## 3 Experiment  
MDI-QKD achieves a better balance between security and practicality, while it generates lower secret key rate than BB84 protocol, especially considering the finite data size effect. For the sake of improving the performance, different approaches and strategies have been proposed and experimentally verified [14; 25; 26; 27; 28]. Here, we investigate the implementation of LSTM model based on a simple three-intensity decoy-state MDI-QKD scheme with biased basis choice [15], in which the decoy pulses are only prepared in X basis. In the biased three-intensity decoy-state MDI-QKD protocol, first, [PERSON] and [PERSON] randomly prepare phase-randomized weak coherent state (WCS) pulses into three different intensities (_u,v,o_) with certain probabilities, each corresponds to the intensity of the signal state, the decoy state, and the vacuum state respectively. The signal pulses are prepared either in \(Z\) or \(X\) basis. Different from standard three-intensity decoy-state schemes [25], the decoy pulses are prepared only in \(X\) basis. Then, [PERSON] performs Bell-state measurements on the pulse pairs from both [PERSON] and [PERSON] and announces the results of the effective events. Figure 2: Diagram of the two-layer LSTM network. Finally, [PERSON] and [PERSON] exchange the basis-choice information and carry out parameter estimation and postprocessing processes. Finally, the lower bound of key rate can be calculated as follows [14; 15; 25]:  
\[R\geqslant p_{\mu,A}P_{Z|\mu,A}p_{\mu B}P_{Z|\mu_{B}}\left\{a_{1}^{n}b_{1}^{n}Y _{11}^{Z,L}[1-H_{2}(e_{11}^{X,U})]-S_{\mu\mu}^{ZZ}fH_{2}(E_{\mu\mu}^{ZZ})\right\} \tag{1}\]  
where \(S_{\mu\mu}^{ZZ}\) and \(E_{\mu\mu}^{ZZ}\) each represents the average counting rate and the average QBER in Z basis; \(Y_{11}^{Z,L}\) and \(e_{11}^{X,U}\) each denotes the yield and the phase-flip error-rate of the single-photon-pair pulses in Z basis, in which the superscript \(L\) and \(U\) each represents the lower bound and upper bound, respectively; \(f\) is the inefficiency of the error correction and we reasonably assume \(f=1.16\); and \(H_{2}(x)\) is the binary Shannon information function, defined as \(H_{2}(x)=-xlog_{2}(x)-(1-x)log_{2}(1-x)\). The MDI-QKD experiment setup is demonstrated in Figure 3. We apply the time-bin phase encoding scheme, and utilize intensity modulators (IMs) and Faraday-Michelson interferometers (FMIs) [29] as the key apparatus for source encoding. The two legitimate users, [PERSON] and [PERSON], which are symmetrical to an unreliable relay Charlie, each owns a narrow linewidth continuous-wave laser whose frequencies are locked to the molecular absorption line with a center wavelength of 1550.0 nm. The light sources, which generate continuous wave, are precisely chopped into pulse trains with a 3 ns temporal width and a repetition rate of 50 MHz by two IMs: the former is used for decoy-state modulation and the latter for extinction ratio improvement. Next, [PERSON] and [PERSON] send their signal laser pulses to [PERSON] for a partial Bell state measurement with two super-conducting nanowire single-photon detectors (SNSPDs). The detectors operate at 2.2 K, providing a 80% detection efficiency at the dark count rate of 10 counts per second. The efficiency could still maintain 60% with the loss of devices at [PERSON]'s side, including an electric polarization controller (EPC), a polarization beam splitter (PBS), a beam splitter (BS), and detectors. The count results from two SNSPDs are recorded by a time-to-digital converter with 4 ns gate. In order to realize the stable Hong-Ou-Mandel (HOM) interference, the indistinguishability of the signal pulses in spectrum, timing, and polarization must be guaranteed. Any difference in these dimensions will bring errors in the X basis. In our experiment, we utilize two narrow line width continuous-wave lasers with high accuracy in the frequency domain. Additionally, we apply an optical delay (OD) in [PERSON]'s station to adjust the arriving time.