A complete characterisation of conditional entropies

TL;DR

This paper provides a comprehensive characterization of conditional entropies satisfying natural axioms, revealing they are exponential averages of Re9nyi entropies, with implications for quantum thermodynamics and information theory.

Contribution

It offers the first complete axiomatic characterization of conditional entropy, generalizing previous definitions and linking them to operational principles.

Findings

Conditional entropy is characterized by exponential averages of Re9nyi entropies.

The family of measures satisfies additivity, invariance, and monotonicity axioms.

These measures determine transformation rates and second laws in quantum thermodynamics.

Abstract

Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of R\'enyi entropies, parameterized by a real number $\alpha$. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterization of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterization of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We prove that the most general form of conditional entropy is captured by a family of measures that are exponential averages of R\'enyi entropies of the conditioned distribution and parameterized by a real parameter and a probability measure on the positive reals. Finally, we show that these quantities determine the rate of transformation under conditional mixing and provide a set of second laws of quantum thermodynamics with side information for states diagonal in the energy eigenbasis.