Large deviations for quantum Markov semigroups on the 2× 2-matrix algebra

Let (T∗t) be a predual quantum Markov semigroup acting on the full 2× 2-matrix algebra and having an absorbing pure state. We prove that for any initial state ω, the net of orthogonal measures representing the net of states (T∗t(ω)) 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 ω. This implies that (T∗t(ω)) is faithful for all t large enough. Examples arising in weak coupling limit are studied.