Towards optimal transport for quantum densities

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators $L^2(\mathbf{R}^d)$, and used to estimate convergence rate various asymptotic theories context quantum mechanics. The present work proves a Kantorovich type duality theorem this variant Monge-Kantorovich or distance, discusses structure optimal couplings. Specifically, we prove that couplings involve gradient similar Brenier transport map (which is convex function), more generally, subdifferential l.s.c. function as Knott-Smith optimality criterion (see Theorem 2.12 [C. Villani: Topics Optimal Transportation, Amer. Soc. 2003]).