Entropy and relative entropy from information-theoretic principles

Abstract

We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these axioms induce sufficient structure to establish continuity in the interior of the probability simplex and meaningful upper and lower bounds, e.g., we find that every relative entropy must lie between the Rényi divergences of order $0$ and $\infty$. We further show simple conditions for positive definiteness of such relative entropies and a characterisation in term of a variant of relative trumping. Our main result is a one-to-one correspondence between entropies and relative entropies.

References in corpus (4)

• Comparison of Quantum Channels by Superchannels
• Necessary and Sufficient Conditions for the Trumping Relation
• Catalytic Conversion Probabilities for Bipartite Pure States
• Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation

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• Continuity bounds for quantum entropies arising from a fundamental entropic inequality
• Inevitable Negativity: Additivity Commands Negative Quantum Channel Entropy
• Matrix Majorization in Large Samples with Varying Support Restrictions
• Entropic and operational characterizations of dynamic quantum resources