Bounds on the smallest sets of quantum states with special quantum nonlocality

TL;DR

This paper investigates the minimal size of quantum state sets exhibiting strong nonlocality, providing bounds and constructions for locally stable sets in multipartite systems, thus answering an open question about quantum nonlocality in high-dimensional spaces.

Contribution

It introduces the concept of locally stable sets, characterizes them via state-dependent spaces, and constructs examples in multipartite systems to bound the smallest such sets, addressing an open problem in quantum nonlocality.

Findings

Locally stable sets coincide with locally indistinguishable sets in two-qubit systems.

Constructed orthogonal sets that are locally stable under all bipartitions.

Provided bounds on the size of the smallest locally stable sets in multipartite systems.

Abstract

An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems \href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.040403}{Phys. Rev. Lett. \textbf{122}, 040403 (2019)}]. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_N} $ for any $d_i \geq 2$ and $1\leq i\leq N$?) raised in a recent paper [\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.105.022209}{Phys. Rev. A \textbf{105}, 022209 (2022)}]. Compared with all previous relevant proofs, our proof here is quite concise.