Research On Local Preserving B-majorization Mapping

With the development of technology,people on the research of quantum information has been more and more deeply.The scholars began to combining quantum information theory with algebra theory to research on majorization problems of all kinds of space,and made some valuable results.In recent years,we began to think about the linear mapping of preserving local majorization relationship.According to the ideas and methods of exploring relationship of preserving local majorization linear mapping and preserving majorization linear mapping on M_n×mspace,we researched on relationship of preserving local B-majorization linear mapping and preserving B-majorization linear mapping on M_n×mspace.This thesis includes five chapters.Firstly,it introduces the concept of majorization,preserving majorization mapping and local preserving majorization mapping in majorization problems of quantum information theory.Based on this,the concept of local preserving B-majorization mapping is given.At the same time,some research backgrounds and research status of this paper are introduced.It expound the main theories,research purposes and significance of this paper.Secondly,it mainly introduces some concepts about preserving B-majorization linear mapping.Then,the equivalent conditions of preserving B-majorization linear mapping on M_n×mare expounded.At the same time,a representation of preserving local B-majorization linear mapping is introduced,which becomes an important lemma for the conclusion of this paper.And it laid the foundation of the following research.Thirdly,it use the knowledge of linear algebra to characterize the linear mapping of preserving Bmajorization.Mainly for the case of n ≥ 3.After simple reasoning and argumentation,it gets a necessary and sufficient condition for the relationship of preserving B-majorization linear mapping and preserving local B-majorization linear mapping.Fourthly,it continues to study the matrix expression of local preserving B-majorization with n = 2 on the basis of the previous chapter.And with this form,it is divided into 49 different situations.It is proved that in all possible cases of n = 2,the corresponding signed permutation matrix can be found to satisfy the required condition.Then,for the nature of preserving B-majorization linear mapping at n = 2,we can get the necessary and sufficient conditions that local preserving B-majorization linear mapping of such a class to be preserving B-majorization linear mapping.Finally it summarizes the whole paper,and puts forwardsome new researchable contents and problems to be solved related to this paper,which provides a new direction for the research of local preserving B-majorization.