From Asymptotic to Finite-Size Security in Decoy-State Quantum Key Distribution

Abstract

Quantum Key Distribution (QKD) promises information-theoretic security, yet bridging the gap between theoretical proofs and practical implementations, specifically those operating with finite resources and imperfect devices against general coherent attacks, remains a critical challenge. This thesis develops a spectrum of efficient security proof techniques within the composable security framework, calculating key rates for both fixed- and variable-length protocols while accounting for realistic imperfections.

We begin by addressing detection setups through an extension of a squashing map, the flag-state squasher, used for reducing the infinite-dimensional Hilbert spaces of optical elements to finite dimensions. This extension accommodates arbitrary passive linear optical setups while allowing for the inclusion of detection inefficiencies and dark counts in the security analysis.

Subsequently, we advance the analysis of decoy-state protocols and introduce two major improvements. First, we reformulate the decoy-state analysis to recover no-decoy key rates, tightening the optimization. Second, we derive a unified framework that performs the key rate optimization and decoy analysis in a single step. This enables the bounding of the relevant entropies with arbitrary precision in the finite-size regime and successfully recovers the Devetak-Winter formula in the asymptotic limit.

Furthermore, we improve the security analysis for generic QKD protocols against independent and identically distributed (IID) collective attacks. Our refined analysis yields finite-size corrections proportional to detected rather than transmitted signals and, by developing sharper concentration inequalities, achieves significantly improved finite-size scaling.

Finally, leveraging the marginal constrained entropy accumulation theorem (MEAT), we establish a flexible numerical Rényi security framework against coherent attacks for both fixed- and variable-length protocols. This approach consistently outperforms existing reference proof techniques, including those based on entropic uncertainty relations, providing significantly higher key rates for both qubit and practically relevant decoy-state protocols. Moreover, we present finite-size key rates for generic QKD protocols accounting for realistic intensity and phase imperfections. Overall, this thesis provides the necessary theoretical framework to bridge the gap between idealized models and experimental reality, offering a scalable path toward secure quantum communication under realistic conditions, as demonstrated by the application of these techniques in experimental collaborations.