# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 4 Transference of Programmed Quantum States Using Double Teleportation
### General Notation for the Control of Post-Selection Problem  
In the current subsection, we are interested in the post-selection by Alice/Bob\({}_{1}\) of certain states as well as in the control and Bob\({}_{1}\) systems. With that purpose, we develop a general notation to solve the problem. First, by defining:  
\[f_{k}(\omega)= \begin{cases}\cos\omega,&k=0\\ \sin\omega,&k=1\end{cases}  \tag{21}\]  
then, clearly \(|\psi_{0}^{j}\rangle=\sum_{k=0}^{1}f_{k}(\omega_{j})|k\rangle_{\theta,p}\). In those terms, \(|\psi_{final}\rangle\) reads:  
\[|\psi_{final}\rangle = \sum_{j=0,1}(-1)^{mj}|b_{j}\rangle\sum_{k=0,1}|k\rangle_{\theta,p }(\sqrt{p_{0}}\beta_{0:0})f_{k}(\omega_{0})-(-1)^{j+m}\sqrt{p_{1}}\beta_{1 \oplus j}f_{k}(\omega_{1})) \tag{22}\]  
If then [PERSON]/[PERSON]\({}_{0}\) pretends to control the post-selection of \(|b_{j}\rangle\) and \(|k\oplus 1\rangle_{\theta,p}\), two conditions should be imposed. The first one is:  
\[\sqrt{p_{0}}\beta_{0\oplus j}f_{k}(\omega_{0})=(-1)^{j+m}\sqrt{p_{1}}\beta_{1 \oplus j}f_{k}(\omega_{1}) \tag{23}\]which post-selects one of \(|k\oplus 1\rangle_{\theta,p},k=0,1\) for certain \(j\). It could be solved by demanding:  
\[0<K\equiv\frac{\sqrt{p_{1}}\beta_{1\oplus j}}{\sqrt{p_{0}}\beta_{0\oplus j}}, \quad K\in\mathbf{R}^{+} \tag{24}\]  
Such a condition states a possible asymmetric treatment (\(K\
eq 1\)) to introduce the secrecy of the QKD procedure [29]. Such condition states an asymmetric treatment to introduce the secrecy of the QKD procedure because the transmitted state to [PERSON]\({}_{1}\) remains uncertain. It immediately implies the fulfilling of:  
\[f_{k}(\omega_{0})=(-1)^{j+m}Kf_{k}(\omega_{1}) \tag{25}\]  
Equation (25) states the way to select \(\omega_{0}\) when \(\omega_{1}\) is first settled choosing certain value for \(K\) stating certain secret asymmetry in the election (together with \(j,m\), all those parameters under the control of [PERSON]/[PERSON]\({}_{0}\)). Figure 2a,b show such process for \(k=0\) and \(k=1\) respectively (remembering that the selected state for [PERSON]\({}_{1}\) is \(k\oplus 1\)). If the restriction \(\omega_{1}\in[0,\frac{\pi}{2}]\) is settled (such condition it is not completely necessary but it eases some further expressions), upon the selection of \(K>0,j+m=0\), then \(\omega_{0}\) could be selected as it is indicated by the green circles in both figures (\(K>1\) or \(K<1\)); otherwise, if \(K>0,j+m=1\), \(\omega_{0}\) could be selected as in the red circles (\(K>1\) or \(K<1\)). There, we restrict \(\omega_{0}\in[0,\pi]\) for \(k=0\) and \(\omega_{0}\in[-\frac{\pi}{2},\frac{\pi}{2}]\) for \(k=1\). Note in any case that \(f_{k\oplus 1}(\omega_{0})>0\) will fulfill. Consequently, the election of \(K\) stated by (24) relates \(p_{0}\) with \(\beta_{0}\). Those relations are shown in Figure 2c,d for \(k=0\) and \(k=1\) respectively. Each red curve corresponds to certain \(K\) value being selected. While \(K>0\), the darkest red curves show the lowest values for \(K\approx 0\), and the lightest ones the largest values for \(K\rightarrow\infty\). Figure 2: (a,b) Plots exhibiting the process of selection of \(\omega_{0}\) departing from the selection of \(\omega_{1}\) through the condition (25) for each case \(k=0\) (left) or \(k=1\) (right). (c,d) Possible values for the combination of \(p_{0}\), \(\beta_{0}\) upon the prior selection of \(K\) (each red curve) for each case \(k=0\) (left) or \(k=1\) (right); blue dots show the values of \(p_{0}=\frac{1}{1+K}\) maximizing \(P_{j}\). The second condition is obtained by substituting the first condition in the probability of success (13) or (14) for \(|b_{j}\rangle\):  
\[P_{j} = p_{0}\beta_{0\oplus j}^{2}+p_{1}\beta_{1\oplus j}^{2}-2\sqrt{p_{0} p_{1}}\beta_{0}\beta_{1}(-1)^{j+m}\cos\Delta_{\omega}^{-} \tag{26}\] \[= p_{0}\beta_{0\oplus j}^{2}(1+K^{2}-2K(-1)^{j+m}\cos\Delta_{\omega} ^{-})\]  
by choosing a high value for \(P_{j}\) (ideally \(P_{j}=1\)).