Decoherence for quantum Markov semi-groups on matrix algebras

In this work we describe a necessary and sufficient condition for decoherence of quantum Markov evolutions acting on matrix spaces (according to the definition introduced by Blanchard and Olkiewicz). This condition is related to the spectral analysis of the generator L of the semi-group and is easily stated: the evolution displays decoherence if and only if the maximal algebra N (T ) on which the semigroup is ∗-automorphic contains all the eigenvalues of L associated with eigenvectors with null real part. Moreover, this condition is surely verified when the semigroup admits a faithful invariant state.