# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 3 Double Teleportation as Superposition and Parallel Post-Processing
### Double Teleportation Process as Superposition  
Figure 1a synthetically depicts the process followed for double teleportation immersed in the context of QKD. In the current section, we develop the double teleportation process as it was originally presented [29]. As it was stated, one party ([PERSON]) generates secretly an arbitrary state \(|\psi_{0}\rangle\) (known or unknown) to then potentially transmit it in superposition by teleportation to other two parties ([PERSON]\({}_{0}\) and [PERSON]\({}_{1}\)). The presence of an eavesdropper ([PERSON]) acting on [PERSON]\({}_{1}\) is possible, so it is shown in Figure 1a,b, but her action will be considered and depicted at the end. In this case, instead of the traditional algorithm, the process intends to virtually teleport such state to those two simultaneous receivers, [PERSON]\({}_{0}\) and [PERSON]\({}_{1}\)[29]. The process begins with the main qubit \(|\psi_{0}\rangle\) to be teleported in possession of [PERSON], where \(|\psi_{0}\rangle=\alpha_{0}|0\rangle+\alpha_{1}|1\rangle\). In this work, the Bell states will be written: \(|\beta_{ij}\rangle=\frac{1}{\sqrt{2}}(|0\;j\rangle+(-1)^{i}|1\;j\oplus 1\rangle)\). Then, a pair of Bell entangled resources \(|\beta_{00}\rangle_{12}\), \(|\beta_{00}\rangle_{34}\) are prepared to be shared with each one of both receivers to implement the teleportation process (subscripts state the numbering of the resources). In the process, a control state is required to rule the quantum transmission on a concrete receiver \(|\psi_{C}\rangle=\sum_{j=0}^{1}\sqrt{p_{i}}|i\rangle_{C}\), with: \(\sum_{j=0}^{1}p_{i}=1\). Thus, \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Process** & **Description** \\ \hline \(|\psi_{0}\rangle\) & DT & Original qubit state to be teleported \\ \(|\psi_{C}\rangle\) & DT & Control state to manage the final receiver in double teleportation \\ \(p_{i}\) & DT & Superposition probabilities for each receiver in double teleportation \\ \(|\beta_{ij}\rangle\) & DT & Entangled resources for teleportation in the for of Bell states \\ \(C_{ll},H_{0}\) & DT & Controlled \(C^{a}NOT_{b}\) and Hadamard gates to manage the double teleportation \\ \(|\psi\rangle,|\psi^{\prime}\rangle\) & DT & Initial state and pre-measurement state during the double teleportation process \\ \(|\psi_{pm}\rangle,|\psi_{thet}|_{reported}\rangle\) & DT & Post-measurement and corrected states at the end of double teleportation process \\ \(\mathcal{U}_{k}\) & PP & Local processing operators on the qubit \(k\) in possession of party \(i\) \\ \(\omega_{i}\) & PP & Parametric continuous characterization of each local processing \\ \(C_{ll}\) & PP & Controlled operation to apply local processing \(\mathcal{U}_{i}\) on each receiver \\ \(|\psi_{ll}^{j}\rangle\) & PP & Output state from the each local processing on \(|\psi^{\prime}\rangle\) \\ \(|\psi_{pmc}\rangle,|\psi_{final}\rangle\) & PP & Local processing operators on the qubit \(k\) in possession of party \(i\) \\ \(\beta_{i},\phi_{m}\) & TC & Parameters for the basis measurement of the control state \\ \(P_{i}\) & TC & Success probability for each outcome of the control measurement \\ \(K,m,j\) & QKD & Key parameters in the QKD process \\ \(p\) & QKD & Absolute rate of the success eavesdropper without reconciliation \\ \(P_{E}\) & QKD & Relative rate of the success eavesdropper with reconciliation \\ \(P_{QBER_{bb}},P_{QBER_{nl}}\) & QKD & Absolute and relative QBER \\ \hline \end{tabular}
\end{table} 
Table 1: States, Gates, and Parameters involved in the analysis through each step (DT, PP, TC, QKD) of the whole QKD protocol. if control is settled in the state \(|0\rangle_{C}\) then \(|\psi_{0}\rangle\) is teleported to Bob\({}_{0}\), otherwise to Bob\({}_{1}\) if the control is \(|1\rangle_{C}\).