| In this paper, we mainly studied the application of majorization theory. Majorization plays animportant role in several mathematical areas such as matrix theory, von Neumann entropy, partiallyordered sets and infinite dimensional Hilbert space. The above discussions enriched the propertiesand application of the majorization theory and promoted the classic result such as the linear mapswhich preserve majorization into abstract algebra and infinite dimensional Hilbert space. Thisstudy provided fundamental results to the developments of quantum information theory and similarsubjects. We divided this paper into five chapters.In Chapter1, we introduced the definitions and related properties of operators, Shannonentropy, von Neumann entropy, partially ordered set, Hilbert space, the basic concept ofmajorization, and so on. Then, we simply reviewed the developments of the recent study on theseconceptions at home and abroad in recent year. Finally, we introduced the main contents of thispaper.In Chapter2, we studied the unitary equivalent quantum states.We given two quantum statewhich satisfied the theory of majorization, we given a sufficient and necessary condition of unitaryequivalence, then improved some important inequality for the von Neumann entropy by the theoryof classical majorization.In Chapter3, we generalized the classical notion of majorization in vectors to a majorizationorder for functions defined on a partially ordered set and studied the connection of classicalmajorization.In Chapter4, we extended the notion of majorization to Banach space of all bounded realsequences, and investigated some of its properties. We defined the doubly stochastic operators onlp(I)and characterized the structure of all bounded linear maps on this space which preservemajorization.In Chapter5, we summarized this paper and gave some open problems. |
