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arXiv:2512.15990

Random coding for long-range continuous-variable QKD

Arpan Akash Ray, Boris Skoric

quant-phcs.CRSubmitted Dec 17, 2025• Announced Dec 19, 2025• 0 citations
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Abstract

Quantum Key Distribution (QKD) schemes are key exchange protocols based on the physical properties of quantum channels. They avoid the computational-hardness assumptions that underlie the security of classical key exchange. Continuous-Variable QKD (CVQKD), in contrast to qubit-based discrete-variable (DV) schemes, makes use of quadrature measurements of the electromagnetic field. CVQKD has the advantage of being compatible with standard telecom equipment, but at long distances has to deal with very low signal to noise ratios, which necessitates labour-intensive error correction. It is challenging to implement the error correction decoding in realtime. In this paper we introduce a random-codebook error correction method that is suitable for long range Gaussian-modulated CVQKD. We use likelihood ratio scoring with block rejection based on thresholding. For proof-technical reasons, the accept/reject decisions are communicated in encrypted form; in this way we avoid having to deal with non-Gaussian states in the analysis of the leakage. The error correction method is highly parallelisable, which is advantageous for realtime implementation. Under conservative assumptions on the computational resources, we predict a realtime key ratio of at least 8% of the Devetak-Winter value, which outperforms existing reconciliation schemes.

