A new class of entropic information measures, formal group theory and information geometry

Abstract

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a grouptheoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated.

Group entropies: from phase space geometry to entropy functionals via group theory

“…We wish to point out that group entropies are an interesting tool in information geometry, since they can be used to define Riemannian structures in statistical spaces via suitable divergences (or relative entropies) [30] associated with them. This has been proved in [29] for Z-entropies and in [31] for the universal-group entropy. Also, a quantum version of these entropies can be used as an entanglement measure for spin chains [4].…”

“…we are naturally lead to the abstract conceptual framework of information theory. We wish to stress that group entropies represent measures of complexity by information geometric means [29] and can characterize limiting probability distributions by means of a maximum entropy procedure for systems where interdependence among its components makes W(N) deviate from the exponential form.…”

Universality Classes and Information-Theoretic Measures of Complexity via Group Entropies

“…For recent applications of the notion of group entropy in Information Geometry and the theory of divergences [1], see e.g. [20].…”

“…Certainly, we can derive other group entropies with similar properties. More precisely, a theorem proved in [20] shows that for a given grouptheoretical structure, one can associate a "tower" of group entropies sharing the same structure. At the same time, the Z-family defined in eq.…”

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

“…The two cases (I) and turns out not to be consistent with additivity, which takes one outside the framework of Boltzmann-Shannon-Khinchin and therefore in accordance with Lieb and Yngvason outside standard thermodynamics [ 8 , 30 , 31 ] i.e., we are naturally lead to the abstract conceptual framework of information theory. We wish to stress that group entropies represent measures of complexity by information geometric means [ 32 ] and can characterise limiting probability distributions by means of a maximum entropy procedure for systems where interdependence among its components makes deviate from the exponential form.…”