# 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
Hence, the amount of entanglement resources are \(2+4\log_{2}3\) ebits in total. The beginning of the protocol for \(1\to 3\) is given in Protocol 3. In Protocol 3, first, asymmetric UQCs of an \(q^{\prime}\)-dimensional input state \(|\psi\rangle\) is created on the source node \(s\) at Step 1. Then, by measuring ancillary systems \(R\) and \(S\), the whole state is splitted into classical information (measurement results \(r\) and \(s\)), which is sent to target nodes and a tripartite quantum state at Step 2. The continuation of the protocol branches depending on whether \(r\
eq s\) or \(r=s\). The continuation for \(r\
eq s\) is given in Protocol 4, and for \(r=s\) is given in Protocol 5. In both Protocols 4 and 5, the tripartite quantum state is compressed into a \(q^{\prime}\)-dimensional state at Step 3. The state is distributed to the target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\) by the protocol of [PERSON] et al. at Step 4. From the GHZ-type state received in the previous step and the classical information received at Step 2, target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\) reconstruct asymmetric UQCs of \(|\psi\rangle\) by LOCC and preshared small entanglement resource; this process is completed in the remaining part of the protocols. As the result of the protocol, \(1\to 3\) asymmetric UQCs given by Equation (12) are created system \(M_{1}M_{2}M_{3}\), where \(M_{1}\), \(M_{2}\), and \(M_{3}\) are on the target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\), respectively. Note that as we explained in the previous subsection, asymmetric UQCs depends on the parameters \(\alpha\), \(\beta\), and \(\gamma\) in Equation (10). We can set these parameters in step 1 of the protocol, when we apply \(U_{1\to 3}^{(\alpha,\beta,\gamma)}\). **Protocol 3**\(1\to 3\) quantum multicast network coding protocol (beginning)  
**Step 1:** At the beginning, the source node \(s\) has an unknown input quantum state \(|\psi\rangle_{A}\) on system \(A\), and makes \(1\to 3\) asymmetric universal clones by applying an isometry \(U_{1\to 3}^{(\alpha,\beta,\gamma)}\) defined by Equation (10) from system \(A\) to system \(ABCRS\). **Step 2:** The source node \(s\) measures the systems \(R\) and \(S\) in the computational basis, where the measurement outcomes of \(R\) and \(S\) are called \(r\) and \(s\), respectively. The source node \(s\) sends the measurement outcomes \(r\) and \(s\) to the target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\). The following steps of the protocol depend on whether \(r\
eq s\) or \(r=s\). **Protocol 4** Continuation of Protocol 3 for \(1\to 3\) quantum multicast network coding (for \(r\
eq s\))  
**Step 3:** The source node \(s\) applies unitary operator \(U_{2}^{(r,s)}\) defined by Equation (43) to system \(ABC\), and then, discards systems \(B\) and \(C\). **Step 4:** The state on system \(A\) is multicast to the target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\) over the quantum network \(G\) using the protocol of [PERSON] et al. The target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\) put the output of the protocol of [PERSON] et al. on system \(DEF\). Then, using \(2\log_{2}3\) ebits of entanglement, the targets nodes share the following state on system \(M_{1}M_{2}M_{3}\):  
\[\big{(}\alpha_{1}^{\prime}(|012\rangle+|021\rangle)+\beta_{1}^{\prime}(|102 \rangle+|201\rangle)+\gamma_{1}^{\prime}(|120\rangle+|210\rangle)\big{)}_{M_{1 },M_{2},M_{3}},\]  
where \(\alpha_{1}^{\prime}\), \(\beta_{1}^{\prime}\), and \(\gamma_{1}^{\prime}\) are defined by Equation (50). Furthermore, by using \(2\) ebits of entanglement, the target nodes share the following state on system \(N_{1}N_{2}N_{3}\):  
\[(\alpha_{1}^{\prime\prime}|001\rangle+\beta_{1}^{\prime\prime}|100\rangle+ \gamma_{1}^{\prime\prime}|010\rangle)_{N_{1},N_{2},N_{3}}, \tag{60}\]  
where \(\alpha_{1}^{\prime\prime}\), \(\beta_{1}^{\prime\prime}\) and \(\gamma_{1}^{\prime\prime}\) are defined by Equation (50).