We derive upper and lower bounds on the convergence behavior of certain classes of one-parameter quantum dynamical semigroups. The classes we consider, consist of tensor product channels and of channels with commuting Liouvillians. We introduce the notion of cutoff phenomenon in the setting of quantum information theory, and show how it exemplifies the fact that the convergence of (quantum) stochastic processes is not solely governed by the spectral gap of the transition map. We apply the new methods to show that graph states can be prepared efficiently, albeit not in constant time, by dissipation, and give the exact scaling behavior of the time to stationarity.
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