Uniqueness and Optimality of Dynamical Extensions of Divergences

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 for the state domain in Gour and Tomamichel [arXiv:2006.11164v1 (2020), arXiv:2006.12408v2 (2020)]. We show that these axioms are sufficient to give enough structure in the channel domain as well, 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.