# 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  
In this sub-subsection, we prove that Protocols 3-5 create \(1\to 3\) asymmetric UQCs given by Equation (12) in system \(M_{1}M_{2}M_{3}\). Proof. : Let the input state at the source node be \(|\psi\rangle=\sum_{j=0}^{d-1}\delta_{j}|j\rangle\). Then, from Equation (10), the state on system \(ABCRS\) after Step 1 can be written as  
\[ \begin{split}\sqrt{\frac{d}{2d+2}}[\alpha|\psi\rangle_{A}(|\[PERSON]^ {+}\rangle_{BR}|\[PERSON]^{+}\rangle_{CS}+|\[PERSON]^{+}\rangle_{BS}|\Phi^{+}\rangle_{CR} )\\ +\beta|\psi\rangle_{B}(|\[PERSON]^{+}\rangle_{AR}|\[PERSON]^{+}\rangle_{CS} +|\[PERSON]^{+}\rangle_{AS}|\[PERSON]^{+}\rangle_{CR})\\ +\gamma|\psi\rangle_{C}(|\[PERSON]^{+}\rangle_{AR}|\[PERSON]^{+}\rangle_{ BS}+|\[PERSON]^{+}\rangle_{AS}|\[PERSON]^{+}\rangle_{BR})]\end{split}  \tag{62}\]  
After Step 2, the protocol branches depending on whether \(r\
eq s\) or \(r=s\), where \(r\) and \(s\) are the measurement outcomes of system \(R\) and \(S\), respectively. The unnormalized state \(\left|\Psi_{2}^{(r,s)}\right\rangle\) after Step 2 for \(r\
eq s\) can be written as  
\[ \begin{split}&\left|\Psi_{2}^{(r,s)}\right\rangle\\ =&\frac{1}{\sqrt{2d(d+1)}}\Big{[}\alpha(|\psi\rangle_{A}| r\rangle_{B}|s\rangle_{C}+|\psi\rangle_{A}|s\rangle_{B}|r\rangle_{C})+\beta(|r\rangle_{A}| \psi\rangle_{B}|s\rangle_{C}+|s\rangle_{A}|\psi\rangle_{B}|r\rangle_{C})\\ &\qquad\qquad+\gamma(|r\rangle_{A}|s\rangle_{B}|\psi\rangle_{C}+| s\rangle_{A}|r\rangle_{B}|\psi\rangle_{C})\Big{]}\\ =&\frac{1}{\sqrt{2d(d+1)}}\Big{[}\delta_{r}\big{(} (\alpha+\beta)|rrs\rangle+(\beta+\gamma)|srr\rangle+(\gamma+\alpha)|rsr\rangle \big{)}_{ABC}\\ &\qquad+\delta_{s}\big{(}(\alpha+\beta)|ssr\rangle+(\beta+\gamma )|rss\rangle+(\gamma+\alpha)|srs\rangle\big{)}_{ABC}\\ &\qquad+\sum_{j\
eq r,s}\delta_{j}\big{(}\alpha|jrs\rangle+ \beta|rjs\rangle+\gamma|r\bar{s}j\rangle+\alpha|jsr\rangle+\beta|jr\rangle+ \gamma|srj\rangle\big{)}_{ABC}\Big{]}.\end{split}  \tag{63}\]  
The unnormalized state \(\left|\Psi_{2}^{(r,r)}\right\rangle\) after Step 2 for \(r=s\) can be written  
\[ \begin{split}&\left|\Psi_{2}^{(r,r)}\right\rangle\\ =&\sqrt{\frac{2}{d(d+1)}}[\alpha|\psi\rangle_{A}|r \rangle_{B}|r\rangle_{C}+\beta|r\rangle_{A}|\psi\rangle_{B}|r\rangle_{C}+ \gamma|r\rangle_{A}|r\rangle_{B}|\psi\rangle_{C}]\\ =&\sqrt{\frac{2}{d(d+1)}}\Big{[}\delta_{r}(\alpha +\beta+\gamma)|rrr\rangle+\sum_{j\
eq r}\delta_{j}(\alpha|jrr\rangle+\beta|rjr \rangle+\gamma|rrj\rangle)\Big{]}.\end{split}  \tag{64}\]  
As for the \(1\to 2\) quantum multicast network coding protocol, \(\Big{\{}\left|\Psi_{2}^{(r,s)}\right\rangle\Big{\}}_{r,s=0}^{d-1}\) satisfies  
\[e_{1\to 3}^{\alpha,\beta,\gamma}(|\psi\rangle\langle\psi|)=\sum_{r,s=0}^{d-1} \left|\Psi_{2}^{(r,s)}\right\rangle\Big{\langle}\Psi_{2}^{(r,s)}\Big{|}, \tag{65}\]  
where \(e_{1\to 3}^{\alpha,\beta,\gamma}\) is a \(1\to 3\) optimal asymmetric UQCM defined by Equation (12). Hence, the purpose of the remaining part of the protocol is to transfer \(\left|\Psi_{2}^{(r,s)}\right\rangle\) to the target nodes. First, we give the continuation of the proof for \(r\
eq s\) (Protocol 4). We compress the state on a \(d\)-dimensional system on step 3. The unnormalized state on system \(A\) after Step 3 can be written as  
\[\sum_{j=0}^{d-1}\kappa_{j}|j\rangle, \tag{66}\]  
where \(\{\kappa_{j}\}_{j=0}^{d-1}\) is defined as  
\[ \begin{split}&\kappa_{j}=\sqrt{\frac{d}{2d+2}}\frac{\delta_{j}}{d} \sqrt{2\alpha^{2}+2\beta^{2}+2\gamma^{2}}\qquad(j\
eq r,s),\\ &\kappa_{r}=\sqrt{\frac{d}{2d+2}}\frac{\delta_{r}}{d}\sqrt{( \alpha+\beta)^{2}+(\beta+\gamma)^{2}+(\gamma+\alpha)^{2}},\\ &\kappa_{s}=\sqrt{\frac{d}{2d+2}}\frac{\delta_{s}}{d}\sqrt{( \alpha+\beta)^{2}+(\beta+\gamma)^{2}+(\gamma+\alpha)^{2}}.\end{split}  \tag{67}\]In Step 4, the protocol of [PERSON] et al. successfully works based on the assumption for the existence of a classical linear multicast network code.