# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## Appendix C Conditional Probability for [PERSON] Success in the Protocol  
Departing from the double teleported state after of the \(\text{Bob}_{1}\) processing but before to the \(\text{Bob}_{0}\) processing and [PERSON]'s measurement of the control system:  
\[|\psi_{1}\rangle\equiv\sqrt{p_{0}}|0\rangle_{C}\otimes|\psi_{0}\rangle_{2} \otimes|0\rangle_{4}+\sqrt{p_{1}}|1\rangle_{C}\otimes|0\rangle_{2}\otimes| \psi_{0}^{1}\rangle_{4} \tag{113}\]  
then, we consider the state \(|k\rangle_{\theta,p}\) stated on the basis generated by the \(\theta,p\) parameters:  
\[|k\rangle_{\theta,p} = (-1)^{pk}(f_{k}(\frac{\theta}{2})|0\rangle+(-1)^{p+k}f_{1\in k}( \frac{\theta}{2})|1\rangle) \tag{114}\]  
so, we get the expressions for the following projections:  
\[{}_{\theta,p}\langle k|0\rangle = (-1)^{pk}f_{k}(\frac{\theta}{2}) \tag{115}\] \[{}_{\theta,p}\langle k|\psi_{0}^{1}\rangle = (-1)^{pk}W_{1} \tag{116}\]  
where we have defined the quantity:  
\[W_{i} \equiv f_{k}(\frac{\theta}{2})\cos\left(\omega_{i}-(-1)^{p}\frac{ \theta}{2}\right)+(-1)^{p+k+1}f_{1\in k}(\frac{\theta}{2})\sin\left(\omega_{i} -(-1)^{p}\frac{\theta}{2}\right) \tag{117}\]  
Then, [PERSON] performs the sifting on the \(\text{Bob}_{1}\) state measuring it and then returning it to \(\text{Bob}_{1}\). In addition, \(\text{Bob}_{0}\) processing is followed, which gives (omitting the tensor product for simplicity, but indicating the systems with a proper subscript):  
\[|\psi_{2}\rangle\equiv C^{C}U_{0_{2}}\cdot|k\rangle_{\theta,p_{4}}\langle k| \cdot|\psi_{1}\rangle=(-1)^{pk}|k\rangle_{\theta,p_{4}}\Big{[}\sqrt{p_{0}}f_{ k}(\frac{\theta}{2})|0\rangle_{C}|\psi_{0}^{0}\rangle_{2}+\sqrt{p_{1}}W_{1}|1 \rangle_{C}|0\rangle_{2}\Big{]} \tag{118}\]  
where the previous expressions have been applied on the corresponding projections on the \(\text{Bob}_{1}\) state. At this point, note that the Eve sifting could be performed equivalently before or after to the \(\text{Bob}_{0}\) processing because measurement and the last processing works on different systems. It implies that Type B and C become equivalent for the current calculation as it was stated in Section 5.2. Thus, in any case Eve obtains \(|k\rangle_{\theta,p}\) as outcome (after selecting the basis defined by \(\theta,p\)). In the following step, the \(C_{SWAP_{2,4}}\equiv\mathbf{X}_{C}\cdot C_{SWAP_{2,4}}^{C}\cdot\mathbf{X}_{C}\) is applied between the [PERSON]'s, giving:  
\[|\psi_{3}\rangle\equiv C_{SWAP_{2,4}}\cdot|\psi_{2}\rangle=(-1)^{pk}\Big{[} \sqrt{p_{0}}f_{k}(\frac{\theta}{2})|0\rangle_{C}|k\rangle_{\theta,p_{2}}|\psi_ {0}^{0}\rangle_{4}+\sqrt{p_{1}}W_{1}|1\rangle_{C}|0\rangle_{2}|k\rangle_{ \theta,p_{4}}\Big{]} \tag{119}\]  
Then, [PERSON] performs the measurement of the control state on the basis stated by the election of \(K\). Here, she hits her selection \(|b_{j}\rangle\) so the next measurement performed by could be performed equivalently after or before to the [PERSON]'s measurement for calculation purposes. Employing such property, we get first:  
\[|\psi_{4}\rangle\equiv{}_{\theta,p_{4}}\langle k|\psi_{3}\rangle=(-1)^{pk}\bigg{[} \sqrt{p_{0}}f_{k}(\frac{\theta}{2})|0\rangle_{C}|k\rangle_{\theta,p_{2}}{}_{ \theta,p_{4}}\langle k|\psi_{0}^{0}\rangle_{4}+\sqrt{p_{1}}W_{1}|1\rangle_{C} |0\rangle_{2}\bigg{]} \tag{100}\]  
where it has been assumed that he hits on the same basis selection and outcome that [PERSON] to then get the success probability of her. Then, finally performing the [PERSON]'s measurement with outcome \(|b_{j}\rangle\):  
\[|\psi_{5}\rangle \equiv {}_{C}\langle b_{j}|\cdot{}_{\theta,p_{4}}\langle k|\psi_{4}\rangle \tag{101}\] \[= \sqrt{p_{0}}\beta_{0\oplus j}W_{0}f_{k}(\frac{\theta}{2})e^{i \phi_{m}j}|k\rangle_{\theta,p_{2}}+(-1)^{j+pk}\sqrt{p_{1}}\beta_{1\oplus j}W_{ 1}e^{-i\phi_{m}(1+j)}|0\rangle_{2}\]  
We will need to switch \(k\to k\oplus 1\) as the outcome obtained by [PERSON] and [PERSON]\({}_{1}\) in the main text.