Finite-key security analysis of the decoy-state BB84 QKD with passive measurement

TL;DR

This work provides a fully analytical finite-key security proof for the decoy-state BB84 QKD protocol with passive, biased basis choice at the receiver, delivering a closed-form secret-key-rate formula expressible in terms of experimentally accessible quantities. The authors implement a phase-error-correction framework that avoids estimating Bob's single-photon rate and bound the overall phase error via observed X-basis error and cross-click data, using Kato's inequality for finite-key concentration. The main contributions are the bound on the virtual X-basis outcome set $|\Omega_{\rm ph}|$ and the decoy-state bound for $N_Z^{1}$, yielding a direct, implementable key-rate formula; simulations show passive and active implementations achieve nearly identical rates under typical parameters, with cross-clicks governing any residual gap. The results support the practical viability of passive measurements in QKD and point toward extending the approach to fully passive BB84 with passive state preparation by Alice.

Abstract

The decoy-state Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol is widely regarded as the de facto standard for practical implementations. On the receiver side, passive basis choice is attractive because it significantly reduces the need for random number generators and eliminates the need for optical modulators. Despite these advantages, a finite-key analytical security proof for the decoy-state BB84 protocol, where the basis is chosen passively with a biased probability, has been lacking. In this work, we present a simple analytical finite-key security proof for this setting, yielding a closed-form secret-key rate formula that can be directly evaluated using experimentally accessible parameters. Numerical simulations show that the key rates of passiveand active-measurement implementations are nearly identical, indicating that passive measurement does not compromise key-generation efficiency in practical QKD systems.

Figures (3)

• Figure 1: Schematic of our passive decoy-state BB84 protocol. For each $i$th emission, Alice sends to Bob a phase randomized coherent state $\hat{\rho}_{S_i}(a_i,\omega_i,\alpha_i)$ with her choices of bit $a_i$, basis $\alpha_i$ and intensity label $\omega_i$. The state is characterized by its intensity $\mu_{\omega_i}$, set by the intensity modulator (IM), and its polarization $\phi(a_i,\alpha_i)$, controlled by the polarization rotator (PR). Bob has a passive measurement setup, where the incoming pulse passes through the beam splitter (BS), whose transmittance is $q$, followed by the $Z$ and $X$ lines. The $Z$ line consists of a polarization beam splitter (PBS) and two threshold photon detectors with detection efficiency $\eta_{{\rm det}}$ and dark count probability $d$. The $X$ line consists of a half-wave plate (HWP), a polarization beam splitter (PBS), and two threshold photon detectors with the same detection efficiency and dark count probability.
• Figure 2: Secret key rate $R:=\ell/N$ per single emitted pulse as a function of the overall channel transmission $\eta$, with the dark count probability $d = 10^{-9}$. From bottom to top, we plot the key rates for the number of pulses emitted $N = 10^8$, $10^{10}$, $10^{12}$, and the asymptotic case, using solid lines. For comparison, we also plot the key rates of the active polarization-encoding decoy-state BB84 protocol with dashed lines in the same manner.
• Figure 3: Secret key rate $R:=\ell/N$ per single emitted pulse as a function of the overall channel transmission $\eta$, with the dark count probability $d = 10^{-6}$. From bottom to top, we plot the key rates for $N = 10^8$, $10^{10}$, $10^{12}$, and the asymptotic case, using solid lines. For comparison, we also plot the key rates of the active polarization-encoding decoy-state BB84 protocol with dashed lines in the same manner.