Phase error rate estimation with basis-efficiency mismatch for decoy-state BB84

We present a finite-size security proof of the BB84 QKD protocol against coherent attacks, using entropic uncertainty relations, in the presence of imperfectly characterized basis-efficiency mismatch. Our proof requires no new modifications to the protocol steps or hardware, and works by obtaining a suitable bound on the phase error rate. It is applicable to imperfectly characterized detectors, and only requires the maximum relative difference in detection efficiencies and dark count rates of the detectors to be characterized. Moreover, our proof allows Eve to use any spatio-temporal degree of freedom in each round to choose detector efficiencies and dark count rates in their allowed ranges, thereby addressing an important problem of detector side channels. We prove security in the variable-length framework, where users are allowed to adaptively determine the length of key to be produced, and number of bits to be used for error-correction, based on observations made during the protocol. We quantitatively demonstrate the effect of basis-efficiency mismatch by applying our results to the decoy-state BB84 protocol.