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

The BB84-based quantum key distribution system is limited in high-speed and chip integration due to the requirement of preparing and measuring four states. Recently, the simplified BB84 protocol with preparation and measurement of only three states has showed the approximate secret key rate with standard BB84. However, the security proof is valid only 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 subtle observation. We expect that the simplified BB84 protocol with the decoy-state method can be implemented widely in reality.