  We show that, for a Quantum Markov Semigroup (QMS) with a faithful normal
invariant state, the atomicity of the decoherence-free subalgebra and
environmental decoherence are equivalent. Moreover, we characterize the set of
reversible states and explicitly describe the relationship between the
decoherence-free subalgebra and the fixed point subalgebra for QMSs with the
above equivalent properties.
