Security of the BB84 protocol with passive biased basis choice by the receiver

The Bennett-Brassard 1984 (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. However, conventional theoretical techniques to prove the security of the BB84 protocol are not applicable if the receiver chooses their measurement basis passively, rather than actively, with a biased probability, followed by measurement with threshold detectors. Here we present a fully analytical security proof against coherent attacks for such a decoy-state BB84 protocol with the receiver’s passive basis choice and measurement with threshold detectors. Numerical simulations under practical situations show that the difference in secure key rate between the active and passive implementations of the protocol is negligible except for long communication distances.