Free Information Theory Techniques in von Neumann Algebras

冯·诺伊曼代数作为量子力学的数学框架出现在20世纪30年代。在经典力学中，同时观察和测量一个物理系统的各种特性是可能的——例如，它所有组成部分的位置和速度。这样的属性通常被称为可观察对象。可观察对象被视为底层系统的函数，形成一个代数——它们可以相加和相乘。在量子力学中，同时测量是不可能的。在数学上，这反映在量子系统的可观测代数的非交换性上。尽管如此，许多可以用普通函数完成的操作都有量子类似物。本文从Voiculescu的自由概率论的角度研究了这类可观测物的非交换代数，该理论将可观测物视为随机变量。这导致了一个极其丰富的理论，导致经典对象的自由概率推广，如偏微分方程和布朗运动，适用于由经典信息论启发的技术分析。该项目将通过研究生和本科生的研究机会来促进人力资源的发展，并将在加州大学洛杉矶分校奥尔加·拉德科捐赠数学圈的赞助下支持学生。本文的研究涉及冯·诺依曼代数中的几个问题，这些问题是用自由概率和自由信息方法来解决的，包括自由熵理论。这包括在非交换环境下进一步发展基于偏微分方程的方法，加强自由概率和随机矩阵理论之间的联系。其中一个研究方向是基于最优运输距离行为的维度概念，以及自由信息论技术在冯·诺伊曼代数理论中的应用。该项目包括各种各样的问题，有些来自现有的研究方向，有些则探索新的研究方向。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。

Von Neumann algebras arose in the 1930s as a mathematical framework for quantum mechanics. In classical mechanics it is possible to simultaneously observe and measure various properties of a physical system — for example, the locations and velocities of all of its components. Such properties are often called observables. Observables be viewed as functions of the underlying system and form an algebra — they can be added and multiplied. In quantum mechanics, simultaneous measurements are no longer possible. Mathematically this is reflected by the non-commutativity of the algebra of observables for quantum systems. Nonetheless, many of the operations that can be done with ordinary functions have quantum analogs. The current proposal studies such non-commutative algebras of observables from the angle of Voiculescu’s free probability theory, which treats observables as random variables. This results in an extremely rich theory that leads to free probability generalizations of classical objects such as partial differential equations and Brownian motion, amenable to analysis by techniques inspired by classical information theory. This project will promote human resource development through graduate and undergraduate research opportunities and will support students under the auspices of the UCLA Olga Radko Endowed Math Circle. The proposed research deals with several questions in von Neumann algebras which are approached by free probability and free information methods, including free entropy theory. This includes further developing PDE based methods in the non-commutative context and strengthening the connection between free probability and random matrix theory. Among the research directions is a notion of dimension that is based on the behavior of optimal transportation distance, as well as applications of free information theory techniques to von Neumann algebra theory. The project includes a mixture of problems, some coming from existing research directions and some exploring new lines of inquiry.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.