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arXiv:2402.04770

Continuous-Variable Quantum Key Distribution with key rates far above the PLOB bound

Arpan Akash Ray, Boris Škorić

Published Feb 7, 2024• 0 citations
arXivHTMLPDF

Abstract

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 $I(X;Y)$ between Alice's quadratures $X$ and Bob's measurements $Y$, 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 $I(X;Y) - I(E;Y)$, 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.

DOI
10.48550/arxiv.2402.04770

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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.
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.
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 an efficient logical layer-based reconciliation method for continuous-variable quantum key distribution (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 low-dimensional (scalar) space, and the reconciliation can be extended to arbitrary dimensions. The CVQKD systems allow an unconditionally secret communication over standard telecommunication networks. To exploit the real potential of CVQKD a robust reconciliation technique is needed. It is currently unavailable, which makes it impossible to reach the real performance of the CVQKD protocols. The reconciliation is a post-processing step separated from the transmission of quantum states, which is aimed to derive the secret key from the raw data. 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 define the low-dimensional reconciliation. We prove that the error probability of one-dimensional reconciliation is zero in any practical CVQKD scenario, and provides unconditional security. The results allow for significantly improving the currently available key rates and transmission distances of CVQKD.
We propose a continuous-variable quantum key distribution (CVQKD) protocol that uses a Shannon- Kotel'nikov expansion map for preparing coherent states. This kind of mapping acts like an analog error correcting code that improves the fidelity between the source signal and the reconstructed one at the receiver. It is well known that a limitation for extending CVQKD to larger distances is the reconciliation efficiency. Such lack of efficiency in reconciliation is mainly due to the difficulty of finding good error correcting codes for low signal-to-noise ratios. Having this limitation in mind, our protocol allows increasing the signal-to-noise ratio between Alice and Bob, thus making reconciliation easier.
Information reconciliation is an essential step of continuous-variable quantum key distribution (CV-QKD) because its efficiency affects both the maximum transmission distance and the final secret key rate of the whole system. Slice reconciliation can achieve high secret key rates for the CV-QKD systems under high signal-to-noise ratios (SNRs). However, it suffers from the high designing cost of the error-correcting codes in the multi-level coding and multi-stage decoding step. To reconcile one set of data with fixed SNR, multiple error-correcting codes with different code rates need to be designed. If the channel environment changed, the code rates need to change along to maintain high reconciliation efficiency, making it complicated and costly. In this paper, we introduce polar codes into slice reconciliation and proposed a scheme that is simply constructed, effectively performed, and well adapted to the systems with wide-range and variational SNRs. Simulation results show that the proposed scheme offers reconciliation efficiency over 95% for the systems with SNRs range from 1 to 10, which can support the systems with high secret key rates. The decrease of complexity and the increase of secret key rate have great significance for a practical CV-QKD system.
The party Alice in a cryptograph system transmits coherent quantum signals to the party Bob,obtaining a series of correlated continuous Gaussian variables.The process to distill a binary key out of the correlated Gaussian variables that they share is called continuous-variable quantum key distribution(CVQKD).Reconciliation in CVQKD,i.e.,to correct differences between the correlated variables,is the key issue discussed in the paper.Based on both sliced error correction(SEC) protocol and multilevel coding/multistage decoding(MLC/MSD),reconciliation is implemented by LDPC code-based Slepian-Wolf CODEC.According to the proposed multilevel Tanner graph which depicts message transition of information,multistage iterative intrinsic and extrinsic information belief-propagation(LLR-BP) update formulae are derived.Three decoding modes of MSD are compared experimentally in convergence and performance.Simulation results show that the proposed algorithm can reconcile reliably 10000 continuous quantum variables with efficiency of 92.3% when SNR of the channel is above 2.5 dB.