# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 1 Introduction
At the beginning, source node \(s_{i}\) has an unknown state \(\ket{\psi_{i}}=\sum_{k=1}^{d}a_{i,k}\ket{k}\), and their purpose is that for a given function \(f_{k}:\mathbb{C}^{N}\rightarrow\mathbb{C}\), after quantum communication over the quantum network, each target node reconstructs a quantum state \(\ket{\phi}\) defined by  
\[\ket{\phi}=\sum_{k=1}^{d}f_{k}(\alpha_{1,k},\cdots,\alpha_{N,k})\ket{k}. \tag{1}\]  
Figure 1: The left-hand side is a multicast classical network coding on the butterfly network, where a single source node \(s\) sends messages \(b_{0}\) and \(b_{1}\) on the finite field \(\mathbb{F}_{q}:=\mathbb{Z}/\mathbb{Z}_{q}\) to both target nodes \(t_{1}\) and \(t_{2}\), where \(q\) is a prime power. The right-hand side is a multiple-unicast classical network coding on the butterfly network, where a source node \(s_{0}\) sends message \(b_{0}\in\mathbb{F}_{q}\) to target node \(t_{0}\), and source node \(s_{1}\) sends message \(b_{1}\in\mathbb{F}_{q}\) to target node \(t_{1}\). They showed that this problem can be considered as a generalization of classical multicast communication and presented protocols using network coding to probabilistically achieve the above goal on the multiple-unicast butterfly network given in the right figure of Figure 1. Although the protocols of [PERSON] et al., [PERSON] et al., and [PERSON] et al. can be considered as generalizations of classical multicast network coding to quantum networks, rigorously speaking, the goal of their protocols is not exactly to achieve a multicast of a quantum state. Recently, [PERSON] et al. proposed a new multicast protocol using quantum network coding [57]. The purpose of the protocol is to probabilistically send an exact copy or an exact orthogonal complement of a known qubit state from a source node to multiple target nodes on a quantum network. Because the sender knows the state, this protocol does not contradict the no-cloning theorem [69]. On the other hand, this task is trivial when free classical communication is allowed. Hence, they consider the situation where each channel on the quantum network can send either one qubit or two classical bits in a single session of the protocol; that is, classical communication is restricted in their problem setting. They give an efficient protocol to achieve this goal on the multicast butterfly network given in the left figure of Figure 1 and also on the extended butterfly network. In this paper, we consider a yet different problem setting on multicast on the quantum network. As we have already mentioned, perfect multicast of an unknown state is impossible. Nevertheless, imperfect multicast of an unknown quantum state is still possible. When we restrict ourselves to imperfect multicast of a quantum state through noiseless quantum channels, the problem of multicast of an unknown quantum state reduces to a problem of cloning an unknown quantum state. This problem is called quantum approximate cloning [70; 71] and has been intensively studied both theoretically [72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85] and experimentally [86; 87; 88; 89; 90; 91]. The performance of a cloning protocol is normally measured in terms of the fidelity between an unknown input state and an output clone. When this fidelity does not depend on an input state, the protocol is called universal cloning. Among them, a protocol achieving the maximum fidelity is called optimal universal cloning [70; 71]. Here, we need to add the following remark. When \(M\) output clones \(\rho_{i}\) (\(i=1\cdots M\)) of an input unknown state, \(|\psi\rangle\) are created by a cloning protocol. We need to treat \(M\) independent fidelities \(F_{i}:=\langle\psi|\rho_{i}|\psi\rangle\) and cannot straightforwardly define an optimal protocol. One way to resolve this problem is to add a constraint that all the clones are equivalent, that is, \(\rho_{i}=\rho_{j}\) for all \(i\) and \(j\).