QLunch: David Jekel
Speaker: David Jekel
Title: Quantum channels and the Wasserstein distance in free probability
Abstract: This talk will present connections between the Wasserstein distance in free probability and quantum channels. While quantum theory uses operators as an analog of densities, free probability uses operators as an analog of random variables. The joint distribution of several non-commutative random variables (X_1,...,X_d) is described by a tracial state on a certain C*-algebra, and Biane and Voiculescu defined an analog of Wasserstein distance for these distributions. I will explain how this Wasserstein distance can be expressed using factorizable quantum channels. This connection allows application of results of Haagerup and Musat as well as Musat and Rordam to construct n x n matrix tuples (X_1,...,X_d) and (Y_1,...,Y_d) for which optimal couplings only exist in high-dimensional algebras. It also suggests that the free Wasserstein distance may be more closely related to quantum Wasserstein distances than it seems at first glance.