# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 2 Materials and Methods
### Classical Multicast Network Coding  
Since our protocol uses the protocol of Kobayashi et al. as a subroutine and since the protocol of Kobayashi et al. is based on a classical linear multicast network code, we introduce classical linear multicast network coding in this subsection. A detailed description of classical multicast network coding can be found in standard textbooks of network coding [37; 38]. A classical network is represented by a directed graph \(G^{\prime}=(V,E^{\prime})\), where a vertex \(v\in V\) represents a node of the network and an edge \(e\in E^{\prime}\) represents a noiseless classical channel. In this paper, we assume that \(G^{\prime}\) is _acyclic_. There exist a source node \(s\in V\) and \(N\) target nodes \(t_{1},\ldots,t_{N}\in V\) on the network. A node that is neither a source node nor a target node is called an intermediate node. In a single session of a classical multicast network coding, an alphabet on the finite field \(\mathbb{F}_{q}\) is sent from node \(u\) to node \(v\) if \((u,v)\in E^{\prime}\), where the order of \(\mathbb{F}_{q}\) is a prime power \(q\). Since \(G^{\prime}\) is an acyclic directed graph, a natural partial ordering can be defined on \(E^{\prime}\). This partial ordering is generated by the condition that \((u,v),(v,w)\in E^{\prime}\Longrightarrow(u,v)\prec(v,w)\). The order of transmissions of classical information can be determined by this partial ordering. That is, an edge \(e\in E^{\prime}\) transmits an alphabet after all edges \(e^{\prime}\in E^{\prime}\) satisfying \(e^{\prime}\prec e\) have transmitted alphabets. We assume that there is no incoming edge to the source node \(s\) and that there is no outgoing edge from any target node. Hence, all edges whose tail node is the source node \(s\) are a local minimum, and all edges whose head node is a target node are a local maximum under the partial ordering. We further assume that all edges whose tail node is not the source node \(s\) are not a local minimum and that all edges whose head node is not a target node are not a local maximum. This is because the edges that do not satisfy these conditions are useless for our purpose. A classical linear multicast network code over \(\mathbb{F}_{q}\) on \(G^{\prime}\) consists of a set of linear maps \(\{f_{e}\}_{e\in E^{\prime}}\). At the beginning of a session, an input message \(\vec{x}:=(x_{1},\ldots,x_{r})\in\mathbb{F}_{q}^{r}\) is chosen on the source node \(s\), where \(r\) is the source rate of the classical multicast network code. Suppose \(e\) is an outgoing edge of \(v\). At the first step of the network coding, an alphabet \(y_{e}\) transmitted through the edge \(e\) is chosen as a linear combination of \(x_{1},\ldots,x_{r}\). In other words, in terms of a linear function \(f_{e}:\mathbb{F}_{q}^{r}\rightarrow\mathbb{F}_{q}\), \(y_{e}\) can be written as  
\[y_{e}:=f_{e}(\vec{x})=f_{e}(x_{1},\cdots x_{r}). \tag{14}\]  
After calculating \(y_{e}\), \(y_{e}\) is transmitted through \(e\). After all edges outgoing from the source node \(s\) transmitted an alphabet, all intermediate nodes transmit alphabet in the order determined by the partial ordering as follows: Suppose an intermediate node \(v\) on the network has \(m\) incoming edges and \(e\) is an outgoing edge from \(v\). After all transmissions of \(m\) incoming edges to \(v\) have finished, the node \(v\) has \(m\)-alphabets \(y_{j}\in\mathbb{F}_{q}\) (\(j=1,\ldots,m\)), where \(y_{j}\) is an alphabet sent through the \(j\)th incoming edge. Then, an alphabet \(y_{e}\) transmitted through the edge \(e\) is chosen as a linear combination of \(y_{1},\ldots,y_{m}\). In other words, there exists a linear function \(f_{e}:\mathbb{F}_{q}^{m}\rightarrow\mathbb{F}_{d}\) such that  
\[y_{e}:=f_{e}(y_{1},\cdots y_{m}). \tag{15}\]  
After the calculation, \(y_{e}\) is transmitted through \(e\). Suppose a target node \(t_{i}\) has \(m_{i}\) incoming edges.