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arXiv:2508.05558 · Physical Review A

Joint parameter estimation and multidimensional reconciliation for continuous-variable quantum key distribution

Anonymous, Xueqin Jiang, Peng Huang, Tao Wang, Guihua Zeng

Published Jul 23, 2026• 1 citation
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Abstract

Accurate quantum channel parameter estimation is essential for effective information reconciliation in continuous-variable quantum key distribution (CV-QKD). However, conventional maximum likelihood (ML) estimators rely on a large amount of disclosed data, leading to a significant loss in symbol efficiency. Moreover, the separation between the estimation and reconciliation phases can introduce error propagation. In this paper, we propose a novel joint message-passing scheme that unifies channel parameter estimation and information reconciliation within a Bayesian framework. By leveraging the expectation-maximization (EM) algorithm, the proposed method simultaneously estimates unknown parameters during decoding, eliminating the need for separate ML estimation. Furthermore, we introduce a hybrid multidimensional rotation scheme that removes the requirement for norm feedback, significantly reducing classical channel overhead. To the best of our knowledge, this is the first work to unify multidimensional reconciliation and channel parameter estimation in CV-QKD, providing a practical solution for high-efficiency reconciliation with minimal information disclosure.

DOI
10.1103/7w6s-93ql

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Accurate quantum channel parameter estimation is essential for effective information reconciliation in continuous-variable quantum key distribution (CV-QKD). However, conventional maximum likelihood (ML) estimators rely on a large amount of disclosed data, leading to a significant loss in symbol efficiency. Moreover, the separation between the estimation and reconciliation phases can introduce error propagation. In this paper, we propose a novel joint message-passing scheme that unifies channel parameter estimation and information reconciliation within a Bayesian framework. By leveraging the expectation-maximization (EM) algorithm, the proposed method simultaneously estimates unknown parameters during decoding, eliminating the need for separate ML estimation. Furthermore, we introduce a hybrid multidimensional rotation scheme that removes the requirement for norm feedback, significantly reducing classical channel overhead. To the best of our knowledge, this is the first work to unify multidimensional reconciliation and channel parameter estimation in CV-QKD, providing a practical solution for high-efficiency reconciliation with minimal information disclosure.
Multidimensional rotation serves as a powerful tool for enhancing information reconciliation and extending the transmission distance in continuous-variable quantum key distribution (CV-QKD). However, the lack of closed-form orthogonal transformations for high-dimensional rotations has limited the maximum reconciliation efficiency to channels with 8 dimensions over the past decade. This paper presents a cross-rotation scheme to overcome this limitation and enable reconciliation in arbitrarily high dimensions, constrained to even multiples of 8. The key treatment involves reshaping the string vector into matrix form and applying orthogonal transformations to its columns and rows in a cross manner, thereby increasing the reconciliation dimension by one order per cross-rotation while significantly reducing the communication overhead over the classical channel. A rigorous performance analysis is also presented from the perspective of achievable sum-rate. Simulation results demonstrate that 64-dimensional cross-rotation nearly approaches the upper bound, making it a recommended choice for practical implementations.
Continuous-variable quantum key distribution (CV-QKD) requires highly efficient reconciliation techniques to operate at low signal-to-noise ratios and long distances. Multidimensional reconciliation addresses this challenge by transforming the physical Gaussian quantum channel into a virtual binary-input additive white Gaussian noise (BIAWGN) channel, enabling the use of modern errorcorrecting codes. In this work, we review the principles of multidimensional reconciliation, with a particular focus on high-dimensional constructions beyond the algebraic dimensions 1, 2, 4, 8. We describe the construction of the virtual channel, discuss practical coding schemes for reverse reconciliation, and analyse their integration with linear error-correcting codes. We also present an opensource simulation framework, HDirac, implementing multidimensional reconciliation for arbitrary dimensions, and use it to evaluate state-of-the-art LDPC codes. The results highlight key trade-offs between dimension, reconciliation efficiency, and frame error rate, providing practical guidance for CV-QKD system design.
Continuous-variable quantum key distribution (CV-QKD) relies on efficient information reconciliation to establish identical secret key sequences from correlated modulation data and measurement outcomes. Accurate quadrature measurements are essential for reliable reconciliation, but residual phase mismatch between the signal and local oscillator distorts these measurements in coherent receivers and degrades decoding performance. To improve robustness against residual phase mismatch in homodyne-based CV-QKD, we propose a phase-aware multidimensional reconciliation method that jointly performs syndrome-based low-density parity-check (LDPC) decoding and expectation-maximization-based residual-phase refinement. We derive a discrete-time detection model that explicitly incorporates the receiver’s pulse-dependent quadrature-selection phase and construct phase-dependent virtual observations through multidimensional orthogonal mapping. The LDPC decoder provides posterior information for calculating the posterior spherical-symbol moments used in residual-phase refinement, while the refined phase is fed back to update the virtual observations and decoding likelihoods for the next iteration. Numerical results show that the proposed method achieves a lower residual-phase-estimation RMSE than disclosure-based cross-correlation estimation, BER and FER close to those under perfect phase compensation, and a higher finite-size secret key rate.
Abstract Information reconciliation is a significant stage in continuous-variable quantum key distribution (CV-QKD) systems as it directly affects the performance of the CV-QKD systems including secret key rate and secure transmission distance. This paper proposes a multidimensional reconciliation scheme using deep learning in CV-QKD systems. Firstly, different neural networks are constructed to obtain the norm information. Secondly, a multidimensional reconciliation scheme with deep learning assisted norm information is proposed which no longer needs to transmit the norm information through the authenticated classical public channel. Finally, simulation results and performance analysis show that, compared with the traditional multidimensional reconciliation scheme, the multidimensional reconciliation scheme with deep learning assisted norm information can decrease the communication traffic to a certain extent.
The well-known multidimensional reconciliation is a significant stage of a continuous-variable quantum key distribution (CVQKD) system, which uses ddd-dimensional rotations to build a virtual channel between authenticated parties, Alice and Bob. Generally, a block low-density parity-check code with a belief-propagation (BP) iterative decoding algorithm, which is optimized for an additive white Gaussian noise channel, is always used in multidimensional reconciliation for the CVQKD system. In this paper, we study the signal-to-noise ratio (SNR) of the virtual channel of multidimensional reconciliation in CVQKD and prove that the noise of the virtual channel follows the Student's ttt-distribution. Therefore, we propose a ttt-BP decoding algorithm, which can be better applied to multidimensional reconciliation. Simulation results show that the frame error rate (FER) with the proposed ttt-BP decoding algorithm is superior to that with the conventional BP decoding algorithm applied to multidimensional reconciliation. Subsequently, the FER improvement results in significant influence on reconciliation efficiency and secret key rate of the CVQKD system.
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.