Uniqueness and Optimality of Dynamical Extensions of Divergences

TL;DR

This paper develops an axiomatic framework for channel divergences, establishing properties like faithfulness and continuity, and proves the uniqueness of the classical Kullback-Leibler extension while identifying the optimality of the amortized quantum Umegaki relative entropy.

Contribution

It introduces an axiomatic approach for channel divergences, proves a uniqueness theorem for classical channels, and demonstrates the optimality of the amortized quantum Umegaki relative entropy.

Findings

The Kullback-Leibler divergence has a unique extension to classical channels.

The amortized quantum Umegaki relative entropy is optimal among channel divergences.

Properties like faithfulness and continuity hold for all channel divergences.

Abstract

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.