# Inequalities in R\'enyi's quantum thermodynamics

## TL;DR

This paper develops a quantum thermodynamics framework based on Renyi entropy, deriving key concepts and inequalities, and explores the implications for quantum measurement uncertainty principles, reducing to von Neumann results as a1 approaches 1.

## Contribution

It introduces a novel Renyi-based quantum thermodynamics theory, defining fundamental quantities and inequalities, and revisits quantum uncertainty principles within this new framework.

## Key findings

- Derived quantum thermodynamic inequalities using Renyi entropy
- Established the Renyi maximum entropy principle and equilibrium conditions
- Revisited Heisenberg and Schrodinger uncertainty principles in this context

## Abstract

A theory of thermodynamics has been recently formulated and derived on the basis of R\'enyi entropy and its relative versions. In this framework, we define the concepts of partition function, internal energy and free energy, and fundamental quantum thermodynamical inequalities are deduced. In the context of R\'enyi's thermodynamics, the variational Helmholtz principle is stated and the condition of equilibrium is analyzed. The R\'enyi maximum entropy principle is formulated and the equality case is discussed. The obtained results reduce to the von Neumann ones when the R\'enyi entropic parameter $\alpha$ approaches 1. The Heisenberg and Schr\"odinger uncertainty principles on the measurements of quantum observables are revisited. The presentation is self-contained and the proofs only use standard matrix analysis techniques.