Reversibility conditions for quantum operations

A family of quantum states can be seen as carrying some information. If a quantum operation is applied, then some information can be lost. However, there are situations when this does not happen and the original family of states can be recovered, we then say that the operation is reversible with respect to the family. Characterization of such situations is an important question in quantum statistics and error correction. We give a list of equivalent reversibility conditions, in the finite dimensional situation. These are given by preservation of certain distinguishability measures on states: a class of f-divergences, L_1 distance and the Chernoff and Hoeffding distances. Another type of conditions is given in terms of a quantum Radon-Nikodym derivative and a factorization condition on the states. Reversibility is also characterized by preservation of a large class of quantum Fisher metrics.