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arXiv:2510.10674 · IEEE Transactions on Communications

Soft-Decoding Reverse Reconciliation in Discrete-Modulation CV-QKD

Marco Origlia, Emanuele Parente, Marco Secondini

Published Jan 1, 2026• 0 citations
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Abstract

In continuous-variable quantum key distribution, information reconciliation is required to extract a shared secret key from correlated random variables obtained through the quantum channel. Reverse reconciliation (RR) is generally preferred, since the eavesdropper has less information about Bob’s measurements than about Alice’s transmitted symbols. When discrete modulation formats are employed, however, soft information is available only at Bob’s side, while Alice has access only to hard information (her transmitted sequence). This forces her to rely on hard-decision decoding to recover Bob’s key. In this work, we introduce a novel RR technique for PAM (and QAM) in which Bob discloses a carefully designed soft metric to help Alice recover Bob’s key, while leaking no additional information about the key to an eavesdropper. We assess the performance of the proposed technique in terms of the achievable secret key rate (SKR) and its bounds, showing that the achievable SKR closely approaches the upper bound, with a significant gain over hard-decision RR. Finally, we implement the scheme at the coded level using binary LDPC codes with belief-propagation decoding, assess its bit-error rate through numerical simulations, compare the observed gain with theoretical predictions from the achievable SKR, and discuss the residual gap.

DOI
10.1109/tcomm.2026.3693195

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In continuous-variable quantum key distribution, information reconciliation is required to extract a shared secret key from correlated random variables obtained through the quantum channel. Reverse reconciliation (RR) is generally preferred, since the eavesdropper has less information about Bob’s measurements than about Alice’s transmitted symbols. When discrete modulation formats are employed, however, soft information is available only at Bob’s side, while Alice has access only to hard information (her transmitted sequence). This forces her to rely on hard-decision decoding to recover Bob’s key. In this work, we introduce a novel RR technique for PAM (and QAM) in which Bob discloses a carefully designed soft metric to help Alice recover Bob’s key, while leaking no additional information about the key to an eavesdropper. We assess the performance of the proposed technique in terms of the achievable secret key rate (SKR) and its bounds, showing that the achievable SKR closely approaches the upper bound, with a significant gain over hard-decision RR. Finally, we implement the scheme at the coded level using binary LDPC codes with belief-propagation decoding, assess its bit-error rate through numerical simulations, compare the observed gain with theoretical predictions from the achievable SKR, and discuss the residual gap.
The performance of the information reconciliation phase is crucial for quantum key distribution (QKD). Reverse reconciliation (RR) is typically preferred over direct reconciliation (DR) because it yields higher secure key rates. However, a significant challenge in continuous-variable (CV) QKD with discrete modulations (such as QAM) is that Alice lacks soft information about the symbol decisions made by Bob. This limitation restricts error correction to hard-decoding methods, with low reconciliation efficiency. This work introduces a reverse reconciliation softening (RRS) procedure designed for CV-QKD scenarios employing discrete modulations. This procedure generates a soft metric that Bob can share with Alice over a public channel, enabling her to perform soft-decoding error correction without disclosing any information to a potential eavesdropper. After detailing the RRS procedure, we investigate how the mutual information between Alice's and Bob's variables changes when the additional metric is shared. We show numerically that RRS improves the mutual information with respect to RR with hard decoding, practically achieving the same mutual information as DR with soft decoding. Finally, we test the proposed RRS for PAM-4 signalling with a rate 1/2 binary LDPC code and bit-wise decoding through numerical simulations, obtaining more than 1dB SNR improvement compared to hard-decoding RR.
The performance of the information reconciliation phase is crucial for quantum key distribution (QKD). Reverse reconciliation (RR\mathbf{R R}RR) is typically preferred over direct reconciliation (DR) because it yields higher secure key rates. However, a significant challenge in continuous-variable (CV) QKD with discrete modulations (such as QAM) is that Alice lacks soft information about the symbol decisions made by Bob. This limitation restricts error correction to hard-decoding methods, with low reconciliation efficiency. This work introduces a reverse reconciliation softening (RRS) procedure designed for CVQKD scenarios employing discrete modulations. This procedure generates a soft metric that Bob can share with Alice over a public channel, enabling her to perform soft-decoding error correction without disclosing any information to a potential eavesdropper. After detailing the RRS procedure, we investigate how the mutual information between Alice's and Bob's variables changes when the additional metric is shared. We show numerically that RRS improves the mutual information with respect to RR with hard decoding, practically achieving the same mutual information as DR with soft decoding. Finally, we test the proposed RRS for PAM-4 signalling with a rate1/2\mathbf{1 / 2}1/2binary LDPC code and bit-wise decoding through numerical simulations, obtaining more than 1dB SNR improvement compared to hard-decoding RR.
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.
Discrete modulation continuous-variable quantum key distribution protocols are compatible with modern coherent optical communication technologies and devices. As a key part of the protocol, the performance of information reconciliation directly affects the secret key rate and transmission distance. Herein, we propose a reconciliation scheme for hard-decision strategy--based discrete modulation continuous-variable quantum key distribution protocol with eight-phase shift keying by combining slice reconciliation and raptor-like low-density parity-check (RL-LDPC) codes. We design the optimal binary sequence encoding of the discretized values of the receiver and sender, which reduce both the number of levels of error correction and the reconciliation complexity. In order to reduce the frame error rate, both the degree distribution and the size of the base matrix of RL-LDPC code is optimized. Simulation results show that the proposed scheme can reach reconciliation efficiencies of 92.05% and 93% with frame error rates of 6.2% and 11.7% at the signal-to-noise ratio of 0.15, respectively. At these points, the key rates can reach 0.0193 and 0.0195 bits/pulse, respectively.
This paper is focused on the problem of Information Reconciliation (IR) for continuous variable Quantum Key Distribution (QKD). The main problem is quantization and assignment of labels to the samples of the Gaussian variables observed at Alice and Bob. Trouble is that most of the samples, assuming that the Gaussian variable is zero mean which is de-facto the case, tend to have small magnitudes and are easily disturbed by noise. Transmission over longer and longer distances increase the losses corresponding to a lower effective Signal to Noise Ratio (SNR) exasperating the problem. In previous work [1], [2] we approach the IR problem in a fundamentally different way than how it is addressed in published literature. At very low SNR, one can focus exclusively on the sign of the Gaussian samples at Alice and Bob and perform Reverse Reconciliation (RR) on sign of the samples given their quantized magnitudes as side information Alice and Bob share in the open. In this paper we demonstrate that the weighted sum capacity of the Binary Symmetric Channels (BSCs) derived from magnitude dependent sign error probabilities of the Gaussian samples shared between Alice and Bob approaches the mutual information between Alice and Bob at very low SNR regimes. Thus, when SNR is very low, one could just as well ignore any common randomness that may be obtained from magnitude of the Gaussian samples and focus exclusively on their sign. The caveat is, to get close to this mutual information, Unequal Error Protection (UEP) must be employed based on magnitude of the observed samples at Bob in RR.
In continuous-variable quantum key distribution (CV-QKD), the performance of the information reconciliation (IR) step is critical for the achievable secret key rate (SKR) and transmission distance. We show how to improve on the recently introduced implementation of an IR-protocol involving multiple decoding attempts (MDA) and validate the method on simulated data in different application scenarios. Throughout, we demonstrate meaningful SKR-gains compared to both the standard protocol of a single decoding attempt and to the original MDA-implementation, even at given decoding complexity.