# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 1 Introduction
Although many studies have considered network coding on noisy classical networks in classical information theory, almost all the studies of quantum network coding consider noise-free quantum networks. This is because quantum network coding is regarded as a protocol implemented on a layer on which the errors have been already corrected. Hence, in this study, we consider noise-free quantum networks. In classical network coding, the majority of the studies have focused on multicast communication, where a single source node transmits the same information to multiple target nodes on a network [37; 38]. The left-hand side of Figure 1 shows the network coding for a the butterfly network. This is one of the simplest examples of classical multicast network coding. Another type of network coding is called multiple-unicast network coding. Here, there are \(k\) pairs of source and target nodes \((s_{0},t_{0}),\ldots,(s_{k-1},t_{k-1})\) on the network, and each source node \(s_{i}\) independently transmits a message to the corresponding target node \(t_{i}\) for all \(i\)[68]. The modified version of the butterfly network in the right-hand side of Figure 1 is one of the simplest examples of classical multiple-unicast network coding. Most of the research on quantum network coding considered multiple-unicast communication, i.e., multiple-unicast quantum network coding, where each source node transmits a quantum state (instead of a classical message) to the corresponding target node [41; 43; 44; 45; 58; 59; 60; 63; 64; 61]. The most important results are those of [PERSON] et al. If classical information (or measurement results) can be freely sent among the nodes on a quantum network, [PERSON] et al. gave a canonical procedure for constructing a quantum multiple-unicast network code from a given classical multiple-unicast network code. Here, the quantum network for the quantum code and the classical network for the classical code must be represented by the same graph [43; 46]. Unlike quantum multiple-unicast network coding, there has been less research on quantum network coding focusing on multicast communication [42; 44; 48; 49; 50; 52; 57]. This is because in quantum information theory, the no-cloning theorem prohibits perfect multicast communication [69], and, thus, it is not straightforward to construct a multicast quantum network coding protocol as an extension of a classical multicast network coding protocol. One paper by [PERSON] et al. is the first to treat quantum multicast network coding [42]. They consider the problem of distributing \(N\)-identical copies of a state \(\ket{\psi}\) from a single source node to \(N\) target nodes. Since the number of copies of \(\ket{\psi}\) is equal to the number of target nodes, \(\ket{\psi}\) can be distributed without cloning the quantum states. [PERSON] et al. showed that coding on intermediate nodes can increase the throughput of the quantum network. The second work treating this topic is one paper by [PERSON] et al. [44]. In this paper, a single copy of a state \(\ket{\psi}=\sum_{i=1}^{d}\alpha_{i}\ket{i}\) is given on the source node and the aim is to share a Greenberger-Horne-Zeilinger (GHZ)-type state \(\sum_{i=1}^{d}\alpha_{i}\ket{i}_{1}\otimes\cdots\otimes\ket{i}_{N}\) among target nodes, where the \(i\)th local system is on the \(i\)th target node. From this GHZ-type state shared among the target nodes, the input state \(\ket{\Psi}\) can be reconstructed at any target node by local operations and classical communication (LOCC). Based on classical multicast network coding, [PERSON] et al. developed a quantum protocol to achieve the above task under the assumption of free classical communication among nodes on the quantum network. The third work treating this topic is one paper by [PERSON] et al. [52]. They proposed a communication protocol on a quantum network called quantum cooperative multicast. In their problem setting, there are multiple source nodes \(s_{1},\cdots,s_{N}\) and target nodes \(t_{1},\cdots,t_{M}\) in a given quantum network.