Formal Groups and $Z$-Entropies

TL;DR

This paper introduces a new family of multi-parametric entropies called Z-entropies, generalizing Rènyi and Boltzmann entropies, with a focus on their composability and underlying group-theoretical structure.

Contribution

It establishes Z-entropies as a novel class of entropies with composability, extending beyond trace-form entropies like Boltzmann and Tsallis, and explores their mathematical foundations.

Findings

Z-entropies generalize Rènyi and Boltzmann entropies.

Z-entropies are composable and form a group law.

They have applications in classical and quantum information theory.

Abstract

We shall prove that the celebrated R\'enyi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the $Z$-entropies. Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and R\'enyi. A crucial aspect is that every $Z$-entropy is composable [1]. This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required, to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the $Z$-entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.