Continuous-mode analysis for practical continuous-variable quantum key distribution

TL;DR

This paper tackles the gap between theory and practice in CV-QKD by moving from a single-mode to a continuous-mode description using temporal modes (TMs). It develops an entanglement-based security framework with a secret-key-rate method tailored to continuous-mode CV-QKD and demonstrates that pulse shaping, detector bandwidth, and sampling-time effects critically influence performance through the mode-matching coefficient $ abla_{ ext{match}}$. The authors validate the model experimentally at 30 km and show that a DSP-based linear weighted-reconstruction approach can boost the key rate by about 50% without extra hardware, highlighting tangible optimization routes for metropolitan-scale deployments. Overall, the TM-based framework generalizes existing analyses, accurately captures practical nonidealities, and provides actionable guidance for optimizing digital CV-QKD systems.

Abstract

Continuous-variable quantum key distribution (CV-QKD) enables two remote parties to establish information-theoretically secure keys and offers high practical feasibility due to its compatibility with mature coherent optical communication technologies. However, as CV-QKD systems progress toward digital implementations, device nonidealities drive the optical field from a single-mode to a continuous-mode region, thereby underscoring the mismatch between theoretical models and practical systems. Here, we introduce temporal modes to construct an entanglement-based scheme that more accurately captures device nonidealities and develop a corresponding secret key rate calculation method applicable to continuous-mode scenarios. We demonstrate that optimizing the pulse-shaping format can significantly improve performance under detector-bandwidth-limited conditions. Experimental results also confirm that the proposed model effectively describes the impact of sampling-time deviations. We further analyze a linear weighted-reconstruction digital signal processing method,which improves the secret key rate by approximately 50% in a 30-km fiber experiment without requiring additional hardware, demonstrating a substantial performance enhancement at metropolitan distances. The proposed theoretical framework accommodates a broader range of experimental conditions and can guide the optimization of digital CV-QKD systems.

Figures (5)

• Figure 1: Comparison between the single-mode and continuous-mode scenarios. (a) PM scheme under the single-mode assumption. (b) EB scheme in the continuous-mode scenario. (c) EB scheme under the single-mode assumption. (d) EB scheme in the continuous-mode scenario. When $T_A=1 / 2$, the scheme is equivalent to Alice sending coherent states; when $T_A=1$, it is equivalent to Alice sending squeezed states. When $T_B=1 / 2$, the scheme is equivalent to Bob performing heterodyne detection; when $T_B=1$, it corresponds to Bob performing homodyne detection. The dark-blue detector represents an ideal detector, while the yellow detector represents a non-ideal detector that includes only detection efficiency and electronic noise. $\xi$ denotes a wavepacket that contains time-domain information.
• Figure 2: The impact of different detector bandwidths and pulse-shaping formats on the TM matching coefficient between the transmitter and receiver. (a) Detector bandwidth: 5-MHz; square pulse with a 5-MHz modulation rate at the transmitter. (b) Detector bandwidth: 10-MHz; square pulse with a 5-MHz modulation rate at the transmitter. (c) Detector bandwidth: 10-MHz; square pulse with a 10-MHz modulation rate at the transmitter. (d) Detector bandwidth: 10-MHz; raised-cosine pulse with a 5-MHz modulation rate at the transmitter.
• Figure 3: Experimental optical setup for the CV-QKD system. We perform a Gaussian-modulated coherent state and homodyne detection experiment. We analyze the impact of sampling-time offsets on the secret key rate based on the proposed theoretical model. We also evaluate the performance improvement provided by the linear weighted-reconstruction DSP method. AM: amplitude modulator. PM: phase modulator. DAC: digital-to-analog converter. BC: bias controller. VOA: variable optical attenuator. MPC: manual polarization controller. BS: beam splitter. BHD: balanced homodyne detector.
• Figure 4: Experimental results for the 30-km fiber experiment. The blue curve represents the wavepacket reconstructed from the receiver data. The orange markers indicate the secret key rates obtained at different sampling points. The gray region corresponds to zero key rate.
• Figure 5: Simulation of the impact of different mode-matching coefficients on system performance.