Duality for optimal couplings in free probability

• Published in: arXiv.org 2022-06
• Main Authors: Gangbo, Wilfrid, Jekel, David, Nam, Kyeongsik, Shlyakhtenko, Dimitri
• Format: Article
• Language: English
• Subjects: Algebra; Couplings; Information theory; Mathematical analysis; Operators (mathematics); Optimization; Quantum phenomena; Random variables; Statistical analysis
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2105.12351

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 \(m > 1\).