we also give the strongest nonlocal sets with size ${d}_{2}{d}_{3}+{d}_{1}\ensuremath{-}1$ in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$ and the strongest nonlocal sets with size ${d}^{3}+d\ensuremath{-}1$ in $d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d$. All the sizes of the strongest nonlocal sets are close to the minimum size and are smaller than all previously known constructions. As an application, our strongest nonlocal sets can be used to construct partially genuinely entangled subspaces in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$ when $2\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3}$ and ${d}_{2}\ensuremath{\ge}3$.