Noncommutative Analysis with Applications to Quantum Information Theory

量子信息理论是一个快速发展的领域，研究如何在量子力学定律下存储、处理和交流信息。它旨在利用量子现象，如纠缠和相干，在密码学，通信和计算能力方面获得实质性的优势。为此，开发数学工具来研究量子信息处理的能力和局限性是非常必要的。由于量子力学的性质，量子信息处理的数学理论往往是非交换的。可交换的数学对象是数字和函数，其中乘法顺序无关紧要。量子物理在很大程度上是由矩阵和运算符建模的，它们的乘法是非交换的。这种固有的非交换性决定了算子代数理论与量子信息论之间的本质联系。基于这种联系，首席研究员将使用算子代数的数学工具来研究量子信息和量子随机过程中的熵，这在量子信息和量子物理中具有理论相关性和应用价值。该项目将提高研究生和本科生的参与，特别是那些来自数学科学中代表性不足的群体的学生，在快速发展的量子信息科学领域。该项目的目标是使用泛函分析方法来研究重要的量子现象，如纠缠和相干。算子代数理论为研究非交换对象的分析提供了许多强有力的工具，如非交换Lp空间和算子空间。本课题的目标之一是研究量子马尔可夫半群的泛函不等式，这是推导开放量子系统收敛性的有力工具。另一个课题是研究一般von Neumann代数的量子渐近均分性质，这可以用于发展一般无限维量子系统中的资源理论和其他信息任务。所提出的研究有望激发非交换分析/概率、非交换几何和非交换最优输运之间的新相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.