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An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F.
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F. Golse, T. Paul: Wave packets and the quadratic Monge-Kantorovich distance in quantum mechanics
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E.A. Carlen, J. Maas: Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
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H. Brezis: “Functional Analysis, Sobolev Spaces and Partial Differential Equations”, Springer Science + Business Media 2011
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E.A. Carlen, J. Maas: An Analog of the 2 2 -Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy
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