# 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
\(\mathrm{Y}^{(r)}\) is defined as a bipartite unitary operator satisfying  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\] \[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]  
for all \(j\in\{0,\ldots,d-1\}\) satisfying \(j\
eq r\), and  
\[\mathrm{Y}^{(r)}|ij\rangle=|ij\rangle \tag{22}\]  
for all \(i,j\in\{0,\ldots,d-1\}\) satisfying \(i,j\
eq r\), where \(\eta\) is defined by  
\[\cos\eta=\frac{a}{\sqrt{1-\frac{2 ab}{d}}}\quad\text{and}\quad\sin\eta=\frac{b} {\sqrt{1-\frac{2 ab}{d}}}. \tag{23}\]The bipartite unitary operator \(V^{(r)}\) is defined by  
\[V^{(r)}:=\sum_{j\
eq r}|j\rangle\langle j|\otimes U_{r,j}+|r\rangle \langle r|\otimes I \tag{24}\]  
where the unitary operator \(U_{r,j}\) is defined by  
\[U_{r,j}=I-|j\rangle\langle j|-|r-1\rangle\langle r-1|+|j\rangle \langle r-1|+|r-1\rangle\langle j|.\]  
The bipartite unitary operator \(\Delta^{(r)}\) is defined by  
\[\Delta^{(r)}:=|r\rangle\langle r|\otimes X_{d}^{-(r-1)}+\sum_{j \
eq r}|j\rangle\langle j|\otimes I. \tag{25}\]  
The unitary operator \(\Gamma^{(r)}\) on \(\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{2}\) is defined by  
\[\Gamma^{(r)}:=|r\rangle\langle r|\otimes\text{swap}+\sum_{j\
eq r }|j\rangle\langle j|\otimes I, \tag{26}\]  
where swap is a unitary operator on \(\mathbb{C}^{d}\otimes\mathbb{C}^{2}\) defined by  
\[\text{swap}:=\sum_{i=2}^{d-1}\sum_{j=0,1}|ij\rangle\langle ij|+ \sum_{i,j=0,1}|ij\rangle\langle ji|.\]  
The unitary operator \(\Theta\) on \(\mathbb{C}^{2}\otimes\mathbb{C}^{2}\) is defined by  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]0 \[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]1 \[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]2  
The bipartite unitary operator \(\Lambda^{(r)}\) is defined by  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]3  
Before starting the protocol, we prepare three \(d\)-dimensional systems \(A\), \(B\), and \(M\) at the source node \(s\), \(d\)-dimensional systems \(C\), \(E\), and \(2\)-dimensional systems \(G\), \(T_{1}\) at the target node \(t_{1}\). Similarly, we prepare \(d\)-dimensional systems \(D\), \(F\), and \(2\)-dimensional systems \(H\), \(T_{2}\) at \(t_{2}\). The entanglement resource \(\cos\eta|0\rangle_{E}|1\rangle_{F}+\sin\eta|1\rangle_{E}|0\rangle_{F}\) is shared between \(E\) and \(F\), and the Bell state \(\frac{1}{\sqrt{2}}(|00\rangle_{T_{1}T_{2}}+|11\rangle_{T_{1}T_{2}})\) is shared between \(T_{1}\) and \(T_{2}\). Thus, the amount of entanglement resources is at most \(2\) ebits. The protocol for \(1\to 2\) is shown as Protocol 2. In Protocol 2, 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 an ancilla, the whole state is splitted into classical information (measurement result), which is sent to target nodes, and a bipartite quantum state at Step 2. The bipartite quantum state is compressed into a \(q^{\prime}\)-dimensional state at Step 3. The state is distributed to the target nodes \(t_{1}\) and \(t_{2}\) 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 \(t_{1}\) and \(t_{2}\) reconstruct asymmetric UQCs of \(|\psi\rangle\) by LOCC and preshared small entanglement resource; this process is completed from Step 5 to Step 12. As a result of the protocol, \(1\to 2\) asymmetric UQCs given by Equation (7) are created in systems \(EF\), where \(E\) and \(F\) are on the target nodes \(t_{1}\) and \(t_{2}\), respectively. Note that as we explained in the previous subsection, optimal asymmetric UQCs depend on the parameters \(a\) and \(b\) in Equation (4).