Wave Packets and the Quadratic Monge-Kantorovich Distance in Quantum Mechanics

TL;DR

This paper extends bounds on a quantum pseudo-distance related to the quadratic Monge-Kantorovich metric, broadening the class of initial data for mean-field convergence in quantum mechanics.

Contribution

It generalizes the pseudo-distance bounds to positive quantizations beyond rank-one cases, improving convergence results for quantum mean-field limits.

Findings

Established bounds for the pseudo-distance in broader quantum settings

Proved wider initial data classes for mean-field convergence

Discussed the pseudo-distance's relation to Schatten norms in semiclassical analysis

Abstract

In this paper, we extend the upper and lower bounds for the "pseudo-distance" on quantum densities analogous to the quadratic Monge-Kantorovich(-Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165-205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank one as in the case of the T\"oplitz quantization. As a corollary, we prove that the uniform (for vanishing h) convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, loc. cit.]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime.