# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 4 Discussion
### Summary  
In this paper, we considered quantum multicast network coding as the multicasting of optimal UQCs over a quantum network. By extending the results of [PERSON] et al. [48; 49; 50] for a multicast of symmetric optimal UQCs, we developed a protocol to multicast asymmetric optimal UQCs over a quantum network. Our results can be summarized as follows. Suppose a quantum network is described by an undirected graph \(G\) with one source node and two (three) target nodes, and each quantum channel on the quantum network \(G\) can transmit one \(q\)-dimensional quantum system in a single session. Furthermore, suppose that there exists a classical solvable multicast network code with source rate \(r\) for a classical network described by an acyclic directed graph \(G^{\prime}\), where \(G\) is an undirected underlying graph of \(G^{\prime}\). We showed that under the above assumptions, our protocol can multicast \(1\to 2\) (\(1\to 3\)) asymmetric optimal UQCs of a \(q^{\prime}\)-dimensional state from the source node to the target nodes by consuming a small amount of entanglement that does not scale with \(q\), which is shared among the target nodes. We further showed that our protocol can be used for efficient preparation of quantum telecloning over a quantum network.  
As we have discussed in the previous subsection, when we use a known scheme of a \(1\to N\) asymmetric optimal UQC for \(N\geq 4\), a protocol which derived straightforward extension of our scheme need \(O(\log d)\) ebits shared among target nodes. Thus, the extension of our protocol for \(1\to n\) asymmetric optimal UQCs for \(n\leq 4\) is not so straightforward. Hence, we leave this study as our future work.  
In this paper, we assumed that all quantum channels on a quantum network are noiseless. This assumption can be justified by considering our protocol as a protocol on the layer on which error correction has been already implemented. However, as is well known, such complete error correction of a quantum channel is beyond the present technology. For example, the long-distance distribution of GHZ-type states, which is nothing but the purpose of the protocol of [PERSON] et al., is a huge challenge in quantum networks. We note that as a practical protocol for this purpose, recently, a protocol to distribute the postselected GHZ state by using entanglement swapping is proposed [106].  
As we have mentioned above, in this paper, we assume that error correction of quantum channels has been applied on a quantum network before our protocol is implemented. In other words, we assume that optimization is applied separately on these two layers. On the other hand, if we optimize these two layers simultaneously, we may derive a better protocol. We leave this study as future work.  
The protocol of [PERSON] et al., which is one of the main subroutines of our protocol, gives an efficient protocol to share GHZ states. On the other hand, as we have explained in Section 1, except GHZ-states, there are many classes of multipartite entangled states that are incomparable to the class of GHZ states under SLOCC-like W states, Dicke states (generalized W states), etc. It may be desirable to find an efficient quantum network protocol to share these classes of states. This may be also our future work.  
Not applicable.  
The authors declare no conflict of interest.