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

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.

💡 Research Summary

The paper addresses a critical gap in the security analysis of decoy‑state quantum key distribution (QKD) when the source intensities are not independent but exhibit correlated drift across successive rounds. Traditional decoy‑state analysis assumes that the yield (the probability that a pulse containing n photons causes a detection) depends only on n and not on the nominal intensity setting a. Under this assumption the parameter‑estimation problem can be cast as a linear program (LP) and solved efficiently. However, in realistic devices the actual mean photon number αₖ of a pulse deviates from the nominal setting aₖ, and these deviations are correlated over a finite memory length ξ. An eavesdropper who learns the intensity history can tailor attacks so that yields Y_{n,a} and error rates H_{n,a} become explicit functions of the intensity setting, breaking the LP formulation.

Previous works have introduced Cauchy‑Schwarz (CS) constraints that couple yields and error rates across different intensities, thereby restoring security. The CS constraints are inherently nonlinear (they involve square‑root terms) and have been handled by an “outer linearisation”: a reference point—typically derived from a channel model—is chosen, the CS constraints are linearised around it, and the resulting LP is solved. This approach is computationally convenient but heavily dependent on the choice of reference point; if the channel model does not accurately reflect the actual quantum channel, the resulting key‑rate lower bound can be overly conservative.

The authors propose a two‑stage optimisation strategy that eliminates the reliance on an a‑priori channel model. In the first stage they keep the CS constraints in their original nonlinear form and solve the full parameter‑estimation problem using the interior‑point nonlinear solver IPOPT. IPOPT efficiently handles the smooth, convex‑like structure of the problem even with dozens of variables (yields and error rates for each photon number up to a cutoff N_cut). The solution obtained, denoted (\hat S), is a candidate optimal set of yields and error rates that already respects all physical constraints.

In the second stage (\hat S) is used as the linearisation point for the CS constraints. By performing a first‑order Taylor expansion of the square‑root terms around (\hat S), the problem becomes a linear program again. Solving this LP yields a certified lower bound on the asymptotic secret‑key rate (K_\infty). Crucially, if the LP solution coincides with the original IPOPT solution, the authors can claim that the global optimum of the original nonlinear problem has been found, thus providing a provable optimal key‑rate bound.

The paper evaluates the method on two families of intensity‑correlation models:

1. Coarse‑grained, model‑independent correlations – only the maximal relative deviation (\delta_{\max}) and the memory length ξ are specified, allowing any underlying distribution of actual intensities that respects these bounds. This model captures worst‑case behaviour without committing to a specific statistical description.

2. Fine‑grained truncated‑Gaussian correlations – parameters (mean, variance, truncation limits) are taken from experimental characterisation of a real source, yielding a more realistic but still non‑trivial correlation structure.

For both models the authors compare three approaches: (i) the canonical reference‑point linearisation used in earlier literature, (ii) the proposed two‑stage method, and (iii) the theoretical optimum obtained by exhaustive search (where feasible). The results consistently show that the two‑stage method delivers tighter key‑rate lower bounds, typically improving the rate by 5 %–15 % over the canonical method. The improvement is most pronounced when the channel model deviates strongly from the actual channel (e.g., high loss, asymmetric error rates) or when the intensity drift is large. In several parameter regimes the IPOPT solution and the subsequent LP solution coincide, thereby certifying optimality.

From a computational standpoint, the method is practical. With a photon‑number cutoff of N_cut≈10–12 (resulting in ~150–200 variables), IPOPT converges within a few seconds on a standard CPU, and the linear programme solves in sub‑second time using off‑the‑shelf solvers such as Gurobi or CPLEX. This demonstrates that real‑time parameter estimation and key‑rate updating—essential for operational QKD systems—are feasible without specialized hardware.

The authors also discuss broader implications. By keeping the CS constraints in their exact nonlinear form, the analysis becomes largely model‑independent; the only assumptions are the Poissonian photon‑number statistics conditioned on the actual intensity and the bounded relative deviation. This reduces reliance on detailed channel modelling and makes the technique applicable to a wide variety of QKD platforms, including free‑space, satellite, and multiplexed systems. Moreover, the two‑stage framework can be extended to other device imperfections (e.g., phase or polarization drifts) that also lead to nonlinear coupling between observable statistics and hidden variables.

In summary, the paper makes three key contributions: (1) it demonstrates that the full CS‑constrained parameter‑estimation problem can be solved directly with a modern nonlinear optimiser, (2) it shows how the solution can be leveraged as an optimal linearisation point to obtain certified, tighter key‑rate lower bounds, and (3) it validates the approach on both worst‑case and experimentally motivated correlation models, achieving demonstrable improvements over existing methods while retaining computational efficiency. This work therefore provides a robust, model‑agnostic tool for enhancing the security analysis of intensity‑correlated decoy‑state QKD.