# Tripartite Quantum Key Distribution Implemented with Imperfect Sources
## Appendix A Estimation of Decoy State Parameters
 \tag{13}\]  
\[\begin{  
\[Y^{\gamma}_{1X;X;X_{1}X} =e^{-2\gamma\eta_{B}t\eta_{C}t}(1-Y_{0})^{2}\Big{(}e^{\gamma\eta_{B}t (\frac{1+\cos\delta}{2})(\frac{1+\cos\delta_{2}}{2})}+Y_{0}-1\Big{)}\Big{(}e^{ \gamma\eta_{C}t(\frac{1+\cos\delta}{2})(\frac{1+\cos\delta_{2}}{2})}+Y_{0}-1 \Big{)}, \tag{111}\] \[Y^{\gamma}_{0X;\emptyset X;X_{1}X} =e^{-2\gamma\eta_{B}t\eta_{C}t}(1-Y_{0})^{2}\Big{(}e^{\gamma\eta_ {B}t(\frac{1+\cos\delta}{2})(\frac{1+\cos\delta_{2}}{2})}+Y_{0}-1\Big{)}\Big{(}e ^{\gamma\eta_{C}t(\frac{1-\cos\delta}{2})(\frac{1-\cos\delta_{2}}{2})}+Y_{0}-1 \Big{)},\] (112) \[Y^{\gamma}_{0X;\emptyset X;X_{1}X} =e^{-2\gamma\eta_{B}t\eta_{C}t}(1-Y_{0})^{2}\Big{(}e^{\gamma\eta_ {B}t(\frac{1-\cos\delta}{2})(\frac{1-\cos\delta_{2}}{2})}+Y_{0}-1\Big{)}\Big{(}e ^{\gamma\eta_{C}t(\frac{1+\cos\delta}{2})(\frac{1+\cos\delta_{2}}{2})}+Y_{0}-1 \Big{)},\] (113) \[Y^{\gamma}_{1X;\emptyset X;X_{1}X} =e^{-2\gamma\eta_{B}t\eta_{C}t}(1-Y_{0})^{2}\Big{(}e^{\gamma\eta_ {B}t(\frac{1-\cos\delta}{2})(\frac{1-\cos\delta_{2}}{2})}+Y_{0}-1\Big{)}\Big{(}e ^{\gamma\eta_{C}t(\frac{1+\cos\delta}{2})(\frac{1+\cos\delta_{2}}{2})}+Y_{0}-1 \Big{)},\] (114) \[Y^{\gamma}_{1X;\emptyset X;\emptyset X;X} =e^{-2\gamma\eta_{B}t\eta_{C}t}(1-Y_{0})^{2}\Big{(}e^{\gamma\eta_ {B}t(\frac{1+\cos\delta}{2})(\frac{1+\cos\delta_{2}}{2})}+Y_{0}-1\Big{)}\Big{(} e^{\gamma\eta_{C}t(\frac{1-\cos\delta}{2})(\frac{1-\cos\delta_{2}}{2})}+Y_{0}-1 \Big{)}. \tag{115}\]  
In the above, it is assumed that the misalignment degree, \(\beta\), is similar for [PERSON]'s and [PERSON]'s channel. Additionally, it is assumed that the source flaws, \(\delta\), are the same for the different bases. The overall gain, \(Q^{\gamma}_{X_{A}X_{B}X_{C}}\), is obtained by combining the results of Equations (10) to (115). Additionally, the overall gain in the \(Z\) basis is provided by  
\[Q^{\mu}_{Z_{A}Z_{B}Z_{C}} = (Y^{\mu}_{0Z;0Z0Z}+Y^{\mu}_{0Z;0Z1Z}+Y^{\mu}_{0Z;1Z0Z}+Y^{\mu}_{0Z ;1Z1Z}+Y^{\mu}_{1Z;02Z}+Y^{\mu}_{1Z;0Z1Z}+Y^{\mu}_{1Z;1Z0Z} \tag{116}\] \[+Y^{\mu}_{1Z;1Z1Z}).\]  
The QBER between [PERSON] and [PERSON] ([PERSON]) is described as follows:  
\[E^{\mu}_{Z_{A}Z_{B(C)}}=\frac{Y^{\mu}_{0Z;1Z}+Y^{\mu}_{1Z;0Z}}{Y^{\mu}_{0Z;0Z}+ Y^{\mu}_{0Z;1Z}+Y^{\mu}_{1Z;0Z}+Y^{\mu}_{1Z;1Z}}. \tag{117}\]  
Note that, contrary to the bipartite case, the channel with the worst QBER in the tripartite case is the one that influences the achievable key length when considering the information that has leaked to [PERSON] during the error correction step. Moreover, the gain for a single-photon component in the \(Z\) basis can be written as  
\[Q^{\mu,1}_{Z_{A}Z_{B}Z_{C}}=\frac{\mu}{(1+\mu)^{2}}Q^{\mu}_{Z_{A}Z_{B}Z_{C}}. \tag{118}\]  
The upper bound, \(I^{U}_{E}\), which is dependent on the parameters \(C\) and \(E_{Z_{A}Z_{B}Z_{C}}\), is obtained from the lower bound \(C^{L}_{1}\) and upper bound for the error rate in the \(Z\) basis, which is \(e^{1,U}_{Z_{A}Z_{B}Z_{C}}\) for single photon components. We define the error rate, \(E_{Z_{A}Z_{B}Z_{C}}\), to be the probability that at least one party obtains a different outcome than [PERSON] in a \(Z\)-basis measurement. Notably, this error rate is different from the bipartite error rate, \(E_{Z_{A}Z_{B}}\) or \(E_{Z_{A}Z_{C}}\), which provides the probability of the \(Z\)-measurement of [PERSON] or [PERSON] disagreeing with [PERSON]. The parameter \(e^{1,U}_{Z_{A}Z_{B}Z_{C}}\) is expressed as  
\[e^{1,U}_{Z_{A}Z_{B}Z_{C}}=\frac{E^{H}_{Z_{A}Z_{B}Z_{C}}Q^{\mu}_{Z_{A}Z_{B}Z_{C} }-e_{0}Y_{0}p(\mu|0)}{Y^{\mu,1;L}_{J;Z\times ZmZ}p(\mu|1)} \tag{119}\]  
where  
\[Y^{\gamma}_{0X;\emptyset X;X_{1}X} =e^{-2\gamma\eta_{B}t\eta_{C}t}(1-Y_{0})^{2}\Big{(}e^{\gamma\eta_ {B}t(\frac{1+\cos\delta}{2})(\frac{1+\cos\delta_{2}}{2})}+Y_{0}-1\Big{)}\Big{(}e ^{\gamma\eta_{C}t(\frac{1-\cos\delta}{2})(\frac{1-\cos\delta_{2}}{2})}+Y_{0}-1 \Big{)},\]0  
In the above equation, \(p(\gamma|k)\) is the probability that signals generated with intensity \(\gamma\in\{\mu,\
u\}\) contain \(k\) photons.