  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.
