# 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  
In this sub-subsection, we present a protocol that multicasts \(1\to 3\) optimal asymmetric UQCs of an unknown input quantum state from the source node \(s\) to two target nodes \(t_{1}\), \(t_{2}\), and \(t_{3}\). The problem setting for the \(1\to 3\) quantum multicast protocol is almost the same as that of the \(1\to 2\) protocol given in the last subsection. Hence, we consider only the difference between these two problem settings. First, the number of target nodes is different. That is, in this subsection, a quantum network \(G\) has three target nodes \(t_{1},t_{2}\), and \(t_{3}\). The purpose of the protocol is to construct \(1\to 3\) optimal asymmetric universal clones given by Equation (12) among target nodes \(t_{1},t_{2}\), and \(t_{3}\) for a unknown \(d\)-dimensional input state \(|\psi\rangle=\sum_{j=0}^{d}\alpha_{j}|j\rangle\in\mathcal{H}_{s}\) on a source node, where \(\mathcal{H}_{s}\) is a \(d\)-dimensional input space. We again assume \(d=q^{\prime}\). In other words, we consider a mulcast quantum network code with source rate \(r\). The assumption for the existence of a classical linear multicast network code is also similar. That is, a classical linear multicast network code is a code on \(\mathbb{F}_{q}\) used to multicast from the node \(s\) to the nodes \(t_{1},t_{2},t_{3}\) on \(G^{\prime}\) with source rate \(r\). The amount of entanglement shared among the target nodes is also different. In \(1\to 3\) case, we assume that at most \(2+4\log_{2}3\) ebits are shared among the target nodes \(t_{1},t_{2}\), and \(t_{3}\). Hence, the amount of this entanglement resource is constant with respect to the dimension \(d\) of the input state. Before we present the protocol, we define the unitary operators used in the protocol. In the following definitions, a subscript of an unitary operator denotes the step of the protocol where it is used. Thus, for example, we do not define \(U_{2}\) since we will not use any unitary operation at Step 2 of the protocol. \(U_{3}^{(r,s)}\) is a tripartite unitary operator satisfying the following conditions:  
\[U_{3}^{(r,s)}\cdot\frac{\alpha|jrs\rangle+\beta|rjs\rangle+ \gamma|rsj\rangle+\alpha|jsr\rangle+\beta|jr\rangle+\gamma|srj\rangle}{\sqrt{2 \alpha^{2}+2\beta^{2}+2\gamma^{2}}}=|j00\rangle,\quad(\forall j\
eq r,s)\] \[U_{3}^{(r,s)}\cdot\frac{(\alpha+\beta)|rrs\rangle+(\beta+\gamma )|srr\rangle+(\gamma+\alpha)|rsr\rangle}{\sqrt{(\alpha+\beta)^{2}+(\beta+ \gamma)^{2}+(\gamma+\alpha)^{2}}}=|r00\rangle, \tag{43}\] \[U_{3}^{(r,s)}\cdot\frac{(\alpha+\beta)|ssr\rangle+(\beta+\gamma )|rss\rangle+(\gamma+\alpha)|sss\rangle}{\sqrt{(\alpha+\beta)^{2}+(\beta+ \gamma)^{2}+(\gamma+\alpha)^{2}}}=|s00\rangle.\]  
\(U_{5}^{(r,s)}\) is a bipartite unitary operator defined by  
\[U_{5}^{(r,s)}:=|r\rangle\langle r|\otimes I+|s\rangle\langle s| \otimes I+\sum_{j\
eq r,s}^{d-1}|j\rangle\langle j|\otimes(\sum_{x=0}^{d-1}| \pi_{jrs}(x)\rangle\langle x|), \tag{44}\]  
where \(\pi_{jrs}\) is a permutation satisfying the following conditions:  
\[\pi_{jrs}(0)=j,\quad\pi_{jrs}(1)=r,\quad\pi_{jrs}(2)=s \tag{45}\]  
\(U_{6}^{(r,s)}\) is a tripartite unitary operator defined by  
\[U_{6}^{(r,s)}:=|r\rangle\langle r|\otimes swap+|s\rangle\langle s| \otimes swap+\sum_{j\
eq r,s}|j\rangle\langle j|\otimes I\otimes I, \tag{46}\]  
where \(swap\) is a swap operator defined by  
\[\text{swap:}=\sum_{i,j=0}^{d-1}|ij\rangle\langle ii|. \tag{47}\]  
\(U_{7}^{(r,s)}\) is a bipartite unitary operator defined by  
\[U_{7}^{(r,s)}:=\sum_{i,j\
eq r,s}|i\rangle\langle i|\otimes I+|r \rangle\langle r|\otimes\sum_{j=0}^{d-1}|\pi_{rs}^{\prime\prime}(j)\rangle \langle j|+|s\rangle\langle s|\otimes\sum_{k=0}^{d-1}|\pi_{sr}^{\prime\prime} (k)\rangle\langle k|, \tag{48}\]where \(\pi^{\prime\prime}_{xy}\) is a permutation satisfying \(\pi^{\prime\prime}_{xy}(0)=x\) and \(\pi^{\prime\prime}_{xy}(1)=y\). \(U_{8}\) is a unitary operator on \(\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}\) satisfying  
\[