Quantum Symmetries: Approximation Properties, Operator Algebras, and Applications to Quantum Information

该研究项目旨在解决数学分析的两个领域中出现的几个基本问题，这些问题起源于量子物理学：算子代数理论和量子信息理论。将这个项目的各个方面联系在一起的共同数学结构被称为量子对称性。量子对称性可以被认为是一种丰富的对称性概念，适用于模拟自然界中出现的复杂结构。在实际环境中，数学家和科学家对研究给定系统的性质感兴趣（例如，机械系统、分子或生物群体）。理解这样一个系统的对称性--保持相关结构的系统变换--提供了对其性质的丰富见解。在许多情况下，人们遇到的对称性是由称为群的数学结构描述的。然而，对于某些复杂系统（特别是那些与量子力学现象或算子代数理论有关的系统），群不足以描述出现的相关对称性。这导致需要一个更一般的对称性数学概念，并导致了量子群的概念。量子群在20世纪80年代被正式引入，作为研究统计物理中出现的某些类型的非经典对称性的工具。从那时起，量子群（以及量子对称性的相关概念）被证明在拓扑学、量子代数、算子代数、量子概率、量子信息论和物质拓扑相的分类中有着显着的应用。本项目旨在利用各种数学和物理系统中出现的量子对称性，为算子代数理论和量子信息理论中的挑战性问题提供新的见解。本项目分为三个部分。第一部分讨论了与一类离散量子群（称为正交自由量子群）相关的冯诺依曼代数的结构理论。这里的原则目标是使用最近令人兴奋的发展自由概率论和几何量子群理论来解决一个猜想，其中指出，这些冯诺依曼代数永远不可能同构的自由群因子。这个项目的第二部分涉及一类冯诺依曼代数的结构，称为q-变形Araki-Woods代数。PI将使用这些代数的某些渐近分布对称来研究有限维近似性质和代数不可分解性结果。该项目的第三部分也是最后一部分提出了量子对称性在量子信息理论（QIT）中的全新应用。PI将研究一种新的方法，用于使用量子群的表示类别来构建量子通道的例子。这些新的量子通道的应用QIT的各种相关问题将进行调查。

This research project aims to address several fundamental problems arising in two areas of mathematical analysis that have have their origins in quantum physics: operator algebra theory and quantum information theory. The common mathematical structures that thread the various aspects of this project together are called quantum symmetries. Quantum symmetry can be thought of as an enriched notion of symmetry, adapted to model the complex structures arising in nature. In practical contexts, mathematicians and scientists are interested in studying properties of a given system (e.g., a mechanical system, a molecule, or a population of organisms). Understanding the symmetries of such a system - transformations of the system that preserve the relevant structure - provides a wealth of insight into its properties. In many contexts, the symmetries that one encounters are described by mathematical structures called groups. However, for certain complex systems (particularly those connected to quantum mechanical phenomena, or operator algebra theory), groups are insufficient to describe the relevant symmetries that arise. This results in the need for a more general mathematical notion of symmetry, and led to the concept of a quantum group. Quantum groups were formally introduced in the 1980's as a tool to study certain types of non-classical symmetries arising in statistical physics. Since that time, quantum groups (and related notions of quantum symmetry) have proved to have remarkable applications in topology, quantum algebra, operator algebras, quantum probability, quantum information theory, and the classification of topological phases of matter. By exploiting the quantum symmetries appearing in various mathematical and physical systems, this project aims to provide new insights into challenging problems in both operator algebra theory and quantum information theory.This project is divided into three components. The first part concerns the structural theory of von Neumman algebras associated to a class of discrete quantum groups, called orthogonal free quantum groups. The principle aim here is to use recent exciting developments in free probability theory and geometric quantum group theory to solve a conjecture which states that these von Neumann algebras can never be isomorphic to free group factors. The second part of this project concerns the structure of a class of von Neumann algebras, called the q-deformed Araki-Woods algebras. The PI will use certain asymptotic distributional symmetries for these algebras to investigate finite dimensional approximation properties and algebraic indecomposability results. The third and final part of this project proposes completely new applications of quantum symmetries in quantum information theory (QIT). The PI will investigate a new method for constructing examples of quantum channels using representation categories of quantum groups. Applications of these new quantum channels to various relevant questions in QIT will be investigated.