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In this work, we incorporate decoy-state analysis into a well-establishednumerical framework for key rate calculation, and apply the numerical frameworkto decoy-state BB84 and measurement-device-independent (MDI) QKD protocols asexamples. Additionally, we combine with these decoy-state protocols what iscalled "fine-grained statistics", which is a variation of existing QKDprotocols that makes use of originally discarded data to get a better key rate.We show that such variations can grant protocols resilience against any unknownand slowly changing rotation along one axis, similar toreference-frame-independent QKD, but without the need for encoding physicallyin an additional rotation-invariant basis. Such an analysis can easily beapplied to existing systems, or even data already recorded in previousexperiments, to gain significantly higher key rate when considerablemisalignment is present, extending the maximum distance for BB84 and MDI-QKDand reducing the need for manual alignment in an experiment.

Numerical Security Proof for Decoy-State BB84 and Measurement-Device-Independent QKD Resistant against Large Basis Misalignment