  We introduce an axiomatic approach for channel divergences and channel
relative entropies that is based on three information-theoretic axioms of
monotonicity under superchannels (i.e. generalized data processing inequality),
additivity under tensor products, and normalization, similar to the approach
given recently for the state domain. We show that these axioms are sufficient
to give enough structure also in the channel domain, leading to numerous
properties that are applicable to all channel divergences. These include
faithfulness, continuity, a type of triangle inequality, and boundedness
between the min and max channel relative entropies. In addition, we prove a
uniqueness theorem showing that the Kullback-Leibler divergence has only one
extension to classical channels. For quantum channels, with the exception of
the max relative entropy, this uniqueness does not hold. Instead we prove the
optimality of the amortized channel extension of the Umegaki relative entropy,
by showing that it provides a lower bound on all channel relative entropies
that reduce to the Kullback-Leibler divergence on classical states. We also
introduce the maximal channel extension of a given classical state divergence
and study its properties.
