# Tripartite Quantum Key Distribution Implemented with Imperfect Sources
## 3 Operation of Proposed Reference-Frame Independent (RFI) Protocol  
### State Preparation  
For each run \(i\), [PERSON] prepares a two-photon entangled state using a Spontaneous Parametric Down-Conversion source (SPDC). The SPDC process creates two identical photons, which are defined by [49, 50]  
\[(\cosh\lambda)^{-1}\sum_{n=0}^{\infty}(\tanh\lambda)^{n}e^{in\chi}|n,n\rangle. \tag{1}\]  
For a source modulated with an intensity \(\mu\), which is equivalent to \(\sinh^{2}\lambda\), the above description simplifies to  
\[\sum_{n=0}^{\infty}\sqrt{\frac{\mu^{n}}{(1+\mu)^{n+1}}}e^{in\chi}|n,n\rangle. \tag{2}\]  
[PERSON] randomly selects a basis \(a_{i}\in\{X,Y,Z\}\) with probabilities \(p_{z}\) and \(p_{c}=1-p_{z}\), respectively. Here, the \(Z\) basis is chosen with probability \(p_{z}>\frac{1}{2}\) and the complementary bases, \(\{X,Y\}\) with probability \(p_{c}=1-p_{z}\). Specifically, the proposed protocol employs time-bin encoding in the \(Z\) basis and phase encoding in the \(X\) and \(Y\) bases. The time-bin eigenstate \(|0_{Z}\rangle\) or \(|1_{Z}\rangle\) is chosen when [PERSON] switches on her longer arm or shorter arm of the interferometer. In addition, the eigenstate \(|0_{Z}\rangle\) or \(|1_{Z}\rangle\) is assigned a bit values, \(r_{i}=0\) or \(r_{i}=1\), respectively. [PERSON] applies phase modulation \(\theta_{A}\in\{0,\pi\}\) and \(\theta_{A}\in\{\frac{\pi}{2},\frac{3\pi}{2}\}\) when she selects the \(X\) and \(Y\) basis, respectively. Here the phase values \(\theta_{A}\in\{0,\frac{\pi}{2}\}\) and \(\theta_{A}\in\{\pi,\frac{3\pi}{2}\}\) are assigned the bit values \(r_{i}=0\) and \(r_{i}=1\), respectively. Note that for each run \(i\) [PERSON] switches on the same arm of the interferometer for each mode of entangled photons when using the \(Z\) basis or performs the same phase shift to both entangled photons when using the \(X\) basis, i.e., both photons are prepared in the same state. Thus, she keeps one bit value, \(r_{i}\), corresponding to the prepared states. The two entangled photons are then delivered to [PERSON] and [PERSON] via insecure quantum channels (see Figure 1). In particular, the three-party RFI protocol follows the loss-tolerant technique presented by [PERSON] et al. [41], where they conjecture that the six-state protocol can be implemented with four states, instead of the usual six in its original version. This technique takes into consideration the imperfections in the intensity and phase modulation of the photons [41]; thus, the actual states of each mode after [PERSON] applies the phase shift can be expressed as  
\[|\phi_{j\beta}\rangle_{B(C)}=|\sqrt{\mu}e^{i\chi}\rangle_{r}|\sqrt{\mu}e^{i( \chi+\theta_{B(C)}+\delta\theta_{B(C)}/\pi)}\rangle_{s} \tag{3}\]  
where \(r(s)\) represents the reference (signal) states, \(\mu=\sinh^{2}\lambda\) corresponds to the intensity of pulses, and \(\delta\) represents the deviation from desired encoding angle \(\theta\) in the actual states; here, \(\beta\in\{X,Y,Z\}\) and \(j\in\{0,1\}\). Notably, each mode created with an SPDC source of intensity \(\mu\) and a global phase \(\chi\) represents the linear superposition of a photon number state \(\{|n\rangle\}\), whereby \(|\sqrt{\mu}e^{i\chi}\rangle=\sum_{n=0}^{\infty}\sqrt{\frac{\mu^{n}}{(1+\mu)^ {n}}}e^{i\pi\chi}|n\rangle\). Thus, the single-photon component corresponding to each mode in Equation (3) that [PERSON] sends to [PERSON] ([PERSON]) is  
\[|\phi_{j\beta}\rangle_{B(C)}=\frac{1}{\sqrt{2}}(|1\rangle_{r}|0\rangle_{s}+e^{ i(\theta_{B(C)}+\delta\theta_{B(C)}/\pi)}|0\rangle_{r}|1\rangle_{s}), \tag{4}\]  
where \(|0\rangle\) and \(|1\rangle\) denote a vacuum state and a single photon state, respectively. Based on the above description, the states are defined as \(|1\rangle_{r}|0\rangle_{s}=|0_{Z}\rangle\), \(|0\rangle_{r}|1\rangle_{s}=|1_{Z}\rangle\)  
Figure 1: The illustration of an RFI-QKD protocol with three participating parties. [PERSON] prepares a two-photon entangled state using a spontaneous parametric down-conversion (SPDC) source.