Operator theory and matrix analysis methods in quantum information theory

量子信息理论（QIT）涉及原子和亚原子粒子相互作用的物理学（量子力学），同时广泛涉及各种方式的信息概念（测量信息，传输信息，保持信息安全，纠正错误等）。理解量子信息技术中概念的潜在数学本质是很重要的，这样才能利用它们的特性，超越（经典）信息理论的可能性。事实上，无论是在高层次的研究还是日常生活活动方面，量子信息技术的进步都有望对人类活动产生深远的影响。该地区的研究和发展为商业部门提供了重要的优势。******我的研究主要集中在量子态转移，量子资源理论（特别是量子相干和纠缠）和正算子值度量（特别是量子概率度量）。量子态转移涉及在量子计算机中精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。相干性和纠缠性是将量子量子技术与经典量子技术区别开来的两个主要资源。量子概率测量在量子力学中自然产生，但为了充分解释物理实验中观察到的现象，有许多开放的理论问题必须得到回答。我的工作经常涉及多数化和一般化，这是用来比较对象和捕捉最佳的感觉。******我提出的研究计划的主要目标是填补关于这些重要概念的数学知识空白，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子理论的角度探讨了量子力学的数学基础；优化理论、凸分析、摄动理论、线性和多线性代数、矩阵理论和近似理论等技术也发挥了作用。

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.