  A quantum channel is sufficient with respect to a set of input states if it
can be reversed on this set. In the approximate version, the input states can
be recovered within an error bounded by the decrease of the relative entropy
under the channel. Using a new integral representation of the relative entropy
in arXiv:2208.12194, we present an easy proof of a characterization of
sufficient quantum channels and recoverability by preservation of optimal
success probabilities in hypothesis testing problems, equivalently, by
preservation of $L_1$-distance.
