Noncommutative Analysis in the Theory of Nonlocal Games

非局部博弈论中的非交换分析

• 批准号: 2154459
• 负责人: Ivan Todorov
• 金额: $ 26.11万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154459&HistoricalAwards=false
• 关键词: Noncommutative; Analysis; Theory; Nonlocal; Games

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.

量子力学最有趣、同时也是最实用的特征之一是量子纠缠。现在我们知道，纠缠可以完成仅使用经典资源无法完成的操作任务。非局域博弈最初是从理论计算机科学的角度进行研究的，已被证明是研究其威力和局限性的有用工具。该项目旨在进一步追求这些组合和概率对象之间的有机联系以及非交换结构的数学分析。该项目将为当前的大规模数学量化计划做出贡献，并重点关注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增强纯数学的跨学科性，同时为目前在全国多个层面上推行的量子计划做出贡献。该项目为本科生和研究生提供研究培训机会。该项目的骨干由有限呈现的算子系统和 C* 代数及其张量积构成。它开发核心算子代数技术并将其应用于量子信息论领域，研究由量子和经典图、超图和偏序产生的新算子代数概念。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概括，并使用算子理论描述其完美策略； (ii) 开发博弈间策略转移的算子代数方法并研究博弈等价的新概念； (iii) 通过算子张量范数提供封闭公式，以获得当玩家能够获得高度纠缠时赢得给定量子游戏的最佳概率。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、算子代数张量理论以及冯·诺依曼代数理论。该项目揭示了具有广泛应用的离散结构的分析特征，例如超图，引入了新的算子空间张量产品，并研究了由等距算子和酉算子产生的新颖算子系统。该奖项反映了 NSF 的法定使命，并被认为是值得的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来获得支持。

项目成果

Synchronicity for quantum non-local games

量子非局域博弈的同步性

• DOI: 10.1016/j.jfa.2022.109738
• 发表时间: 2021-06-22
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Michael Brannan;Samuel J. Harris;I. Todorov;L. Turowska
• 通讯作者: L. Turowska