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Long Distance Continuous-Variable Quantum Key Distribution with a Gaussian Modulation

Paul Jouguet, Sébastien Kunz-Jacques, Anthony Leverrier

arXiv:1110.0100v2quant-phcs.IT

TL;DR

Long-distance Gaussian-modulated CVQKD is limited by inefficient reconciliation at low SNR. The paper designs high-efficiency error-correcting codes for a binary-input AWGN channel and combines them with multidimensional reconciliation. The resulting procedure supports secret-key distribution over distances above 150 km against collective attacks in the asymptotic regime.

  • Problem

    Reconciliation efficiency limits the secret-key rate and secure range of Gaussian-modulated CVQKD, especially at the low SNRs associated with long distances.

  • Method

    The paper designs low-rate multi-edge LDPC codes for the BIAWGNC and combines them with multidimensional reconciliation of Gaussian variables.

  • Results

    Above 150 km secure distance is obtained against collective attacks in the asymptotic regime, with codeword efficiency within 1% of asymptotic efficiency at length 220.

  • Takeaways & Limitations

    The codes enable Gaussian-modulated CVQKD secret-key distribution over long distances and can be implemented through software modifications to existing experimental setups.

Abstract

from arXiv · show

We designed high-efficiency error correcting codes allowing to extract an errorless secret key in a continuous-variable quantum key distribution 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 AWGN 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 continuous-variable quantum key distribution with a Gaussian modulation, giving a secret key rate of about 10^{-3} bit per pulse at a distance of 120 km for reasonable physical parameters.

I. INTRODUCTION

Gaussian-modulated CVQKD offers theoretically optimal secret-key rates and uses standard telecommunication components, but reconciliation inefficiency limits long-distance operation. The paper addresses this bottleneck with efficient low-SNR codes combined with multidimensional reconciliation.

  • Gaussian-modulated coherent states measured by homodyne or heterodyne detection use standard telecommunication components and are theoretically optimal for secret-key rate.
  • Existing reconciliation inefficiency limits Gaussian-modulated CVQKD, especially at the low SNRs associated with long distances.
  • Non-Gaussian protocols may increase achievable secure distance theoretically, but they had not yet been demonstrated experimentally and generally require lower modulation variance at long distances.
  • High-efficiency error-correcting codes combined with multidimensional reconciliation enable secret-key distillation with Gaussian modulation at very low SNR.
  • The reconciliation problem is translated into BIAWGNC channel coding, followed by the design of very low-rate error-correcting codes and evaluation of long-distance Gaussian-protocol performance.

II. THEORY OF RECONCILIATION OF GAUSSIAN VARIABLES

The paper develops a multidimensional reconciliation framework that maps correlated Gaussian variables to a virtual binary-input AWGN channel. Its efficiency depends on both code quality and the approximation dimension, while the construction has algebraic dimensionality limits.

  • Reconciliation efficiency affects both the secret-key rate and secure range, making correlated Gaussian-variable reconciliation central to CVQKD.
  • The multidimensional scheme maps d physical Gaussian-channel instances to d approximate copies of a virtual BIAWGNC for error correction and secret-key distillation.
  • Overall efficiency is determined by the error-correcting code’s BIAWGNC efficiency and the quality of the virtual-channel approximation.
  • Increasing the code efficiency and rotation dimension improves the global reconciliation efficiency.
  • The correlated Gaussian model represents direct reconciliation as y = tx + z and reverse reconciliation as x = t′y + z′ with Gaussian noise.
  • The virtual-channel noise becomes a fading channel with known side information, approaching BIAWGNC behavior as dimension increases.
  • The required division operator exists only in dimensions 1, 2, 4, and 8, preventing this construction from being used in arbitrary dimension.

III. RECONCILIATION OF GAUSSIAN VARIABLES: IMPLEMENTATION WITH LDPC CODES

The implementation uses optimized low-rate multi-edge LDPC codes for BIAWGNC reconciliation at very low SNR. These codes approach Shannon-limit performance and retain near-asymptotic efficiency at finite block length.

  • LDPC codes use sparse parity-check matrices and can be decoded efficiently through iterative belief propagation.
  • Differential Evolution and Discretized Density Evolution optimize LDPC ensembles by maximizing the correctable channel threshold.
  • Multi-edge-type LDPC codes provide low-rate, high-efficiency BIAWGNC codes suited to very low SNR, with degree-1 edges improving the threshold.
  • Reconciliation efficiency β is defined as R/C(s), and prior Gaussian-modulation techniques achieved at most 90% efficiency.
  • 95.9% efficiency is reported for a rate 1/10 BIAWGNC code, while the paper introduces lower-rate codes with higher asymptotic thresholds.
  • The finite-length efficiency of codewords of length 220 is within 1% of the asymptotic efficiency.

A. Simulation Results with Rotations on S1, S3 and S7

The simulations evaluate multidimensional reconciliation with rotations at dimensions d = 1, 2, 4, and 8, using low-rate multi-edge LDPC codes. Increasing dimension brings the virtual channel closer to the BIAWGNC, while successful decoding yields zero bit errors.

  • Simulation setup: The simulations compare dimensions d = 1, 2, 4, and 8 using multidimensional reconciliation with low-rate multi-edge LDPC codes.The evaluated codes use block size 2^20.
  • Decoding outcomes: A frame error rate of about 1/3 coexists with a null bit error rate on successfully decoded blocks.Residual-error removal with concatenated BCH codes is therefore unnecessary for these blocks.
  • Reconciliation efficiency: The rotated channels always achieve lower efficiencies than BIAWGNC density-evolution predictions because they are not exact BIAWGNCs.The efficiency gap reflects the approximation between the physical Gaussian channel and its virtual binary-input channel.
  • Reconciliation efficiency: Increasing rotation dimension improves efficiency by making the virtual channel closer to the BIAWGNC.The input-vector norm follows a χ(d) distribution that approaches a Dirac distribution as d increases.
  • Capacity comparison: Figure 1 compares multidimensional-channel capacities with BIAWGNC capacity and BIAWGNC capacity with AWGNC capacity across SNR values.The multidimensional cases shown are d = 1, 2, 4, and 8.

