# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 2 Materials and Methods
### Optimal Asymmetric Quantum Universal Cloning Machine
This protocol uses three systems: \(A\), \(B\), and \(M\), whose Hilbert spaces are \(\mathcal{H}_{A}\), \(\mathcal{H}_{B}\), and \(\mathcal{H}_{M}\), respectively. Here, \(\mathcal{H}_{A}\) works as an input system and the first output system, \(\mathcal{H}_{B}\) is the second output system, and \(\mathcal{H}_{M}\) is an ancilla system. The dimensions of all three systems are the same, and we denote this dimension as \(d\); that is, \(\mathrm{dim}\ \mathcal{H}_{A}=\mathrm{dim}\ \mathcal{H}_{B}=\mathrm{dim}\ \mathcal{H}_{M}=:d\). Then, for an input state \(\ket{\psi}\) on system \(A\), a \(1\to 2\) optimal asymmetric UQCM is given by an isometry \(U_{1\to 2}^{(a,b)}\) from \(\mathcal{H}_{A}\) to \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{M}\) satisfying [78]:  
\[U_{1\to 2}^{(a,b)}\ket{\psi}_{A}=a\ket{\psi}_{A}\ket{\Phi_{A}^{+}}_{BM}+b\ket{ \psi}_{B}\ket{\Phi_{A}^{+}}_{AM}. \tag{4}\]  
where \(\ket{\Phi_{d}^{+}}\) is a standard \(d\)-dimensional maximally entangled state:  
\[\ket{\Phi_{d}^{+}}:=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\ket{k}\ket{k}, \tag{5}\]  
and \(a\) and \(b\) are positive real parameters satisfying  
\[a^{2}+b^{2}+\frac{2 ab}{d}=1. \tag{6}\]  
Note that in order to simplify the formulas, the parameters \(a,b\) are often used instead of the weight \(\{w_{i}\}_{i=1}^{2}\) in the definition of the \(1\to 2\) optimal asymmetric UQCM. Using \(U_{1\to 2}^{(a,b)}\) defined above, the optimal asymmetric UQCM \(\varepsilon_{1\to 2}^{(a,b)}\) is  
\[\varepsilon_{1\to 2}^{(a,k)}(\ket{\psi}\bra{\psi}):=\mathrm{Tr}_{M}\Big{(}U_{1 \to 2}\ket{\psi}\bra{\psi}_{A}U_{1\to 2}^{\dagger}\Big{)}. \tag{7}\]The reduced density matrices of the output states can be written as  
\[ \begin{split}\rho_{A}&:=\operatorname{Tr}_{B}\varepsilon _{1\to 2}^{(a,b)}(|\psi\rangle\langle\psi|)=(1-b^{2})|\psi\rangle\langle\psi|+b^{2} \frac{I}{d}\\ \rho_{B}&:=\operatorname{Tr}_{A}\varepsilon_{1\to 2}^{(a,b)}(|\psi \rangle\langle\psi|)=(1-a^{2})|\psi\rangle\langle\psi|+a^{2}\frac{I}{d}.\end{split}  \tag{8}\]  
Thus, the fidelity of the reduced density matrices, which have been proved to be optimum [78], is given by  
\[ \begin{split} F_{A}&:=\langle\psi|\operatorname{ Tr}_{B}\Bigl{(}\varepsilon_{1\to 2}^{(a,b)}(|\psi\rangle\langle\psi|)\Bigr{)}|\psi \rangle=1-b^{2}\frac{d-1}{d},\\ F_{B}&:=\langle\psi|\operatorname{Tr}_{A}\Bigl{(} \varepsilon_{1\to 2}^{(a,b)}(|\psi\rangle\langle\psi|)\Bigr{)}|\psi \rangle=1-a^{2}\frac{d-1}{d},\end{split}  \tag{9}\]  
where \(I\) is an identity operator on a \(d\)-dimensional system. The well-known formula \(F_{A}=F_{B}=\frac{d+3}{2(d+1)}\) of the fidelity of an optimal symmetric UQCM is derived from the above equations by substituting \(a=b\)[71]. Next, we give an optimal asymmetric UQCM with \(M=1\) and \(N=3\) (we call this protocol the \(1\to 3\) optimal asymmetric UQCM). This protocol use five systems \(A\), \(B\), \(C\), \(R\), and \(S\) whose Hilbert spaces are \(\mathcal{H}_{A}\), \(\mathcal{H}_{B}\), \(\mathcal{H}_{C}\), \(\mathcal{H}_{R}\), and \(\mathcal{H}_{S}\), respectively. Here, \(\mathcal{H}_{A}\) is an input system that is also the first output system. \(\mathcal{H}_{B}\) and \(\mathcal{H}_{C}\) are the second and third output systems, respectively. \(\mathcal{H}_{R}\) and \(\mathcal{H}_{S}\) are ancilla systems. The dimensions of all systems are the same, which we denote as \(d\). For an input state \(|\psi\rangle\) on system \(A\), \(1\to 3\) optimal asymmetric UQCM is given by an isometry \(U_{1\to 3}^{(\alpha,\beta,\gamma)}\) from \(\mathcal{H}_{A}\) to \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}\otimes\mathcal{H }_{R}\otimes\mathcal{H}_{S}\) satisfying the following equation:  
\[