Strongest quantum nonlocality in $N$-partite systems

A set of orthogonal states possesses the strongest quantum nonlocality if only a trivial orthogonality-preserving positive operator-valued measure (POVM) can be performed for each bipartition of the subsystems. This concept originated from the strong quantum nonlocality proposed by Halder $et~al.$ [Phy. Rev. Lett. $\textbf{122}$, 040403 (2019)], which is a stronger manifestation of nonlocality based on locally indistinguishability and finds more efficient applications in quantum information hiding. However, demonstrating the triviality of orthogonality-preserving local measurements (OPLMs) is not straightforward. In this paper, we present a sufficient and necessary condition for trivial OPLMs in $N$-partite systems under certain conditions. By using our proposed condition, we deduce the minimum size of set with the strongest nonlocality in system $(\mathbb{C}^{3})^{\otimes N}$, where the genuinely entangled sets constructed in Ref. [Phys. Rev. A $\textbf{109}$, 022220 (2024)] achieve this value. As it is known that studying construction involving fewer states with strongest nonlocality contribute to reducing resource consumption in applications. Furthermore, we construct strongest nonlocal genuinely entangled sets in system $(\mathbb{C}^{d})^{\otimes N}~(d\geq4)$, which have a smaller size than the existing strongest nonlocal genuinely entangled sets as $N$ increases. Consequently, our results contribute to a better understanding of strongest nonlocality.