# Quantum R\'enyi relative entropies on density spaces of $C^*$-algebras: their symmetries and their essential difference

## TL;DR

This paper extends quantum Re9nyi relative entropies to $C^*$-algebras, revealing their fundamental differences in non-commutative cases and analyzing their symmetry groups, with implications for quantum information theory.

## Contribution

It introduces a generalized framework for quantum Re9nyi entropies on $C^*$-algebras and characterizes their symmetries, highlighting essential differences from classical cases.

## Key findings

- Quantum Re9nyi entropies are essentially different on non-commutative algebras.
- Symmetry groups of density spaces are identical for all considered Re9nyi entropies.
- Similar symmetry results are obtained for Umegaki and Belavkin-Staszewski relative entropies.

## Abstract

We extend the definitions of different types of quantum R\'enyi relative entropy from the finite dimensional setting of density matrices to density spaces of $C^*$-algebras. We show that those quantities (which trivially coincide in the classical commutative case) are essentially different on non-commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces. Besides, we determine the symmetry groups of density spaces corresponding to each of those quantum R\'enyi relative entropies and find that they are identical. Similar results concerning the Umegaki and the Belavkin-Staszewksi relative entropies are also presented.