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 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 W ∗ -algebra A , then W ∗ ( ( 1 - t ) X + t Y ) = W ∗ ( X , Y ) for all t ∈ ( 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 .