Geometric & Functional Analysis GAFA

, Volume 11, Issue 6, pp 1125-1138

First online:

A free probability analogue of the Wasserstein metric on the trace-state space

  • P. BianeAffiliated withCNRS, DMA, École Normale Supérieure, 45 Rue d'Ulm, 75005 Paris, France, e-mail: biane@dmi.ens.fr
  • , D. VoiculescuAffiliated withDepartment of Mathematics, University of California, Berkeley, California 94720—3840, USA, e-mail: dvv@math.berkeley.edu

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Abstract.

We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semi-circle distribution is majorized by a modified free entropy quantity.