Simple security analysis of phase-matching measurement-device-independent quantum key distribution

Variations of phase-matching measurement-device-independent quantum key distribution (PM-MDI QKD) protocols have been investigated before, but it was recently discovered that this type of protocol (under the name of twin-field QKD) can beat the linear scaling of the repeaterless bound on secret key capacity. We propose a variation of PM-MDI QKD protocol, which reduces the sifting cost and uses non-phase-randomized coherent states as test states. We provide a security proof in the infinite key limit. Our proof is conceptually simple and gives tight key rates. We obtain an analytical key rate formula for the loss-only scenario, confirming the square root scaling and also showing the loss limit. We simulate the key rate for realistic imperfections and show that PM-MDI QKD can overcome the repeaterless bound with currently available technology.

Figure 1

(a) Schematic setup of the PM-MDI QKD protocol. Alice and Bob send coherent states to the untrusted third party Charlie in the middle, who performs measurements and broadcasts outcomes. BS: 50-50 beam splitter. $D_{+}$, $D_{-}$: single-photon detectors. (b) Equivalent view of the protocol. Eve is assumed to perform the measurements in the middle. Effectively, Eve performs a four-element POVM, denoted as $\{F^{+},F^{-},F^{?},F^d\}$, corresponding to four possible announcements $\{D_{+}$ clicks, $D_{-}$ clicks, no detectors click, both detectors $\text{click}\},$ which are abbreviated as $\{+,-,?,d\}.$

Figure 2

A log-linear plot for the key rate as a function of transmission distance. The solid line is the fundamental repeaterless secret key capacity bound $-{log}_2(1-{10}^{-\frac{0.2L}{10}})$ for a transmission distance $L$ (in km) [5]. The short-dashed line is the key rate for PM-MDI QKD in the loss-only scenario in Eq. (24). The long-dashed line is the key rate for PM-MDI QKD with experimentally feasible parameters listed in Table 2. The dash-dotted line is phase-matching QKD key rate provided by authors of Ref. [12] with the same set of experimental parameters. The intensity $\mu$ is optimized for each distance in these curves.

Figure 3

Explanation of labels for input and output modes of the beamsplitter.