Operator theory and matrix analysis methods in quantum information theory

量子信息理论(QIT)涉及相互作用的原子和亚原子粒子的物理学(量子力学)，同时广泛地涉及以各种方式(测量信息、传输信息、确保信息安全、纠正错误等)的信息概念。重要的是要理解QIT中概念的基本数学性质，以便能够利用它们的性质并超越(经典)信息论中可能的东西。事实上，QIT的进步对人类活动具有深远的影响，无论是在高水平的研究方面，还是在日常生活活动方面。该领域的研究和开发在商业领域提供了重要的优势。我的研究重点是量子态转移、量子资源理论(特别是量子相干和纠缠)和正算符值度量(特别是量子概率度量)。量子态转移是指在量子计算机中准确地将量子态从一个位置传输到另一个位置的能力，这是量子计算机充分发挥其功能所必需的过程。相干和纠缠是QIT区别于经典QIT的两个主要资源。量子概率测量在量子力学中自然而然地产生，但为了完全解释在物理实验中观察到的现象，必须回答许多悬而未决的理论问题。我的工作经常涉及优化和泛化，它们被用来比较对象并捕捉到一种最佳感。我提出的研究计划的主要目标是填补有关这些重要概念的数学知识的空白，以便在纯数学和QIT方面取得重大进展。我的研究从矩阵分析和算符理论的角度探索量子力学的数学基础，来自最优化理论、凸分析、微扰理论、线性和多线性代数、矩阵理论和逼近理论的技术也发挥了作用。

Quantum information theory (QIT) relates to the physics of interacting atoms and subatomic particles (quantum mechanics), while broadly concerning the notion of information in various ways (measuring information, transmitting it, keeping it secure, correcting errors, etc). It is important to understand the underlying mathematical nature of concepts in QIT so as to be able to take advantage of their properties and go beyond what is possible in (classical) information theory. Indeed, the advancement of QIT promises far-reaching implications on human activities both in terms of high level research as well as daily life activities. Research and development in the area is offering important advantages in the business sector. My research focuses on quantum state transfer, quantum resource theory (in particular, quantum coherence and entanglement), and positive operator-valued measures (in particular, quantum probability measures). Quantum state transfer concerns the ability to accurately transmit a quantum state from one location to another within a quantum computer, a process necessary for quantum computers to function to their full capacity. Coherence and entanglement are the two major resources that set QIT apart its classical counterpart. Quantum probability measures arise naturally in quantum mechanics, yet there are many open theoretical questions that must be answered in order to fully explain the phenomena observed in physical experiments. My work often involves majorization and generalizations thereof, which are used to compare objects and capture a sense of optimality. The major goal of my proposed research program is to fill in the gaps of mathematical knowledge with respect to these important concepts in order to make significant advances in both pure mathematics and QIT.  My research explores the mathematical foundations of quantum mechanics from the point of view of matrix analysis and operator theory; techniques from optimization theory, convex analysis, perturbation theory, linear and multilinear algebra, matrix theory, and approximation theory also play a role.