# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## Appendix A Control on Faraway Non-Local Resources  
Controlled operations on faraway non-local parties as that in the second factor of (7), \(C^{C}\mathcal{U}_{1_{4}}=|0\rangle_{C}\langle 0|\otimes\mathbf{1_{4}}+|1 \rangle_{C}\langle 1|\otimes\mathcal{U}_{1_{4}}\), are not possible to be performed directly (assuming that the control and qubit 4 cannot be moved from their locations). Nevertheless, they can be achieved via LOCC with the support of an entangled pair \(|\beta_{00}\rangle_{ab}\) where qubit \(a\) is in possession of [PERSON] and \(b\) is sent to [PERSON]\({}_{1}\):  
\[|\psi\rangle = (\sqrt{p_{0}}|0\rangle_{C}|\phi_{0}\rangle_{2}|\psi_{0}\rangle_{ 4}+\sqrt{p_{1}}|1\rangle_{C}|\phi_{1}\rangle_{2}|\psi_{1}\rangle_{4})\otimes| \beta_{00}\rangle_{ab} \tag{1}\]A direct calculation shows that a such state can be written in the basis of the Bell states for the qubits \(C\) and \(a\) as:  
\[|\psi\rangle = \frac{1}{2}\sum_{x,y\in\{0,1\}}|\beta_{xy}\rangle_{Ca}(\sqrt{p_{0}} |0\oplus y\rangle_{b}|\phi_{0}\rangle_{2}|\psi_{0}\rangle_{4}+(-1)^{x}\sqrt{p_ {1}}|1\oplus y\rangle_{b}|\phi_{1}\rangle_{2}|\phi_{1}\rangle_{4}) \tag{22}\] \[= \frac{1}{2}\sum_{x,y\in\{0,1\}}|\beta_{xy}\rangle_{Ca}\mathbf{Z}_ {b}^{x}\mathbf{X}_{b}^{y}(\sqrt{p_{0}}|0\rangle_{b}|\phi_{0}\rangle_{2}|\psi_{ 0}\rangle_{4}+\sqrt{p_{1}}|1\rangle_{b}|\phi_{1}\rangle_{2}|\phi_{1}\rangle_{4}) \tag{23}\]  
then, [PERSON] applies the operation \(H_{C}\cdot C^{C}NOT_{a}\) on her qubits, getting:  
\[|\psi^{(1)}\rangle = \frac{1}{2}\sum_{x,y\in\{0,1\}}|xy\rangle_{Ca}\mathbf{Z}_{b}^{x} \mathbf{X}_{b}^{y}(\sqrt{p_{0}}|0\rangle_{b}|\phi_{0}\rangle_{2}|\psi_{0} \rangle_{4}+\sqrt{p_{1}}|1\rangle_{b}|\phi_{1}\rangle_{2}|\psi_{1}\rangle_{4}) \tag{24}\]  
The following development could be achieved using delayed measurements [35] or still just controlled operations. Despite, they commonly require interactions between faraway resources, which implies some of them will be moved from their locations using extra classical communication operations. Instead, we will use projective measurements and corrections. Thus, [PERSON] measures their qubits \(C,a\) obtaining the outcomes \(|x\rangle_{C}\) and \(|y\rangle_{a}\) respectively. Using classical communication, [PERSON] shares those outcomes with [PERSON]\({}_{1}\) who applies the controlled operation \(C^{C}\mathbf{X}_{b}^{y}\cdot C^{a}\mathbf{Z}_{b}^{x}\). The outcome is:  
\[|\psi^{(2)}\rangle = |xy\rangle_{Ca}(\sqrt{p_{0}}|0\rangle_{b}|\phi_{0}\rangle_{2}| \psi_{0}\rangle_{4}+\sqrt{p_{1}}|1\rangle_{b}|\phi_{1}\rangle_{2}|\psi_{1} \rangle_{4}) \tag{25}\]  
thus, [PERSON]\({}_{1}\) applies the controlled operation \(C^{b}\mathcal{U}_{1_{4}}\):  
\[|\psi^{(3)}\rangle = |xy\rangle_{Ca}(\sqrt{p_{0}}|0\rangle_{b}|\phi_{0}\rangle_{2}| \psi_{0}\rangle_{4}+\sqrt{p_{1}}|1\rangle_{b}|\phi_{1}\rangle_{2}\mathcal{U}_{ 1_{4}}|\psi_{1}\rangle_{4}) \tag{26}\]  
Finally, qubit \(b\) is sent to Alice to perform the \(SWAP_{Cb}\) operation:  
\[|\psi^{(4)}\rangle = |yx\rangle_{ab}(\sqrt{p_{0}}|0\rangle_{C}|\phi_{0}\rangle_{2}| \psi_{0}\rangle_{4}+\sqrt{p_{1}}|1\rangle_{C}|\phi_{1}\rangle_{2}\mathcal{U}_{ 1_{4}}|\psi_{1}\rangle_{4}) \tag{27}\]  
which, disregarding the qubits \(a\) and \(b\), is the same state obtained by \(C^{C}\mathcal{U}_{1_{4}}\).