# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 2 Materials and Methods
### Optimal Asymmetric Quantum Universal Cloning Machine  
The no-cloning theorem states that quantum mechanics prohibits a quantum operation that makes perfect copies of an unknown quantum state [69]. That is, there is no quantum channel (or a completely positive and trace preserving map) satisfying \(\epsilon(\rho)=\rho\otimes\rho\) for all pure states \(\rho\) on a Hilbert space \(\mathcal{H}\), where \(\mathfrak{B}(\mathcal{H})\) is a space of all linear operators on the Hilbert space \(\mathcal{H}\). This immediately leads to the impossibility of a perfect multicast of an unknown quantum state. On the other hand, quantum mechanics does not completely prohibit approximate cloning of a quantum state. Thus, there may exist a quantum channel \(\epsilon:\mathfrak{B}(\mathcal{H})\to\mathfrak{B}(\mathcal{H}^{\otimes 2})\) such that \(\operatorname{Tr}_{2}\epsilon(\rho)\) and \(\operatorname{Tr}_{1}\epsilon(\rho)\) are close to \(\rho\) for  
Figure 2: Schematic diagram of a protocol for multicasting asymmetric optimal UQCs from one source node to two target nodes. The asymmetric optimal cloning protocol for the input state \(|\psi\rangle\) is implemented at the source node. The resulting state is compressed into a \(d\)-dimensional system and transmitted to the two target nodes using the quantum multicast network coding protocol of [PERSON] et al. [44]. Finally, the asymmetric optimal clones of the input state are reconstructed by LOCC on target nodes with the help of a small amount of entanglement. an arbitrary pure state \(\rho\), where \(\mathrm{Tr}_{i}\) is the partial trace of the \(i\)th subsystem. Hence, many studies have focused on quantum protocols to make an approximate copy of unknown states (so-called quantum cloning machines) [72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85]. In general, a quantum cloning machine (QCM) that produces \(N\) approximate clones based on \(M\) copies of a given quantum state \(\ket{\psi}\in\mathcal{H}\) is a quantum channel \(\varepsilon\) from \(\mathfrak{B}\left(\mathcal{H}^{\otimes M}\right)\) to \(\mathfrak{B}\left(\mathcal{H}^{\otimes N}\right)\). Suppose that \(\rho_{i}\) is a reduced density matrix of the output state on the \(i\)th subsystem: \(\rho_{i}=\mathrm{Tr}_{-i}\varepsilon\left(\left(\ket{\psi}\bra{\psi}\right)^{ \otimes M}\right)\), where \(\mathrm{Tr}_{-i}\) is a partial trace of all subsystems except the \(i\)th subsystem. Since the purpose of a QCM is to make \(\rho_{i}\) as closed as the input state \(\ket{\psi}\bra{\psi}\), the performance of a QCM can be described by the output fidelity \(F_{i}\) between \(\rho_{i}\) and \(\ket{\psi}\bra{\psi}\):  
\[F_{i}=\bra{\psi}\rho_{i}\ket{\psi},\ \ (i=1,...,M). \tag{2}\]  
A QCM is called universal if \(F_{i}\) does not depend on the input state \(\ket{\psi}\). Furthermore, a universal QCM (UQCM) is called symmetric if the all clones are the same: \(\rho_{i}=\rho_{j}\) for all \(i\) and \(j\). A UQCM that is not symmetric is asymmetric. Since the output states of an asymmetric UQCM satisfy \(\rho_{i}\
eq\rho_{j}\), the output fidelity \(F_{i}\) also depends on \(i\). Thus, to discuss the optimality of an asymmetric UQCM, we need to define a weight \(\{w_{i}\}_{i=1}^{M}\) satisfying \(w_{i}\geq 0\) and \(\sum_{i}^{M}w_{i}=1\). Then, the mean output fidelity \(\overline{F}\) of an asymmetric UQCM \(\varepsilon\) with respect to the weight \(\{w_{i}\}_{i=1}^{M}\) is defined by  
\[\overline{F}:=\sum_{i}^{M}w_{i}F_{i}. \tag{3}\]  
For a given weight \(\{w_{i}\}_{i=1}^{M}\), an asymmetric UQCM that attains the optimum of the mean output fidelity \(\overline{F}\) is called an asymmetric optimal UQCM w.r.t \(\{w_{i}\}\). For existing results on asymmetric optimal UQCM, we refer to [71; 84]. Here, we only give known facts on asymmetric optimal UQCM that are necessary for our research. First, we give an optimal asymmetric UQCM with \(M=1\) and \(N=2\) (we call this protocol a \(1\to 2\) optimal asymmetric UQCM).