# Tripartite Quantum Key Distribution Implemented with Imperfect Sources
## 6 Simulation Results
### Discussion of Results  
The trend in the results portrayed by the curves in Figure 2a shows that as \(\delta\) increases, which is associated with encoding source flaws, the achievable secret key rate and the maximum attainable transmission distance decrease. Furthermore, the simulation results show that despite increased encoding source flaws, i.e., an increase in the parameter \(\delta\), the achievable key rates are comparable to those obtained via perfect encoding, where \(\delta=0\). These results clearly demonstrate that the three-party RFI QKD is loss-tolerant to signal state preparation flaws. Furthermore, in Figure 2b it can be observed that the increase in the parameter \(\beta\), associated with the relative rotation of reference frames, leads to a reduction in both the secret key rate and maximum attainable transmission distance. Additionally, it can  
Figure 3: Plot of the expected secret key rate for our proposed protocol (solid blue line) in terms of the transmission distance (km) based on fixed encoding source flaws \(\delta=0.10\). A case where source flaws are the same in different bases is assumed. The rotation angle for the reference frames is \(\beta=\pi/5\). For comparison purposes, the plot of the curve for the PM-QCC protocol [14] is denoted by the red dashed line. The parameters used in the experiment are as follows: dark counts rate \(P_{d}=1.7\times 10^{-6}\), channel loss coefficient 0.2 km/dB, and detection efficiency \(\eta=14.5\%\). Figure 2: (**a**) Plot of the expected secret key rate for our proposed protocol in terms of the transmission distance (km) based on fixed encoding source flaws \(\delta=0.35\), \(\delta=0.20\), \(\delta=0.10\), and \(\delta=0\). A case where source flaws are the same in different bases is assumed. The rotation angle for the reference frames is \(\beta=\pi/5\). (**b**) The expected secret key rate for our proposed protocol in terms of distance (km) of the transmission for a fixed misalignment degree of \(\beta=\pi/5\), \(\beta=\pi/6\), \(\beta=\pi/7\), and \(\beta=0\); it is assumed that misalignment is the same in both channels. The encoding source flaws are fixed at \(\delta=0.10\), and the parameters used in the experiment are as follows: dark counts rate \(P_{d}=1.7\times 10^{-6}\), channel loss coefficient 0.2 km/dB, and detection efficiency \(\eta=14.5\%\). For comparison, plots for the two-party RFI QKD protocol are provided. be noted that regardless of the increase in the reference frame misalignment, the achieved key rates are comparable to a scenario without any misalignment in the reference frames (i.e., \(\beta=0\)). This is a clear indication that the three-party RFI QKD is robust against drift in reference frames, which makes it a suitable candidate for practical multipartite QKD. For further comparison, the key rate is simulated for a two-party RFI QKD protocol using similar parameters in Figure 2a,b. It is evident from both figures that the two-party RFI QKD protocol performs better than the three-party RFI QKD protocol based on the achievable secret key rate for different encoding source flaws, \(\delta\), and misalignment degrees of \(\beta\). However, it is noteworthy that despite this observation, the three-party RFI protocol is preferable in scenarios that involve secure communications involving three communicating parties, as a single run of the protocol produces a secret key for all the parties, which can then be used for communication among themselves. On the contrary, the two-party QKD protocol requires the three communicating parties to run the protocol three times to produce secret keys that can be used for secure communication amongst themselves. Furthermore, the trend in results obtained in Figure 3 can be attributed to the fact that the three-party PM-QCC protocol neglects the imperfections in the signal state preparation devices and does not account for misalignment in the channels linking communicating parties to the measurement station.