Strongest nonlocal sets with minimum cardinality in tripartite systems

Strong nonlocality, proposed by Halder {\it et al}. \[\href{https://doi.org/10.1103/PhysRevLett.122.040403}{Phys. Rev. Lett. \textbf{122}, 040403 (2019)}\], is a stronger manifestation than quantum nonlocality. Subsequently, Shi {\it et al}. presented the concept of the strongest nonlocality \[\href{https://doi.org/10.22331/q-2022-01-05-619}{Quantum \textbf{6}, 619 (2022)}\]. Recently, Li and Wang \[\href{https://doi.org/10.22331/q-2023-09-07-1101}{Quantum \textbf{7}, 1101 (2023)}\] posed the conjecture about a lower bound to the cardinality of the strongest nonlocal set $\mathcal{S}$ in $\otimes _{i=1}^{n}\mathbb{C}^{d_i}$, i.e., $|\mathcal{S}|\leq \max_{i}\{\prod_{j=1}^{n}d_j/d_i+1\}$. In this work, we construct the strongest nonlocal set of size $d^2+1$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$. Furthermore, we obtain the strongest nonlocal set of size $d_{2}d_{3}+1$ in $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}\otimes \mathbb{C}^{d_3}$. 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.