One of the highlights in optimal transport is the interpretation of diffusion equations as gradient flows of the entropy in the Wasserstein space of probability measures. In this talk we show that this interpretation extends to the setting of non-commutative probability. We construct a class of Riemannian metrics on the space of density matrices, which may be regarded as non-commutative analogues of the 2-Wasserstein metric. These metrics allow us to formulate quantum Markov semigroups as gradient flows of the von Neumann entropy. We present transportation inequalities in this setting and obtain non-commutative versions of results by Bakry--Emery and Otto--Villani.