Finite-size security of QKD: comparison of three proof techniques

TL;DR

This paper compares three proof techniques for finite-size security in quantum key distribution, analyzing their effectiveness at practical block lengths for the BB84 protocol, highlighting the strengths and limitations of each approach.

Contribution

It provides a comparative analysis of EUR, AEP, and FME methods for finite-size security proofs in QKD, with insights into their performance at different block sizes.

Findings

EUR-based bounds are most favorable across parameter ranges.

AEP bounds are asymptotically tight but can be overly pessimistic at small sizes.

FME remains effective in small-block regimes, providing positive key rates where AEP fails.

Abstract

We compare three proof techniques for composable finite-size security of quantum key distribution under collective attacks, with emphasis on how the resulting secret-key rates behave at practically relevant block lengths. As a benchmark, we consider the BB84 protocol and evaluate finite-size key-rate estimates obtained from entropic uncertainty relations (EUR), from the asymptotic equipartition property (AEP), and from a direct finite-block analysis based on the conditional min-entropy, which we refer to as the finite-size min-entropy (FME) approach. For BB84 we show that the EUR-based bound provides the most favorable performance across the considered parameter range, while the AEP bound is asymptotically tight but can become overly pessimistic at moderate and small block sizes, where it may fail to certify a positive key. The FME approach remains effective in this small-block regime, yielding nonzero rates in situations where the AEP estimate vanishes, although it is not asymptotically optimal for BB84. These results motivate the use of FME-type analyses for continuous-variable protocols in settings where tight EUR-based bounds are unavailable, notably for coherent-state schemes where current finite-size analyses typically rely on AEP-style corrections.