# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 3 Results
### \(1\to 3\) Optimal Asymmetric Quantum Universal Clones Multicast Protocol
#### 3.2.1 \(1\to 3\) Quantum Multicast Protocol
\begin{split}& U_{8}(\alpha^{\prime\prime}_{1}|001)+\beta^{\prime \prime}_{1}|100)+\gamma^{\prime\prime}_{1}|010))=|000)\\ & U_{8}(\alpha^{\prime}_{1}(|012)+|021))+\beta^{\prime}_{1}(|102 )+|201))+\gamma^{\prime}_{1}(|120)+|210)))=|100),\end{split}  \tag{49}\]  
where \(\alpha^{\prime}_{1}\), \(\beta^{\prime}_{1}\), \(\gamma^{\prime}_{1}\), \(\alpha^{\prime\prime}_{1}\), \(\beta^{\prime\prime}_{1}\), and \(\gamma^{\prime\prime}_{1}\) are defined by  
\[ \begin{split}&\alpha^{\prime}_{1}=\frac{\alpha}{\sqrt{2\pi^{2}+2 \beta^{2}+2\gamma^{2}}},\quad\beta^{\prime}_{1}=\frac{\beta}{\sqrt{2\alpha^{2}+ 2\beta^{2}+2\gamma^{2}}},\quad\gamma^{\prime}_{1}=\frac{\gamma}{\sqrt{2\alpha ^{2}+2\beta^{2}+2\gamma^{2}}}\\ &\alpha^{\prime\prime}_{1}=\frac{\alpha+\beta}{\sqrt{(\alpha+ \beta)^{2}+(\beta+\gamma)^{2}+(\gamma+\alpha)^{2}}},\quad\beta^{\prime\prime}_ {1}=\frac{\beta+\gamma}{\sqrt{(\alpha+\beta)^{2}+(\beta+\gamma)^{2}+(\gamma+ \alpha)^{2}}},\\ &\gamma^{\prime\prime}_{1}=\frac{\gamma+\alpha}{\sqrt{(\alpha+ \beta)^{2}+(\beta+\gamma)^{2}+(\gamma+\alpha)^{2}}}.\end{split}  \tag{50}\]  
\(U^{(r,s,k)}_{9}\) is a unitary operator on \(\mathbb{C}^{d}\) defined by  
\[U^{(r,s,k)}_{9}:=\sum_{j\
eq r,s}|j\rangle\langle j|+(-1)^{k}|r\rangle\langle r |+(-1)^{k}|s\rangle\langle s|. \tag{51}\]  
\(U^{r(r)}_{2}\) is a tripartite unitary operator satisfying  
\[U^{r(r)}_{2}\cdot\frac{\alpha|j\tau r\rangle+\beta|rjr\rangle+ \gamma|r\tau j\rangle}{\sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}} =|j00\rangle,\quad(\forall j\
eq r)\] \[U^{r(r)}_{2}|rrr\rangle =|r00\rangle \tag{52}\]  
\(U^{r(r)}_{5}\) is a bipartite unitary operator defined by  
\[U^{r(r)}_{5}:=|r\rangle\langle r|\otimes I+\sum_{j\
eq r}|j\rangle\langle j| \otimes\left(\sum_{x=0}^{d-1}|\pi_{jr}(x)\rangle\langle x|\right), \tag{53}\]  
where \(\pi_{jr}\) is a permutation satisfying  
\[\pi_{jr}(1)=r,\quad\pi_{jr}(0)=j \tag{54}\]  
\(U^{r(r)}_{6}\) is a tripartite unitary operator defined by  
\[U^{r(r)}_{6}:=|r\rangle\langle r|\otimes\text{swap}+\sum_{j\
eq r}^{d-1}|j \rangle\langle j|\otimes I\otimes I, \tag{55}\]  
where swap is an operator defined by Equation (47). \(U^{r(r)}_{7}\) is a tripartite unitary operator defined by  
\[U^{r(r)}_{7}:=\sum_{i\
eq\tau}|i\rangle\langle i|\otimes I+|r\rangle\langle r |\otimes X^{r}_{d}, \tag{56}\]  
where \(X_{d}\) is the Pauli \(X\) operator defined by Equation (20). \(U^{r}_{8}\) is a unitary operator on \(\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}\) defined by  
\[ \begin{split} U^{r}_{8}|000\rangle&=|000\rangle\\ U^{r}_{8}(\alpha^{\prime}_{2}|011\rangle+\beta^{\prime}_{2}|101 \rangle+\gamma^{\prime}_{2}|110\rangle)&=|100\rangle\end{split}  \tag{57}\]Finally, \(U_{9}^{\prime(r,k)}\) is a unitary operator on \(\mathbb{C}^{d}\) defined by  
\[U_{9}^{\prime(r,k)}:=\sum_{j\
eq r}|j\rangle\langle j|+(-1)^{k}|r\rangle\langle r|. \tag{58}\]  
We will also use in the protocol the projective measurement \(\{P_{k}\}_{k=0}^{2}\) defined by the following equations:  
\[P_{0}:=\big{|}\tilde{0}\big{\rangle}\big{\langle}\tilde{0}\big{|},P_{1}:= \big{|}\tilde{1}\big{\rangle}\big{\langle}\tilde{1}\big{|},P_{2}:=I-\big{|} \tilde{0}\big{\rangle}\big{\langle}\tilde{0}\big{|}-\big{|}\tilde{1}\big{\rangle} \big{\langle}\tilde{1}\big{|}, \tag{59}\]  
where \(\big{|}\tilde{0}\big{\rangle}:=\frac{|0\rangle+|1\rangle}{\sqrt{2}},|\tilde{ 1}\rangle:=\frac{|0\rangle-|1\rangle}{\sqrt{2}}\). At the beginning of the protocol, the source node \(s\) has five \(d\)-dimensional systems \(A\), \(B\), \(C\), \(R\), and \(S\). The target node \(t_{1}\) has three \(d\)-dimensional systems \(D\), \(M_{1}\), and \(N_{1}\). The target node \(t_{2}\) has three \(d\)-dimensional systems \(E\), \(M_{2}\), and \(N_{2}\). The target node \(t_{3}\) has three \(d\)-dimensional systems \(F\), \(M_{3}\), and \(N_{3}\). Furthermore, the target nodes \(t_{1}\) and \(t_{2}\) share \(1+2\log_{2}3\) ebits of entanglement, and the target nodes \(t_{1}\) and \(t_{2}\) share \(1+2\log_{2}3\) ebits of entanglement in the form of maximally entangled states.