Reversibility conditions for quantum operations

We give a list of equivalent conditions for reversibility of the adjoint of a unital Schwarz map with respect to a set of quantum states. A large class of such conditions is given by preservation of distinguishability measures: f-divergences, L_1 -distance, quantum Chernoff and Hoeffding distances; here we summarize and extend the known results. Moreover, we prove a number of conditions in terms of the properties of a quantum Radon-Nikodym derivative and factorization of states in the given set. Finally, we show that reversibility is equivalent with preservation of a large class of quantum Fisher informations and \chi^2-divergences.