【摘要】：近些年來,在量子信息理論中,越來越多的學者對各類空間上的控制間題進行了研究,也獲得了很多有價值的研究成果.其中,比較熱門的是對各種空間上的保控映射的研究.隨著這一研究領域的逐步發(fā)展,我們開始考慮對局部保持受控關系(majorization)的線性映射的研究,包括對Rn空間上的局部�？赜成渑c�？赜成涞年P系進行探究,以及分別對Rn空間上局部�？赜成浜蚆n×m空間上局部保多元受控映射的等價條件的探究.全文分為五個章節(jié).第一章首先介紹了量子信息理論中的控制問題所涉及到的受控關系、保控映射和保多元受控映射的相關概念,并且簡單介紹了與本文相關的研究背景及其研究現(xiàn)狀,最后簡要闡述了本文的主要內(nèi)容、研究目的及意義.第二章主要研究了Rn空間上的局部�？赜成渑c�？赜成涞年P系,�？赜成湟欢ㄊ蔷植勘？赜成�,并且通過反例說明其逆命題不成立.最后給出了一個局部�？赜成涑蔀楸？赜成涞某浞謼l件.第三章是在第二章的基礎上利用線性代數(shù)的知識繼續(xù)研究了Rn上的局部保控映射的矩陣刻畫形式.首先證明了如果Aφ是局部�？赜成洇账鶎木仃�,則對任意P,Q∈Pn,PAφQ也是Rn上的局部�？赜成�.然后給出了線性映射φ∈L(Rn)成為局部�？赜成涞某浞直匾獥l件,即φ是局部�？赜成洚斍覂H當存在a∈Rn,α∈R以及P∈Pn使得對任意x∈Rn,滿足φ(x)=αPx+tr(x)a.第四章將R”空間上的研究結(jié)果做了一個推廣,在矩陣空間Mn×m上定義了局部保多元受控映射這一概念,給出了線性映射Φ∈L(Mn×m)成為局部保多元受控映射的充分必要條件,即Φ是局部保多元受控映射當且僅當存在P∈Pn,R∈Mm以及A1,A2,…,Am∈Mn×m使得對任意X∈Mn×m,有其中xj表示X的第j列.第五章對全文進行了總結(jié),并提出了和本文相關的仍有待解決的問題.
[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，

本文編號：2215486