局部保持受控关系的线性映射
发布时间：2018-08-31 16:05
【摘要】：近些年来,在量子信息理论中,越来越多的学者对各类空间上的控制间题进行了研究,也获得了很多有价值的研究成果.其中,比较热门的是对各种空间上的保控映射的研究.随着这一研究领域的逐步发展,我们开始考虑对局部保持受控关系(majorization)的线性映射的研究,包括对Rn空间上的局部保控映射与保控映射的关系进行探究,以及分别对Rn空间上局部保控映射和Mn×m空间上局部保多元受控映射的等价条件的探究.全文分为五个章节.第一章首先介绍了量子信息理论中的控制问题所涉及到的受控关系、保控映射和保多元受控映射的相关概念,并且简单介绍了与本文相关的研究背景及其研究现状,最后简要阐述了本文的主要内容、研究目的及意义.第二章主要研究了Rn空间上的局部保控映射与保控映射的关系,保控映射一定是局部保控映射,并且通过反例说明其逆命题不成立.最后给出了一个局部保控映射成为保控映射的充分条件.第三章是在第二章的基础上利用线性代数的知识继续研究了Rn上的局部保控映射的矩阵刻画形式.首先证明了如果Aφ是局部保控映射φ所对应的矩阵,则对任意P,Q∈Pn,PAφQ也是Rn上的局部保控映射.然后给出了线性映射φ∈L(Rn)成为局部保控映射的充分必要条件,即φ是局部保控映射当且仅当存在a∈Rn,α∈R以及P∈Pn使得对任意x∈Rn,满足φ(x)=αPx+tr(x)a.第四章将R”空间上的研究结果做了一个推广,在矩阵空间Mn×m上定义了局部保多元受控映射这一概念,给出了线性映射Φ∈L(Mn×m)成为局部保多元受控映射的充分必要条件,即Φ是局部保多元受控映射当且仅当存在P∈Pn,R∈Mm以及A1,A2,…,Am∈Mn×m使得对任意X∈Mn×m,有其中xj表示X的第j列.第五章对全文进行了总结,并提出了和本文相关的仍有待解决的问题.
[Abstract]:In recent years, in the quantum information theory, more and more scholars have carried on the research on various kinds of spatial control problems, and also obtained many valuable research results. Among them, the research of preserving mapping on various spaces is more popular. With the development of this research field, we begin to study the linear mapping of locally controlled relation (majorization), including the relationship between locally preserving mapping and preserving mapping in Rn space. The equivalent conditions of locally conserved mappings on Rn spaces and locally conserved multivariate controlled mappings on Mn 脳 m spaces are studied respectively. The full text is divided into five chapters. In the first chapter, we introduce the controlled relation, the concepts of conserved mapping and multivariate controlled mapping in quantum information theory, and briefly introduce the research background and research status of this paper. At last, the main contents, purpose and significance of this paper are briefly described. In the second chapter, we study the relationship between the locally guaranteed mapping and the preserving mapping in Rn space. The preserving mapping must be a locally protected map, and the inverse proposition is not true by counterexample. Finally, a sufficient condition for a locally guaranteed map to be a preserving map is given. In chapter 3, we use the knowledge of linear algebra to study the matrix characterization of locally guaranteed mappings on Rn based on the second chapter. Firstly, it is proved that if A 蠁 is a matrix corresponding to locally guaranteed mapping 蠁, then for any PQ 鈭，
本文编号：2215486
[Abstract]:In recent years, in the quantum information theory, more and more scholars have carried on the research on various kinds of spatial control problems, and also obtained many valuable research results. Among them, the research of preserving mapping on various spaces is more popular. With the development of this research field, we begin to study the linear mapping of locally controlled relation (majorization), including the relationship between locally preserving mapping and preserving mapping in Rn space. The equivalent conditions of locally conserved mappings on Rn spaces and locally conserved multivariate controlled mappings on Mn 脳 m spaces are studied respectively. The full text is divided into five chapters. In the first chapter, we introduce the controlled relation, the concepts of conserved mapping and multivariate controlled mapping in quantum information theory, and briefly introduce the research background and research status of this paper. At last, the main contents, purpose and significance of this paper are briefly described. In the second chapter, we study the relationship between the locally guaranteed mapping and the preserving mapping in Rn space. The preserving mapping must be a locally protected map, and the inverse proposition is not true by counterexample. Finally, a sufficient condition for a locally guaranteed map to be a preserving map is given. In chapter 3, we use the knowledge of linear algebra to study the matrix characterization of locally guaranteed mappings on Rn based on the second chapter. Firstly, it is proved that if A 蠁 is a matrix corresponding to locally guaranteed mapping 蠁, then for any PQ 鈭，
本文编号：2215486
本文链接：https://www.wllwen.com/kejilunwen/yysx/2215486.html
