  In this paper, we use the Carnot-Caratheodory distance from sub-Riemanian
geometry to prove entropy decay estimates for all finite dimensional symmetric
quantum Markov semigroups. This estimate is independent of the environment size
and hence stable under tensorization. Our approach relies on the transference
principle, the existence of $t$-designs, and the sub-Riemannian diameter of
compact Lie groups and implies estimates for the spectral gap.
