# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 5 A QKD Protocol Based on a Shared Collaboration among Three Parties
### QKD Protocol Description Based on a Shared Generation
Thus, when the control measurement stochastically fits with the selection of \(j\), [PERSON] has successfully transferred a qubit in one specific state from the respective set expanding the Hilbert space on the basis selected by [PERSON]. Then, [PERSON]\({}_{1}\) should measure his state by selecting one of the two agreed basis, \(B_{B}\in\{H,D\}\). Thus, if [PERSON]\({}_{1}\) selects the same basis \(H\) or \(D\) to measure his state (characterized by \(\alpha\) instead \(\theta\) in (16) or (17)), the outcome is already known previously by [PERSON], thus sharing secretly a common element of the key. It only happens if the control measurement fits with \(j\) and if both bases coincide (\(\theta=\alpha\)), precisely as the BB84 protocol works. At this point, still they can infer the key until [PERSON] publishes her basis (\(\theta\) or \(B_{A}\)) also as [PERSON]\({}_{1}\) (\(\alpha\) or \(B_{B}\)). Still, [PERSON] also should skip the failed control measurements (they are expected to be a minimum as she is closer to \(P_{j}=1\)) but still communicate it to [PERSON]\({}_{1}\) as a failed outcome. Table 2 shows an illustrative sequence of such procedure until the information sharing, thus getting the useful key using QKD based on double teleportation. It skips the technical details dealt with in the previous subsection. The first two columns state the \(\theta\) value selected by [PERSON], thus implying the basis \(B_{A}\) selected, \(H\) or \(D\), used to state the final transferred state. Third to fifth columns state the elections of [PERSON]: \(j,m\), and the \(|k\otimes 1\rangle_{\theta,p}\) selected to be delivered to [PERSON]\({}_{1}\). The following four columns state the related information regarding the measurement outcomes: \(|b_{j}\rangle\) performed by [PERSON] on the control; \(\alpha\) or \(B_{B}\) the measurement basis selection of [PERSON]\({}_{1}\), as well as the corresponding outcome \(|k\oplus 1\rangle_{\alpha,p}\). The tenth and eleventh columns exhibit information sharing through a public channel. Note that [PERSON] should report the failed events marked there with \(\times\). Two last columns show the understood decisions of usefulness, acceptation, or rejection (A/R), of each event with (success) or (fail), as well as the useful bit keys: 0110... Note that the key just arises with their non-communicated outcomes when the registers of bases coincide and the [PERSON] measurement on the control succeeds her predictions. As in the BB84 protocol, the key still should pass an error correction reconciliation by sharing and comparing part of the key to detect transmission errors, otherwise the presence of an eavesdropper through the quantification of the QBER. Another relevant  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multicolumn{2}{c}{**Setup**} & \multicolumn{2}{c}{**Selection**} & \multicolumn{2}{c}{**Measurement**} & \multicolumn{2}{c}{**Sharing**} & \multicolumn{2}{c}{**Decision**} \\ \hline [PERSON] & & & [PERSON] & [PERSON] & [PERSON]\({}_{1}\) & [PERSON] & [PERSON]\({}_{1}\) & [PERSON] & [PERSON]\({}_{1}\) & [PERSON]/[PERSON]\({}_{1}\) \\ \(\theta\) & \(B_{A}\) & \(j\) & \(m\) & \(|k\oplus 1\rangle_{\theta,p}\) & \(|b_{j}\rangle\) & \(\alpha\) & \(B_{B}\) & \(|k\oplus 1\  
characteristic of this procedure is that [PERSON] can switch the BB84 four-state protocol into a six-state protocol [40; 47] adapting the post-processing (12) and selecting \(\theta,\phi\) conveniently.