Operator theory and matrix analysis methods in quantum information theory

量子信息论中的算子理论和矩阵分析方法

• 批准号: RGPIN-2019-05276
• 负责人: Plosker, Sarah
• 金额: $ 1.38万
• 依托单位: Brandon University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738163
• 关键词: Operator; theory; matrix; analysis; methods

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.

量子信息论（QIT）涉及原子和亚原子粒子的物理学（量子力学），同时以各种方式广泛涉及信息的概念（测量信息、传输信息、保持信息安全、纠正错误等），这一点很重要。理解 QIT 中概念的潜在数学本质，以便能够利用它们的属性并超越（经典）信息理论的可能性。 事实上，QIT 的进步有望对人类活动产生深远的影响。高的条件该领域的研究和开发为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（特别是量子相干性和纠缠）和正算子值。测量（特别是量子概率测量）涉及在量子计算机内将量子状态从一个位置准确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。设置 QIT 的两大资源除了其经典对应物之外，量子概率测量自然出现在量子力学中，但为了充分解释物理实验中观察到的现象，必须回答许多开放的理论问题，我的工作经常涉及其主要化和概括。我提出的研究计划的主要目标是填补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。量子力学的基础矩阵分析和算子理论的观点；优化理论、凸分析、微扰理论、线性和多重线性代数、矩阵理论和逼近理论等技术也发挥了作用。