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Preprints, Working Papers, ... Year : 2018

TOWARDS OPTIMAL TRANSPORT FOR QUANTUM DENSITIES

Abstract

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on R d has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on L 2 (R d), and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: "Topics in Optimal Transportation", Amer. Math. Soc. 2003].
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Dates and versions

hal-01963667 , version 1 (21-12-2018)
hal-01963667 , version 2 (06-02-2021)

Identifiers

  • HAL Id : hal-01963667 , version 1

Cite

Emanuele Caglioti, François Golse, Thierry Paul. TOWARDS OPTIMAL TRANSPORT FOR QUANTUM DENSITIES. 2018. ⟨hal-01963667v1⟩

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