Side-channel security of practical quantum key distribution

We present general security proof for quantum key distribution (QKD) with imperfect states in whole space including both the operational space and the other space (side-channel space). We show that in a protocol where Alice (Bob) uses two states, the secure key rate for a real protocol using imperfect (mixed) states is the same as that of a perfect protocol if the real protocol can be mapped from the perfect protocol. The calculation of the secure key rate for imperfect sources is reduced to the lower bound of projection probability to vacuum for each state. This paves the operable way to side-channel-secure QKD under more realistic conditions, including imperfections of whole-space states, mixed states, and finite data size. Taking the sending-or-not-sending encoding as an example in calculation, we obtain good key rates and long secure distance for QKD that is both measurement-device-independent and secure with imperfect whole-space source states.