  We study a class of ergodic quantum Markov semigroups on finite-dimensional
unital $C^*$-algebras. These semigroups have a unique stationary state
$\sigma$, and we are concerned with those that satisfy a quantum detailed
balance condition with respect to $\sigma$. We show that the evolution on the
set of states that is given by such a quantum Markov semigroup is gradient flow
for the relative entropy with respect to $\sigma$ in a particular Riemannian
metric on the set of states. This metric is a non-commutative analog of the
$2$-Wasserstein metric, and in several interesting cases we are able to show,
in analogy with work of Otto on gradient flows with respect to the classical
$2$-Wasserstein metric, that the relative entropy is strictly and uniformly
convex with respect to the Riemannian metric introduced here. As a consequence,
we obtain a number of new inequalities for the decay of relative entropy for
ergodic quantum Markov semigroups with detailed balance.
