Continuous-variable quantum key distribution (CV-QKD) offers information-theoretically secure key exchange using the continuous properties of quantized electromagnetic light. In modern CV-QKD implementations, the local oscillator (LO) at the receiver is independent of the transmitter laser, simplifying the system and enhancing security. However, this necessitates carrier recovery to compensate for frequency and phase noise, which is crucial as these impairments are indistinguishable from eavesdropper-induced noise. Traditional telecommunication carrier recovery algorithms perform poorly in the low signal-to-noise ratio (SNR) regime typical of CV-QKD and with Gaussian modulation formats. Pilot-aided techniques, employing a reference signal (pilot tone) transmitted alongside the quantum signal, have been developed to address this. The challenge lies in minimizing pilot tone power to reduce interference with the quantum signal. This work proposes a machine learning framework based on Bayesian inference, specifically an unscented Kalman filter (UKF), for efficient phase noise estimation using a pilot tone. The UKF's adaptive nature allows it to adjust to the system's behavior and thus overcome limitations of traditional approaches.

Previous research has explored phase tracking in CV-QKD using Bayesian inference with extended Kalman filters or smoothers, primarily focusing on systems with discrete modulation formats. Gaussian modulation, while having more mature security proofs, is more susceptible to phase noise due to higher mean photon numbers. Bayesian inference methods have also shown promise in measuring and characterizing laser phase noise, particularly in low-power regimes. The UKF is expected to improve upon the extended Kalman filter due to its enhanced performance with nonlinear systems.

The researchers conducted experiments on a CV-QKD system using commercially available telecom equipment over a 20-km SMF-28 fiber link. The transmitter prepared a 50 MBaud quantum signal and a frequency-multiplexed pilot tone, both modulated using electro-optic modulators. After transmission, a balanced heterodyne coherent receiver with a free-running LO detected the signal. Digital signal processing (DSP) steps included the UKF-based carrier recovery algorithm for phase noise compensation. Channel parameter estimation (excess noise mean photon number *e*, optical efficiency *n*, and mean photon number *N*) was performed to calculate the achievable secret key rate. Two different transmitter lasers were used: nominally 100 Hz linewidth fiber lasers and a 10 kHz linewidth telecom laser. The pilot tone SNR was varied by adjusting the filter bandwidth, keeping the pilot tone power constant to isolate other potential effects such as receiver saturation or nonlinearity. The reference method involved calculating the phase from the pilot signal using a Hilbert transform, removing the linear trend to compensate for frequency offset. The UKF, on the other hand, employed a state-space model describing the phase evolution and a measurement model of the pilot signal, using a Gaussian approximation of the process noise and a Metropolis-Hastings algorithm for iterative estimation and updating.

Experimental results demonstrated the superior performance of the UKF compared to both the reference method and the EKF. Using the 100 Hz laser, the UKF maintained low excess noise (*e*) even at pilot tone SNRs as low as 4 dB, reaching 2 × 10⁻³ at high SNRs, while the reference method performed significantly worse at low SNRs. The EKF showed comparable performance at high SNRs but deteriorated at lower SNRs. With the 10 kHz laser, the UKF's performance degraded at SNRs below 7 dB, still achieving *e* < 0.01 in the best case. Both the reference method and EKF performed worse than the UKF. Graphs displaying the secret key rate in the asymptotic regime further highlight the UKF's advantage, enabling secret key generation in scenarios where the reference method failed. The UKF consistently outperformed the other methods across a wide SNR range (over 10 dB).

The study demonstrates a substantial performance improvement by using a machine learning approach (UKF) for laser phase noise compensation in a Gaussian modulation CV-QKD system over a 20-km link. The success in achieving secret key generation with both low-linewidth (100 Hz) and higher-linewidth (10 kHz) lasers, even with relatively low pilot power, highlights the UKF's robustness. This robustness is particularly valuable in real-world scenarios where fiber attenuation and noise can degrade SNR. The feasibility of real-time UKF implementation given the moderate symbol rates in CV-QKD suggests its potential as a crucial component in future CV-QKD systems employing independent LOs. The relatively simple implementation compared to polarization multiplexing is another substantial advantage.

This work successfully demonstrates the effectiveness of an unscented Kalman filter based machine learning framework for compensating laser phase noise in CV-QKD. The UKF consistently outperforms standard methods, enabling secure key generation even under challenging conditions (low SNR, high laser linewidth). Its robustness and relative simplicity make it a promising candidate for integration into future practical CV-QKD systems.

The study's focus on the asymptotic regime limits its direct applicability to finite-key scenarios. Further investigation is needed to fully understand the performance degradation observed with the 10 kHz laser at low SNRs, potentially due to the faster changing beat mode frequency. While the study demonstrates the superiority of the UKF in the given experimental setup, extensive testing across diverse environments and conditions is recommended for broader validation.