Interactions between noncommutative geometry and quantum information

这项研究计划将研究非对易几何的数学领域与理论量子信息科学之间的相互作用。非对易几何(NCG)是一个广泛而成熟的数学领域，它使用非对易代数、算子代数、表示理论和范畴理论中的工具来研究几何结构及其代数量子化。另一方面，量子信息理论(QIT)起源于计算机科学和量子物理，致力于量化使用量子力学系统执行信息处理任务的能力和局限性。在数学水平上，NCG和QIT场之间的相互作用是非常自然的，因为在NCG中起核心作用的非对易“坐标函数”代数也被证明是描述与现实世界量子系统相关的可观测和关联的正确的数学对象。QIT和NCG之间相互作用的一些最重要的例子包括：(1)完全正映射理论在描述开放量子系统演化中的应用；(2)两人非局域博弈理论与NCG中映射的量子空间研究之间的相互作用；以及(3)量子对称群在量子通道理论和非局域博弈中的应用。QIT和NCG之间最引人注目的互动可能是Ji-Natarajan-Vidick-Wright-袁[arxiv：2001.04383]最近的工作，其中来自量子复杂性理论和非局部对策的强大思想被用来寻找von Neumann代数中Connes嵌入问题的负解。这项提议将研究围绕NCG和QIT之间的这些互动的丰富多样的主题，包括：-两人非本地博弈的理论，特别强调与诸如图和有限度量空间等代数结构相关联的博弈。这里将研究与量子对称群和子因子理论的有趣联系。-将非局域对策推广到量子投入-量子产出对策的框架中。这里的项目与新兴的量子图理论以及量子图同态和同构的新兴概念密切相关。-作为NCG和量子组合学中感兴趣的基本对象的量子图的研究。特别是，我们将研究量子图的量子对称群，以及与量子图相关的量子Cuntz-Krieger C*-代数。-研究使用(量子)图/度量空间及其量子对称群作为von Neumann代数的Connes嵌入问题的显式反例的可能来源。该研究计划的所有方面都将支持各级高素质人才的培训(本科生研究机会、研究生学习和博士后培训)。

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).