# Tripartite Quantum Key Distribution Implemented with Imperfect Sources
## 7 Conclusions  
A tripartite quantum key distribution method that can tolerate imperfections in state preparation and be implemented without aligning reference frames between the legitimate communicating parties is presented. Notably, the performance of the proposed protocol is examined when the encoding flaws are set at \(\delta=0.35\), \(\delta=0.20\), and \(\delta=0.10\), denoting a change of \(20.05^{\circ}\), \(11.46^{\circ}\), and \(5.73^{\circ}\) from the desired phase encoding angle, respectively. The results significantly demonstrate that the achievable key rates compare to perfect encoding regardless of state preparation flaws. Moreover, the simulation shows the secret key rate for the proposed protocol in terms of transmission distance for various degrees of misalignment (i.e., \(\beta=\pi/5,\pi/6\), and \(\pi/7\)) and investigates the impact on the statistical quantity \(C\) and protocol stability. The simulation results confirm that the proposed protocol is insignificantly affected by increased misalignment of reference frames, as the achievable distances in transmission remain similar to scenarios without misalignment of reference frames (i.e., \(\beta=0\)). Most significantly, this work finds applications in quantum communication network environments involving unknown and slowly varying reference frames, such as earth-to-satellite quantum communication networks, chip-to-chip communications in free space, and chips connected by fiber optics. Other applications include web conferences, online communications, and scenarios where multiple parties must communicate securely using secret keys. Lastly, using the formalism developed in [44,45] guarantees unconditional security, although this does not mean that the bounds developed in the proposed protocol are very tight; possible improvements might be achieved by considering two-way classical post-processing. However, in order to fulfill the objectives of this work, a restriction of the security bounds for the case when the secret key is extracted through one-way post-processing is undertaken. In the future, as an extension of the decoy state-based protocol, it will be worthwhile to consider other single-photon source techniques which are robust against photon number splitting attacks, such as the deterministic single-photon sourcesproposed in [54], to improve the performance of the three-party RFI protocol. Furthermore, in the future, it would be interesting to investigate the performance of the three-party RFI QKD through underwater channels under the influence of diverse factors such as signal attenuation and noise ratio, which has previously been considered in [22].  
Conceptualization, C.S. and M.M.; methodology, C.S.; software, C.S.; validation, C.S. and M.M.; formal analysis, C.S.; investigation, C.S.; writing--original draft preparation, C.S.; writing--review and editing, M.M.; visualization, C.S.; supervision and project administration, M.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.