Strong quantum nonlocality in $N$-partite systems

A set of multipartite orthogonal quantum states is strongly nonlocal if it is locally irreducible for every bipartition of the subsystems [S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Phys. Rev. Lett. 122, 040403 (2019)]. Although this property has been shown in three-, four-, and five-partite systems, the existence of strongly nonlocal sets in $N$-partite systems remains unknown when $N\ge 6$. In this paper, we successfully show that a strongly nonlocal set of orthogonal entangled states exists in ${\left({\mathbb{C}}^d\right)}^{\otimes N}$ for all $N\ge 3$ and $d\ge 2$, which reveals the strong quantum nonlocality in general $N$-partite systems. For $N=3$ or 4 and $d\ge 3$, we present a strongly nonlocal set consisting of genuinely entangled states, which has a smaller size than any known strongly nonlocal orthogonal product set. Finally, we connect strong quantum nonlocality with local hiding of information as an application.

Figure 1

We represent two orbits by two cycles. For the left cycle, the orbit is ${\mathcal{O}}_{(0,0,0,1)}=\{(0,0,0,1),$ $(0,0,1,0),(0,1,0,0),(1,0,0,0)\}$, which yields the set of states ${\mathcal{S}}_{(0,0,0,1)}={\left\{\right|0\rangle }_{A_1}{|0\rangle }_{A_2}{|0\rangle }_{A_3}{|1\rangle }_{A_4}+w_4^s{|0\rangle }_{A_1}{|0\rangle }_{A_2}{|1\rangle }_{A_3}{|0\rangle }_{A_4}+w_4^{2s}{|0\rangle }_{A_1}{|1\rangle }_{A_2}{|0\rangle }_{A_3}{|0\rangle }_{A_4}+w_4^{3s}{|1\rangle }_{A_1}{|0\rangle }_{A_2}{|0\rangle }_{A_3}{|0\rangle }_{A_4}:s\in {\mathbb{Z}}_4\}$. For the right cycle, the orbit is ${\mathcal{O}}_{(0,1,0,1)}=\{(0,1,0,1),(1,0,1,0)\}$, which yields the set of states ${\mathcal{S}}_{(0,1,0,1)}={\left\{\right|0\rangle }_{A_1}{|1\rangle }_{A_2}{|0\rangle }_{A_3}{|1\rangle }_{A_4}\pm {|1\rangle }_{A_1}{|0\rangle }_{A_2}{|1\rangle }_{A_3}{|0\rangle }_{A_4}\}.$

Figure 2

Local hiding of information. The information is encoded in a strongly nonlocal set of $N$-partite orthogonal states. The so-called “boss” sends the information to his $N$ “subordinates,” $A_1,A_2,...,A_N$ through some quantum channels. If any $k$ “subordinates” are collusive, $k<N$, and they can only perform OPLMs, then the information will not be accessible at all.