# Uniqueness and characterization theorems for generalized entropies

## TL;DR

This paper proves that, under mild conditions, the Tsallis entropy is the only composable generalized entropy in trace form, and characterizes a broad class of non-trace-form entropies like Rényi's.

## Contribution

It establishes the uniqueness of Tsallis entropy among trace-form generalized entropies and characterizes non-trace-form entropies with similar composability properties.

## Key findings

- Tsallis entropy is the only trace-form composable generalized entropy.
- Characterization of a broad class of non-trace-form entropies like Rényi's.
- Implications for studying complex systems with generalized entropies.

## Abstract

The requirement that an entropy function be composable is key: it means that the entropy of a compound system can be calculated in terms of the entropy of its independent components. We prove that, under mild regularity assumptions, the only composable generalized entropy in trace form is the Tsallis one-parameter family (which contains Boltzmann-Gibbs as a particular case). This result leads to the use of generalized entropies that are not of trace form, such as R\'enyi's entropy, in the study of complex systems. In this direction, we also present a characterization theorem for a large class of composable non-trace-form entropy functions with features akin to those of R\'enyi's entropy.