# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 3 Results
### \(1\to 2\) Asymmetric UQC Multicast Protocol
#### 3.1.1 \(1\to 2\) Quantum Multicast Protocol
We can set these parameters in Step 2 of the protocol, when we apply \(U_{1\to 2}^{(a,b)}\). As we already explained,  
**Step 1:** At the beginning, the source node \(s\) has an unknown input quantum state \(|\psi\rangle_{A}\) on system \(A\) and makes \(1\to 2\) asymmetric universal clones by applying an isometry \(U_{1\to 2}^{(a,b)}\) defined by Equation (4) from the system \(A\) to the system \(ABM\). **Step 2:** The source node \(s\) measures system \(M\) in the computational basis and sends the measurement outcome \(r\) to the two target nodes \(t_{1}\) and \(t_{2}\). **Step 3:** The source node \(s\) applies the unitary operator \(Y_{AB}\) defined by Equations (21) and (22) to the systems \(AB\), then discards the system \(B\). **Step 4:** The state on system \(A\) is multicast to the target nodes \(t_{1}\) and \(t_{2}\) over the quantum network \(G\) using the protocol of [PERSON] et al. The target nodes \(t_{1}\) and \(t_{2}\) put the output GHZ-type state of the protocol of [PERSON] et al. on system \(CD\). **Step 5:** The target nodes \(t_{1}\) and \(t_{2}\) apply \(X_{d}^{r-1}\otimes X_{d}^{r-1}\) to system \(EF\) using the measurement outcome \(r\) sent from the source node \(s\). **Step 6:** The target node \(t_{1}\) applies \(V_{C,E}^{(r)}\) defined by Equation (24) to system \(CE\), and the target node \(t_{2}\) applies \(V_{D,F}^{(r)}\) to system \(DF\). Then, the target node \(t_{1}\) applies \(\Delta_{C,E}^{(r)}\) defined by Equation (25) to system \(CE\), and the target node \(t_{2}\) applies \(\Delta_{D,F}^{(r)}\) to system \(DF\). **Step 7:** The target node \(t_{1}\) initializes \(G\) in \(|0\rangle\) and applies \(\Gamma_{C,E,G}^{(r)}\) defined by Equation (26) on system \(CE\). The target node \(t_{2}\) initializes \(H\) in \(|0\rangle\) and applies \(\Gamma_{C,E,G}^{(r)}\) to system \(DFH\). **Step 8:** The target node \(t_{2}\) sends the state on system \(H\) to system \(T_{1}\) at the target node \(t_{1}\) using the Bell state on system \(T_{1}T_{2}\) by the quantum teleportation. **Step 9:** The target node \(t_{1}\) applies \(\Theta_{GT_{1}}\) defined by Equation (27) to systems \(G\) and \(T_{1}\) and discards \(T_{1}\). **Step 10:** The target node \(t_{1}\) measures system \(G\) in  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]4  
and derives the measurement outcome \(k\). Then, \(t_{1}\) performs \((I-2|r\rangle\langle r|)^{k}\) on system \(C\). **Step 11:** The target node \(t_{1}\) applies \(\Lambda_{CE}^{(r)}\) defined by Equation (28) on the system \(CE\), and the target node \(t_{2}\) applies \(\Lambda_{DF}^{(r)}\) the system \(DF\). **Step 12:** The target nodes \(t_{1}\) and \(t_{2}\) measure system \(C\) and \(D\) in the Fourier basis  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]5  
respectively, and derive the measurement outcomes \(p_{1}\) and \(p_{2}\), respectively. Then, \(t_{1}\) applies \(Z_{d}^{p_{1}+p_{2}}\) to system \(E\), and \(t_{2}\) applies \(Z_{d}^{p_{1}+p_{2}}\) to system \(F\). **3.1.2. ** Proof of \(1\to 2\) Quantum Multicast Protocol  
In this sub-subsection, we present the proof that Protocol 2 creates \(1\to 2\) asymmetric UQCs given by Equation (7) in system \(EF\) shared by target nodes \(t_{1}\) and \(t_{2}\). Proof.