Generalized R\'enyi entropy accumulation theorem and generalized quantum probability estimation

TL;DR

This paper introduces a new entropy accumulation bound that improves finite-size performance, simplifies computation via convex optimization, and extends applicability to fully-Rényi security proofs in quantum cryptography.

Contribution

It derives a novel entropy accumulation bound that is easier to compute, more optimal, and applicable directly at the Rénnyi entropy level, advancing quantum cryptography security analysis.

Findings

Provides significantly better finite-size bounds.

Enables computation through convex optimization without affine min-tradeoff functions.

Applies directly to prepare-and-measure protocols without additional procedures.

Abstract

The entropy accumulation theorem, and its subsequent generalized version, is a powerful tool in the security analysis of many device-dependent and device-independent cryptography protocols. However, it has the drawback that the finite-size bounds it yields are not necessarily optimal, and furthermore it relies on the construction of an affine min-tradeoff function, which can often be challenging to construct optimally in practice. In this work, we address both of these challenges simultaneously by deriving a new entropy accumulation bound. Our bound yields significantly better finite-size performance, and can be computed as an intuitively interpretable convex optimization, without any specification of affine min-tradeoff functions. Furthermore, it can be applied directly at the level of R\'enyi entropies if desired, yielding fully-R\'enyi security proofs. Our proof techniques are based on elaborating on a connection between entropy accumulation and the frameworks of quantum probability estimation or $f$-weighted R\'enyi entropies, and in the process we obtain some new results with respect to those frameworks as well. In particular, those findings imply that our bounds apply to prepare-and-measure protocols without the virtual tomography procedures or repetition-rate restrictions previously required for entropy accumulation.