Positive Linear Maps And Inequalities On Operator Systems

In this dissertation, we study positive linear maps and inequalities on operator sys-tems. It contains four chapters. In Chapter1, we introduce the background and basic definitions of this dissertation.In Chapter2, we construct a class of linear maps "(?)(n,σ)[a; c1,c2,..., cn]" on finite di-mensional operator system Mn. The conditions when these maps are positive, completely positive, atomic and decomposable are given. We obtain new atomic and decomposable positive linear maps. As applications in quantum information theory, we provide con-ditions when the structure physical approximations of these maps are separable. Using these maps, we obtain new optimal entanglement witnesses.In Chapter3, we give the definition of entanglement breaking map on operator sys-tems, which is a generalization of matrix cases. Then we give the definition of strong entanglement breaking map. We develop key properties of these maps, and obtain condi-tions when the completely positive linear maps between operator systems coincide with the strong entanglement breaking maps. Especially, we show that a linear map from a nuclear operator system to the maximal operator system structure of an Archimedean ordered*-vector space is completely positive if and only if it is strong entanglement break-ing. We also discuss the relationships between strong entanglement breaking map and weak*-entanglement breaking map, and give a nuclear characterization of strong entangle-ment breaking map. At the end of this Chapter, we discuss the relations between mapping cones consisting of positive linear maps and operator system structures on matrices.In Chapter4, we discuss operator-valued Jensen’s inequalities on unital C*-algebras which are operator systems with multiplication. Let21be a unital C*-algebra. Firstly, we give the definition of21-convex function. We show a discrete operator-valued Jensen’s inequality on21. Using the Bochner integral of vector-valued functions, we obtain a continuous operator-valued Jensen’s inequality on21. Finally, as the main theorem of this Chapter, we obtain a continuous operator-valued Jensen’s inequality on unital21-fields.