Noncommutative Analysis with Applications to Quantum Information Theory

非交换分析及其在量子信息论中的应用

• 批准号: 2154903
• 负责人: David Blecher
• 金额: $ 22.98万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154903&HistoricalAwards=false
• 关键词: Noncommutative; Analysis; Applications; Quantum; Information

Quantum information theory is a rapidly growing area studying how information is stored, processed and communicated under the laws of quantum mechanics. It aims to utilize quantum phenomena such as entanglement and coherence to gain substantial advantages in cryptography, communication, and computational power. To this end, developing mathematical tools to study the capability and limitations of quantum information processing is very much desired. Due to the nature of quantum mechanics, the mathematical theory of quantum information processing is often noncommutative. Commutative mathematical objects are numbers and functions, where the multiplication order does not matter. Quantum physics is largely modeled by matrices and operators whose multiplication is noncommutative. Such inherent non-commutativity decides the essential connection between the theory of operator algebras and quantum information theory. Based on this connection, the Principal Investigator will use mathematical tools from operator algebras to study entropic quantities in quantum information and quantum stochastic processes, which has theoretical relevance as well as applications in quantum information and quantum physics. This project will enhance the participation of graduate and undergraduate students, especially those from underrepresented group in the mathematical sciences, in the fast-growing area of quantum information science.The objective of the project is to use functional analytic approaches to study important quantum phenomena such as entanglement and coherence. The theory of operator algebras provides many powerful tools, such as noncommutative Lp spaces and operator spaces, for the study of analysis of noncommutative objects. One goal of the project is to investigate the functional inequalities of quantum Markov semigroups, which are powerful tools in deriving convergence property of open quantum systems. Another topic is to study the quantum asymptotic equipartition properties on general von Neumann algebras, which can be used to develop resource theories and other information tasks in general infinite dimensional quantum systems. The proposed research is expected to inspire new interactions between noncommutative analysis/probability, noncommutative geometry, and noncommutative optimal transport.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

量子信息理论是一个快速增长的领域，研究了如何根据量子力学定律存储，处理和传达信息。它旨在利用量子现象，例如纠缠和连贯性，以在加密，通信和计算能力方面获得实质性优势。为此，非常需要开发数学工具来研究量子信息处理的能力和局限性。由于量子力学的性质，量子信息处理的数学理论通常是非共同的。交换数学对象是数字和函数，其中乘法顺序无关紧要。量子物理学在很大程度上是由乘法不相同的矩阵和算子建模的。这种固有的非交换性决定了操作者代数理论与量子信息理论之间的基本联系。基于这一联系，首席研究者将使用操作员代数的数学工具研究量子信息和量子随机过程中的熵量，该过程具有理论上的相关性以及量子信息和量子物理学中的应用。该项目将增强研究生和本科生的参与，尤其是数学科学中代表性不足的人群的参与，在量子信息科学的快速增长领域。运算符代数理论提供了许多强大的工具，例如非交流性LP空间和操作员空间，用于研究非交换对象的分析。该项目的一个目标是研究量子马尔可夫半群的功能不平等，这些功能不平等是在推导开放量子系统的收敛属性方面的强大工具。另一个主题是研究一般冯·诺伊曼（Von Neumann）代数的量子渐近线路特性，该特性可用于开发资源理论和其他无限尺寸量子系统中的其他信息任务。拟议的研究预计将激发非交通性分析/概率，非交易性几何形状和非交易性最佳运输之间的新相互作用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响审查标准通过评估来进行评估的。