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Soft Reverse Reconciliation for Discrete Modulations

Marco Origlia, Marco Secondini

TL;DR

This paper addresses the bottleneck of information reconciliation in CV-QKD with discrete modulations by introducing reverse reconciliation softening (RRS). Bob broadcasts a soft metric $N$ derived from his channel observation, designed to satisfy $I(N;\hat{X})=0$ so that Eve gains no additional information, while enabling Alice to compute log-likelihood ratios for soft decoding. The authors derive the transformation functions $g_i$, establish the zero-leakage constraint, formulate LAPPRs for bit-wise decoding, and validate the approach with simulations on PAM-4 showing significant MI and BER gains over hard RR and approaching the performance of direct soft RR. The results indicate that RRS can substantially improve reconciliation efficiency and key rates in discrete-modulation CV-QKD and can be extended to higher-order modulations and different coding schemes in practical systems.

Abstract

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.

Soft Reverse Reconciliation for Discrete Modulations

TL;DR

This paper addresses the bottleneck of information reconciliation in CV-QKD with discrete modulations by introducing reverse reconciliation softening (RRS). Bob broadcasts a soft metric derived from his channel observation, designed to satisfy so that Eve gains no additional information, while enabling Alice to compute log-likelihood ratios for soft decoding. The authors derive the transformation functions , establish the zero-leakage constraint, formulate LAPPRs for bit-wise decoding, and validate the approach with simulations on PAM-4 showing significant MI and BER gains over hard RR and approaching the performance of direct soft RR. The results indicate that RRS can substantially improve reconciliation efficiency and key rates in discrete-modulation CV-QKD and can be extended to higher-order modulations and different coding schemes in practical systems.

Abstract

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.

Paper Structure

This paper contains 11 sections, 15 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reconciliation Softening scheme overview
  3. Theory
  4. Constraint on information leakage
  5. Transformation functions
  6. Mutual Information
  7. Construction of the LAPPRs
  8. Simulations and Results
  9. Mutual information
  10. Bit Error Rate after Error Correction
  11. Conclusions

Figures (4)

  • Figure 1: System overview of the scheme. Bob generates a soft metric $N$ from the symbol $Y$ he received and Alice uses it to generate the log-a posteriori probability ratios (LAPPR) of the corresponding bits.
  • Figure 2: Alice hypothesis on $Y$ given the received side information $n$
  • Figure 3: Mutual information for a PAM-4 modulation with different reconciliation directions and configurations.
  • Figure 4: BER characteristic curves for PAM-4, LDPC rate 1/2 and different reconciliation directions and configurations.