# 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
The unnormalized state on systems \(DEF\) shared by the target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\) can be written as  
\[\sum_{j=0}^{d-1}\kappa_{j}|j\rangle_{D}|j\rangle_{E}|j\rangle_{F}.\]  
Hence, the unnormalized state after Step 4 can be written as  
\[ \begin{split}&\sum_{j=0}^{d-1}\kappa_{j}|jjj\rangle_{DEF}\otimes \left(\alpha_{1}^{\prime}|012\rangle+\beta_{1}^{\prime}|102\rangle+\gamma_{1 }^{\prime}|120\rangle+\alpha_{1}^{\prime}|021\rangle+\beta_{1}^{\prime}|201 \rangle+\gamma_{1}^{\prime}|210\rangle\right)_{M_{1}M_{2}M_{3}}\\ &\otimes(\alpha_{1}^{\prime\prime}|001\rangle+\beta_{1}^{\prime \prime}|100\rangle+\gamma_{1}^{\prime\prime}|010\rangle)_{N_{1}N_{2}N_{3}} \end{split}  \tag{68}\]  
The purpose of the remaining part of the protocol is to reconstruct \(\ket{\Psi_{2}^{(r,s)}}\) from the above state. The unnormalized state after Step 5 can be written as  
\[ \begin{split}&\Big{(}\sum_{j\
eq r,s}^{d-1}\kappa_{j}|jjj\rangle_{DEF} \otimes(\alpha_{1}^{\prime}|jrs\rangle+\beta_{1}^{\prime}|rjs\rangle+\gamma_{1 }^{\prime}|rsj\rangle+\alpha_{1}^{\prime}|jsr\rangle+\beta_{1}^{\prime}|sjr \rangle+\gamma_{1}^{\prime}|srj\rangle)_{M_{1}M_{2}M_{3}}\\ &+(\kappa_{r}|rrr\rangle_{DEF}+\kappa_{s}|sss\rangle_{DEF})\\ &\qquad\otimes(\alpha_{1}^{\prime}|012\rangle+\beta_{1}^{\prime }|102\rangle+\gamma_{1}^{\prime}|120\rangle+\alpha_{1}^{\prime}|021\rangle+ \beta_{1}^{\prime}|201\rangle+\gamma_{1}^{\prime}|210\rangle)_{M_{1}M_{2}M_{3} }\Big{)}\\ &\otimes(\alpha_{1}^{\prime\prime}|001\rangle+\beta_{1}^{\prime \prime}|100\rangle+\gamma_{1}^{\prime\prime}|010\rangle)_{N_{1}N_{2}N_{3}} \end{split}  \tag{69}\]  
The unnormalized state after Step 6 can be written as  
\[ \begin{split}&\sum_{j\
eq r,s}^{d-1}\kappa_{j}|jjj\rangle_{DEF} \otimes(\alpha_{1}^{\prime}|jrs\rangle+\beta_{1}^{\prime}|rjs\rangle+\gamma_{ 1}^{\prime}|rsj\rangle+\alpha_{1}^{\prime}|jsr\rangle+\beta_{1}^{\prime}|sjr \rangle+\gamma_{1}^{\prime}|srj\rangle)_{M_{1}M_{2}M_{3}}\\ &\qquad\otimes(\alpha_{1}^{\prime\prime}|001\rangle+\beta_{1}^{ \prime\prime}|100\rangle+\gamma_{1}^{\prime\prime}|010\rangle)_{N_{1}N_{2}N_{3} }\\ &+(\kappa_{r}|rrr\rangle+\kappa_{s}|sss\rangle)_{DEF}\otimes( \alpha_{1}^{\prime\prime}|001\rangle+\beta_{1}^{\prime\prime}|100\rangle+ \gamma_{1}^{\prime\prime}|010\rangle)_{M_{1}M_{2}M_{3}}\\ &\otimes(\alpha_{1}^{\prime}|012\rangle+\beta_{1}^{\prime}|102 \rangle+\gamma_{1}^{\prime}|120\rangle+\alpha_{1}^{\prime}|021\rangle+ \beta_{1}^{\prime}|201\rangle+\gamma_{1}^{\prime}|210\rangle)_{N_{1}N_{2}N_{3} }\end{split}  \tag{70}\]  
Then, the unnormalized state after Step 7 can be written as  
\[ \begin{split}&\sum_{j\
eq r,s}^{d-1}\kappa_{j}|jjj\rangle_{DEF} \otimes(\alpha_{1}^{\prime}|jrs\rangle+\beta_{1}^{\prime}|rjs\rangle+\gamma_{ 1}^{\prime}|rsj\rangle+\alpha_{1}^{\prime}|jsr\rangle+\beta_{1}^{\prime}|sjr \rangle+\gamma_{1}^{\prime}|srj\rangle)_{M_{1}M_{2}M_{3}}\\ &\qquad\otimes(\alpha_{1}^{\prime\prime}|001\rangle+\beta_{1}^{ \prime\prime}|100\rangle+\gamma_{1}^{\prime\prime}|010\rangle)_{N_{1}N_{2}N_{3} }\\ &\qquad+\kappa_{r}|rrr\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|rsrs\rangle+\beta_{1}^{\prime\prime}|srr\rangle+\gamma_{1}^{\prime \prime}|rsr\rangle)_{M_{1}M_{2}M_{3}}\\ &\qquad\otimes(\alpha_{1}^{\prime}|012\rangle+\beta_{1}^{\prime }|102\rangle+\gamma_{1}^{\prime}|120\rangle+\alpha_{1}^{\prime}|021\rangle+ \beta_{1}^{\prime}|201\rangle+\gamma_{1}^{\prime}|210\rangle)_{N_{1}N_{2}N_{3} }\\ &\qquad+\kappa_{s}|sss\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|ssr\rangle+\beta_{1}^{\prime\prime}|rss\rangle+\gamma_{1}^{\prime \prime}|ssr\rangle)_{M_{1}M_{2}M_{3}}\\ &\qquad\otimes(\alpha_{1}^{\prime}|012\rangle+\beta_{1}^{\prime }|102\rangle+\gamma_{1}^{\prime}|120\rangle+\alpha_{1}^{\prime}|021\rangle+ \beta_{1}^{\prime}|201\rangle+\gamma_{1}^{\prime}|210\rangle)_{N_{1}N_{2}N_{3} }\end{split}  \tag{71}\]  
The unnormalized state after Step 8 can be written as  
\[