Strongest nonlocal sets with minimum cardinality in tripartite systems

Strong nonlocality, proposed by Halder { et al}. [{https://doi.org/10.1103/PhysRevLett.122.040403}{Phys. Rev. Lett. 122, 040403 (2019)}], is a stronger manifestation than quantum nonlocality. Subsequently, Shi { et al}. presented the concept of the strongest nonlocality [{https://doi.org/10.22331/q-2022-01-05-619}{Quantum 6, 619 (2022)}]. Recently, Li and Wang [{https://doi.org/10.22331/q-2023-09-07-1101}{Quantum 7, 1101 (2023)}] posed the conjecture about a lower bound to the cardinality of the strongest nonlocal set S in ⊗ ᵢ₌₁ⁿCdᵢ, i.e., |S|≤ maxᵢ\∏ⱼ₌₁ⁿdⱼ/dᵢ+1\. In this work, we construct the strongest nonlocal set of size d²+1 in Cᵈ⊗ Cᵈ⊗ Cᵈ. Furthermore, we obtain the strongest nonlocal set of size d₂d₃+1 in Cd₁⊗ Cd₂⊗ Cd₃. Our construction reaches the lower bound, which provides an affirmative solution to Li and Wang's conjecture. In particular, the strongest nonlocal sets we present here contain the least number of orthogonal states among the available results.