Separability Of Multipartite Quantum States

This paper mainly studies the separable criterion and genuine multipartite entanglement for multipartite quantum states.Firstly,the new layers method is presented according to the theory of graphs and Laplacian matrices.In the tripartite quantum systems,several properties of degree symmetric graph and partially symmetric graph are shown,and these properties are generalized to multipartite systems.We give the separability of degree symmetric graph for the tripartite and multipartite quantum systems.The relationship between partially symmetric graph and degree symmetric graph is analyzed.It is proved that every partially symmetric graph is degree symmetric.In the tripartite quantum systems,the separable criterion of partially symmetric graph is presented by the new layers method.It is proved that the density matrix of class of partially symmetric graphs is fully separable,and the separable criterion is generalized to multipartite systems.Secondly,separable criteria of two classes of density matrices are given by studying two classes of special graphs in multipartite systems.We present the separability of union graphs of simple graph in multipartite quantum systems by the definition of union graph,the method of graph s layer and the property of Laplacian matrix.We construct a class of graph,this class of graph and the property of relevant Laplacian matrices are analyzed combining the property of graph s layer.It is proved that the class of graph represents a separable state in tripartite quantum systems,and these conclusions are generalized to multipartite systems.Finally,we study the detection of genuine multipartite entanglement for multipartite qudit quantum states.In the four-partite quantum systems,we define density matrix according to special unitary group,and define some tensors for different bipartitions.We study the relation of the norm of the correlation tensors under the different bipartitions and these bounds are given.Moreover,we present an analytical sufficient criterion for detecting the genuine four-partite entanglement.Some conclusions of four-partite quantum systems are generalized to multipartite systems.We define some tensors for multipartite quantum states by the density matrix,and give some bounds of the norm of the correlation tensors under the different bipartitions in the multipartite quantum systems.