# Randomness extraction via a quantum generalization of the conditional collision entropy

## TL;DR

This paper introduces a quantum generalization of collision entropy for randomness extraction against quantum side information, optimizing extractable key length and providing asymptotic optimality with a novel eigenvalue-based evaluation.

## Contribution

It presents a new quantum collision entropy measure for randomness extraction, optimizing it with respect to side information, and proves its asymptotic optimality.

## Key findings

- Derived a lower bound expressed as the difference of two entropies
- Reduced evaluation to an eigenvalue problem of two states
- Established asymptotic optimality of the proposed entropy measure

## Abstract

Randomness extraction against side information is the art of distilling from a given source a key which is almost uniform conditioned on the side information. This paper provides randomness extraction against quantum side information whose extractable key length is given by a quantum generalization of the collision entropy, which is smoothed and conditioned differently from how this is done in existing schemes. Based on the fact that the collision entropy is not subadditive, its optimization with respect to additional side information is introduced, and is shown to be asymptotically optimal. The lower bound derived there for general states is expressed as the difference between two unconditional entropies and its evaluation reduces to an eigenvalue problem of two states, which are the entire state and the marginal state of side information.