Afterpulsing Effect on the Baseline System Error Rate and on the Decoy-State Quantum Key Distribution Protocols

There is considerable interest in predicting the efficiency of Quantum Key Distribution (QKD) protocols when one of their implementation quantities is modified. One significant imperfection that affects the efficiency of the setup is the afterpulse phenomenon which consists in the spontaneous detections triggered by trapped carriers after previous avalanches at the detectors. While it is widely studied in bibliography for various QKD protocols, it has been reported much more scarcely for the well-known decoy-state QKD protocols and for dual detectors only. We develop a theoretical analysis of afterpulsing effect on the decoy-state QKD protocols for multiple detectors, delivering results which can be used as a guide for every practical decoy-state QKD protocol implementation in real-world deployments. A new formula connecting the baseline system error rate and the afterpulse probability is derived which may hold for all protocols as it is consisted of only setup-related quantities. Numerical simulations addressing the significance of breaking down the quantities pertaining to the decoy-state QKD protocols are being made, focusing on the weak+vacuum decoy-state QKD protocol as a characteristic subcase.