Quantum Optimal Transport and Weak Topologies

• URL: http://arxiv.org/abs/2306.12944v3
• Date: Fri, 22 Dec 2023 16:21:11 GMT
• Title: Quantum Optimal Transport and Weak Topologies
• Abstract summary: We investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017. These pseudometrics serve as a quantum analogue of the Monge-Kantorovich-Wasserstein distances of order $2$ on the phase space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Several extensions of the classical optimal transport distances to the quantum setting have been proposed. In this paper, we investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017]. These pseudometrics serve as a quantum analogue of the Monge-Kantorovich-Wasserstein distances of order $2$ on the phase space. We prove that they are comparable to negative Sobolev norms up to a small term due to a positive "self-distance" in the semiclassical approximation, which can be bounded above using the Wigner-Yanase skew information. This enables us to improve the known results in the context of the mean-field and semiclassical limits by requiring less regularity on the initial data.

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