Speaker
David Jekel, University of Copenhagen
Abstract
This talk will relate Wasserstein distances in two different noncommutative settings, that of quantum systems and that of free probability, as well as explaining several ways in which these noncommutative settings are qualitatively different than the classical setting, using results from operator algebras. All the distances considered here are generalizations of the classical Wasserstein distance of two probability measures on $\mathbb{R}^d$, or more general metric spaces, which measures the minimal $L^2$ distance of two random variables with the given distributions. Several types of Wasserstein distance have been defined in the quantum setting, notably by Carlen and Maas, by de Palma and Trevisan, and by Duvenhage. The distance of Duvenhage is defined for faithful normal states on general von Neumann algebras, and as we will see, it is very closely related to the distance that Biane and Voiculescu defined in the free probability context. After putting these Wasserstein distances in a common framework, I will explain several properties that make the non-commutative setting more challenging, including non-separability of the Wasserstein space, and a stark difference between the Wasserstein and weak-* topologies on certain state spaces.
This talk uses material from joint work with Gangbo, Nam, and Shlyakhtenko, as well as joint work in progress with Melchior Wirth.