Strongest quantum nonlocality in N -partite systems

The strongest quantum nonlocality is a stronger form of nonlocality than that manifested by locally indistinguishable quantum states. It has efficient applications in quantum information hiding and quantum secret sharing. Constructing a smaller-size set with the strongest nonlocality contributes to reducing resource consumption. In this paper we establish a sufficient and necessary condition for the strongest nonlocality in general N-partite systems. Then we derive the minimum size of the strongest nonlocal set with symmetry in the system (C³)^N. Notably, the genuinely entangled sets constructed in our previous work [Phys. Rev. A 109, 022220 (2024)] achieved this minimum size. In addition, we construct the strongest nonlocal genuinely entangled sets in the system (Cᵈ)^N for d≥4, which have a smaller size than the existing sets as N increases. Our results provide a meaningful complement to the current understanding of the strongest nonlocality.