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Tighter Asymptotic Key Rates for Intensity-Correlated Decoy-State QKD via Nonlinear Programming

Matej Pivoluska, Mateus Araújo

TL;DR

This work addresses decoy-state QKD with intensity-correlated PRWCP, where standard decoy-state analysis fails due to intensity leakage across rounds. It advances a practical, reproducible two-stage approach: first solve the full CS-constrained nonlinear parameter-estimation problems with IPOPT to obtain high-quality linearization points, then perform a linearised outer optimization to certify a lower bound on the asymptotic key rate $K_ abla$. The authors develop coarse-grained model-independent CS formulations and fine-grained truncated-Gaussian models, demonstrating substantial improvements in key-rate bounds over canonical reference points, and in some cases proving optimality when both stages align. Simulations across model types show that IPOPT-derived reference points give tighter bounds, especially when channel models poorly reflect the actual device, and the method provides a drop-in upgrade to existing CS-based analyses with provable security guarantees.

Abstract

Decoy-state QKD with phase-randomized weak coherent pulses is typically analyzed assuming independent, precisely prepared intensities. Real sources, however, can exhibit correlated intensity drift across rounds, potentially leaking intensity information and breaking the standard decoy-state reduction to linear programs. Cauchy--Schwarz (CS) constraints can restore security by coupling $n$-photon yields across intensities, but they introduce nonlinear square-root constraints that are commonly handled via outer linearisation around channel-model-based reference points. We propose a reproducible alternative: first solve the full CS-constrained parameter-estimation problems using the interior-point nonlinear solver IPOPT, then use the resulting candidate solution as the linearisation point for the outer optimisation that certifies a valid lower bound on the asymptotic key rate. Simulations for both coarse-grained model-independent correlations and fine-grained truncated-Gaussian models show consistently tighter key-rate bounds than canonical reference points, and in some cases allow certifying optimality when both optimisation stages coincide.

Tighter Asymptotic Key Rates for Intensity-Correlated Decoy-State QKD via Nonlinear Programming

TL;DR

This work addresses decoy-state QKD with intensity-correlated PRWCP, where standard decoy-state analysis fails due to intensity leakage across rounds. It advances a practical, reproducible two-stage approach: first solve the full CS-constrained nonlinear parameter-estimation problems with IPOPT to obtain high-quality linearization points, then perform a linearised outer optimization to certify a lower bound on the asymptotic key rate . The authors develop coarse-grained model-independent CS formulations and fine-grained truncated-Gaussian models, demonstrating substantial improvements in key-rate bounds over canonical reference points, and in some cases proving optimality when both stages align. Simulations across model types show that IPOPT-derived reference points give tighter bounds, especially when channel models poorly reflect the actual device, and the method provides a drop-in upgrade to existing CS-based analyses with provable security guarantees.

Abstract

Decoy-state QKD with phase-randomized weak coherent pulses is typically analyzed assuming independent, precisely prepared intensities. Real sources, however, can exhibit correlated intensity drift across rounds, potentially leaking intensity information and breaking the standard decoy-state reduction to linear programs. Cauchy--Schwarz (CS) constraints can restore security by coupling -photon yields across intensities, but they introduce nonlinear square-root constraints that are commonly handled via outer linearisation around channel-model-based reference points. We propose a reproducible alternative: first solve the full CS-constrained parameter-estimation problems using the interior-point nonlinear solver IPOPT, then use the resulting candidate solution as the linearisation point for the outer optimisation that certifies a valid lower bound on the asymptotic key rate. Simulations for both coarse-grained model-independent correlations and fine-grained truncated-Gaussian models show consistently tighter key-rate bounds than canonical reference points, and in some cases allow certifying optimality when both optimisation stages coincide.
Paper Structure (27 sections, 44 equations, 2 figures)

This paper contains 27 sections, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Key-rate optimisation with intensity-correlated sources
  3. Protocol description
  4. Modelling intensity correlations
  5. Assumption 1 (Poisson photon-number statistics).
  6. Assumption 2 (bounded relative deviation).
  7. Assumption 3 (finite correlation range).
  8. Photon-number distribution with correlated intensities
  9. Key-rate formula
  10. Yields, error rates, and observable quantities
  11. Standard decoy-state method without intensity correlations
  12. Breakdown of the standard method with intensity correlations
  13. Coarse-grained model-independent CS constraintsformulation
  14. Problem (P1): lower bound on $Z_{1,\mu}$.
  15. Problem (P2): lower bound on $X_{1,\mu}$.
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Coarse grained key rate calculated using formulation described in Section \ref{['subsec:coarseGrainedFormulation']}. It is apparent that choosing IPOPT to calculate the reference points for outer linearization outperforms the canonical reference points derived from the noise model. In addition in this particular instance IPOPT essentially finds the optimal points, which can be verified by comparing IPOPT results to outer optimization results using IPOPT results as reference points.
  • Figure 2: Here we compare lower bounds on key rates for System B from Trefilov2025 obtained by using canonical linearization points and IPOPT. In Trefilov2025 the Gaussian correlation model could be modified by adding additional global attenuation to control the intensity of the signal state $\mu_1$ while keeping the intensity ratios constant. We illustrate the power of our new method by considering system B from Trefilov2025 with three choices of global attenuation $\mu_1\in\{0.43,0.22,0.11\}$. In all three cases our method significantly outperformed the canonical choice of the reference points.