Duality for optimal couplings in free probability

TL;DR

This paper extends optimal transport theory to free probability, establishing a duality for non-commutative laws and revealing unique properties of non-commutative couplings in operator algebras.

Contribution

It introduces a Monge-Kantorovich duality for non-commutative laws using new convex functions, advancing the understanding of optimal couplings in free probability.

Findings

Characterizes optimal couplings via a new duality principle.

Shows invariance of generated von Neumann algebras under optimal couplings.

Highlights complexities of non-commutative couplings, including infinite-dimensional requirements.

Abstract

We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on $\mathbb{R}^m$ are replaced by non-commutative laws of $m$-tuples. We prove an analog of the Monge-Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu's non-commutative $L^2$-Wasserstein distance using a new type of convex functions. As a consequence, we show that if $(X,Y)$ is a pair of optimally coupled $m$-tuples of non-commutative random variables in a tracial $\mathrm{W}^*$-algebra $\mathcal{A}$, then $\mathrm{W}^*((1 - t)X + tY) = \mathrm{W}^*(X,Y)$ for all $t \in (0,1)$. Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of $m$-tuples is not separable with respect to the Wasserstein distance for $m > 1$.