# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 2 Materials and Methods
### Quantum Multicast Network Coding  
In this subsection, we review the protocol of [PERSON] et al. [44]. First, we give a problem setting for multicast quantum network coding that is common between our protocol and the protocol of [PERSON] et al. A quantum network is described by an _undirected_ graph \(G=(V,E)\), where \(V\) represents a set of nodes and \(E\) represents a set of quantum channels. There exist a source node \(s\in V\) and \(N\) target nodes \(t_{1},\ldots,t_{N}\in V\) on the network. 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 (or classical information) 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 \(s\). Here, the reason a quantum channel is represented by an undirected edge is that the direction of a quantum channel can be effectively reversed by quantum teleportation under the condition of free classical communication [45]. The purpose of both protocols is to multicast an input state \(|\psi\rangle\) from the source node to all target nodes in a single session. Here, we should note that the meaning of \"multicast\" in the protocol of [PERSON] et al. is different from that in our protocol. As we have explained in the introduction, the purpose of our protocol is to construct optimal asymmetric universal clones among target nodes for a given \(q^{\prime}\)-dimensional input state \(|\psi\rangle=\sum_{j=0}^{q^{\prime}-1}\alpha_{j}|j\rangle\in\mathcal{H}_{s}\) on a source node, where \(\mathcal{H}_{s}\) is a \(q^{\prime}\)-dimensional input space. In other words, we consider multicast quantum network coding with source rate \(r\). On the other hand, the purpose of the protocol of [PERSON] et al. is to construct a GHZ-type state \(\sum_{j=0}^{q^{\prime}-1}\alpha_{j}|j\rangle_{1}\otimes\cdots\otimes|j\rangle_{N}\) among target nodes, where the \(i\)th local system is on the \(i\)th target node. Both the protocol of [PERSON] et al. and our protocol are constructed under the assumption 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 \(G^{\prime}\). In other words, \(G\) can be derived by replacing all directed edges on \(G^{\prime}\) by undirected edges. Using this replacement, a directed edge \(e^{\prime}\in E^{\prime}\) is naturally mapped to an undirected \(e\in E\), and this map is a bijection. Hence, in the following part of this subsection, we will not distinguish \(e^{\prime}\) from \(e\) and will write \(e^{\prime}\) as \(e\). The protocol of [PERSON] et al. imitates a classical linear multicast network code \(\{f_{e}\}_{e\in E^{\prime}}\) and corresponding decoding operations \(\{g_{i}\}_{i=1}^{N}\) by unitary operators. Each linear map \(f_{e}\) is imitated by a unitary operator \(U_{e}\), and each recovery operator \(g_{i}\) is imitated by a unitary operator \(V_{i}\), where \(U_{e}\) and \(V_{i}\) are defined as follows: Since \(\dim\mathcal{H}_{s}=q^{\prime}\), due to the bijection between \(\{0,1\ldots q^{\prime}-1\}\) and \(\mathbb{F}_{q}^{\prime}\), an input state \(|\psi\rangle\in\mathcal{H}_{s}\) can be written as \(|\psi\rangle=\sum_{\vec{x}\in\mathbb{F}_{q}^{\prime}}\alpha_{\vec{x}}|\vec{x}\rangle\).