  We generalize Holley-Stroock's perturbation argument from commutative to
quantum Markov semigroups. As a consequence, results on (complete) modified
logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for
self-adjoint quantum Markov process can be used to prove estimates on the
exponential convergence in relative entropy of quantum Markov systems which
preserve a fixed state. This leads to estimates for the decay to equilibrium
for coupled systems and to estimates for mixed state preparation times using
Lindblad operators. Our techniques also apply to discrete time settings, where
we show that the strong data processing inequality constant of a quantum
channel can be controlled by that of a corresponding unital channel.
