  We study the convergence of states under continuous-time depolarizing
channels with full rank fixed points in terms of the relative entropy. The
optimal exponent of an upper bound on the relative entropy in this case is
given by the log-Sobolev-1 constant. Our main result is the computation of this
constant. As an application we use the log-Sobolev-1 constant of the
depolarizing channels to improve the concavity inequality of the von-Neumann
entropy. This result is compared to similar bounds obtained recently by Kim et
al. and we show a version of Pinsker's inequality, which is optimal and tight
if we fix the second argument of the relative entropy. Finally, we consider the
log-Sobolev-1 constant of tensor-powers of the completely depolarizing channel
and use a quantum version of Shearer's inequality to prove a uniform lower
bound.
