Free Information Theory Techniques in von Neumann Algebras

冯·诺依曼代数出现在20世纪30年代，作为量子力学的数学框架。在经典力学中，可以同时观察和测量物理系统的各种属性-例如，所有组件的位置和速度。这些性质通常被称为可观测量。 可观测量被视为底层系统的函数，并形成一个代数-它们可以被添加和相乘。 在量子力学中，同时测量不再可能。在数学上，这反映在量子系统的观测量代数的非交换性上。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。 目前的建议是从Voiculescu的自由概率论的角度研究这种非交换的观测代数，它把观测作为随机变量。这导致了一个非常丰富的理论，导致自由概率的经典对象，如偏微分方程和布朗运动，服从于分析技术的启发，经典信息论。 该项目将通过研究生和本科生的研究机会促进人力资源开发，并将在UCLA Olga Radko捐赠数学圈的主持下支持学生。 本文主要研究冯·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，包括自由熵理论来处理的。 这包括在非交换的背景下进一步发展基于PDE的方法，并加强自由概率和随机矩阵理论之间的联系。 其中的研究方向是一个概念的尺寸是基于行为的最佳运输距离，以及应用自由信息理论技术冯诺依曼代数理论。该项目包含了各种各样的问题，有些来自现有的研究方向，有些探索新的调查路线。该奖项反映了NSF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.