# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 4 Transference of Programmed Quantum States Using Double Teleportation
### Control Prescriptions for the Quantum State Transference  
In fact, we can analyse \(c_{\omega}^{-}\) in terms of \(k,\omega_{1}\) and \(K\) (using the fact \(f_{k\oplus 1}(\omega_{0})>0\) with \(\omega_{1}\in[0,\frac{7}{2}]\)):  
\[\frac{c_{\omega}^{-}}{(-1)^{j+m}}=\cos\Delta_{\omega}^{-}=\cos(\omega_{0}- \omega_{1})=f_{k}^{2}(\omega_{1})(-1)^{j+m}K+f_{k\oplus 1}(\omega_{1})\sqrt{1-K^{2 }f_{k}^{2}(\omega_{1})} \tag{27}\]  
Thus, when each upper contour intersects to their lower partner, it generates the affordable solutions. Such solutions are shown in the Figure 4 but in the variables to be selected, \(\omega_{1},K,j,k,m\) (remembering that the selected state for Bob\({}_{1}\) is \(k\oplus 1\)): (a) \(k=0,j+m=0\), (b) \(k=0,j+m=1\), (c) \(k=1,j+m=0\), and (d) \(k=1,j+m=1\) (in fact, \(j\oplus m=0,1\), but we will maintain just those simpler expressions in the following). Curves in each plot show some affordable solutions (black region) for each \(c_{\omega}^{-}\) value in color from red (\(c_{\omega}^{-}=-1\)) to blue (\(c_{\omega}^{-}=1\)). We have plotted the region only in the more convenient interval for \(\omega_{1}\in[0,\frac{7}{2}]\) being congruent with the previous remark. \(K>0\) values are not restricted in their strength, but clearly \(K>1\) reduces the possible solutions. Solving (24) for \(\beta_{0\oplus j}^{2}\) and introducing the overall restrictions in (26), we get the following expression for \(P_{j}\):  
\[P_{j}=\frac{p_{0}(1-p_{0})}{1-p_{0}(1-K^{2})}\Big{[}1+K^{2}-2K(f_{k}^{2}(\omega_{ 1})K-(-1)^{j+m}f_{k\oplus 1}(\omega_{1})\sqrt{1-K^{2}f_{k}^{2}(\omega_{1})} \Big{]} \tag{28}\]  
The coefficient there, \(C_{0}(p_{0},K)\equiv\frac{p_{0}(1-p_{0})}{1-p_{0}(1-K^{2})}\), depends only on \(p_{0}\) and \(K\). An easy analysis shows that such coefficient reaches its maximum for \(p_{0_{max}}=\frac{1}{1+K}\) becoming \(C_{0_{max}}=\frac{1}{(1+K)^{2}}\). Such optimal values are also shown for each \(K\)-curve in Figure 2c,d with blue dots. Because it is zero in their edges \(p_{0}=0,1\), then the values of such coefficient are folded in the intervals \(p_{0}\in[0,p_{0_{max}}]\) and \(p_{0}\in[p_{0_{max}},1]\). Figure 5 depicts \(P_{j}\) for (a) \(k=0,j+m=0\), (b) \(k=0,j+m=1\), (d) \(k=1,j+m=0\), and (e) \(k=1,j+m=1\) (remembering that the selected state transmitted to [PERSON]\({}_{1}\) is \(k\oplus 1\)). They are three-dimensional regions (transparent clear gray regions) under the main maximal surface plotted in dark gray, which corresponds to the two folded points generated vertically by \(p_{0}\in[0,1]\) (shown by the arrows in the Figure 5a,b,d,e). Figure 5c,f show the comparison between the corresponding maximum values in each case, (c) \(k=0\) and (f) \(k=1\) respectively, remarking the advantage for \(j+m=1\) (green) against \(j+m=0\) (red), thus it is better to choose \(m\) with a different parity of the selected \(j\). In such cases, \(\omega_{1}\) could be selected almost openly, to then select those \(K\) values reaching \(P_{j}\) at least near from \(1\), thus controlling better the stochastic selection of \(|b_{j}\rangle\). Note in the figures, that the regions have been maintained in \(\omega_{1}\in[0,\frac{\pi}{2}]\) as it was initially recommended in the procedure. In addition, we have not extended the interval for \(K\) further than \(K=1\), because plot regions there become restricted to narrower non-rectangular regions as it was shown in the Figure 4, becoming unpractical because of the restricted combinations of \(\omega_{1}\) and \(K\) values able to be selected. Still, in the practice, \(K>1\) also provides valuable solutions. Figure 5: Plots for \(P_{j}\) in (28) as function of \(\omega_{1}\) and \(K\) for the cases (a) \(k=0\), \(j+m=0\), (b) \(k=0\), \(j+m=1\), (c) \(k=1\), \(j+m=0\), and (d) \(k=1\), \(j+m=1\). Comparisons for the cases (e) \(k=0\), and (f) \(k=1\) exhibiting the advantage for \(j+m=1\) to reach higher values for \(P_{j}\). Another important remark should be stated--Despite the election of \(p_{0_{max}}\) is quite recommendable to maximize \(P_{j}\).