We describe an analog of Monge-Kantorovich duality for optimal couplings in free probability with respect to Biane and Voiculescu's non-commutative $L^2$ Wasserstein distance. Random variables are replaced by $m$-tuples of self-adjoint operators from a tracial von Neumann algebra. Classical convex functions are replaced by $E$-convex functions, convex functions which are defined on self-adjoint $m$-tuples in tracial von Neumann algebras and satisfy a certain monotonicity property with respect to conditional expectations. By connecting the free probabilistic optimal coupling problem with quantum information theory, we deduce from results of Musat and Rordam that there exist $m$-tuples of $n \times n$ matrices which require an infinite-dimensional algebra to optimally couple. In fact, they sometimes require a non-Connes-embeddable algebra thanks to the failure of the Connes embedding problem (Ji-Natarajan-Vidick-Wright-Yuen) and another result of Haagerup and Musat.

This is joint work with Wilfrid Gangbo, Kyeongsik Nam, and Dimitri Shlyakhtenko. Jekel was supported by NSF postdoc grant DMS-2002826.