# Tripartite Quantum Key Distribution Implemented with Imperfect Sources
## 4 Security Analysis
For simplicity, the estimate of the phase error rate, \(E_{X_{A}X_{B}X_{C}}\) and other parameters in Equation (15) arederived similarly. The term \(E_{X_{A}X_{B}X_{C}}\) is evaluated based on a virtual protocol whereby [PERSON] prepares an entangled state \(|\Psi_{Z}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}|\phi_{0Z}\rangle_{B(C)}+|1 \rangle_{A(B)}|\phi_{1Z}\rangle_{B(C)})\), where \(B\) and \(C\) represent systems sent to [PERSON] and [PERSON]. The three parties then select the \(Z\) basis and measure their corresponding subsystems in the \(X\) basis (instead of the selected \(Z\) basis). Therefore, the error rate can be expressed as  
\[E_{X_{A}X_{B}X_{C}}=\frac{Y_{0X_{0}X_{1}}^{Z,\text{vir}}+Y_{0X_{1}X_{1}}^{Z, \text{vir}}+Y_{1X_{0}X_{1}}^{Z,\text{vir}}+Y_{1X_{0}X_{1}}^{Z,\text{vir}}+Y_{1X _{0}X_{0}}^{Z,\text{vir}}+Y_{1X_{1}X_{0}X_{1}}^{Z,\text{vir}}+Y_{0X_{1}X_{0}X_ {1}}^{Z,\text{vir}}}{Y_{0X_{1}X_{1}}^{Z,\text{vir}}+Y_{0X_{1}X_{0}X_{1}}^{Z, \text{vir}}+Y_{1X_{1}X_{1}X_{0}}^{Z,\text{vir}}+Y_{0X_{1}X_{0}X_{0}}^{Z,\text{ vir}}+Y_{1X_{1}X_{1}X_{1}}^{Z,\text{vir}}}, \tag{16}\]  
where \(Y_{jX,kym_{Y}}^{Z,\text{vir}}\) represents the joint probability that [PERSON], [PERSON] and [PERSON] measured \(|j_{X}\rangle\), \(|k_{X}\rangle\), and \(|m_{X}\rangle\), respectively. In this hypothetical protocol, the state of the pulses received by [PERSON] ([PERSON]) can be expressed as  
\[\hat{\sigma}_{B(C);j_{X}}^{\text{vir}}=\text{Tr}_{A}[\hat{P}(|j_{X}\rangle_{A} )\otimes\mathds{1}_{B(C)}\hat{P}(|\Psi_{Z}\rangle_{AB(C)})], \tag{17}\]  
where \(\hat{P}(|x\rangle)=|x\rangle\langle x|\) is a projection operator for a specific pure state, \(|x\rangle\). The resultant normalized state is \(\hat{\sigma}_{B(C);j_{X}}^{\text{vir}}=\hat{\sigma}_{B(C);j_{X}}^{\text{vir}}/ \text{Tr}(\hat{\sigma}_{B(C);j_{X}}^{\text{vir}})\). Accordingly, the joint probability that the three parties respectively measure \(|j_{X}\rangle\), \(|k_{X}\rangle\), and \(|m_{X}\rangle\) is provided by  
\[ \begin{split} Y_{j_{X};k_{X}m_{X}}^{Z,\text{vir}}& =p(j_{X})\text{Tr}(\hat{D}_{k_{X}}\hat{\sigma}_{B;j_{X}}^{\text{vir }})\text{Tr}(\hat{D}_{m_{X}}\hat{\sigma}_{C;j_{X}}^{\text{vir}})\\ &=p(j_{X})Y_{j_{X};k_{X}}^{Z,\text{vir}}Y_{j_{X};m_{X}}^{Z,\text{ vir}},\end{split}  \tag{18}\]  
where \(\hat{D}_{k_{X}(m_{X})}\) represents the operator which contains [PERSON]'s operation and [PERSON] ([PERSON])'s POVM measurement and \(p(j_{X})\) represents the probability that [PERSON] measures her subsystem in the \(X\) basis. Because the virtual state \(\hat{\sigma}_{B(C);j_{X}}^{\text{vir}}\) can be expressed in terms of identity and Pauli operators as  
\[\hat{\sigma}_{B(C);j_{X}}^{\text{vir}}=\frac{1}{2}\Big{(}\mathds{1}+\sum_{s(t )=x,y,z}\mathbf{n}_{s(t)}^{j_{X}}\hat{\sigma}_{s(t)}\Big{)}, \tag{19}\]  
it follows that Equation (18) can be rewritten as  
\[Y_{j_{X};k_{X}m_{X}}^{Z,\text{vir}}=p(j_{X})\sum_{s=X,Y,Z}\mathbf{n}_{s}q_{k_{ X}|s}\sum_{t=X,Y,Z}\mathbf{n}_{t}q_{m_{X}|t}. \tag{20}\]  
Therefore, in order to obtain \(Y_{j_{X};k_{X}m_{X}}^{Z,\text{vir}}\) it is sufficient to evaluate the transmission rate of Pauli operators, defined by  
\[q_{k(m)_{X}|s(t)}=\text{Tr}(\hat{D}_{k(m)_{X}}\sigma_{s(t)})/2, \tag{21}\]  
where \(s,t\in\{\mathds{1},X,Y,Z\}\) and the parameters \(\mathbf{n}_{s}\) and \(\mathbf{n}_{t}\) denote the coefficients of Pauli matrices. Note that the transmission rate of operators can be evaluated from the yield of signal states used in the actual protocol. In order to evaluate the yield of these states, the entanglement description where [PERSON] prepares the entangled states is employed, as follows:  
\[ \begin{array}{ll}|\Psi_{Z}\rangle&=\frac{1}{\sqrt{2}}(|0_{Z}\rangle_{A}|\phi _{0Z}\rangle_{B(C)}+|1_{Z}\rangle_{A}|\phi_{1Z}\rangle_{B(C)}),\\ |\Phi_{X}\rangle&=|0_{X}\rangle_{A}|\phi_{0X}\rangle_{B(C)},\\ |\Phi_{Y}\rangle&=|0_{Y}\rangle_{A}|\phi_{0Y}\rangle_{B(C)}.\end{array}