(Seminar) Optimal transport and quantum mechanics: more facts and applications

Abstract: After showing that the extension of the Monge-Kantorovich-Wasserstein distance introduced in the talk by F. Golse is more convenient to separate density matrices than the usual Schatten topologies usually used in quantum mechanics, we shall show how (and explain why) they produce a cost for the quantum bipartite matching problem which is cheapper than the corresponding classical one. We shall then show that a quantum version of the Kantorovich duality provides a form of Knott-Smith-Brenier theorem in quantum mechanics, under technical conditions on the density matrices involved, with a suitable quantum definition of the gradient of an observable, naturally constructed on the classical one. The finite rank case, always tractable, will give rise itself to a non-gradient «flow » without classical counterpart. Finally, we will study transport associated to a semiquantum analogue of the Wasserstein distances and show that they involve a generalization the Legendre transform between classical and quantum densities. (Based on a series of works with E. Caglioti, F. Golse and C. Mouhot)

analysis of PDEs

Audience: researchers in the topic

Open PDE and analysis seminar and lectures