# Dimensionally sharp inequalities for the linear entropy

## TL;DR

This paper establishes sharp bounds for the linear entropy of finite-dimensional quantum systems using generalized Bloch decompositions, extending entropy inequalities to finite regimes and aiding in entanglement detection and quantum state characterization.

## Contribution

It introduces a new inequality for linear entropy that provides strict bounds for all finite-dimensional quantum states, improving upon previous asymptotic results.

## Key findings

- Derived sharp bounds for linear entropy in finite dimensions
- Extended entropy inequalities from asymptotic to finite regimes
- Potential applications in entanglement detection and quantum state characterization

## Abstract

We derive an inequality for the linear entropy, that gives sharp bounds for all finite dimensional systems. The derivation is based on generalised Bloch decompositions and provides a strict improvement for the possible distribution of purities for all finite dimensional quantum states. It thus extends the widely used concept of entropy inequalities from the asymptotic to the finite regime, and should also find applications in entanglement detection and efficient experimental characterisations of quantum states.