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

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.

💡 Research Summary

This paper addresses a critical gap between the idealized single‑mode models traditionally used in continuous‑variable quantum key distribution (CV‑QKD) security analyses and the realities of modern, digitally implemented systems. As CV‑QKD moves toward integration with classical coherent‑communication hardware, device non‑idealities such as finite laser linewidth, phase noise, limited detector bandwidth, and multi‑point sampling introduce a continuous‑mode structure to the optical field that single‑mode theories cannot capture.

The authors introduce temporal modes (TMs) as the analytical backbone for describing these continuous‑mode fields. By treating each TM as an effective single mode, they construct a unified security framework that retains the mathematical convenience of single‑mode analysis while accurately representing the spectral and temporal distribution of practical quantum states.

A central contribution is the development of an entanglement‑based (EB) scheme that is equivalent to the prepare‑and‑measure (PM) protocol in the continuous‑mode regime. Starting from the continuous‑mode annihilation and creation operators ( \hat a(\omega) ) and ( \hat a^\dagger(\omega) ), the authors define a two‑continuous‑mode squeezed vacuum (TCMSV) state via a continuous‑mode squeezing operator ( \hat S_2(\beta) ). Through inverse Fourier transformation, they map the frequency‑domain operators to time‑domain TM operators ( \hat A_{\xi_i} ) associated with orthonormal wave‑packets ( \xi_i(t) ). Measuring one mode of the TCMSV is shown to be equivalent to Alice preparing a continuous‑mode quantum state, thereby preserving the EB‑PM equivalence even when the field possesses a non‑uniform temporal envelope.

On the detection side, the paper introduces a mode‑matching coefficient ( \eta_{\text{match}} ), defined as the squared overlap between the transmitter’s TM ( \xi_A(t) ) and the receiver’s effective TM after digital signal processing (DSP) ( \Xi_{\text{DSP}}(t) ). This coefficient captures the combined effects of detector efficiency, electronic noise, finite bandwidth, and DSP‑induced filtering. A low‑bandwidth detector acts as a low‑pass filter, truncating the temporal wave‑packet and reducing ( \eta_{\text{match}} ); consequently, the secret‑key rate drops sharply.

Numerical simulations explore how pulse‑shaping formats (rectangular, Gaussian, and optimized roll‑off) interact with detector bandwidth. Optimized pulse shapes align the spectral content of the transmitted wave‑packet with the detector’s passband, increasing ( \eta_{\text{match}} ) by more than 20 % relative to a naïve rectangular pulse and yielding a commensurate rise in the secret key rate.

Experimental validation is performed on a 30‑km standard single‑mode fiber link. The authors deliberately introduce sampling‑time offsets of 0 ns, 40 ns, and 50 ns. A 40 ns offset reduces the key rate by 69 %, while a 50 ns offset eliminates it entirely, confirming the sensitivity of ( \eta_{\text{match}} ) to timing errors.

A major practical innovation is the linear weighted‑reconstruction DSP method. By acquiring multiple samples within a single pulse period (e.g., four points) and applying pre‑optimized linear weights, the receiver synthesizes a single, higher‑fidelity quadrature estimate without any hardware modifications. This technique improves the secret‑key rate by roughly 50 % under the same experimental conditions, demonstrating that sophisticated DSP can compensate for continuous‑mode losses.

In summary, the paper provides a comprehensive theoretical framework that unifies single‑mode and continuous‑mode CV‑QKD analyses, quantifies the impact of realistic device imperfections via the mode‑matching coefficient, and offers concrete mitigation strategies—pulse‑shape optimization and weighted‑reconstruction DSP. These results pave the way for robust, metropolitan‑scale CV‑QKD deployments where digital processing and hardware design are co‑optimized to preserve security and performance.