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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715256
• 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.

量子信息理论（QIT）与相互作用原子和亚原子颗粒（量子力学）的物理学有关，同时以各种方式（测量信息，传输信息，确保其安全，纠正错误等）广泛地涉及信息的概念）。重要的是要了解QIT中概念的基本数学性质，以便能够利用其属性并超越（经典）信息理论中的可能性。的确，QIT的进步有望在高水平研究和日常生活活动方面对人类活动的深远影响。该地区的研发正在提供商业领域的重要优势。 我的研究重点是量子状态转移，量子资源理论（尤其是量子相干性和纠缠）以及积极的操作员值测量（尤其是量子概率度量）。量子状态转移涉及将量子状态从一个位置传输到量子计算机中另一个位置的能力，这是量子计算机充分发挥其功能所需的过程。连贯性和纠缠是将其经典的两个主要资源设置为QIT。量子概率措施自然出现在量子力学中，但是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。我的工作通常涉及大量化和概括，用于比较对象并捕获最佳感。 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差距，以在纯数学和QIT方面取得重大进步。 我的研究从矩阵分析和操作者理论的角度探讨了量子力学的数学基础。优化理论，凸分析，扰动理论，线性和多线性代数，矩阵理论以及近似理论的技术也起作用。