B. Use of rotations in higher dimension spaces

The paper examines higher-dimensional rotations for multidimensional reconciliation and develops a more efficient representation of random orthogonal transformations. The proposed representation reduces complexity to O(d^2), while the available reconciliation scheme is restricted to dimensions 1, 2, 4, and 8.

  • Dimensional scope: The multidimensional reconciliation construction is limited to dimensions 1, 2, 4, and 8 because only these dimensions support the required division structure.
  • Original construction: The standard construction draws a random orthogonal transformation and composes it with a reflection, with complexity O(d^3).The resulting transformation maps Alice’s vector x to u while preserving Euclidean distance.
  • Improved construction: The proposed representation reduces the transformation-drawing complexity to O(d^2) without revealing information about u beyond the constraint R(x) = u.The method avoids revealing the transformation in matrix form.
  • Improved construction: Householder decomposition represents a Haar-distributed orthogonal transform recursively through reflections described by vectors.The decomposition uses a fixed orthogonal basis and recursively transforms the subspace spanned by e2 through ed.
  • Constrained sampling: The constrained sampling procedure chooses g so that u · g = x · e1, enabling the required relation Q(x) = u and Q(e1) = g.For d > 1, g is selected from unit vectors satisfying the constraint; the resulting vector is computed in linear time.
  • Higher-dimensional performance: For the rate 1/2 multi-edge LDPC code, increasing dimension above 8 at high SNR significantly increases reconciliation efficiency and the resulting QKD key rate.The code’s BIAWGNC threshold is s* = 1.074, corresponding to 98.2% efficiency.

C. Dealing with a continuous range of SNR with puncturing, shortening and repetition

A finite family of low-rate codes is extended across a continuous SNR range using repetition, puncturing, and shortening, while retaining high efficiency for small rate changes.

  • Repetition: A rate-0.02 code with 98% efficiency at SNR 0.03 yields 97% efficiency at SNR 0.01 using repetition length 3.Repetition factors 2 and 4 produce the rate-0.01 and rate-0.005 codes listed in Table V.
  • Puncturing and shortening: Puncturing deletes p symbols and shortening deletes s symbols, adapting an (n, k) LDPC code to nearby rates.Puncturing changes the code to (n−p, k), while shortening changes the rate according to the stated transformed expression.
  • Puncturing and shortening: A 5% rate decrease through shortening or 10% increase through puncturing incurs an efficiency loss smaller than 1%.The loss remains small for small relative variations of the code rate.

IV. PRACTICAL USE FOR A CONTINUOUS-VARIABLE QUANTUM KEY DISTRIBUTION SYSTEM

The proposed reconciliation codes are evaluated in Gaussian-modulated CVQKD under practical channel and detector assumptions, showing improved key-rate robustness across SNR and distance.

  • Code evaluation: The reconciliation efficiency is computed relative to AWGNC capacity, β = R / C_AWGNC, because the physical quantum channel is Gaussian.For small SNR values, BIAWGNC and AWGNC capacities are very close.
  • Code evaluation: The codes achieve about 95% efficiency at selected low SNRs, and the analysis omits finite-size effects.The key-rate plots therefore represent the asymptotic regime described in the section.
  • Experimental assumptions: The practical model restricts modulation variance to [1, 100], assumes 0.2 dB/km attenuation, 0.6 homodyne efficiency, and 1% electronic noise.The lower modulation bound reflects compatibility with synchronization and phase-tracking signals under limited modulator extinction ratios.
  • Modulation choice: Good reconciliation efficiency permits high modulation variance, whereas previous discrete-modulation schemes require variances 10 times lower.Figure 2 plots optimal modulation variance against distance for β = 95% and β = 90%.
  • Key-rate performance: Improved reconciliation efficiency provides a wider SNR range with a close-to-optimum secret key rate at every distance.Figures 3 and 4 consider excess noise of 1% and 4% of shot noise, respectively.
  • Key-rate performance: Above 150 km secure distance is obtained with 1% excess noise, and above 140 km with 4% excess noise using the same codes.This exceeds the approximately 50 km distance associated with 90% efficiency at SNR 0.5 in the cited prior scheme.

V. CONCLUSION

The paper presents high-efficiency error-correcting codes for long-distance Gaussian-modulated CVQKD and reports secure operation beyond 150 km against collective attacks asymptotically.

  • Conclusion: The codes enable secret-key distribution with Gaussian modulation over long distances in a continuous-variable quantum key distribution system.The conclusion states that implementation requires only software modifications to the experimental setups of references and.
  • Conclusion: Above 150 km secure distance is achieved against collective attacks in the asymptotic regime.

Appendix A: A rate 1/50 multi-edge LDPC code (σ∗= 5.91 on the BIAWGNC)

The appendix describes a rate-0.02 multi-edge LDPC code through variable- and check-node multidegree distributions.

  • Code construction: The rate-0.02 multi-edge LDPC ensemble is specified by multidegree distributions for variable nodes and check nodes.The left half of the array describes variable-node distributions, while the right half describes check-node distributions.
  • Code construction: A variable node has multidegree [2, 57, 0] with probability 0.0225 in the illustrated ensemble.
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