Noncommutative Analysis in the Theory of Nonlocal Games

量子纠缠是量子力学最耐人寻味、同时也是最实用的特征之一。现在已经知道，纠缠可以完成仅使用经典资源无法完成的操作任务。非本地游戏最初是从理论计算机科学的角度进行研究的，事实证明它是研究其力量和局限性的有用工具。这个项目的目的是进一步探索这些组合和概率对象之间的有机联系，以及对非对易结构的数学分析。该项目将为目前数学上的大规模量子化计划做出贡献，并专注于从经典到量子非局域博弈的过渡。该项目的工作将直接服务于加强纯数学的跨学科，同时促进目前在全国许多层面上开展的量子倡议。该项目为本科生和研究生提供研究培训机会。该项目的骨干是由有限表示的算子系统和C*-代数及其张量积组成的。它发展了核心算符代数技术，并将其应用于量子信息理论领域，研究由量子图和经典图、超图和偏序产生的新算符代数概念。主要目标是：(I)识别图同构博弈的量子版本及其一些有用的推广，并利用算子论刻画其完美策略；(Ii)发展博弈之间策略转移的算子代数方法，并研究博弈等价的新概念；以及(Iii)通过算子张量范数提供闭合公式，以求当参与者获得强纠缠程度时，给定量子博弈的最优获胜概率。实现这些目标的技术包括算子理论、完全有界和完全正映射、算子代数的张量理论以及von Neumann代数理论。该项目揭示了具有广泛应用范围的离散结构的分析特征，如超图，引入了一种新的算子空间张量积，并研究了由等距算子和么正算子产生的新型算子系统。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.