Local discrimination of lattice states via adjacent matrix

We investigate the distinguishability of lattice states by local operations and classical communication (LOCC) in \({\mathbb {C}}^{p^{r}}\otimes {\mathbb {C}}^{p^{r}}\) ${\mathbb{C}}^{p^r}\otimes {\mathbb{C}}^{p^r}$ , where p is a prime. Firstly, for all the lattice matrices, we present that there are \(\prod _{a=1}^{r}(p^{a}+1)\) ${\prod }_{a=1}^r(p^a+1)$ number of distinct maximal commuting sets. Secondly, we give a criterion to determine the local discrimination of lattice states via adjacent matrix. The previous results (Phys Rev A 92:042320, 2015; Phys Scr 98:115102, 2023) can be covered by our result. Finally, we give a sufficient condition for LOCC indistinguishability of \(p^{r}\) $p^r$ lattice states.