Separability of Two Classes of Special Graphs in Multipartite Quantum Systems

Quantum entanglement is one of the most fascinating features of quantum theory and has numerous applications in quantum information processing and communication. Many unsolved problems in classical information theory can be solved by bipartite entanglement and multipartite entanglement. In this paper, the separable criterion of classes of density matrices was studied. The separable criterion of two classes of graphs was presented by studying two classes of special graphs in multipartite systems. Firstly, the concept of union graph was generalized. The separability of union graphs of simple graph in multipartite quantum systems was proven by the method of graph's layer and the property of Laplacian matrices. Secondly, a class of graph was constructed by the concept of partially symmetric and graph's layer. Combining the properties of graph and graph's layer, this class of graph and the properties of the relevant Laplacian matrices were analyzed. The research shows that the classes of graph represent a separable state in multipartite quantum systems.