Strongest nonlocal sets with minimum cardinality in tripartite systems

TL;DR

This paper constructs the smallest known strongest nonlocal sets in tripartite quantum systems, confirming a conjecture and advancing understanding of nonlocality with minimal orthogonal states.

Contribution

It provides explicit constructions of the smallest strongest nonlocal sets in tripartite systems, affirming a conjecture and improving previous bounds.

Findings

Constructed strongest nonlocal set of size d^2+1 in tripartite systems.

Achieved the lower bound for the size of strongest nonlocal sets, confirming the conjecture.

Presented the smallest known strongest nonlocal sets with minimal orthogonal states.

Abstract

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.