  In this article we introduce a complete gradient estimate for symmetric
quantum Markov semigroups on von Neumann algebras equipped with a normal
faithful tracial state, which implies semi-convexity of the entropy with
respect to the recently introduced noncommutative 2-Wasserstein distance. We
show that this complete gradient estimate is stable under tensor products and
free products and establish its validity for a number of examples. As an
application we prove a complete modified logarithmic Sobolev inequality with
optimal constant for Poisson-type semigroups on free group factors.
