Noncommutative Analysis in the Theory of Nonlocal Games

University Of Delaware, Newark DE

Investigators

Abstract

One of the most intriguing and, at the same time, practically useful features of quantum mechanics is quantum entanglement. It is now known that entanglement allows the accomplishment of operational tasks that are impossible to perform by using classical resources alone. Nonlocal games, originally studied from the perspective of theoretical computer science, have proved to be a useful tool for the study of its power and limitations. This project is aimed at pursuing further the organic links between these combinatorial and probabilistic objects and mathematical analysis on noncommutative structures. The project will contribute to the current large-scale quantization program in mathematics and focuses on the passage from classical to quantum nonlocal games. The work of the project will serve directly the enhancement of interdisciplinarity in pure mathematics, while contributing to the quantum initiatives currently pursued at a number of levels nationally. The project provides research training opportunities for undergraduate and graduate students. The backbone of the project is formed by finitely presented operator systems and C*-algebras, and their tensor products. It develops core operator algebraic techniques and applies them in areas of quantum information theory, studying new operator algebraic concepts arising from quantum and classical graphs, hypergraphs and partial orders. The main objectives are to (i) identify quantum versions of the graph isomorphism games and some of their useful generalizations, and characterize their perfect strategies using operator theory; (ii) develop operator algebraic methods of strategy transfer between games and study a new notion of game equivalence; and (iii) provide closed formulas via operator tensor norms for the optimal probability of winning a given quantum game when the players have access to a strong degree of entanglement. The techniques that will be used to achieve these goals include operator theory, completely bounded and completely positive maps, tensor theory for operator algebras as well as von Neumann algebra theory. The project reveals analytical features of discrete structures with a broad spectrum of applications, such as hypergraphs, introduces a new operator space tensor product, and studies novel operator systems arising from isometric and unitary operators. 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.