# 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
Hence, for a given input state \(|\psi_{l}\rangle\in\mathcal{H}_{I}\), the output state \(|\Psi_{O}\rangle\in\mathcal{H}_{O}\) can be approximated as  
\[|\Psi_{O}\rangle\approx|\psi_{l}\rangle\otimes|GHZ_{N-1}\rangle, \tag{91}\]  
where \(|GHZ_{N-1}\rangle\) is the \(N-1\) partite GHZ state on \(\left(\mathbb{C}^{d}\right)^{\otimes N-1}\) defined as  
\[|GHZ_{N-1}\rangle:=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|ii\cdots i\rangle.\]  
Let us consider a quantum network described by an undirected graph \(G\) with one source node \(s\) and \(N\) target nodes \(t_{1},\cdots t_{N}\), and assume that the protocol can multicast a GHZ-type state \(\sum_{i=0}^{d-1}a_{i}|ii\cdots i\rangle\) from the source node to the \(N\) target nodes for a given input state \(|\psi_{l}\rangle=\sum_{i=0}^{d-1}a_{i}|i\rangle\). Now, we consider the similar type of multicast protocol used in this paper. Thus, the protocol consists of the following three steps:  
1. A quantum operation possibly including measurements are applied to \(|\psi_{l}\rangle\) on \(s\), where the measurement result (if they exist) is sent to \(t_{1},\cdots t_{N}\). 2. The output of the quantum operation, which should be \(d\)-dimensional system, is multicasted to the target nodes \(t_{1},\cdots t_{N}\) by the protocol of [PERSON] et al. 3. The state \(|\Psi_{O}\rangle\) defined by Equation (89) is constructed by LOCC with an additional entanglement resource on \(t_{1},\cdots t_{N}\). Now, we give a rough discussion that strongly suggests that the necessary entanglement resource on the target node is \(O(\log d)\)  ebit: In order to make the state \(|\Psi_{O}\rangle\) given by Equation (89) on the target nodes, the protocol of [PERSON] et al. may be necessary to multicast the whole \(d\)-dimensional output space of the quantum operation on \(s\) in Step 2. In other words, any quantum state \(\sum_{i=0}^{d-1}a_{i}^{\prime}|i\rangle\) on \(\mathbb{C}^{d}\) is necessary to multicast by the protocol of [PERSON] et al. from \(s\) to \(t_{1},\cdots t_{N}\), where \(a_{i}^{\prime}\) is not equal to \(a_{i}\) in general. Then, when \(a_{0}^{\prime}=1\) and \(a_{i}^{\prime}=0\) for all \(i\geq 1\), the output state of the protocol of [PERSON] et al.is a product state \(|00\cdots 0\rangle\). On the other hand, Equation (91) guarantees that when \(d>>N\), the entanglement of the target state \(|\Psi_{O}\rangle\) is \(O(\log d)\) ebit for any bipartition on \(\{1,\cdots,N\}\) except \(1|2,\cdots N\). Therefore, we need an additional entanglement resource with an amount of at least \(O(\log d)\) ebit for any bipartite except \(1|2,\cdots N\). This conclusion is completely different from the results in Sections 3.1 and 3.2, where \(1\to 2\) and \(1\to 3\) asymmetric optimal UQCs are constructed with one use of the protocol of [PERSON] et al. and an entanglement resource, _which is a constant with respect to the dimension \(d\)_. Hence, we conclude that by using existing asymmetric optimal UQC protocol, there might not exist a straightforward extension of our protocol for an arbitrary large number of terminal nodes.