# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 1 Introduction
This immediately leads \(F_{i}=F_{j}\) for all \(i\) and \(j\), and the optimal universal cloning satisfying this condition is called symmetric optimal universal quantum cloning. On the other hand, when we do not use the constraint of symmetry, we need to define a weight \(w_{i}\) satisfying \(w_{i}\geq 0\) and \(\sum_{i}w_{i}=1\) and a weighted mean fidelity \(\overline{F}:=\sum_{i}w_{i}F_{i}\). The universal cloning which is optimal in terms of \(\overline{F}\) is called asymmetric optimal universal quantum cloning. By the definition, symmetric optimal universal cloning is a special case of asymmetric optimal universal quantum cloning such that the weight \(w_{i}=1/M\) for all \(i\). In the viewpoint of multiparty communication, optimal universal quantum cloning is nothing but a multicast quantum channel that is optimal in terms of fidelity. Therefore, when we consider a quantum network communication, sending an optimal universal quantum clone (UQC) from a single source node to multiple target nodes is a multicast network communication that is optimal in terms of fidelity. Furthermore, this problem may be considered one of the most natural quantum extensions of classical multicast network coding. Hence, our research problem to study in which condition this communication task is achievable is information-theoretically important. Furthermore, it is known that quantum approximate cloning is useful for several information processing tasks including eavesdropping on quantum key distribution protocols [92; 93; 94; 95; 96], broadcasting quantum coherence [97], and multiple-unicast quantum network coding over the butterfly network without classical communication [40]. This fact strongly suggests that our research problem is practically important as well. Based on this idea, [PERSON] et al. constructed a protocol to exactly share a symmetric optimal UQC of an input state on the target nodes under the conditions that classical information can be sent freely among nodes on a quantum network and that a small amount of entanglement is shared on target nodes at the beginning of the protocol [49; 50]. They also gave a protocol to approximately share a symmetric optimal UQC without entanglement shared among target nodes [48]. We further note that although the references [48; 49; 50] are written in Japanese, their English version is now planned to be written [98]. In this paper, we focus on extending [PERSON] et al. 's results [49; 50] to asymmetric optimal universal quantum cloning [75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85], which is a generalization of symmetric optimal universal quantum cloning. Thus, we construct a protocol to efficiently and _exactly_ multicast an asymmetric optimal clone of a \(q^{\prime}\)-dimensional input quantum state from one source node to two (three) target nodes, _where \(q\) is assumed to be a prime power_. Our problem setting is given by the following four assumptions, which are almost in common with those used in [49; 50]:  
* The noise-free quantum network can be described by an undirected graph \(G\) with one source node and two (three) target nodes. * Each quantum channel on the quantum network can transmit one \(q\)-dimensional quantum system in a single session. * Measurement results (or classical information) can be sent freely from one node to another node on the quantum network. * A small amount of entanglement that does not scale with \(q\), is shared among the target nodes. The amount of entanglement is at most \(2\,\mathrm{ebit}\) for two target nodes, and at most \((2+4\log_{2}3)\) ebit for the case of three target nodes. Under these assumptions, we prove that multicasting of \(1\to 2\) (\(1\to 3\)) asymmetric optimal UQCs of a \(q^{\prime}\)-dimensional state is possible, if there exists a classical solvable linear multicast network code with source rate \(r\) for a noise-free classical network described by an acyclic directed graph \(G^{\prime}\), where \(G\) is an undirected underlying graph of \(G^{\prime}\).