Locally stable sets with minimum cardinality

TL;DR

This paper constructs minimal cardinality locally stable sets in multipartite quantum systems, demonstrating their strong nonlocality and resolving an open problem about their minimal size.

Contribution

It provides explicit constructions of minimal locally stable sets in bipartite and multipartite systems, confirming the lower bounds and advancing understanding of quantum nonlocality.

Findings

Constructed locally stable sets with minimum size in bipartite systems.

Established the minimal cardinality bounds for locally stable sets.

Resolved an open problem on the minimal size of such sets.

Abstract

The nonlocal set has received wide attention over recent years. Shortly before, Li and Wang arXiv:2202.09034 proposed the concept of a locally stable set: the only possible orthogonality preserving measurement on each subsystem is trivial. Locally stable sets present stronger nonlocality than those sets that are just locally indistinguishable. In this work, we focus on the constructions of locally stable sets in multipartite quantum systems. First, two lemmas are put forward to prove that an orthogonality-preserving local measurement must be trivial. Then we present the constructions of locally stable sets with minimum cardinality in bipartite quantum systems $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ $(d\geq 3)$ and $\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}$ $(3\leq d_{1}\leq d_{2})$. Moreover, for the multipartite quantum systems $(\mathbb{C}^{d})^{\otimes n}$ $(d\geq 2)$ and $\otimes^{n}_{i=1}\mathbb{C}^{d_{i}}$ $(3\leq d_{1}\leq d_{2}\leq\cdots\leq d_{n})$, we also obtain $d+1$ and $d_{n}+1$ locally stable orthogonal states respectively. Fortunately, our constructions reach the lower bound of the cardinality on the locally stable sets, which provides a positive and complete answer to an open problem raised in arXiv:2202.09034 .