Research About Separability Of Quantum Mixed States

In the decades entanglement of quantum becomes one of the key resources in quan-tum computation and quantum information, it has played a remarkable role in manyapplications in the rapidly expanding ?elds of quantum information and quantum com-putation. A quantum state is called separable (or not entangled)if it can be written asa convex composition of tensor product of product states. There are two aspects in thequestion regarding quantum entanglement: the ?rst is to judge a general quantum stateis entangled or not, and the second is to answer how much entanglement remained aftersome noisy quantum process. Recently, problem about entanglement of pure states hasbeen solved, but it is still a complicated problem about mixed states.In the third chapter of this paper, for a class of mixed states with Schmidt rank in a(2k +1)×(2k +1)(k∈N) quantum system, we prove that this class of states is separableif and only if the PPT condition holds. According to PPT condition, we get 2k + 1independent linear systems, then we have the so called PPT conditions, and construct aoperational method to decompose this mixed state into separable pure states by the PPTconditions.In the forth chapter, we put emphases on the relations about Schmidt decompositionof pure state, basis change and coordinate matrix. We ?nd a su?cient condition for twoperpendicular pure states such that they have Schmidt decomposition simultaneously onthe same basis. Moreover, in a quantum system with size (2k+1)×(2k+1)(k∈N), at most2k + 1 vectors which are perpendicular with each other have the Schmidt decompositionsimultaneously (on the same basis).In the ?fth chapter, for a given density matrix, we give an exact solution for theoptimization problem of approximating any Hermitian matrix by positive semi-de?nitematrices. Then we transform the Hermitian matrices decomposition to density matricesdecomposition and a constant matrix, thus we introduce a separable indicator, and provethat a mixed state is separable if and only if the separable indicator is non-negative. Butit is di?cult to compute the separable indicator, we provide some bounds of separableindicator, and hope it can help to solve the problem of the quantum entanglement.In the last chapter, we consider the optimization problem of Hermitian matricesdecomposition of density matrix. If the Hermitian matrices in the decomposition compute with each other, we provide a method approximating the Hermitian matrix decompositionby a positive semi-de?nite matrix decomposition. At the same time, we give a appropriatereason for the existence of the semi-deonite matrix decomposition by Lie theory.