A Decoy-like Protocol for Quantum Key Distribution: Enhancing the Performance with Imperfect Single Photon Sources

TL;DR

This work addresses the security bottleneck of QKD with imperfect solid-state single-photon sources by proposing a decoy-like protocol that continuously monitors $g^{(2)}(0)$ to detect photon-number-splitting attacks. It derives a secret-key-rate bound that includes single- and two-photon contributions, using a bound on Eve's information and assuming worst-case for $n \ge 3$, and shows $g^{(2)}(0)$ is invariant to linear loss while PNS attacks perturb higher-order statistics. The authors experimentally characterize an $h$BN$ defect emitter to obtain $g^{(2)}(0)$ and higher-order correlators, and perform Monte Carlo simulations to demonstrate attack signatures and the resulting SKR advantages over the Gottesman–Lo–Lütkenhaus–Preskill framework, including satellite-relevant channel losses. The method requires no extra hardware and relaxes the need for ultra-pure SPSs, enabling secure QKD with $g^{(2)}(0)$ values above 0.1 and practical post-processing on existing BB84-like setups.

Abstract

Quantum key distribution (QKD) relies on single photon sources (SPSs), e.g. from solid-state systems, as flying qubits, where security strongly requires sub-Poissonian photon statistics with low second-order correlation values (\$g^{(2)}(0)\$). However, achieving such low \$g^{(2)}(0)\$ remains experimentally challenging. We therefore propose a decoy-like QKD protocol that relaxes this constraint while maintaining security. This enables the use of many SPSs with \$g^{(2)}(0) > \$0.1, routinely achieved in experiments but rarely considered viable for QKD. Monte Carlo simulations and our experiment from defects in hexagonal boron nitride show that, under linear loss, \$g^{(2)}(0)\$ remains constant, whereas photon-number-splitting (PNS) attacks introduce nonlinear effects that modify the measured \$g^{(2)}(0)\$ statistics. Exploiting this \$g^{(2)}(0)\$ variation as a diagnostic tool, our protocol detects PNS attacks analogously to decoy-state methods. Both single- and two-photon pulses consequently securely contribute to the secret key rate. Our protocol outperforms the Gottesman--Lo--Lutkenhaus--Preskill (GLLP) framework under high channel loss across various solid-state SPSs and is applicable to the satellite-based communication. Since \$g^{(2)}(0)\$ can be extracted from standard QKD experiments, no additional hardware is required. The relaxed \$g^{(2)}(0)\$ requirement simplifies the laser system for SPS generation. This establishes a practical route toward high-performance QKD without the need for ultra-pure SPSs.

Figures (4)

• Figure 1: Measured photon statistics of our hBN quantum emitter.a Second-order ($g^{(2)}(\tau)$). b Third-order ($g^{(3)}(\tau_1,\tau_2)$) photon correlation functions under pulsed excitation at a 25 MHz repetition rate and 80.5 $\mu W$ excitation power. The delay range of $\pm20~\mu s$ is chosen in the calculation to ensure that the side peaks become flattened; however, here we show the zoomed $\pm 500~ns$ window for the sake of clarity.
• Figure 2: Simulated photon number distributions $P_n$ after soft and hard PNS attacks, obtained via Monte Carlo simulation. The attack strength $x$ is varied from 0 (no attack) to 1 (full PNS attack). a is for our hBN. b is for hBN with high quantum efficiency 10.1364/OPTICA.6.001084. c is for QD 10.1103/PhysRevA.90.023846. The solid and dashed lines represent the soft and hard PNS attacks, respectively. Each simulation was performed $100$ times with $10^7$ samples per run. The reported $P_n$ values represent the mean with standard deviations.
• Figure 3: Absolute changes in photon statistics of different physical systems under varying PNS attack ratios. a,c the second-order correlation function at zero delay, $|g^{2}(0)' - g^{2}(0)|$, and b,d the mean photon number, $|\mu' - \mu|$, as functions of the PNS attack strength $x$. These results quantify how photon number splitting attacks progressively alter the photon statistics of the source under both soft and hard PNS attack models. a,b is for soft PNS attack while c,d is for hard PNS attack. Each simulation was performed $100$ times with $10^7$ samples per run. The reported $|g^{2}(0)' - g^{2}(0)|$ and $|\mu' - \mu|$ values represent the mean with standard deviations.
• Figure 4: Estimation of $g^{(2)}(0)$ and computed secret key rate as functions of channel loss for two scenarios.a Convergence of $g^{(2)}(0)$ values across Monte Carlo samples for the hBN emitter. Each simulation round was repeated $100$ times. Reported $g^{(2)}(0)$ values represent the mean with standard deviations. b Differences in $g^{(2)}(0)$ across Monte Carlo samples for our hBN emitter, compared with the reference $g^{(2)}(0)$ obtained from $10^8$ samples. Each simulation round was repeated $100$ times. c Comparison of computed secret key rates between the conventional GLLP protocol and our proposed scheme for our hBN emitter. The secondary $y$-axis shows the waiting time required to receive $10^5$ photons at each channel loss. The orange star indicates the flyover duration and the 38-dB channel loss for the Micius satellite–ground link at 645 km 10.1038/nature23655. d Simulations for various solid-state quantum emitters using experimental parameters from Ref. 10.1002/qute.202200059 for hBN-1, Ref. 10.1002/qute.202300038 for hBN-2, Ref. 10.1088/1367-2630/16/2/023021 for NV and SiV in diamond, Ref. 10.1093/nsr/nwaf147 for GaN-1, Ref. 10.1103/PhysRevApplied.23.054022 for GaN-2, and Ref. 10.48550/arXiv.2506.15520 for QD. All QKD experiments are assumed to follow the BB84 protocol.