Large deviations for quantum markov semigroups on the 2 x 2-matrix algebra

Let (T(*)t) be a predual quantum Markov semigroup acting on the full 2 x 2-matrix algebra and having an absorbing pure state. We prove that for any initial state., the net of orthogonal measures representing the net of states (T(*)t(.)) satisfies a large deviation principle in the pure state space, with a rate function given in terms of the generator, and which does not depend on.. This implies that (T(*)t(.)) is faithful for all t large enough. Examples arising in weak coupling limit are studied.