Research On Majorization And Quantum Entropy

As a mathematical relation, Majorization determines the partial ordering rela-tion in two probability distributions. The formers combine majorization with von Neumann entropy to give a sequence of fundamental properties and simple appli-cations. In the article, we will give a easy proof for the necessary and sufficient condition about S(ρAB)=S(ρA)-S(ρB) under the basic knowledge of majoriza-tion. After the article [2] giving the space structure of the equality holding, on the basis, Lieb also supplies another equivalence proof for the equality, in the paper, we will prove that the three conditions are equivalent.The proposition of unified entropy and unified relative entropy provides new research topics for quantum mechanic, the perserving of the two entropies is another lightspot for this paper. We will give the mapping structure which remains the unified entropy and unified relative entropy unchanged with the help of the method of article [4] and [5].This paper is divided into three chapters, the concrete structure is as follows:Chapter1First, the basic concepts-density operators is introduced. Second, we will give some knowledge about the matrix, the theory of majorization and Schur-convex function theory, which are the fundamental theories of this paper.Chapter2First, we will give the necessary and sufficient condition about S(ρAB)=S(ρA)-S(ρB) and prove it is equivalent to the proof of [2] and [3] Then, under the basis of majorization, we will make further research about the preserving problems of unified quantum entropy.Chapter3First, the definition of Quantum relative entropy and its properties will be given; Second, we will also give the structure of maps which keeps the unified relative entropy.