Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification

TL;DR

This paper introduces a new framework for quantum privacy amplification using smooth conditional entropies, leading to tighter bounds and optimal second-order asymptotics in one-shot quantum information tasks.

Contribution

It develops a novel class of smooth entropies based on measurement lifting, improving bounds on randomness extraction and decoupling in quantum settings.

Findings

Established a tightened leftover hash lemma with improved bounds.

Derived a one-shot bound for decoupling, the quantum analogue of privacy amplification.

Provided a matching one-shot converse bound, proving the optimality of the results.

Abstract

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth R\'enyi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured R\'enyi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering the state-of-the-art asymptotic i.i.d. error exponents shown there. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.