One important insight optimal transport has brought to the study of partial differential equations is that the heat flow can be interpreted as gradient flow of the entropy when the space of probability measures is endowed with the 2-Wasserstein metric. In this talk a noncommutative version of this result will be discussed. Following the work of Carlen-Maas and Mielke in the finite-dimensional case, we construct for a given (tracially symmetric) quantum Markov semigroup an analog of the Wasserstein metric on the space of density operators. If the QMS satisfies a certain gradient estimate, then the entropy is convex and the trajectories of the QMS are metric gradient flow curves of the entropy. As applications one obtains several functional inequalities for the QMS such as a modified logarithmic Sobolev inequality.