  The paper provides a systematic characterization of quantum ergodic and
mixing channels in finite dimensions and a discussion of their structural
properties. In particular, we discuss ergodicity in the general case where the
fixed point of the channel is not a full-rank (faithful) density matrix.
Notably, we show that ergodicity is stable under randomizations, namely that
every random mixture of an ergodic channel with a generic channel is still
ergodic. In addition, we prove several conditions under which ergodicity can be
promoted to the stronger property of mixing. Finally, exploiting a suitable
correspondence between quantum channels and generators of quantum dynamical
semigroups, we extend our results to the realm of continuous-time quantum
evolutions, providing a characterization of ergodic Lindblad generators and
showing that they are dense in the set of all possible generators.
