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

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 [Phys. Rev. Lett. 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. Although the concept of locally stable set was proposed from the interest in mathematical properties, it also has its physical significance. One finds that locally stable sets of orthogonal product states could not be perfectly distinguishable even with the use of asymptotic local operations and classical communication (LOCC), wherein an error is allowed but must vanish in the limit of an infinite number of rounds. 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 C-d1 circle times C-d1 circle times center dot center dot center dot circle times C-dN for any d(i) >= 2 and 1 <= i <= N?) raised in a recent paper [Phys. Rev. A 105, 022209 (2022)]. Compared with all previous relevant proofs, our proof here is quite concise.