Secure BB84-type quantum key distribution with simple phase error formula

BB84-based quantum key distribution system is limited in high speed and chip integration due to the requirement of four states preparation and measurement. Recently, the simplified BB84 protocol with only three states preparation and measurement has showed the approximate secret key rate with standard BB84. However, the security proof is only valid under the collective attack and requires the basis-independent detection efficiency condition. Here, we provide a security proof against the coherent attack, which simultaneously removes the basis-independent detection efficiency condition. Importantly, the phase error rate formula is quite simple through a subtle observation. We expect that the simplified BB84 protocol with decoy-state method can be widely implemented in reality.