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Article Dans Une Revue Journal of Functional Analysis Année : 2022

Quantum and Semiquantum Pseudometrics and applications

François Golse
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Thierry Paul
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Résumé

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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hal-03136855 , version 1 (09-02-2021)

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  • HAL Id : hal-03136855 , version 1

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