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 Rényi entropies, parameterized by a real number $α$. 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 characterisation of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterisation 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 show that the most general conditional entropy is a generalized convex combination of conditional entropies defined as exponential averages of Rényi entropies of the conditioned distribution, parameterized by a real number and a probability measure on the positive reals.

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