Strongest nonlocal sets with minimum cardinality in tripartite systems
Xiao-Fan Zhen, Mao-Sheng Li, Hui-Juan Zuo · Physical Review A · 2024
Strong nonlocality, proposed by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)], is a stronger manifestation than quantum nonlocality. Subsequently, Shi et al. presented the concept of the strongest nonlocality [Quantum 6, 619 (2022)]. Recently, Li and Wang [Quantum 7, 1101 (2023)] posed the conjecture about a lower bound to the cardinality of the strongest nonlocal set $\mathcal{S}$ in ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$, i.e., $|\mathcal{S}|\ensuremath{\le}{max}_{i}{{\ensuremath{\prod}}_{j=1}^{n}{d}_{j}/{d}_{i}+1}$. In this work, we construct the strongest nonlocal set of size ${d}^{2}+1$ in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$. Furthermore, we obtain the strongest nonlocal set of size ${d}_{2}{d}_{3}+1$ in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$. Our construction reaches the lower bound, which provides an affirmative solution to Li and Wang's conjecture. ln particular, the strongest nonlocal sets we present here contain the least number of orthogonal states among the available results.