Decoherence Time Scales and the H\"ormander condition

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.