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.Comment: 20 pages, no figure
