算子矩阵的谱及其在量子信息学的应用

Spectral theory of linear operators is one of the most important topics in functional analysis, while quantum information is a new interdisciplinary field combining physics, information science and mathematics. Based on our previous results on these two fields, this project will focus on the following two aspects: . First of all, based on the results in the literature about the spectrum, Weyl spectrum and essential spectrum for the operator matrices, we will devote to exploring the perturbations and Fredholm perturbations of Browder spectrum, Drazin spectrum of operator matrices; Meanwhile, we will explore the further applications of Samuel multiplicities in the spcetral theory of linear operators and quantum information; Also, we will discuss the common properties of operators RS and SR.. Secondly, making use of the method of operator matrices, we propose to study the characterizations of the generalized quantum gates and restricted allowable generalized quantum gates and the relationship between them. Also, we will further consider some open problems of fixed points of quantum operations. . We will make great efforts to effectively combine the two fields of spectral theory of linear operators and quantum information theory.

算子谱理论是泛函分析的核心研究内容之一，而量子信息论是物理学、信息论、数学等学科结合而产生的新型交叉学科。在申请人对这两方面中若干问题都有一定研究成果的基础上，本项目继续研究以下两方面问题：. 首先，在前人关于算子矩阵的谱、Weyl 谱、本质谱等谱种的研究基础上，进一步研究包括 Browder 谱和 Drazin 谱在内的其它品种谱的C扰动和 Fredholm 扰动问题；探讨 Samuel 重数在算子谱理论乃至量子信息学上的应用；进一步研究 RS 和 SR 间的共同性质。. 其次，以算子矩阵为工具，研究广义量子门和限制可允许广义量子门的内在联系及其新刻画；进一步探讨量子操作不动点的若干公开问题。. 我们将努力实现算子矩阵谱理论与量子信息理论这两个领域的有机结合。

我们顺利地完成了预期的研究目标，取得如下研究成果：.1，用Samuel 移位重数给出了左（右）半Browder谱的谱扰动，本性扰动和可逆扰动刻画；.2，给出了广义量子门的刻画，给证明了大多数半Fredholm算子类为广义量子门，并给出了广义量子门的许多有用性质；.3，给出了左右可分解正则算子、左右可分解Fredholm正则算子集以及这些结合拓扑内点和闭包的完全刻画；.4，给出了算子RS和SR之间的更多共同性质，特别是它们的局部谱性质，并研究更为一般的AC与BA间的性质。.上述结果我们整理成多篇文章发表，均被SCI收录，详见附件。