Quantum optimal transport with convex regularization

The goal of this paper is to settle the study of non-commutative optimal transport problems with convex regularization, in their static and finite-dimensional formulations. We consider both the balanced and unbalanced problem and show in both cases a duality result, characterizations of minimizers (for the primal) and maximizers (for the dual). An important tool we define is a non-commutative version of the classical (c,ψ)-transforms associated with a general convex regularization, which we employ to prove the convergence of Sinkhorn iterations in the balanced case. Finally, we show the convergence of the unbalanced transport problems towards the balanced one, as well as the convergence of transforms, as the marginal penalization parameters go to +∞.