Local distinguishability of lattice states in arbitrary dimensional system

In this paper, we investigate the local distinguishability of lattice states in an arbitrary dimension system. Suppose \(d=\prod _{j=1}^{f}p_{j}^{r_{j}}\) is the prime factorization of a positive integer d. For all the lattice matrices in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d}\), we present a useful characterization of the maximal commuting set (MCS). We also show that each MCS contains exactly d matrices and there are a total of \(\prod _{j=1}^{f}\prod _{i=1}^{r_{j}}(p_{j}^{i}+1)\) distinct MCSs. Using these results of MCS, we present two methods to determine the local discrimination of lattice states. The previous results [Sci. China-Phys. Mech. Astron. 63, 280312 (2020)] can be covered by our results.