Wasserstein distance between noncommutative dynamical systems

TL;DR

This paper explores quadratic Wasserstein distances on spaces of generalized dynamical systems within von Neumann algebras, analyzing symmetry properties and illustrating with examples from reduced dynamics.

Contribution

It introduces a framework for Wasserstein distances on noncommutative dynamical systems, highlighting symmetry aspects and providing illustrative examples.

Findings

Symmetry of Wasserstein distances can be characterized in this setting.

Asymmetric cases are also studied and understood.

Examples demonstrate the application to reduced dynamics.

Abstract

We study a class of quadratic Wasserstein distances on spaces consisting of generalized dynamical systems on a von Neumann algebra. We emphasize how symmetry of such a Wasserstein distance arises, but also study the asymmetric case. This setup is illustrated in the context of reduced dynamics, and a number of simple examples are also presented.