Rényi security framework against coherent attacks applied to decoy-state QKD

TL;DR

A flexible and robust framework for finite-size security proofs of quantum key distribution protocols under coherent attacks, applicable to both fixed- and variable-length protocols, and extends the analysis to accommodate realistic device imperfections, such as independent intensity and phase imperfections.

Abstract

We develop a flexible and robust framework for finite-size security proofs of quantum key distribution (QKD) protocols under coherent attacks, applicable to both fixed- and variable-length protocols. Our approach achieves high finite-size key rates across a broad class of protocols while imposing minimal requirements. In particular, it eliminates the need for restrictive conditions such as limited repetition rates or the implementation of virtual tomography procedures. To achieve this goal, we introduce new numerical techniques for the evaluation of sandwiched conditional Rényi entropies. In doing so, we also find an alternative formulation of the "QKD cone" studied in previous work. We illustrate the versatility of our framework by applying it to several practically relevant protocols, including decoy-state protocols. Furthermore, we extend the analysis to accommodate realistic device imperfections, such as independent intensity and phase imperfections. Overall, our framework provides both greater scope of applicability and better key rates than existing techniques, especially for small block sizes, hence offering a scalable path toward secure quantum communication under realistic conditions

Figures (7)

• Figure 1: Representation of the tensor product structure of the MEAT theorem arqand_marginal-constrained_2025.
• Figure 2: Results for variable-length secret key rates of the qubit BB84 protocol plotted against the channel loss in $\unit{dB}$ and a constant depolarization with $p^\mathrm{hon}_\mathrm{depol} = 0.03$ with a varying number of total signals sent $N=10^5, \dots, 10^{10}$. The security parameters, and error correction efficiency were chosen as $\varepsilon_\mathrm{PA} = \varepsilon_\mathrm{EV} = \frac{1}{2}10^{-80}$ and $f_{\mathrm{EC}}=1.1$. As a comparison, we showcase results using the EUR approach of tupkary_phase_2024 incorporating the improvements from mannalath_sharp_2025. For both approaches, we optimize the probability $\gamma$ of using the $Z$-basis choice in each round, and for our work we additionally optimize the Rényi parameter $\alpha$.
• Figure 3: Results for variable-length secret key rates of the active decoy BB84 protocol with two decoy intensities plotted against the channel loss in $\unit{dB}$ with a varying number of total signals sent $N=10^5, \dots, 10^{11}$. The security parameters, and error correction efficiency were chosen as $\varepsilon_\mathrm{PA} = \varepsilon_\mathrm{EV} = \frac{1}{2}10^{-80}$ and $f_{\mathrm{EC}}=1.1$. As a comparison, we showcase results using the EUR approach of tupkary_phase_2024 incorporating the improvements from mannalath_sharp_2025, omitting any curve leading to zero key rate. For both approaches, we optimize the probability $\gamma$ of using the $Z$-basis choice in each round and the signal intensity $\mu_s$. Additionally, for our work we optimize the Rényi parameter $\alpha$.
• Figure 4: Results for variable-length secret key rates of the active decoy BB84 protocol with one decoy intensity plotted against the channel loss in $\unit{dB}$ with a varying number of total signals sent $N=10^5, \dots, 10^{11}$. The security parameters, and error correction efficiency were chosen as $\varepsilon_\mathrm{PA} = \varepsilon_\mathrm{EV} = \frac{1}{2}10^{-80}$ and $f_{\mathrm{EC}}=1.1$. As a comparison, we showcase results using the EUR approach of Refs. rusca_finite-key_2018wiesemann_consolidated_2024, omitting any curve leading to zero key rate. For both approaches, we optimize probability $\gamma$ of using the $Z$-basis choice in each round and the signal intensity $\mu_s$. Additionally, for our work we optimize the Rényi parameter $\alpha$.
• Figure 5: Results for variable-length secret key rates of the passive decoy BB84 protocol with two decoy intensities and intensity imperfection plotted against the channel loss in $\unit{dB}$ with a varying number of total signals sent $N=10^6, 10^8, 10^{10}$. The security parameters, and error correction efficiency were chosen as $\varepsilon_\mathrm{PA} = \varepsilon_\mathrm{EV} = \frac{1}{2}10^{-80}$ and $f_{\mathrm{EC}}=1.1$.

Theorems & Definitions (96)

• Definition 1
• Definition 2
• Definition 3: Variable-Length $\varepsilon$-security
• Remark 1
• Remark 2
• Theorem 4: Rényi Leftover-hashing Lemma (Theorem 9 restated from dupuis_privacy_2023)
• Definition 5
• Theorem 6
• proof
• Theorem 7