Reverse Reconciliation with Soft Information

Table of Contents

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1. Our papers on the topic (in chronological order)

  1. Soft Reconciliation for Discrete Modulations [1] (2025) is the first paper where we introduced this technique. It was presented at the 14th International ITG Conference on Systems, Communications and Coding (SCC) [2] in Karlsruhe, Germany.
  2. Soft-Decoding Reverse Reconciliation in Discrete-Modulation CV-QKD (2026) [3], the most formal introduction of the RRS scheme, has been published in the IEEE Transactions on Communications.
  3. In Reverse Reconciliation with Soft Information for Discrete-Modulation CV-QKD at Long Range [4], published at the present conference, shows that the RRS scheme works well also in the low-SNR regime.

2. Software

  • `Re2often` [5] is the library that deals with the reverse reconciliation softening: computation of the entropic quantities and random data generation and processing for simulations.
  • `fldpc` [6] is the LDPC library that implements belief propagation, that has been employed for Montecarlo BER estimation.

The parity check matrices we empoyed in our simulations are described in [7], and used in another information reconciliation library [8].

3. Related topics

The well known reconciliation schemes in the literature are:

  • Multi-dimensional reconciliation (MDR) [9]
  • Slice reconciliation (SR) [10]

3.1. On Arithmetic Reconciliation

Recently, arithmetic reconciliation (AR) [11] was also introduced. It is based on the distributional transform expansion (DTE) [12]. As per my understanding, the DTE is made up of 2 fundamental steps:

  1. the channel output is transformed through its own CDF, yielding a uniform random variable in the interval \([0, 1]\);
  2. the resulting uniform random variable is discretized with uniformly spaced thresholds, so to get the symbols (and therefore the bits) that make up the raw key.

As a result, the bits of the raw key are uniformly distributed, and the entropy of the key is maximised.

Relationship of Arithmetic Reconciliation with RRS

AR and RRS relate in two ways. The first way the two works are related is the idea to use the CDF as a transformation function. Here we have a difference, though:

  • in AR the transformation is applied to the channel output before discretization;
  • in RRS the transformation is applied to the channel output after discretization, and the CDF is conditioned to the discretization level.

However, in RRS the threshold selection is a degree of freedom, and one could also select the threshold adaptively, as introduced in [3], to have uniform probability for the discretised channel output. This particular threshold selection is exactly equivalent to the threshold selection in AR. In fact one could apply the CDF to these adaptive thresholds of RRS, and obtain the thresholds as described in AR. This represents the second way AR and RRS are related: the (possibly) uniform threshold selection.

3.2. RRS generalises both AR and SR

Firstly, RRS generalises the threshold selection:

  • SR uses equally spaced thresholds in the space of the channel output.
  • AR applies a distributional transform on the channel output, then applies equally spaced thresholds on the result, obtaining a discrete random variable with uniform distribution.

Conversely, in RRS the choice of the thresholds is free: not only the choices above could be taken, but thresholds could be optimised to increase reconciliation efficiency.

Secondly, RRS further processes the channel output besides (and together with) its discretisation, which neither of the above does.

4. References

[1] M. Origlia and M. Secondini, "Soft Reverse Reconciliation for Discrete Modulations," 2025 14th International ITG Conference on Systems, Communications and Coding (SCC), Karlsruhe, Germany, 2025, pp. 1-6, doi: 10.1109/IEEECONF62907.2025.10949121.
[2] https://scc2025.net
[3] M. Origlia, E. Parente and M. Secondini, "Soft-Decoding Reverse Reconciliation in Discrete-Modulation CV-QKD," in IEEE Transactions on Communications, vol. 74, pp. 8873-8884, 2026, doi: 10.1109/TCOMM.2026.3693195.
[4] M. Origlia, E. E. Cil, L. Schmalen and M. Secondini, "Reverse Reconciliation with Soft Information for Discrete-Modulation CV-QKD at Long Range," in Optica Quantum 2.0 Conference and Exhibition, Glasgow, 2026, https://doi.org/10.48550/arXiv.2603.23585
[5] https://codeberg.org/moriglia/re2often
[6] https://codeberg.org/moriglia/fldpc
[7] E. E. Cil and L. Schmalen, “An Open-Source Library for Information Reconciliation in Continuous-Variable QKD,” Aug. 01, 2024, arXiv: arXiv:2408.00569. doi: 10.48550/arXiv.2408.00569.
[8] https://github.com/erdemeray/IR_for_CVQKD
[9] A. Leverrier, R. Alléaume, J. Boutros, G. Zémor, and P. Grangier, “Multidimensional reconciliation for a continuous-variable quantum key distribution,” Phys. Rev. A, vol. 77, no. 4, p. 042325, Apr. 2008, doi: 10.1103/PhysRevA.77.042325.
[10] G. Van Assche, J. Cardinal, and N. Cerf, “Reconciliation of a quantum-distributed gaussian key,” IEEE Transactions on Information Theory, vol. 50, no. 2, pp. 394–400, 2004.
[11] R. R. S. Leite, J. M. de Assis, M. A. Dias, and F. M. de Assis, “Arithmetic Reconciliation for CVQKD: Challenges and Feasibility,” 2026, arXiv. doi: 10.48550/ARXIV.2602.05526.
[12] M. A. Dias and F. M. de Assis, “Distributional Transform Based Information Reconciliation,” Journal of Communication and Information Systems, vol. 39, no. 1, pp. 74–81, May 2024, doi: 10.14209/jcis.2024.7.

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