  In finite dimensions, we provide characterizations of the quantum dynamical
semigroups that do not decrease the von Neumann, the Tsallis and the Renyi
entropies, as well as a family of functions of density operators strictly
related to the Schatten norms. A few remarkable consequences --- in particular,
a description of the associated infinitesimal generators --- are derived, and
some significant examples are discussed. Extensions of these results to
semigroups of trace-preserving positive (i.e., not necessarily completely
positive) maps and to a more general class of quantum entropies are also
considered.
