Generalized Numerical Framework for Improved Finite-Sized Key Rates with R\'enyi Entropy

Abstract

Quantum key distribution requires tight and reliable bounds on the secret key rate to ensure robust security. This is particularly so for the regime of finite block sizes, where the optimization of generalized R\'enyi entropic quantities is known to provide tighter bounds on the key rate. However, such an optimization is often non-trivial, and the non-monotonicity of the key rate in terms of the R\'enyi parameter demands additional optimization to determine the optimal R\'enyi parameter as a function of block sizes. In this work, we present a tight analytical bound on the R\'enyi entropy in terms of the R\'enyi divergence and derive the analytical gradient of the R\'enyi divergence. This enables us to generalize existing state-of-the-art numerical frameworks for the optimization of the key rate. With this generalized framework, we show improvements in regimes of high loss and low block sizes, which are particularly relevant for long-distance satellite-based protocols.