Interactions between noncommutative geometry and quantum information

这项研究计划将研究非对易几何和理论量子信息科学的数学领域之间的相互作用。非交换几何（NCG）是一个广泛而成熟的数学领域，它使用非交换代数，算子代数，表示论和范畴论的工具研究几何结构及其代数量化。另一方面，量子信息理论（QIT）起源于计算机科学和量子物理学，并关注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。在数学层面上，NCG和QIT领域之间的相互作用是非常自然的，因为在NCG中起核心作用的“坐标函数”的非交换代数也被证明是描述与真实的世界量子系统相关的可观测量和相关性的正确数学对象。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）两个参与者非定域博弈理论和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定域博弈中的应用。也许QIT和NCG之间最引人注目的相互作用是Ji-Natarajan-Vidick-Wright-Yuen最近的工作[arXiv：2001.04383]，其中量子复杂性理论和非局部博弈的强大思想被用来找到冯诺依曼代数中Connes嵌入问题的负解。该提案将调查围绕NCG和QIT之间的这些相互作用为中心的丰富多样的主题，包括：-两个玩家非局部游戏的理论，特别强调与代数结构相关的游戏，如图和有限度量空间。有趣的联系与量子对称群和子因子的理论将在这里进行调查。- 将非定域博弈推广到量子输入-量子输出博弈框架。这里的项目与量子图的新兴理论以及量子图同态和同构的新兴概念有着深刻的联系。- 量子图作为NCG和量子组合学中基本感兴趣对象的研究。特别地，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（量子）图/度量空间的使用，以及它们的量子对称群作为冯·诺依曼代数的康纳斯嵌入问题的明确反例的可能来源。该研究计划的各个方面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).