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Feb 19, 2014•20 citations
Quantum Key Distribution (QKD) schemes are key exchange protocols based on the physical properties of quantum channels. They avoid the computational-hardness assumptions that underlie the security of classical key exchange. Continuous-Variable QKD (CVQKD), in contrast to qubit-based discrete-variable (DV) schemes, makes use of quadrature measurements of the electromagnetic field. CVQKD has the advantage of being compatible with standard telecom equipment, but at long distances has to deal with very low signal to noise ratios, which necessitates labour-intensive error correction. It is challenging to implement the error correction decoding in realtime.
In this paper we introduce a random-codebook error correction method that is suitable for long range Gaussian-modulated CVQKD. We use likelihood ratio scoring with block rejection based on thresholding. For proof-technical reasons, the accept/reject decisions are communicated in encrypted form; in this way we avoid having to deal with non-Gaussian states in the analysis of the leakage. The error correction method is highly parallelisable, which is advantageous for realtime implementation. Under conservative assumptions on the computational resources, we predict a realtime key ratio of at least 8% of the Devetak-Winter value, which outperforms existing reconciliation schemes.
Continuous-Variable Quantum Key Distribution (CVQKD) at large distances has such high noise levels that the error-correcting code must have very low rate. In this regime it becomes feasible to implement random-codebook error correction, which is known to perform close to capacity. We propose a reverse reconciliation scheme for CVQKD in which the first step is advantage distillation based on random-codebook error correction operated above the Shannon limit. Our scheme has a novel way of achieving statistical decoupling between the public reconciliation data and the secret key. We provide an analysis of the secret key rate for the case of Gaussian collective attacks, and we present numerical results. The best performance is obtained when the message size exceeds the mutual information I(X;Y)I(X;Y)I(X;Y) between Alice's quadratures XXX and Bob's measurements YYY, i.e. the Shannon limit. This somewhat counter-intuitive result is understood from a tradeoff between code rate and frame rejection rate, combined with the fact that error correction for QKD needs to reconcile only random data. We obtain secret key rates that lie far above the Devetak-Winter value I(X;Y)−I(E;Y)I(X;Y) - I(E;Y)I(X;Y)−I(E;Y), which is the upper bound in the case of one-way error correction. Furthermore, our key rates lie above the PLOB bound for Continuous-Variable detection, but below the PLOB bound for Discrete-Variable detection.
We designed high-efficiency error correcting codes allowing us to extract an errorless secret key in a continuous-variable quantum key distribution (CVQKD) protocol using a Gaussian modulation of coherent states and a homodyne detection. These codes are available for a wide range of signal-to-noise ratios on an additive white Gaussian noise channel with a binary modulation and can be combined with a multidimensional reconciliation method proven secure against arbitrary collective attacks. This improved reconciliation procedure considerably extends the secure range of a CVQKD with a Gaussian modulation, giving a secret key rate of about ${10}^{\ensuremath{-}3}$ bit per pulse at a distance of 120 km for reasonable physical parameters.
We propose a long-distance continuous-variable measurement-device-independent quantum key distribution (CV-MDI-QKD) protocol with discrete modulation. This kind of discrete-modulated scheme has good compatibility with efficient error correction code, which leads to higher reconciliation efficiency even at low signal-to-noise ratio. Security analysis shows that the proposed protocol is secure against arbitrary collective attacks in the asymptotic limit with the proper use of decoy states. With the use of discrete modulation, the proposed CV-MDI-QKD protocol has simpler implementation and outperforms previous protocols in terms of achievable maximal transmission distance, which precisely solved the bottleneck of the original Gaussian-modulated CV-MDI-QKD protocol.
The discrete-modulated continuous-variable quantum key distribution (DM-CV-QKD) could break the distance limitation of Gaussian-modulated CV-QKD. In practice, the high-performance error correction code plays an important role in DM-CV-QKD and affects the secure transmission distance. However, DM-CV-QKD usually works under low signal-to-noise ratio (SNR) and the design of high-performance error correction code under this condition is difficult, so that it would impose a limitation on further improvement of the secure distance. We propose a DM-CV-QKD with the machine-learning-based detector to further improve the secure distance. The numerical result shows that the proposed scheme could validly improve the system performance. Viewed from another perspective, the proposed scheme could be employed to overcome various impairments induced by the channel and thereby lower the demand of error correction codes on the SNR threshold of the quantum channel without compromising the system performance. The proposed scheme opens the door to applying machine leaning to directly process the raw secret key and improve the performance for CV-QKD systems.
Summary form only given. Continuous-variable quantum key distribution (CVQKD) is a promising alternative to commonly used discrete-variable QKD protocols, because this technology can be implemented with standard telecom components and recent results suggest that it will be compatible with standard wavelength division multiplexed telecommunication networks [1]. Security proofs against general eavesdropping attacks are available for CVQKD protocols using Gaussian modulation [2], and field implementations over deployed telecommunication networks have been demonstrated over the past few years [3].Long-distance operation of CVQKD systems, however, has been hindered until now by the limited efficiency of the complex error-correction (or reconciliation) algorithms that are required to extract the secret key from the correlated data shared by Alice and Bob. We have designed high-efficiency error correcting-codes [4] allowing to extract an errorless secret key in a CVQKD protocol using a Gaussian modulation of coherent states and a homodyne detection system. These codes are available for a wide range of signal-to-noise ratios on a Gaussian noise channel with a binary modulation and can be combined with a multidimensional reconciliation method [5]. This improved reconciliation procedure considerably extends the secure range of a continuous-variable quantum key distribution with Gaussian modulation.
The two-way continuous-variable quantum key distribution (CVQKD) systems allow higher key rates and improved transmission distances over standard telecommunication networks in comparison to the one-way CVQKD protocols. To exploit the real potential of two-way CVQKD systems a robust reconciliation technique is needed. It is currently unavailable, which makes it impossible to reach the real performance of a two-way CVQKD system. The reconciliation process of correlated Gaussian variables is a complex problem that requires either tomography in the physical layer that is intractable in a practical scenario, or high-cost calculations in the multidimensional spherical space with strict dimensional limitations. To avoid these issues, we propose an efficient logical layer-based reconciliation method for two-way CVQKD to extract binary information from correlated Gaussian variables. We demonstrate that by operating on the raw-data level, the noise of the quantum channel can be corrected in the scalar space and the reconciliation can be extended to arbitrary high dimensions. We prove that the error probability of scalar reconciliation is zero in any practical CVQKD scenario, and provides unconditional security. The results allow to significantly improve the currently available key rates and transmission distances of two-way CVQKD. The proposed scalar reconciliation can also be applied in oneway systems as well, to replace the existing reconciliation schemes.