Security of decoy-state quantum key distribution with correlated bit-and-basis encoders

Practical quantum key distribution (QKD) modulators inevitably introduce correlations, causing the state emitted in a given round to depend on the setting choices made in previous rounds. These correlations break the round-by-round independence structure on which many widely used security proof techniques rely, leaving a significant gap between available theoretical guarantees and the reality of practical implementations. In this work, we develop a finite-key security proof for decoy-state BB84 against general coherent attacks that rigorously incorporates correlations introduced by Alice's bit-and-basis encoder, while requiring only partial characterization of such correlations.