Fig. 1. Schematic diagram of phase encoding polarization multiplexing MDI-QKD protocol. BS, beam splitter; PBS, polarization beam splitter; PMZI, polarization-multiplexing Mach–Zehnder interferometer; PC, polarization controller and compensation. The polarization maintaining fibers separated by PBS are colored red and green to represent the orthogonal polarization H and V, respectively. The case is depicted by pulse H (red) and pulse V (green) when |ϕ_0X〉_A and |ϕ_0Z〉_B are prepared.

Fig. 2. Detector result. The figure illustrates how the double click of Charlie’s site detector (DH0, DH1, DV0, DV1 shown in Fig. 1) corresponds to the gains or error rate, when Alice and Bob prepare their states |ϕ_iα〉_A and |ϕ_jβ〉_B. The case is classified by their basis choices α, β and whether their bits are equal or not.

Fig. 3. Comparison of RFI-MDI-QKD with loss-tolerant (LT, solid line) and quantum coin (QC, dashed line) methods with different data sizes ( N^M_aM_b = 10¹¹, 10¹², 10¹³) and the source flaw (δ = 0 or 0.075). The intensities of signal and decoy states are optimized. Other simulation parameters are provided in Table 1.

Fig. 4. Key rate versus source flaw δ and phase drift ω in the RFI-MDI-QKD protocol with LT (solid line) and QC (dashed line) methods at the fixed distance of 2 km (red), 20 km (purple), and 40 km (blue).

Fig. 5. (a) Key rate of loss-tolerant RFI-MDI-QKD under the joint impact of source flaw δ and phase drift ω ∈ [0, π/2] at 2 km. (b) Key rate ratio (R₁/R₂) of MDI-QKD to RFI-MDI-QKD protocol with the LT method, which is denoted by R₁ and R₂, respectively.