Fig. 1. Schematic of a time-multiplexed MDI-QKD and a star-like MDI-QKD network. (a) Schematic of a time-multiplexed MDI-QKD with optimal BSM. Alice and Bob send time-bin encoded qubits to Charlie for exchanging keys. By detecting the coincidence (red) between the early (e) and late (l) pulses with two detectors (D1 and D2), or with one detector (D1 or D2). Charlie projects two incoming photons onto |Ψ−⟩ or |Ψ+⟩ to facilitate the key exchanges between Alice and Bob. The full-recovery time of the single-photon detector sets the lower limit of the temporal separation (Δt) between e and l pulses for realizing optimal BSM. We insert ISBs between e and l for realizing time-multiplexing and hence increase the key rate by reducing the bin separation from Δt to τR. (b) A star-like MDI-QKD network with the untrusted relay server. A series of Alice (A1,A2,…,An) and Bob (B1,B2,…,Bn) prepare modulated weak coherent pulses and send to the routers. Two routers select a pair of Alice and Bob and send their pulses to an untrusted relay server controlled by Charlie.

Fig. 2. Experimental device and setup. (a) Schematic of the experiment setup. Alice (Bob) employs a CW laser as the LS and encodes the keys into optical pulses with an encoder module. In this module, one intensity modulator (IM1) chops out early (|e⟩) and late (|l⟩) temporal modes to generate time-bin qubits with a 370 ps duration and separated by 12 ns with a 41.7 MHz repetition rate. IM2 implements intensity modulation for the decoy-state protocol. A PM applies a π-phase to the late temporal modes for |−⟩ and 0-phase for |+⟩ in X-basis. This PM also implements the phase randomization required for MDI-QKD. A variable attenuator prepares weak coherent pulses and simulates the propagation loss in fibers. An EPC adjusts the polarization of the input pulses. The pulses travel through fibers and are coupled into the integrated chip of the relay server (Charlie) for BSM. On the chip, we use a multi-mode interferometer acting as a 50:50 BS and two SNSPDs. (b) False-color scanning electron micrograph (SEM) of the SNSPD. A 80-nm-wide, 80-μm-long U-shaped NbN nanowire is integrated on a 500-nm-wide silicon optical waveguide and connected with two gold pads for electrical readout. The inset shows the zoomed part of the nanowire. (c) Optical and SEM graphs of the high-efficiency photonic-crystal grating coupler with a back-reflected mirror. (d) The averaged amplified response pulses of the 80-nm-wide SNSPD with different lengths. The 1/e-decay time of different SNSPDs is obtained by fitting: 20μm to 0.96 ns; 40μm to 1.56 ns; 80μm to 3.39 ns. (e) Normalized coincidence counts of one detector consecutively detecting both early and late time bins as a function of time separation Δt between them. PBS, polarization beam splitter; PD, photodiode; PS, power sensor; and EPC, electrical polarization controller.

Fig. 3. Experimental results of optimal BSM and QBER. (a) BSM results of |Ψ−⟩. When Alice and Bob send the same states (|++⟩/|--⟩, blue dots), or different states (|+−⟩/|−+⟩, red dots), we obtain destructive and constructive interference in coincidence counts as functions of relative temporal delay, respectively. (b) BSM results of |Ψ+⟩. Note that the correlations between Alice and Bob are inverted comparing to |Ψ−⟩. (c), (d) The QBER in X-basis (blue) and Z-basis (red) for |Ψ−⟩ and |Ψ+⟩, respectively. (e), (f) The measured QBER in X-basis and Z-basis as a function of the wavelength detuning between two lasers for two different Bell states.

Fig. 4. Enhanced key rate by time-multiplexing. (a) The sifted key rate as a function of the inserted pulse number within the full-recovery time of SNSPD (12 ns). Red squares are the results of optimal BSM and blue squares are the results of |Ψ−⟩ only measurement. To compare fairly, in all the results presented here, Alice and Bob send the weak coherent pulses with the average photon number of 0.66 per pulse, and the total loss is 35.0 dB (including chip insertion loss ∼4.5dB). (b) QBERX and QBERZ versus inserting pulse number, indicating that time-multiplexing has little influence on error rate.

Fig. 5. The key rate at different losses including chip insertion loss. The solid lines show theoretical simulations and the triangle symbols show experimental results with a loss of 24.0, 35.0, and 44.0 dB, respectively. For different losses, the parameters (the intensities, s, μ, ν, and the probabilities of intensities, Ps, Pμ, Pν) are different (see Supplementary Material for detailed parameters of theoretical simulations). The gray solid line: PLOB bond⁵⁵ and the gray dotted line: decoy-state MDI-QKD are numerical simulations (see Supplementary Material for details).