Security of the BB84 protocol with receiver's passive biased basis choice

Abstract: The Bennett-Brassard 1984 protocol (BB84 protocol) is one of the simplest protocols for implementing quantum key distribution (QKD). In the protocol, the sender and the receiver iteratively choose one of two complementary measurement bases. Regarding the basis choice by the receiver, a passive setup has been adopted in a number of its implementations including satellite QKD and time-bin encoding one. However, conventional theoretical techniques to prove the security of BB84 protocol are not applicable if the receiver chooses his measurement basis passively, rather than actively, with a biased probability, followed by the measurement with threshold detectors. Here we present a fully analytical security proof for such a decoy-state BB84 protocol with receiver's passive basis choice and measurement with threshold detectors. Thanks to the analytic nature, our security proof can straightforwardly be lifted up to the one over coherent attacks. The numerical simulations under practical situations show that the difference in secure key rate between the active and the passive implementations of the protocol is negligible except for long communication distances.