Locally stable sets with minimum cardinality

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: only possible orthogonality preserving measurement on each subsystem is trivial. Locally sets present stronger nonlocality than those that are just indistinguishable. In this work, we focus constructions in multipartite quantum systems. First, two lemmas put forward to prove an orthogonality-preserving local must be Then with minimum cardinality bipartite systems ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}3)$ ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2})$. Moreover, for ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}2)$ ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n})$, also obtain $d+1$ ${d}_{n}+1$ orthogonal states, respectively. Fortunately, our reach lower bound sets, which provides positive complete answer open problem raised arXiv:2202.09034.