Locally stable sets without entanglement with minimum cardinality

Over the past two decades, much progress has been made in the construction of nonlocal sets without entanglement. Recently, Li and Wang [Quantum 7, 1101 (2023)] proposed an open question that an orthogonal product set (OPS) $\mathcal{S}$ is locally stable with minimum cardinality in ${⨂}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$, and the numerical lower bound for such a set was given by Wang et al. [Quantum Inf. Process. 22, 15 (2023)], $(|\mathcal{S}|\ensuremath{-}1)|\mathcal{S}|\ensuremath{\ge}{\ensuremath{\sum}}_{i=1}^{n}({d}_{i}^{2}\ensuremath{-}1)$, but the existence of the construction remains unresolved. In this work we construct locally stable sets with minimum cardinalities ${d}^{2}$ and ${d}^{2}+1$ via 1-orthogonal cyclic product sets in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}}$ for odd $d$ and ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}+1}$ for even $d$, respectively, thereby partly addressing Li and Wang's open question and confirming the existence of the locally stable OPS with the lower bound. Furthermore, we present a locally stable orthogonal cyclic product set of size $n$ in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$ and a locally stable OPS with smaller size in general multipartite systems. Our results show a perfectly cyclic structure of locally stable OPSs and provide new insight into the role of local stability without entanglement in quantum information theory.