# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 3 Double Teleportation as Superposition and Parallel Post-Processing
### Double Teleportation Process as Superposition
Then, the global initial state being considered becomes:  
\[|\psi\rangle=|\psi_{0}\rangle\otimes|\psi_{C}\rangle\otimes|\beta_{00}\rangle_{ 12}\otimes|\beta_{00}\rangle_{34} \tag{1}\]  
Then, the following controlled gate is applied: \(C_{U}=|0\rangle_{C}\langle 0|\otimes C^{0}NOT_{1}+|1\rangle_{C}\langle 1|\otimes C ^{0}NOT_{3}\), which is clearly unitary [29]. In fact, this gate is basically a pair of Toffoli gates each one followed by another:  
\[C_{U} = \mathrm{Toff}_{C,0,3}\cdot\mathbf{X}_{C}\cdot\mathrm{Toff}_{C,0, 1}\cdot\mathbf{X}_{C} \tag{2}\] \[= (|0\rangle_{C}\langle 0|\otimes\mathbf{1}_{3}+|1\rangle_{C} \langle 1|\otimes C^{0}NOT_{3})\cdot(|0\rangle_{C}\langle 0|\otimes C^{0}NOT_{1} +|1\rangle_{C}\langle 1|\otimes\mathbf{1}_{1})\]  
Then, remarking that \(C^{a}NOT_{b}=|0\rangle_{a}\langle 0|\otimes\mathbf{1}_{b}+|1\rangle_{a} \langle 1|\otimes\mathbf{X}_{b}\) (there, \(\mathbf{X}\) is the NOT gate), and considering that \(\mathbf{X}_{a}|\beta_{00}\rangle_{ab}=|\beta_{01}\rangle_{ab}\), the double teleportation process follows by applying a Hadamard gate on the qubit to be teleported, \(|\psi^{\prime}\rangle=H_{0}\cdot C_{U}|\psi\rangle\):  
\[|\psi^{\prime}\rangle = \sqrt{p_{0}}|0\rangle_{C}[\alpha_{0}|+\rangle_{0}|\beta_{00} \rangle_{12}+\alpha_{1}|-\rangle_{0}|\beta_{01}\rangle_{12}||\beta_{00}\rangle_ {34}\] \[\qquad+\sqrt{p_{1}}|1\rangle_{C}|\beta_{00}\rangle_{12}[\alpha_{ 0}|+\rangle_{0}|\beta_{00}\rangle_{34}+\alpha_{1}|-\rangle_{0}|\beta_{01} \rangle_{34}]\] \[= \frac{|0\rangle_{0}}{2}\Big{(}\sqrt{p_{0}}|0\rangle_{C}(|0 \rangle_{1}\mathbf{1}_{2}+|1\rangle_{1}\mathbf{X}_{2})|\psi_{0}\rangle_{2}| \beta_{00}\rangle_{34}\] \[\qquad+\sqrt{p_{1}}|1\rangle_{C}(|0\rangle_{3}\mathbf{1}_{4}+|1 \rangle_{3}\mathbf{X}_{4})|\psi_{0}\rangle_{4}|\beta_{00}\rangle_{12}\Big{)}\] \[+\frac{|1\rangle_{0}}{2}\Big{(}\sqrt{p_{0}}|0\rangle_{C}(|0 \rangle_{1}\mathbf{Z}_{2}+|1\rangle_{1}\mathbf{X}_{2}\mathbf{Z}_{2})|\psi_{0 }\rangle_{2}|\beta_{00}\rangle_{34}\] \[\qquad+\sqrt{p_{1}}|1\rangle_{C}(|0\rangle_{3}\mathbf{Z}_{4}+|1 \rangle_{3}\mathbf{X}_{4}\mathbf{Z}_{4})|\psi_{0}\rangle_{4}|\beta_{00}\rangle _{12}\Big{)}\]  
dropping the tensor products for the sake of simplicity, remarking that \(H|0\rangle=|+\rangle\) and \(H|1\rangle=|-\rangle\), and expanding some Bell states to express it in terms of \(|\psi_{0}\rangle\)[29]. Clearly, those steps state an extension of the traditional teleportation algorithm [34] for qubits. Finally, performing measurements on the original qubit to teleportate, as well as the qubits 1 and 3 (being part of the entangled resources) with respective outcomes \(s_{0},s_{1}\) and \(s_{3}\), we get the un-normalized post-measurement state:  
\[|\psi\rangle=|\psi_{0}\rangle\otimes|\psi_{C}\rangle\otimes|\beta_{00}\rangle_{ 12}\otimes|\beta_{00}\rangle_{34} \tag{1}\]0  
where if the exponent on an operator is zero, it implies that such operator is omitted. It is easy to notice that each one of the eight possible measurement outcomes occur with  
\[|\psi\rangle=|\psi_{0}\rangle\otimes|\psi_{C}\rangle\otimes|\beta_{00}\rangle_{ 12}\otimes|\beta_{00}\rangle_{34} \tag{1}\]1 (**a**) Three main parties performing double controlled teleportation with a central control accessible for all of them; a possible eavesdropper is present; (**b**) Quantum circuit representing the main elements of the double teleportation process. probability of \(\frac{1}{8}\). Thus, normalizing and finally, as a function of the outcomes, applying the generic correction \(\mathbf{Z}_{2}^{s_{0}}\mathbf{X}_{2}^{s_{1}}\mathbf{Z}_{4}^{s_{0}}\mathbf{X}_{4}^ {s_{3}}\) (\(\mathbf{X},\mathbf{Y},\mathbf{Z}\) are the Pauli operators), we obtain [29]:  
\[|\psi\rangle=|\psi_{0}\rangle\otimes|\psi_{C}\rangle\otimes|\beta_{00}\rangle_{ 12}\otimes|\beta_{00}\rangle_{34} \tag{1}\]2  
where we dropped the states for the qubits 1 and 3, as well as the original qubit for the sake of simplicity. This state represents the virtual teleportation to both [PERSON]'s.