Geometric & Functional Analysis GAFA

, Volume 11, Issue 6, pp 1125–1138

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


  • P. Biane
    • CNRS, DMA, École Normale Supérieure, 45 Rue d'Ulm, 75005 Paris, France, e-mail:
  • D. Voiculescu
    • Department of Mathematics, University of California, Berkeley, California 94720—3840, USA, e-mail:

DOI: 10.1007/s00039-001-8226-4

Cite this article as:
Biane, P. & Voiculescu, D. GAFA, Geom. funct. anal. (2001) 11: 1125. doi:10.1007/s00039-001-8226-4


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.

Copyright information

© Birkhäuser Verlag, Basel 2001