The Research On Bound Entangled States In Higeh Dimensional Bipartite And Multipartite Quantum Systems

This paper mainly studies the construction of bound entangled states in high dimension bipartite systems and the entanglement properties of multipartite quantum states. First of all we construct a class of mixed states in 3k(?)3k quantum systems. By applying the properties of the partitioned matrix, the positive matrix and the diagonally dominant matrix, the range criterion, and the separable criterion, the structure and the partial transpositions of their density matrices were analyzed. The research shows that these states are PPT(positive partial transposition) and entangled, thus a class of bound entangled states in high dimension bipartite systems is given. Secondly, we present a construction of new bound entangled states from given bound entan-gled for arbitrary dimensional bipartite systems. One way to construct bound entangled states is to show that these states are positive partial transpose (PPT) and violate the range criterion at the same time. By applying certain operators to given bound entangled states or to one of the subsystems of the given bound entangled states, we obtain a set of new states which are both PPT and violate the range criterion. We show that the derived bound entangled states are not local unitary equivalent to the original bound entangled states by detail examples. At last, in order to study the entanglement properties of multipartite quantum states, we construct a class of tripartite quantum states in 3(?)3(?)3 systems. Based on the above methods, combining with the multipartite quantum system partitioning knowledge, we analyzed the structure and the partial transpositions of their density matrices, the entanglement properties of different bipartite splits of these states, thus we give a detailed description about the entanglement with respect to different bipartite splits, we show that the states are bound entangled for the splits A-BC and B-AC, separable for the split AB-C. where A, B, C respectively represents subsystems of tri-partite quantum systems. In addition, a class of PPT states in 4(?)4(?)4 quantum systems was provided, applying the range criterion, then we concluded that the states are bound entangled for any bipartite splits.