Long-distance continuous-variable measurement-device-independent quantum key distribution with post-selection

We introduce a robust scheme for long-distance continuous-variable (CV) measurement-device-independent (MDI) quantum key distribution (QKD) in which we employ post-selection between distant parties communicating through the medium of an untrusted relay. We perform a security analysis that allows for general transmissivity and thermal noise variance of each link, in which we assume an eavesdropper performs a collective attack and controls the excess thermal noise in the channels. The introduction of post-selection enables the parties to sustain a secret key rate over distances exceeding those of existing CV-MDI protocols. In the worst-case scenario in which the relay is positioned equidistant between them, we find that the parties may communicate securely over a range of \SI{14}{\km} in standard optical fiber. Our protocol helps to overcome the rate-distance limitations of previously proposed CV-MDI protocols while maintaining many of their advantages.