Quantum and Semiquantum Pseudometrics and Applications

Abstract

We establish a Kantorovich duality for the pseudometric ${\mathcal{E}}_{\hslash }$ introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223 (2017), 57--94], obtained from the usual Monge-Kantorovich distance $d_{MK,2}$ between classical densities by quantization of one of the two densities involved. We show several type of inequalities comparing $d_{MK,2}$, ${\mathcal{E}}_{\hslash }$ and $M{K}_{\hslash }$, a full quantum analogue of $d_{MK,2}$ introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165--205], including an up to $\hslash$ triangle inequality for $M{K}_{\hslash }$. Finally, we show that, when nice optimal Kantorovich potentials exist for ${\mathcal{E}}_{\hslash }$, optimal couplings induce classical/quantum optimal transports and the potentials are linked by a semiquantum Legendre type transform.

33 pages, no figure

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