The Entanglement Criterion And Property Of Quantum State

This paper mainly studies the entanglement criterion and property of quantum state.First-ly,the separable criterion of a class of density matrix is presented by studying a special graph.Using graph theory,the property of Laplacian matrix,the positive partial transpose criterion and the relationship of degree between the vertices of graph and the corresponding vertices of partial transpose of the graph,the separable criterion of PE-matching graph in Cp(?)Cq and C3(?)C4 is given respectively.In Cp(?)Cq quantum systems,it is proven that if the partial transpose of a PE-matching graph on n = pq vertices is not a PE-matching,the density matrix of this graph is entanglement,otherwise it is PPT(positive partial transpose).It is also presented that,in C3(?)C4 quantum systems,if the density matrix of PE-matching graph on n = 3 x 4 vertices is separable.the necessary and sufficient condition is that the partial transpose of this graph is also a PE-matching graph.Secondly,we study the unextendible maximally entangled basis(UMEB)and its existence in multipartite quantum systems.We generalize the definition of UMEB in bipartite systems to multipartite quantum systems,and then since the property of maximally multipartite entangled states,we prove that there are not exist UMEB in C2(?)C2(?)C2 sys-tem.Moreover based on two types of UMEB in C2(?)C3,we construct two types of UMEB in C2(?)C3(?)C3 respectively,and prove that they are not mutually unbiased.Based on the UMEB in Cd1(?)Cd2,we construct an UMEB in Cd1(?)Cd2(?)Cd3,and when d1 = 2 and d2 =d3 = 3,they are not mutually unbiased either with above two types.