# Tripartite Quantum Key Distribution Implemented with Imperfect Sources
## 6 Simulation Results  
In this section, the performance of the proposed protocol on a fiber-based QKD system model is simulated. The following experimental parameters are used; the loss coefficient of the fiber is 0.2 dB/km, [PERSON] and [PERSON]'s detection efficiencies are \(\eta=14.5\%\), and the dark count rate of a single-photon detector is \(P_{d}=1.7\times 10^{-6}\). It is assumed that the error correction efficiency is \(f_{EC}=1.22\) and the expected photon number for signal states is set at \(\mu=0.6\). The optimal probability, \(p_{z}\), for the key basis is set at 0.95, as per Ref. [48]. It is important to note that the performance of the three-party RFI protocol can, in principle, be simulated with a free-space channel link; however, this is left for further studies.  
The curves in Figure 2a are obtained from Equation (29) using the values \(\delta=0.35\), \(\delta=0.20\), and \(\delta=0.10\). These values correspond to a deviation of \(20.05^{\circ}\), \(11.46^{\circ}\), and \(5.73^{\circ}\) from the desired phase encoding angles, respectively. The parameter \(\delta\) is related to the extinction ratio of the phase modulators according to the definition \(|\tan(\delta/2)|^{2}=\eta_{ex}\)[53]. The non-zero extinction ratio arises owing to imperfections present in phase modulators; in typical experiments, it is of order \(10^{-3}\). Based on this extinction ratio, \(\delta\approx 0.063\); however, in this simulation, pessimistic values are opted for in order to estimate encoding imperfections and show the robustness of the proposed protocol against source flaws. The same parameters were chosen for the states prepared in the \(Z\) basis, which correspond to the 16 dB [\(\tan^{2}(\frac{0.35}{2})=10^{-16/10}\)], 20 dB [\(\tan^{2}(\frac{\text{20}}{2})=10^{-20/10}\)], and 26 dB [\(\tan^{2}(\frac{\text{10}}{2})=10^{-16/10}\)] extinction ratios of the practical optical attenuator, or the intensity modulator in a system with time-bin encoding. For comparison, a plot of the curve when \(\delta=0\) is shown, which is analogous to the perfect encoding scenario. The maximum signal transmission distances between Alice and Bob(Charlie) are 266.9 km (with a secret key rate of \(4.8\times 10^{-15}\) bit/pulse), 262.1 km (with a secret key rate of \(4.9\times 10^{-15}\) bit/pulse), and 258.0 km (with a secret key rate of \(3.6\times 10^{-15}\) bit/pulse) for the encoding source flaws \(\delta=0.10\), \(\delta=0.20\) and \(\delta=0.35\), respectively.  
Figure 2b illustrates the results for the simulation of the secret key rate, \(r\), for the three-party RFI protocol as a function of the transmission distance where the degree of misalignment of reference frames is fixed at \(\beta=0,\pi/5,\pi/6\), and \(\pi/7\), respectively. Our primary goal was to investigate the impact of reference frame misalignment on the stability of the proposed protocol. The parameter \(\beta\) corresponds to the drift of reference frames in the \(X\) and \(Y\) bases. A transmission distance of 301.4 km, 284.6 km, and 268.5 km is obtained for the relative rotation of reference frames at an angle of \(\beta=\pi/7\), \(\beta=\pi/6\), and \(\beta=\pi/5\), respectively. In addition, in order to show how the proposed protocol fares against the other alternative multipartite QKD protocols, the simulation results of the secret key rate are provided as a function of the transmission distance in Figure 3 for a three-party RFI QKD with encoding source flaws \(\delta=0.10\) and the three-party phase matching (PM) QCC protocol proposed in [14]. The results indicate that the three-party PM-QCC protocol outperforms our proposed three-party RFI protocol in terms of maximum attainable transmission distance and the achievable secret key rate. A maximum distance of 284 km (with a secret key rate of \(4.4\times 10^{-12}\) bit/pulse)  
is obtained for the three-party PM-QCC protocol and maximum distance of 266 km (with a secret key rate of \(4.8\times 10^{-15}\) bit/pulse) for the three-party RFI QKD.