# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 3 Double Teleportation as Superposition and Parallel Post-Processing
### Post-Processing Following to Double Teleportation  
In the last expression, each outcome still can be managed by the control state to perform different processing on each virtual teleported qubit. Applying the operator (see Figure 1b on the right in the form of a pair of controlled gates):  
\[C_{U} = (|0\rangle_{C}\langle 0|\otimes\mathcal{U}_{0_{2}}+|1\rangle_{C} \langle 1|\otimes\mathbf{1}_{2})\cdot(|0\rangle_{C}\langle 0|\otimes\mathbf{1 }_{4}+|1\rangle_{C}\langle 1|\otimes\mathcal{U}_{1_{4}}) \tag{7}\] \[= |0\rangle_{C}\langle 0|\otimes\mathcal{U}_{0_{2}}\otimes\mathbf{1 }_{4}+|1\rangle_{C}\langle 1|\otimes\mathbf{1}_{2}\otimes\mathcal{U}_{1_{4}}\]  
Such asymmetric post-processing following to the double teleportation will set the successful secrecy for the QKD procedure. Thus, by defining \(|\psi_{0}^{i-1}\rangle_{2i}=\alpha_{0}^{i}|0\rangle_{2i}+\alpha_{1}^{i}|1 \rangle_{2i}\) (with \(i=1,2\)) as the output of each processing \(\mathcal{U}_{(i-1)_{2i}}\), then we will get:  
\[|\psi_{proc}\rangle=C_{U}|\psi_{teleported}\rangle = \sqrt{p_{0}}|0\rangle_{C}\otimes\mathcal{U}_{0_{2}}|\psi_{0} \rangle_{2}\otimes|0\rangle_{4}+\sqrt{p_{1}}|1\rangle_{C}\otimes|0\rangle_{2} \otimes\mathcal{U}_{1_{4}}|\psi_{0}\rangle_{4} \tag{8}\] \[= \sqrt{p_{0}}|0\rangle_{C}\otimes|\psi_{0}^{0}\rangle_{2}\otimes| 0\rangle_{4}+\sqrt{p_{1}}|1\rangle_{C}\otimes|0\rangle_{2}\otimes|\psi_{0}^{ 1}\rangle_{4} \tag{9}\]  
As a useful possibility for further applications, we consider the final transference of the state from Bob\({}_{0}\) to Bob\({}_{1}\) by applying a controlled \(SWAP\) to send the processed output state on the qubit 4: \(C_{SWAP2A}=|0\rangle_{C}\langle 0|\otimes SWAP_{2A}+|1\rangle_{C}\langle 1| \otimes\mathbf{1}_{2}\otimes\mathbf{1}_{4}\) (see Figure 1b on the right).  
Note operations \(C_{U}\) and \(C_{SWAP2A}\) are few practical because qubits \(C,2\) are far away from qubit 4. We show them in such last synthetic forms, but they can be equivalently achieved using additional entangled resources between Alice/Bob\({}_{0}\) and Bob\({}_{1}\). The details about the equivalence of such processes are given in the Appendices A and B. There, related techniques required are delayed measurements [35] and quantum controlled measurements [36; 37]. In any case, it gives the following state settled on the qubit 4 in possession of Bob\({}_{1}\):  
\[|0\rangle_{2}\otimes|\psi_{final}\rangle\equiv C_{SWAP2A}|\psi_{proc}\rangle= |0\rangle_{2}\otimes(\sqrt{p_{0}}|0\rangle_{C}\otimes|\psi_{0}^{0}\rangle_{4} +\sqrt{p_{1}}|1\rangle_{C}\otimes|\psi_{0}^{1}\rangle_{4}) \tag{10}\]  
disregarding the separable qubit 2. In addition, [PERSON] will decide to measure the control state on an eligible orthogonal basis \(\{|b_{0}\rangle_{C},|b_{1}\rangle_{C}\}\) given by \(|0\rangle_{C}=\beta_{0}|b_{0}\rangle_{C}+e^{i\phi_{m}}\beta_{1}|b_{1}\rangle_{C},|1\rangle_{C}=e^{-i\phi_{m}}\beta_{1}|b_{0}\rangle_{C}-\beta_{0}|b_{1}\rangle_ {C}\), with \(\phi_{m},\beta_{0},\beta_{1}\in\mathbf{R},\beta_{0}^{2}+\beta_{1}^{2}=1\). It means, \(|\psi_{final}\rangle\) can be written as:  
\[|\psi_{final}\rangle=\sqrt{p_{0}}(\beta_{0}|b_{0}\rangle_{C}+e^{i\phi_{m}}\beta _{1}|b_{1}\rangle_{C})\otimes|\psi_{0}^{0}\rangle_{4}+\sqrt{p_{1}}(e^{-i\phi_{ m}}\beta_{1}|b_{0}\rangle_{C}-\beta_{0}|b_{1}\rangle_{C})\otimes|\psi_{0}^{1}\rangle_ {4} \tag{11}\]  
to ease the identification of the measurement outcomes in such basis. In the following, we will commonly drop the labels \(C\) for the control and 4 for the qubit 4, which now become clear from the development.