# A Formulation of R\'enyi Entropy on $C^*$-Algebras

## TL;DR

This paper extends the concept of Renyi entropy to $C^*$-algebras using Ohya's $\\mathcal{S}$-mixing entropy, establishing inequalities for state uncertainties in different systems.

## Contribution

It introduces a formulation of Renyi entropy on $C^*$-algebras based on $\\mathcal{S}$-mixing entropy, advancing the mathematical framework of quantum information theory.

## Key findings

- Established inequalities for uncertainties of states in various reference systems.
- Extended Renyi entropy formulation to $C^*$-algebras.
- Provided mathematical tools for analyzing quantum state uncertainties.

## Abstract

The entropy of probability distribution defined by Shannon has several extensions. R\'enyi entropy is one of the general extensions of Shannon entropy and is widely used in engineering, physics, and so on. On the other hand, the quantum analogue of Shannon entropy is von Neumann entropy. Furthermore, the formulation of this entropy was extended to on $C^*$-algebras by Ohya ($\mathcal{S}$-mixing entropy). In this paper, we formulate Renyi entropy on $C^*$-algebras based on $\mathcal{S}$-mixing entropy and prove several inequalities for the uncertainties of states in various reference systems.