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 Rényi entropic quantities is known to provide tighter bounds on the key rate. However, such an optimization is often nontrivial. In this work, we present an analytical bound on the Rényi entropy in terms of the Rényi divergence and derive the analytical gradient of the Rényi divergence. This enables us to generalize existing state-of-the-art numerical frameworks for the optimization of key rates. 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.

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• Quantum communication
• Quantum information theory