# Sharp Continuity Bounds for Entropy and Conditional Entropy

## TL;DR

This paper derives tight continuity bounds for the Renyi entropy in quantum information theory, generalizing the Fannes inequality for von Neumann entropy and relating entropy differences to trace norm distances.

## Contribution

It introduces a sharp inequality connecting Renyi entropy differences to trace norm distance, which is tight and encompasses the Fannes inequality as a special case.

## Key findings

- Derived a tight inequality for Renyi entropy differences
- Included the Fannes inequality as a special case
- Established the inequality's attainability for all trace norm distances

## Abstract

The Renyi entropy plays an essential role in quantum information theory. We study the continuity estimation of the Renyi entropy. An inequality relating the Renyi entropy difference of two quantum states to their trace norm distance is derived. This inequality is shown to be tight in the sense that equality can be attained for every prescribed value of the trace norm distance. It includes the sharp Fannes inequality for von Neumann entropy as a special case.