Interactions between noncommutative geometry and quantum information

点击翻译按钮获取中文摘要

This research proposal will study interactions between the mathematical fields of noncommutative geometry and theoretical quantum information science. Noncommutative geometry (NCG) is a broad and well-established area of mathematics that studies geometric structures and their algebraic quantizations using tools from noncommutative algebra, operator algebras, representation theory, and category theory. On the other hand, quantum information theory (QIT) has its origins in computer science and quantum physics, and is concerned with quantifying the power and limitations of using quantum mechanical systems to perform information processing tasks. At a mathematical level, the interaction between the fields of NCG and QIT is quite natural because of the fact that the noncommutative algebras of ``coordinate functions'' which play a central role in NCG also turn out to be the right mathematical objects to describe the observables and correlations associated to real world quantum systems. Some of the most important examples of these interactions between QIT and NCG include (1) applications of the theory of completely positive maps in describing the evolution of open quantum systems, (2) interactions between the theory of two player nonlocal games and the study of quantum spaces of maps in NCG, and (3) applications of quantum symmetry groups to quantum channel theory and nonlocal games. Perhaps the most spectacular interaction between QIT and NCG is the recent work of Ji-Natarajan-Vidick-Wright-Yuen [arXiv:2001.04383], where powerful ideas from quantum complexity theory and nonlocal games were used to find a negative solution to the Connes Embedding Problem in von Neumann algebras. This proposal will investigate a rich variety of topics centered around these interactions between NCG and QIT, including: - The theory of two player nonlocal games, with a particular emphasis on games associated to algebraic structures such as graphs and finite metric spaces. Interesting connections with the theory of quantum symmetry groups and subfactors will be investigated here. - Generalizations of nonlocal games to the framework of quantum input - quantum output games. The projects here are deeply connected to the emerging theory of quantum graphs and the emerging notions of quantum graph homomorphisms and isomorphisms. - The study of quantum graphs as fundamental objects of interest in NCG and quantum combinatorics. In particular, the we will study quantum symmetry groups of quantum graphs, and quantum Cuntz-Krieger C*-algebras associated to quantum graphs. - Investigations of the use of (quantum) graphs/metric spaces, and their quantum symmetry groups as a possible source of explicit counter-examples to the Connes Embedding Problem for von Neumann algebras. All aspects of this research program will support the training of Highly Qualified Personnel at all levels (undergraduate research opportunities, graduate study, and postdoctoral training).