Security of continuous-variable measurement-device-independent quantum key distribution with imperfect state preparation

The state preparation operation of continuous-variable measurement-device-independent quantum key distribution (CV-MDI-QKD) protocol may become imperfect in practical applications. We address the security of the CV-MDI-QKD protocol based on imperfect preparation of the coherent state under realistic conditions of lossy and noisy quantum channel. Specifically, we assume that the imperfection of Alice's and Bob's practical state preparations equal to the amplification of ideal modulators and lasers at both Alice's and Bob's sides by untrusted third-parties Fred and Gray employing phase-insensitive amplifiers (PIAs), respectively. The equivalent excess noise introduced by the imperfect state preparation is comprehensively and quantitatively calculated by adopting the gains of PIAs. Security analysis shows that CV-MDI-QKD is quite sensitive to the imperfection of practical state preparation, which inevitably deteriorates the performance and security of CV-MDI-QKD system. Moreover, a lower bound of the secret key rate is derived under arbitrary collective attacks, and the upper threshold of this imperfection tolerated by the system is obtained in the form of the specific gains of PIAs. In addition, the methods presented will improve and perfect the practical security of CV-MDI-QKD protocol.