Abstract
Quantum nonlocality based on state discrimination describes the global property of the set of orthogonal states and has a wide range of applications in quantum cryptographic protocols. Strongest nonlocality is the strongest form of quantum nonlocality recently presented in multipartite quantum systems: a set of orthogonal multipartite quantum states is strongest nonlocal if the only orthogonality-preserving local measurements on the subsystems in every bipartition are trivial. In this work, we found a construction of strongest nonlocal sets in C^d₁⊗ C^d₂⊗ C^d₃ (2≤ d₁≤ d₂≤ d₃) of size d₂d₃+1 without stopper states. Then we obtain the strongest nonlocal sets in four-partite systems with d³+1 orthogonal states in C^d⊗ C^d⊗ C^d⊗ C^d (d≥2) and d₂d₃d₄+1 orthogonal states in C^d₁⊗ C^d₂⊗ C^d₃⊗ C^d₄ (2≤ d₁≤ d₂≤ d₃≤ d₄). Surprisingly, the number of the elements in all above constructions perfectly reaches the recent conjectured lower bound and reduces the size of the strongest nonlocal set in C^d⊗ C^d⊗ C^d⊗ C^d of [\href{https://doi.org/10.1103/PhysRevA.108.062407}{Phys. Rev. A \textbf{108}, 062407 (2023)}] by d-2. In particular, the general optimal construction of the strongest nonlocal set in four-partite system is completely solved for the first time, which further highlights the theory of quantum nonlocality from the perspective of state discrimination.