Noncommutative Analysis in the Theory of Nonlocal Games

量子力学中最有趣，同时也是最实用的特征之一是量子纠缠。现在已知的是，纠缠允许完成单独使用经典资源无法完成的操作任务。非局部博弈，最初是从理论计算机科学的角度来研究的，已经被证明是研究其能力和局限性的有用工具。本项目旨在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交换结构的数学分析。该项目将为当前数学中的大规模量化计划做出贡献，并侧重于从经典到量子非局部对策的过渡。该项目的工作将直接有助于加强纯数学的跨学科性，同时有助于目前在国家若干层面上进行的量子倡议。本项目为本科生和研究生提供研究训练机会。项目的主干是由有限表示算子系统和C*-代数以及它们的张量积构成的。它开发了核心算子代数技术，并将其应用于量子信息论领域，研究了量子和经典图、超图和偏序中产生的新的算子代数概念。主要目标是(i)识别图同构博弈的量子版本和它们的一些有用的推广，并使用算子理论描述它们的完美策略；（ii）发展博弈间策略迁移的算子代数方法，研究博弈等价的新概念；（iii）通过算子张量规范提供了当玩家获得强纠缠度时赢得给定量子游戏的最佳概率的封闭公式。将用于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冯·诺伊曼代数理论。该项目揭示了具有广泛应用的离散结构的解析特征，例如超图，引入了一个新的算子空间张量积，并研究了由等距算子和酉算子产生的新的算子系统。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。

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.