Quantum Separability And Equivalence Of Quantum States Under Local Unitary Transformation

As the key physical resources in quantum information processing and quantum computation, the quantum entangled states have been investigated with a great deal of effort in past years. So far for generic mixed states only partial solutions are known on detection and quantification of entanglement in an operational way. In this thesis, we mainly study the separability and the equivalence under local unitary transformations (LUT) of quantum states.In Sec.3 we investigate the separability of mixed states which have some special properties on bipartite and multipartite quantum systems and present some sufficient and necessary conditions or sufficient conditions for separability of mixed states. We first study the sufficient and necessary conditions for separability of higher-dimensional quantum states with rank two on bipartite and multipartite quantum systems. Then we investigate the canonical form and separability of PPT (positive partial transpose) states on some multipartite quantum systems.For generic mixed states, concurrence is one of the well defined quantitative measures of entanglement. Nevertheless, calculation of the concurrence is a formidable task for higher dimensional case. We investigate the lower bounds of concurrence for tripartite, and even multipartite quantum mixed states in Sec. 4. By establishing functional relations between concurrence and the generalized partial transpositions (GPT), we derive analytical lower bounds of concurrence for multipartite systems.One of the important questions in quantum information theory is how to judge whether two mixed states are equivalent under LUT. One approach in dealing with the equivalence of quantum states under LUT is to find the complete set of invariants under LUT. Two states are equivalent under LUT if and only if they have the same values of all these invariants. However, the number of invariants increases quickly when the subsystems increase, so it is rather difficult to deal with the equivalenceproblem of multipartite states in terms of invariants approach. By using the analysis of fixed point subgroup and tensor decomposability of certain matrices, we first study the equivalence of non-degenerate tripartite quantum mixed states under LUT and provide an operational criterion for the equivalence of two non-degenerate mixed quantum states under LUT. Moreover the result is straightforwardly generalized to the case of multipartite non-degenerate mixed states. Finally, we give a convex decomposition of some bipartite separable Werner states in arbitrary dimensions.