# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 4 Discussion
### Relation to Quantum Teleconing  
In this subsection, we discuss the relationship with our protocol and quantum telecloning [99; 100; 103; 104; 105]. We first give a short review of quantum telecloning. Quantum telecloning is a protocol to multicast an optimal UQC from a sender to multiple receivers by local operation and classical communication with help of a preshared entangled state. There are various different variations of telecloning protocols [99; 100; 103; 104; 105]. Among them, the one that is strongly related to our problem is [PERSON]'s protocol to multicast \(1\to 2\) asymmetric optimal UQCs [99]. In [PERSON]'s protocol, the state \(\ket{\gamma}_{RABM}\) defined by  
\[\ket{Y}_{RABM}:=I_{R}\otimes U_{1\to 2}^{(a,b)}\ket{\Phi_{d}^{+}}_{RA}, \tag{84}\]  
is shared among a sender and the first and second receivers, where \(U_{1\to 2}^{(a,b)}\) is defined by Equation (4) and \(\ket{\Phi_{d}^{+}}\) is a standard maximally entangled state. The sender, the first receiver, and the second receiver possess the system \(R\), \(A\), and \(B\), respectively. The system \(M\) can be possessed by anyone since we do not need to apply any operation on \(M\). The sender further possesses an unknown state \(\ket{\psi}\) on the additional system \(S\). At the first step of [PERSON]'s protocol, the sender applies a generalized Bell measurement on \(RS\), and sends the measurement outcome to the receivers. Then, the resulted state on \(ABM\) is an asymmetric optimal UQCs given by Equation (4) with an error depending on the measurement outcomes. [PERSON] proved that there exist local unitary operations on \(ABM\) that corrects this error [99]. As a result of the local unitary operations, the asymmetric optimal UQCs are shared between the target nodes. Note that since \(M\) is just an ancillary system, we do not necessarily correct the error. If we consider a way to use [PERSON]'s protocol for our problem on the multicast butterfly network, the problem reduces to finding an efficient way to share \(\ket{Y}_{RABM}\) among the source node and the two target nodes. A conventional strategy to share this state may be as follows: First, maximally entangled states are shared among the source node and target nodes by using quantum channels on the network. Second, \(\ket{Y}_{RABM}\) is prepared on the source node. Finally, the system \(A\) and \(B\) are teleported to the target nodes by using the maximally entangled states. However, this protocol is almost the same as the conventional protocol presented in the previous subsection, which is the protocol just teleporting asymmetric optimal QCMs using shared maximally entangled states. Thus, the rate of this protocol is also half of the rate of our protocol using network coding. We can easily see that the system \(M\) does not need to be used on [PERSON]'s protocol, and his telecloning protocol works just using the state \(\rho_{RAB}\) defined by  
\[\rho_{RAB}:=\mathrm{Tr}_{M}\ket{Y}\bra{Y}_{RABM}. \tag{85}\]  
Our protocol can be used to share the state \(\rho_{RAB}\) among the source node and the two target nodes as follows. Suppose that there is a quantum network satisfying the assumption used in our \(1\to 2\) protocol. That is, it has a source node and two target nodes, the target nodes share 2 ebit, and the corresponding classical network has a classical solvable linear network code with rate \(r\). Suppose each quantum channel on the quantum network can transmit \(q\)-dimensional system in a single session. Then, first, the source node prepares \(\ket{\Phi_{q^{\prime}}^{+}}\) on the system \(RA\). Second, the source node applies our protocol using the system \(A\) as an input system. Then, we can easily see that as a result of our protocol, \(\rho_{RAB}\) defined by Equation (85) is shared among the source node and the two target nodes. Here, we emphasize that the dimension of \(R\) ( as well as \(A\) and \(B\)) is \(q^{\prime}\). That is, the rate of this protocol is again \(r\).