  We study quantum Dirichlet forms and the associated symmetric quantum Markov
semigroups on noncommutative $L^2$ spaces. It is known from the work of
Cipriani and Sauvageot that these semigroups induce a first order differential
calculus, and we use this differential calculus to define a noncommutative
transport metric on the set of density matrices. This construction generalizes
both the $L^2$-Wasserstein distance on a large class of metric spaces as well
as the discrete transport distance introduced by Maas, Mielke, and
Chow-Huang-Li-Zhou. Assuming a Bakry-\'Emery-type gradient estimate, we show
that the quantum Markov semigroup can be viewed as a metric gradient flow of
the entropy with respect to this transport metric. Under the same assumption we
also establish that the set of density matrices with finite entropy endowed
with the noncommutative transport metric is a geodesic space and that the
entropy is semi-convex along these geodesics.
