Towards Optimal Transport for Quantum Densities
An analogue of the quadratic Wasserstein (or Monge Kantorovich) distance between Borel probability measures on R has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015) 165 205] for density operators on L2(Rd) and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function) or more generally, to the subdifferential of a l.s.c. convex function as in the Knott Smith optimality criterion (see Theorem 2.12 in [C. Villani: "Topics in Optimal Transportation" Amer. Math. Soc. 2003]