  We consider an open quantum system described by the GKLS equation and we are
interested in the onset of decoherence. We are in particulary interested in
situations where only some degrees of freedom of the system are coupled to the
environment, and we want to understand if, and how fast, the noise travels
through the system and eventually affects all degrees of freedom. We find that
this can be understood in terms of the H\"ormander condition, a condition on
the commutators of the Hamiltonian vectorfields of the Lindblad operators and
the internal Hamiltonian, which is a condition for hypoellipticity known from
the theory of PDE's. We show that for Gaussian quantum channels this condition
leads to a delay in the onset of decoherence and can as well be used to detect
decoherence free subsystems.
