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Afterpulsing is a factor limiting the distance over which discrete-variablequantum key distribution systems are secure, and a common feature insingle-photon detectors. The relevance of this phenomenon stems from itsstochastic, self-interacting nature and the fact that its rate rises with thenumber of avalanche events, which increases the quantum bit error rate. Here weintroduce an effective analytic model, including dead-time and afterpulsingcorrections, where afterpulsing correction depends on dead-time value. Thismodel is useful to evaluate the performance of prepare and measure quantum keydistribution protocols (standard and decoy versions) that use gated singlephoton detectors. The model provides an expression to numerically optimize thesecret key rate over the full distance range for secure communication, enablingin this way the calculation of quantum bit error rate and secure key rate. Inthe conventional procedure, the dead-time value is fixed regardless ofdistance, limiting the distance range of the channel due to remainingafterpulsing effects, which are more relevant at higher operating frequencies.Here we demonstrate that optimizing the dead-time values increases the distancerange of the channel to share secret keys.

Dead-time optimization to increase secure distance range in prepare and measure quantum key distribution protocols