# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 3 Results
### \(1\to 3\) Optimal Asymmetric Quantum Universal Clones Multicast Protocol
#### 3.2.2 Proof of \(1\to 3\) Quantum Multicast Protocol
Hence, the unnormalized state after Step 4 can be written as  
\[\sum_{j=0}^{d-1}\kappa_{j}^{\prime}|jj\rangle_{DEF}\otimes(\alpha_{2}^{\prime }|011\rangle+\beta_{2}^{\prime}|101\rangle+\gamma_{2}^{\prime}|110\rangle)_{M _{1}M_{2}M_{3}}\otimes|000\rangle_{N_{1}N_{2}N_{3}} \tag{77}\]  
Then, the unnormalized state after Step 5 can be written as  
\[\sum_{j\
eq r}^{d-1}\kappa_{j}^{\prime}|jj\rangle_{DEF}\otimes( \alpha_{2}^{\prime}|jrr\rangle+\beta_{2}^{\prime}|rjr\rangle+\gamma_{2}^{ \prime}|rrj\rangle)_{M_{1}M_{2}M_{3}}\otimes|000\rangle_{N_{1}N_{2}N_{3}} \tag{78}\] \[\qquad+\kappa_{r}|rrr\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|rrs\rangle+\beta_{1}^{\prime\prime}|srr\rangle+\gamma_{1}^{\prime \prime}|rsr\rangle)_{M_{1}M_{2}M_{3}}\]0  
The unnormalized state after Step 6 can be written as  
\[\qquad+\kappa_{r}|rrr\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|rrs\rangle+\beta_{1}^{\prime\prime}|srr\rangle+\gamma_{1}^{\prime \prime}|rsr\rangle)_{M_{1}M_{2}M_{3}}\]1 \[\qquad+\kappa_{r}|rrr\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|rrs\rangle+\beta_{1}^{\prime\prime}|srr\rangle+\gamma_{1}^{\prime \prime}|rsr\rangle)_{M_{1}M_{2}M_{3}}\]2The unnormalized state after Step 7 can be written as  
\[ \begin{split}&\sum_{j\
eq r}^{d-1}\kappa^{\prime}_{j}|jj\rangle_{ DEF}\otimes(\alpha^{\prime}_{2}|jrr\rangle+\beta^{\prime}_{2}|rjr\rangle+\gamma^{ \prime}_{2}|rrj\rangle)_{M_{1}M_{2}M_{3}}\otimes|000\rangle_{N_{1}N_{2}N_{3}}\\ &\quad+\kappa^{\prime}_{j}|rrr\rangle_{DEF}\otimes|rrr\rangle_{M _{1}M_{2}M_{3}}(\alpha^{\prime}_{2}|011\rangle+\beta^{\prime}_{2}|101\rangle+ \gamma^{\prime}_{2}|110\rangle)_{N_{1}N_{2}N_{3}}\end{split}  \tag{80}\]  
Then, the unnormalized state after Step 8 can be written as  
\[ \begin{split}&\sum_{j\
eq r}^{d-1}\kappa^{\prime}_{j}|jj\rangle_{ DEF}\otimes\left(\alpha^{\prime}_{2}|jrr\rangle+\beta^{\prime}_{2}|rjr\rangle+ \gamma^{\prime}_{2}|rrj\rangle\right)_{M_{1}M_{2}M_{3}}\otimes|0\rangle_{N_{1}} \\ &\quad+\kappa^{\prime}_{r}|rrr\rangle_{DEF}\otimes|rrr\rangle_{M _{1}M_{2}M_{3}}|1\rangle_{N_{1}}\end{split}  \tag{81}\]  
The unnormalized state after Step 9 can be written as  
\[ \begin{split}&\sum_{j\
eq r}^{d-1}\kappa^{\prime}_{j}|jj\rangle_{ DEF}\otimes(\alpha^{\prime}_{2}|jrr\rangle+\beta^{\prime}_{2}|rjr\rangle+ \gamma^{\prime}_{2}|rrj\rangle)_{M_{1}M_{2}M_{3}}+\kappa^{\prime}_{r}|rrr \rangle_{DEF}\otimes|rrr\rangle_{M_{1}M_{2}M_{3}}\end{split}  \tag{82}\]  
Finally, the unnormalized state after Step 10 can be written as  
\[ \begin{split}&\omega^{(p^{\prime}_{1}+p^{\prime}_{2}+p^{\prime}_{3 })2r}\Big{\{}\sum_{j=0,j\
eq r}^{d-1}\kappa^{\prime}_{j}(\alpha^{\prime}_{2}| jrr\rangle+\beta^{\prime}_{2}|rjr\rangle+\gamma^{\prime}_{2}|rrj\rangle)_{M_{1}M_{2 }M_{3}}+\kappa^{\prime}_{r}|rrr\rangle_{M_{1}M_{2}M_{3}}\Big{\}}\end{split}  \tag{83}\]  
We can easily see that the above state is equivalent to \(\left|\Psi^{(r,r)}_{2}\right\rangle\) as defined by Equation (64) except for a global phase. Hence, the proof is complete for \(r=s\). Thus, we have achieved multicasting of asymmetric optimal clones for the systems \(M_{1}\), \(M_{2}\), and \(M_{3}\) to the three target nodes.