# Tripartite Quantum Key Distribution Implemented with Imperfect Sources
## 2 Review of the Literature on QCC Protocols  
Several QCC protocols have been proposed recently in efforts to close the gap between theory and practice. For instance, in order to address low key rate and transmission distance challenge, protocols inspired by the idea of twin-field QKD [25] have been proposed [12; 14; 26]. More precisely, the authors of [12] evaluated the security of a QCC by exploiting the multipartite entanglement of a W-class state; their protocol relied on single-photon interference in an untrusted node, following the idea of twin field QKD [27]. Based on the single photon interference of optical fields at the untrusted relay, their protocol is capable of surpassing the secret key capacity of bound for repeaterless QKD schemes. To address the problem of detector side-channel attacks, a device-independent (DI) QCC protocol has been proposed in which security is based on the violation of a Parity-CHSH inequality [28]. Unfortunately, the DI QCC requires loophole-free Parity-CHSH experiments, which means that it is not feasible with current technology. A more practical solution is measurement device-independent (MDI) QCC, which is inherently immune to all side-channel attacks targeting the measurement device and removes all detection-related security loopholes. [PERSON] et al. [11] proposed an MDI protocol based on the postselection of the GHZ state. In this protocol, the measurement device is controlled by an untrusted third party, making it immune to all detector side-channel attacks. The primary basis of the protocol is that after a successful detection event occurs, a GHZ state is shared among multiple parties. Thus, multiple parties can distribute secret key bits among themselves via the post-selected entanglement states. Furthermore, [PERSON] et al. [14] presented a QCC network protocol by combining the ideas of phase-matching weak coherent pulses (WCPs) interference and post-selecting GHZ states. In addition, the protocol is measurement device-independent, thus rendering it immune to all detector side-channel attacks. While the results in the aforementioned QCC protocols constitute an essential step toward realization and guaranteeing the security of QCC systems, both analyses neglect the case of imperfect photon sources and misalignment in the reference frames. This means that the results cannot be directly applied to real-life QCC implementations. A serious challenge in the practical implementation of QCC schemes is achieving a well-aligned reference frame between communicating parties. This task is complex because of unstable fiber communication links and imperfections in the measuring devices, which results in obscure and changing reference frames [29]. [PERSON] et al., (2010) introduced a reference-frame independent (RFI) QKD protocol to solve this alignment problem [30]. Subsequently, several variants of their protocol have been proposed, both theoretically [31; 32; 33; 34; 35; 36; 37] and experimentally [38; 39; 40], to advance the merits of RFI QKD. Furthermore, security proofs of existing QCC protocols assume perfect state preparation. However, this assumption fails during practical implementations because of the natural deficiencies of photon sources. [PERSON] et al., (2014) invented a loss-tolerant protocol that is immune against losses in the channel caused by state preparation flaws. The protocol employs only three statesout of the four permissible states present in the BB84 protocol [41]. In addition, the protocol takes into consideration errors due to phase modulation by imperfect phase modulators (PM), and it has been analyzed recently in [42, 43] in order to evaluate the security of QKD systems in the presence of information leakage from Alice's phase modulator. Therefore, inspired by the ideas of a loss-tolerant (LT) protocol and reference frame independence, we determine the security bounds in the presence of imperfect state preparation and misalignment in reference frames for a three-party QKD protocol.