# 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  
In this sub-subsection, we present the protocol for multicasting \(1\to 2\) asymmetric optimal UQCs of an input quantum state from the source node \(s\) to two target nodes \(t_{1}\) and \(t_{2}\). As we have explained in Section 2.3, the problem settings for the protocol of [PERSON] et al. and our protocol are essentially the same, and only their purposes are different. Here, we summarize the problem setting of our quantum multicast network coding: A quantum network is described by an _undirected_ graph \(G=(V,E)\). There exist a source node \(s\in V\) and \(N\) target nodes \(t_{1},\ldots,t_{N}\in V\) on the network. In this subsection, since we consider multicasting \(1\to 2\) asymmetric UQCs, we set \(N=2\). In a single session, any quantum channel \((u,v)\in E\) can send a \(q\)-dimensional quantum system \(\mathcal{H}_{e}\) just once, either from \(u\) to \(v\) or from \(v\) to \(u\), where \(q\) is assumed to be a prime power. Furthermore, any quantum operations can be implemented on any node \(v\in V\), and measurement outcomes can be freely sent among nodes. At the beginning of a session, a single copy of input state \(|\psi\rangle\) is given on the source node \(s\). Under these problem settings, the purpose of our protocol is to construct optimal asymmetric universal clones given by Equation (7) between target nodes \(t_{1}\) and \(t_{2}\) for a given \(d\)-dimensional unknown 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. In other words, we consider multicast communication protocols that are optimal in terms of fidelity. Thus, our research problem is to construct a protocol with a smaller communication cost under the above condition. We assume \(d=q^{r}\). In other words, we consider multicast quantum network coding with source rate \(r\). Here, note that since we assumed \(q\) is a prime power, \(d\) is also a prime power. For this purpose, we use two additional assumptions: The first assumption is the same assumption that [PERSON] et al. used. That is, we assume that there exists a solvable classical linear multicast network code \(\{f_{e}\}_{e\in E^{\prime}}\) with source rate \(r\) on an acyclic directed graph \(G^{\prime}=(V,E^{\prime})\) over a finite field \(\mathbb{F}_{q}\), where \(G\) is an undirected underlying graph of \(G^{\prime}\). Hence, we can use the quantum multicast network coding protocol of [PERSON] et al. with source rate \(r\) on this quantum network \(G\). We further assume that at most 2 ebits of entanglement resource are shared between target node \(t_{1}\) and \(t_{2}\). Hence, the amount of this entanglement resource is constant with respect to the dimension \(d\) of the input state and is negligible for large \(d\) in comparison to \(d\). Before we present the protocol, we define the unitary operators used in it. Pauli operators \(X_{d}\) and \(Z_{d}\) are defined as  
\[X_{d}:=\sum_{k=0}^{d-1}|k\oplus 1\rangle\langle k|,\quad Z_{d}:=\sum_{k=0}^{d-1 }\omega^{k}|k\rangle\langle k|, \tag{20}\]  
where \(\omega:=e^{\frac{2\pi i}{d}}\). In the following part of the paper, unitary operators defined on \(\mathbb{C}^{d}\otimes\mathbb{C}^{d}\) and \(\mathbb{C}^{d}\otimes\mathbb{C}^{d}\) are called bipartite and tripartite unitary operators, respectively.