  Gaussian quantum channels are well understood and have many applications,
e.g., in Quantum Information Theory and in Quantum Optics. For more general
quantum channels one can in general use semiclassical approximations or
perturbation theory, but it is not easy to judge the accuracy of such methods.
We study a relatively simple model case, where the quantum channel is generated
by a Lindblad equation where one of the Lindblad operators is a multiple of the
internal Hamiltonian, and therefore the channel is not Gaussian. For this model
we can compute the characteristic function of the action of the channel on a
Gaussian state explicitly and we can as well derive a representation of the
propagator in an integral form. This allows us to compare the exact results
with semiclassical approximations and perturbation theory and evaluate their
accuracy. We finally apply these results to the study of the evolution of the
von Neumann entropy of a state.
