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Random coding for long-range continuous-variable QKD

Arpan Akash Ray, Boris Skoric

TL;DR

The paper tackles the bottleneck of realtime error correction in long-range CVQKD by introducing a random-codebook reconciliation method with MAP-like, likelihood-ratio scoring and block-level acceptance that is securely hidden via OTP encryption. This decouples Eve’s information from the reconciliation data, enabling a Gaussian-leakage-based security analysis and allowing highly parallel decoding suitable for real-time operation. The authors derive an explicit asymptotic secret key ratio and show that, under conservative resource assumptions, several percent of the Devetak-Winter bound can be achieved (e.g., about 8% at q = 2^15), while also presenting practical pseudorandom variants and implementation considerations. The work provides a path toward scalable, real-time CVQKD reconciliation at very long distances, albeit with trade-offs related to OTP overhead and communication needs, and outlines multiple avenues for extending the approach to discrete modulation and finite-size regimes.

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.

Random coding for long-range continuous-variable QKD

TL;DR

The paper tackles the bottleneck of realtime error correction in long-range CVQKD by introducing a random-codebook reconciliation method with MAP-like, likelihood-ratio scoring and block-level acceptance that is securely hidden via OTP encryption. This decouples Eve’s information from the reconciliation data, enabling a Gaussian-leakage-based security analysis and allowing highly parallel decoding suitable for real-time operation. The authors derive an explicit asymptotic secret key ratio and show that, under conservative resource assumptions, several percent of the Devetak-Winter bound can be achieved (e.g., about 8% at q = 2^15), while also presenting practical pseudorandom variants and implementation considerations. The work provides a path toward scalable, real-time CVQKD reconciliation at very long distances, albeit with trade-offs related to OTP overhead and communication needs, and outlines multiple avenues for extending the approach to discrete modulation and finite-size regimes.

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.

Paper Structure

This paper contains 21 sections, 5 theorems, 26 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Error correction in Continuous-Variable QKD
  3. Contributions and outline
  4. Preliminaries
  5. Notation
  6. CVQKD with Gaussian modulation, homodyne detection, and reverse reconciliation
  7. 'Decoupled' reconciliation data
  8. Useful lemmas
  9. The proposed information reconciliation method
  10. Hiding accept/reject decisions
  11. Protocol steps
  12. Motivation of the score function
  13. The 'decoupling' property
  14. Statistical properties of the scores; error probabilities
  15. Asymptotic Secret Key Ratio
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $U$ and $P$ be independent classical random variables. Let $E$ be a quantum system. Then $I(U;PE)\leq I(UP;E)$.

Figures (5)

  • Figure 1: The mutual informations $I(X_i;Y_i)$ (\ref{['IXiYi']}) and $I(E_i;Y_i)$ (\ref{['leakFull']}) as a function of the modulation variance $\sigma_X^2$, for $T\ll 1$, $\xi\ll \sigma_X^2$.
  • Figure 2: Protocol steps.
  • Figure 3: Probability density function of the score for a fake table entry and for the correct $(u$'th) table entry. Contributions of order ${\mathcal{O}}(\sqrt\varepsilon)$ are neglected. The pdfs are almost Gaussian and have variance 1. The plot is for $q=2^{10}$, $\gamma=-0.28$
  • Figure 4: Secret key rate in bits per second, according to Table \ref{['table:SKRDW']}, at $q=2^{15}$, given a pulse rate of $10^6$ pulses per second. The loss is taken to be 0.2 dB/km. Excess noise is neglected.
  • Figure 5: SKR$_\infty$/DW as a function of $\gamma$ and $\delta$, for various combinations of $q$ and $\sigma_X^2$ (see Table \ref{['table:SKRDW']}). $T=10^{-6}$, $\xi=10^{-5}$.

Theorems & Definitions (5)

  • Lemma 2.1: Lemma 1 in LABZG2008
  • Corollary 2.2: Adapted from LABZG2008 to the case of reverse reconciliation
  • Lemma 2.3: Neyman-Pearson
  • Lemma 2.4
  • Lemma 4.1