Fig. 2: Schematic diagram of the mode-pairing MDI-QKD scheme and the simple pairing strategy with maximal-pairing interval l.

The solid and dashed pulses are those with and without successful detection, respectively. Orange and blue pulses are, respectively, the front and rear pulses that succeed in pairing within l pulses, while grey pulses are the ones fail in pairing. a In the mode-pairing MDI-QKD scheme, Alice and Bob, first prepare coherent pulses with random intensities chosen from {0, μ} and random phases \({\phi }_{i}^{a(b)}\in [0,2\pi )\) and send them to Charlie. After interference measurement, Charlie announces the detection results, based on which Alice and Bob pair the pulses and determine their encoding bases. For X-pairs, they announce the alignment angles θ^a and θ^b and keep data for which θ^a = θ^b. They use Z-pairs to generate keys and other data for parameter estimation. b We set l = 2 in the simple pairing strategy for example. The labels F_k and R_k represent the front and rear pulses, respectively, in the k-th successful pair.