# Quantum and Semiquantum Pseudometrics and applications

Abstract : We establish a Kantorovich duality for he pseudometric $\cE_\hb$ introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. \textbf{223} (2017), 57--94], obtained from the usual Monge-Kantorovich distance $\MKd$ between classical densities by quantization of one side of the two densities involved. We show several type of inequalities comparing $\MKd$, $\cE_\hb$ and $MK_\hb$, a full quantum analogue of $\MKd$ introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. \textbf{343} (2016), 165--205], including an up to $\hbar$ triangle inequality for $MK_\hb$. Finally, we show that, when nice optimal Kantorovich potentials exist for $\cE_\hb$, optimal couplings induce classical/quantum optimal transports and the potentials are linked by a semiquantum Legendre type transform.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-03136855
Contributor : Thierry Paul Connect in order to contact the contributor
Submitted on : Tuesday, February 9, 2021 - 11:42:59 PM
Last modification on : Friday, August 5, 2022 - 12:01:58 PM
Long-term archiving on: : Monday, May 10, 2021 - 7:15:42 PM

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• HAL Id : hal-03136855, version 1

### Citation

François Golse, Thierry Paul. Quantum and Semiquantum Pseudometrics and applications. Journal of Functional Analysis, Elsevier, In press. ⟨hal-03136855⟩

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