Finite-key analysis of loss-tolerant quantum key distribution based on random sampling theory

The core of security proofs quantum key distribution (QKD) is the estimation a parameter that determines amount privacy amplification users need to apply in order distill secret key. To estimate this using observed data, one needs concentration inequalities such as random sampling theory or Azuma's inequality. latter can be straightforwardly employed wider class QKD protocols, including those do not rely on basis-independent sources loss-tolerant (LT) protocol. However, when applied real-life finite-length experiments, inequality typically results substantially lower secret-key rates. Here we propose an alternative analysis LT protocol against general attacks, for both its prepare-and-measure and measurement-device-independent versions, based theory. Consequently, our proof provides considerably higher rates than previous finite-key This work opens up possibility provide other protocols.