Environment Induced Decoherence is a physical concept which provides a dynamical explanation to the disappearance of quantum phenomenon in the real world. Intuitively, it states that a quantum system is never per- fectly isolated, so that quantum correlations disappear due to the action of the environment on the system. Focusing on finite-dimensional quantum systems undergoing Markovian evolutions, we propose natural generaliza- tions of certain non-commutative functional inequalities that are adapted to the study of decoherence. We obtain in this way a natural framework to generalize functional inequalities to non-primitive evolutions. We will also discuss possible applications to fault-tolerant quantum computation on decoherence-free subspaces and subsystems.