Security of quantum key distribution with virtual mutually unbiased bases

In a perfect quantum key distribution(QKD) protocol, quantum states should be prepared and measured with mutually unbiased bases(MUBs). However, in a practical QKD system, quantum states are generally prepared and measured with imperfect MUBs using imperfect devices, possibly reducing the secret key rate and transmission distance. To analyze the security of a QKD system with imperfect MUBs, we propose virtual MUBs to characterize the quantum channel against collective attack, and analyze the corresponding secret key rate under imperfect state preparation and measurement conditions. More generally, we apply the advantage distillation method for analyzing the security of QKD with imperfect MUBs, where the error tolerance and transmission distance can be sharply improved. Our analysis method can be applied to benchmark and standardize a practical QKD system,elucidating the security analysis of different QKD protocols with imperfect devices.