Non-Gaussianity and security of entanglement-based QKD

TL;DR

This work investigates whether quantum non-Gaussianity can serve as a practical pre-check for channel suitability in entanglement-based QKD under depolarizing noise and detector imperfections. By analyzing entanglement-based BB84 and DI-QKD with SPAD and PNRD detectors, and modeling thermal and Poissonian noise, the authors establish non-Gaussianity witnesses and compare them to Devetak-Winter key-rate bounds, using Werner-state depolarization and Bell-CHSH correlations with S = 2 sqrt(2) (1 - 2Q). The study reveals cross-regions where non-Gaussianity witnesses are satisfied and the key rate is positive, with robustness to detection inefficiencies and dark counts, though the overlap diminishes at low transmittance (notably T ≲ 0.3). These findings offer a practical pre-check to gauge channel suitability for secret-key distribution and guide experimental setups prior to full QKD deployment with realistic detectors and noise. $S = 2 \sqrt{2} (1 - 2Q)$ and the Devetak-Winter bounds $r_{dw}^{DI-QKD} \ge 1 - h(Q) - h\left(\frac{1 + \sqrt{(S/2)^2 - 1}}{2}\right)$, $r_{dw}^{BB84} \ge 1 - h(Q) - h\left(Q + \frac{S}{2\sqrt{2}}\right)$ are central to linking non-Gaussianity to security across the considered scenarios.

Abstract

We theoretically analyse the relation between non-Gaussianity and security of entanglement-based quantum key distribution (QKD) protocols, namely device-independent (DI) and entanglement-based BB$84$. A similar analysis has already been made for prepare-and-measure (P\&M) protocols \cite{Lasota2017}. In addition, we consider imperfect detection with dark counts and limited efficiency. We assume a perfect source of entangled Bell states as produced by quantum-dot type sources, depolarisation in the channel and different noise statistics, namely thermal and Poissonian. We consider single-photon avalanche photodiodes (SPAD) and photon number resolving detectors (PNRD) and use their respective criteria for non-Gaussianity. The results show cross-regions for both security and non-Gaussianity, hence, the possibility to conclude about the suitability of a given channel for secret key distribution. Our results can be useful as a pre-check for the implementation of QKD protocols.

Figures (5)

• Figure 1: Schematics are shown for (a) the typical setup used in entanglement-based QKD protocols, consisting of polarization analyzers on each side comprising a half-wave plate (HWP), polarizing beam splitters (PBS), and two detectors $D_T, D_R$, where $T$ and $R$ stand for transmitted and reflected lights accordingly; (b) the setup for detecting non-Gaussian coincidences, which includes a balanced beam-splitter (BS) with transmittance $T=1/2$ and a pair of SPAD detectors on both Alice's and Bob's sides; (c) the setup for detecting non-Gaussian coincidences using PNRD on each side Lachman2021liu2023experimental. Different criteria, as given by \ref{['non-gauss-crit-spad']} and \ref{['non-gauss-crit-pnrd']}, are applied depending on the type of detectors.
• Figure 2: Model illustrating noise addition to the signal, incorporating two distinct noise types: (a) thermal noise characterized by thermal statistics with mean photon number $\nu$, and (b) multi-mode thermal noise which converges to Poissonian statistics in the limit of infinite number of modes. The effective coupling is modeled by the beamsplitter with transmittance $T$.
• Figure 3: The maximum mean number of photons in a thermal mode $\nu$, compatible with non-Gaussianity and security of QKD, versus the coupling ratio $T$ between the signal and the noise modes upon perfect PNRD detection. The non-Gaussianity region is plotted by the yellow area bounded by the blue line. The Dewetak-Winter key rates for entanglement-based BB$84$ and DI-QKD protocols are plotted by the green and red lines, respectively. The cross-region of non-Gaussianity and security is clearly observed for coupling $T$ values larger than approx. $0.3$.
• Figure 4: The maximum mean number of photons in a thermal mode $\nu$, compatible with non-Gaussianity and security of QKD, versus the coupling ratio $T$ between the signal and noise modes upon imperfect PNRD detection with efficiency $\eta = 0.7$ and dark count rate $d = 0.001$. The non-Gaussianity region is highlighted by the yellow area bounded by the blue curve. Dewetak-Winter key rates for entanglement-based BB84 and DI-QKD protocols are shown as green and red lines, respectively. The regions corresponding to non-Gaussianity and security nearly overlap, except at low values of coupling $T \lesssim 0.3$.
• Figure 5: The maximum mean number of photons in a thermal mode $\nu$, compatible with non-Gaussianity and security of QKD, versus the coupling ratio $T$ upon imperfect SPAD detection with efficiency $\eta$ and dark count rate $d$. The results illustrate nearly complete overlap between the regions of security and non-Gaussianity of the detected light, confirming the robustness of the non-Gaussianity criterion under realistic detection conditions. Inconclusive behavior is observed in the region of low coupling $T \lesssim 0.3$.