Linear Mappings Of Local Preserving-majorization

In recent years, more and more mathematicians are fired with an enthusiasm for majorization on all kinds of spaces in the area of quantum information, and they have made valuable achievements. The researches on linear mappings that preserve majorization on various spaces are the most popular ones. With the development of the majorization subject, we begin to focus on the study of linear mappings of local preserving-majorization, including the relationship between the linear mappings of local preserving-majorization and the linear mappings preserving majorization on Rn as well as the equivalent of a linear mapping locally preserves majorization on Rn and the equivalent of a linear mapping locally preserving multivariate majorization on Mnxm. This thesis includes five chapters.In Chapter 1, firstly it explains the relevant concepts about majorization, multivariate majorization, linear mappings preserving majorization and linear mappings preserving multivariate majorization in quantum information theory. Then it reviews the background and development of the recent studies. Lastly it introduces the main content, purpose and meaning of this thesis.Chapter 2 shows that a linear mapping of preserving-majorization on Rn must be of local preserving-majorization and gives an example to illustrate that its contrary is not true. In addition, it discusses the sufficient condition for a linear mapping of local preserving-majorization on Rn is of preserving-majorization.Chapter 3 goes ahead to delve into the matrix structure for the linear mappings of local preserving-majorization on Rn by use of the knowledge of linear algebra. At first, it proves that if Aφ is the corresponding matrix of a linear mapping of local preserving-majorization, then PAφQ is also a linear mapping that locally preserves majorization on Rn for any P,Q∈Tn. Then it gives the necessary and sufficient condition for a linear mapping of local preserving-majorization on Rn:φ∈L(Rn) is a linear mapping of local preserving-majorization if and only if there exist α∈Rn, α∈R and P∈P, such that φ(x)= αPx+tr(x)a for any x ∈ RnChapter 4 extends the above result from Rn to Mnxm. Firstly it defines the concept of linear mappings that locally preserve multivariate majorization. Then it gives the necessary and sufficient condition for a linear mapping of local preserving- multivariate-majorization on Mn×m: φ∈L(Mn×m) locally preserves multivarite majorization if and only if there exist P∈Pn,R∈Mm and A1,A2,…,Am∈Mn×m such that φ(x)=PXR+∑mtr(x)Aj for any X∈Mn×m,where xj is the j-column Of X.Chapter 5 summarizes the thesis and gives some open problems in the future research.