# Variational representations related to Tsallis relative entropy

## TL;DR

This paper develops variational representations for deformed logarithmic and exponential functions, extending quantum Tsallis relative entropy and trace inequalities to new deformation parameter ranges, enriching the mathematical framework of quantum information theory.

## Contribution

It introduces new variational representations for deformed functions and extends the Golden-Thompson inequality to additional parameter ranges, broadening theoretical tools in quantum information.

## Key findings

- Extended Golden-Thompson inequality to q in [0,1]
- Provided variational representations for Tsallis relative entropy
- Enhanced mathematical understanding of deformed exponential functions

## Abstract

We develop variational representations for the deformed logarithmic and exponential functions and use them to obtain variational representations related to the quantum Tsallis relative entropy. We extend Golden-Thompson's trace inequality to deformed exponentials with deformation parameter $ q\in[0,1], $ thus complementing the second author's previous study of the cases with deformation parameter $ q \in [1,2] $ or $ q \in [2,3]. $