  We prove that the complete modified logarithmic Sobolev constant of a quantum
Markov semigroup is bounded by the inverse of its complete positivity mixing
time. For classical Markov semigroups, this implies that every sub-Laplacian
given by a H\"ormander system on a compact manifold satisfies a uniform
modified log-Sobolev inequality for matrix-valued functions. For quantum Markov
semigroups, we obtain that the complete modified logarithmic Sobolev constant
is comparable to spectral gap up to a constant as logarithm of dimension
constant. This estimate is asymptotically tight for a quantum birth-death
process. Our results and the consequence of concentration inequalities apply to
GNS-symmetric semigroups on general von Neumann algebras.
