# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 4 Discussion
### Possibility of the Extension of Our Protocol for More than \(3\) Receivers  
In this subsection, we consider a possible extension of our protocol for more than \(3\) receivers. As a result, we suggest that there might not exist a straightforward extension of the protocol for an arbitrary number of terminal nodes using existing asymmetric optimal UQC schemes. In other words, we estimate that if we use an existing scheme, in order to multicast a \(d\)-dimensional unknown state, target nodes need to share \(O(\log d)\) ebit of entanglement. At first, we note that there are not so many existing works about asymmetric optimal UQCs outputting more than \(3\) copies. That is, [PERSON] et al. and [PERSON] et al. studied \(1\to 4\) asymmetric optimal UQCs [82, 83], and [PERSON] et al. studied \(1\to N\) asymmetric optimal UQCs [81, 84]. Since only existing \(1\to N\) asymmetric optimal UQC scheme is one given by [PERSON] et al., we focus on applying their scheme for multicast communication on quantum networks in this subsection. Suppose \(\mathcal{H}_{I}\) and \(\mathcal{H}_{O}\) are a \(d\)-dimensional input space and a \(d^{N}\) dimensional output space, respectively, and \(N\) satisfies \(N\geq 3\). Then, the \(1\to N\) asymmetric optimal UQC of [PERSON] et al. [84] is given by the following isometry \(U_{I\to O}\) from \(\mathcal{H}_{I}\) to \(\mathcal{H}_{O}\):  
\[U_{I\to O}:=\sum_{i,j=0}^{d-1}\beta_{1}|ji\cdots ii\rangle\langle j|+\beta_{2 }|ij\cdots ii\rangle\langle j|+\cdots+\beta_{N}|ii\cdots ij\rangle\langle j|, \tag{86}\]  
where \(\left\{\beta_{n}\right\}_{n=1}^{N}\) is an eigenvector corresponding to the maximum eigenvalue of matrix \(A\) defined by  
\[A:=\sum_{n,m=1}^{N}\alpha_{n}|n\rangle\langle m|+(d-1)\sum_{n=1}^{N}\alpha_{n }|n\rangle\langle n|, \tag{87}\]  
and satisfies the following normalization condition:  
\[\left(\sum_{n=1}^{N}\beta_{n}\right)^{2}+(d-1)\sum_{n=1}^{N}\beta_{n}^{2}=d. \tag{88}\]  
In the above equation, non-negative real parameters \(\left\{\alpha_{n}\right\}_{n=1}^{N}\) represent an asymmetry of the UQC and satisfy \(\sum_{n=1}^{N}\alpha_{n}=1\). Equation (86) leads that an output state \(|\Psi_{O}\rangle\) of this UQC protocol corresponding to a given input state \(|\psi_{I}\rangle:=\sum_{i=0}^{d-1}a_{i}|i\rangle\) can be written as  
\[|\Psi_{O}\rangle=\sum_{i,j=0}^{d-1}a_{j}(\beta_{1}|ji\cdots i\rangle+\beta_{2 }|ij\cdots i\rangle+\cdots+\beta_{N}|ii\cdots j\rangle). \tag{89}\]  
We should note that even if \(N=3\), \(U_{I\to O}\) defined by Equation (86) does not coincide the cloning map given by Equation (10); that is, the UQC protocol of [PERSON] et al. is not a straightforward extension of the UQC protocol used in this paper. Let us assume \(d>>N\), that is, the dimension \(d\) of the input space \(\mathcal{H}_{I}\) is much larger than the number of clones \(N\), and give a rough estimation of the communication cost that is necessary to multicast \(|\Psi_{O}\rangle\) to \(N\) distinct terminal nodes on quantum networks. Here, we also assume \(\alpha_{1}>\alpha_{n}\) for all \(n\geq 2\) for simplicity. Under this condition, \(A\) can be approximated the diagonal matrix as  
\[A\approx(d-1)\sum_{n=1}^{N}\alpha_{n}|n\rangle\langle n|. \tag{90}\]Then, the maximum eigenvector \(\{\beta_{n}\}_{n=1}^{N}\) of \(A\) satisfying Equation (88) can be approximately given by \(\beta_{1}=1\) and \(\beta_{n}=0\) for all \(n\geq 2\).