Noncommutative Analysis with Applications to Quantum Information Theory

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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.