Local discrimination of lattice states via complete graph

Abstract

In this paper, we use graph theory to study the distinguishability of lattice states under local operations and classical communication (LOCC) in . Firstly, we present that for the basis of lattice unitary matrices, there are (p 2 + 1)(p + 1) distinct maximal commuting sets in p 2-dimensional system. Secondly, for any set of lattice states, we can obtain a graph G which consists of some k-complete subgraphs. Using the graph G, we show that the set can be distinguished by one-way LOCC if , where n is the number of vertices of a graph G and s k is the number of k-complete subgraphs. Moreover, we also proved that this conclusion is necessary and sufficient in . For system, our result can cover the results in (2015, Phys. Rev. A 92, 042 320); (2020, Sci. China-Phys. Mech. Astron. 63, 280 312).