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

TL;DR

This paper investigates composable finite-size security bounds for QKD under collective attacks and compares three methodologies: EUR, AEP, and FME. Using BB84 as a DV benchmark, it derives and contrasts finite-size key-rate expressions, with EUR yielding the strongest bounds at practical block lengths, AEP being asymptotically tight but pessimistic at moderate $N$, and FME maintaining nonzero rates when AEP vanishes though not asymptotically optimal. A closed-form expression for the optimal guessing probability $P_g$ is obtained in the symmetric QBER case via a fidelity-based SDP, illustrating when the direct min-entropy route is advantageous. The results inform security analyses for continuous-variable QKD and coherent-state schemes where EUR-tight bounds are not available, highlighting the complementary role of FME in finite-key regimes.

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.

Figures (4)

• Figure 1: Key rates against block size of the protocol computed exploiting: 1) Guessing Probability, 2) AEP, 3) Entropic uncertainty relation. The rates are estimated for a distance of communication of $10 km$ and for $\text{QBER}=3\%$. The probabilities $\epsilon_{PE}$, $\epsilon_{EC}$, $\epsilon_{h}$, $\epsilon_{h}$ and $\epsilon_{s}$ are all set to $10^{-10}$. For details see gioscaBB84.
• Figure 2: Key rates against block size of the protocol computed exploiting: 1) FME, 2) AEP, 3) EUR. The rates are estimated for a distance of communication of $10 km$ and for $\text{QBER}=6\%$. The probabilities $\epsilon_{PE}$, $\epsilon_{EC}$, $\epsilon_{h}$, $\epsilon_{h}$ and $\epsilon_{s}$ are all set to $10^{-10}$.
• Figure 3: Key rates against QBER computed exploiting 1) Guessing Probability, 2) AEP, 3) Entropic uncertainty relation. The rates are estimated for a distance of communication of $10 km$ and for a block size $N=10^5$.
• Figure 4: Key rates against $Qber$ in the asymptotic regime $N\longrightarrow\infty$ computed exploiting 1) Guessing Probability, 2) AEP. In the asymptotic limit AEP rate coincides with the one obtained through entropic uncertainty relation. The rates are estimated for a distance of communication of $10 km$.