Mean and variance for the visit time of absorbing quantum walks and channels

While the expected time of first visit to a subspace under some quantum evolution is a relevant statistic, it is also important to confront this information with the associated variance, so that one may have a better understanding of how representative such means are. In this work, we study these matters in the context of quantum channels acting on finite-dimensional Hilbert spaces, and we provide practical methods for calculating the above data with respect to goal subspaces of interest. By considering an assumption which is strictly weaker than irreducibility, we are also able to consider unitary quantum walks. Our approach is based on the monitored evolution of positive, trace-preserving operators given by quantum Markov chains as described by Gudder. Then, given any channel and some goal subspace of interest, one may induce an absorbing chain in a natural way so that the monitored evolution of the original map can be examined.