  Von Neumann entropy rate for open quantum systems is, in general, written in
terms of entropy production and entropy flow rates, encompassing the second law
of thermodynamics. When the open-quantum-system evolution corresponds to a
quantum dynamical semigroup, we find a decomposition of the infinitesimal
generator of the dynamics, that allows to relate the von Neumann entropy rate
with the divergence-based quantum Fisher information, at any time. Applied to
quantum Gaussian channels that are dynamical semigroups, our decomposition
leads to the quantum analog of the generalized classical de Bruijn identity,
thus expressing the quantum fluctuation-dissipation relation in that kind of
channels. Finally, from this perspective, we analyze how stationarity arises.
