  In this work we make use of generalized inverses associated with quantum
channels acting on finite-dimensional Hilbert spaces, so that one may calculate
the mean hitting time for a particle to reach a chosen goal subspace. The
questions studied in this work are motivated by recent results on quantum
dynamics on graphs, most particularly quantum Markov chains. We focus on
describing how generalized inverses and hitting times can be obtained, with the
main novelties of this work with respect to previous ones being that a) we are
able to weaken the notion of irreducibility, so that reducible examples can be
considered as well, and b) one may consider arbitrary arrival subspaces for
general positive, trace preserving maps. Natural examples of reducible maps are
given by unitary quantum walks. We also take the opportunity to explain how a
more specific inverse, namely the group inverse, appears in our context, in
connection with matrix algebraic constructions which may be of independent
interest.
