# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 3 Results
### \(1\to 2\) Asymmetric UQC Multicast Protocol
#### 3.1.1 \(1\to 2\) Quantum Multicast Protocol
: As we explained in the previous subsection, an input state at the source node \(s\) can be written as  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]6  
Then, from Equation (4), the state on system \(ABM\) after step 1 can be written as:  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]7The unnormalized state \(\left|\Psi_{2}^{(r)}\right\rangle_{AB}\) on system \(AB\) after deriving measurement outcome \(r\) in Step 2 can be written as:  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]8  
where \(\eta\) is defined by Equation (23), and \(\left\{\beta_{j}\right\}_{j=0}^{d-1}\) is defined by  
\[\mathrm{Y}^{(r)}(\cos\eta|jr\rangle+\sin\eta|rj\rangle)=|jr\rangle,\quad\mathrm{Y}^{(r)}|rr\rangle=|rr\rangle,\]9 \[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]0  
here, \(\left\|\left|\Psi_{2}^{(r)}\right\rangle_{AB}\right\|^{2}=\sum_{j}|\beta_{j}|^ {2}\) is a probability in which outcome \(r\) is derived in Step 2. Since measuring system \(M\) without seeing the outcome is mathematically equivalent to tracing out system \(M\), \(\{\left|\Psi_{2}^{(r)}\right\rangle\}_{r=0}^{d-1}\) satisfies  
\[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]1  
where \(\varepsilon_{1\to 2}\) is a \(1\to 2\) optimal asymmetric UQCM defined by Equation (7). Hence, the purpose of the remaining part of the protocol is to transfer \(\left|\Psi_{2}^{(r)}\right\rangle\) to the target nodes. However, in our problem settings, the throughput of the quantum network is too small to send \(\left|\Psi_{2}^{(r)}\right\rangle\) directly to the target nodes. Hence, first, we compress the state on the \(d\)-dimensional system in Step 3. Then, the unnormalized state of system \(AB\) after Step 3 can be written as  
\[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]2  
In Step 4, the protocol of [PERSON] et al. successfully works based on the assumption for the existence of a classical linear multicast network code. Since the (unnormalized) input state for the protocol of [PERSON] et al. is \(\left|\Psi_{3}\right\rangle_{A}\), the unnormalized state on the system \(C\) at the target node \(t_{1}\) and on system \(D\) at the target node \(t_{2}\) can be written as \(\sum_{j=0}^{d-1}\beta_{j}|j\rangle_{C}|j\rangle_{D}\). The purpose of the remaining part of the protocol is to reconstruct \(\left|\Psi_{2}^{(r)}\right\rangle\) from this state. Since system \(EF\) is initially on \(\cos\eta|0\rangle_{E}|1\rangle_{E}+\sin\eta|1\rangle_{E}|0\rangle_{F}\), the unnormalized state on system \(CDEF\) can be written as  
\[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]3  
Then, the unnormalized state on \(CDEF\) after Step 5 can be written as  
\[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]4  
The unnormalized state on \(CDEF\) after Step 6 is  
\[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]5 \[\mathrm{Y}^{(r)}(\sin\eta|jr\rangle-\cos\eta|rj\rangle)=|rj\rangle \tag{21}\]6Then, the unnormalized state on \(CDEFGH\) after Step 7 can be written as  
\[ \begin{split}\sum_{j\
eq r}\beta_{j}|jj\rangle_{CD}\otimes(\cos\eta| j\rangle_{E}|r\rangle_{F}+\sin\eta|r\rangle_{E}|j\rangle_{F})\otimes|00\rangle_{GH}\\ +\beta_{r}|rr\rangle_{CD}\otimes|00\rangle_{EF}\otimes(\cos\eta| 0\rangle_{G}|1\rangle_{H}+\sin\eta|1\rangle_{G}|0\rangle_{H}).\end{split}  \tag{37}\]  
Next, in Step 8, the state on the system \(H\) is transferred to system \(T_{1}\) by quantum teleportation [102]. Thus, the unnormalized state on \(CDEFG\) after Step 9 can be written as  
\[ \begin{split}\sum_{j\
eq r}\beta_{j}|jj\rangle_{CD}\otimes(\cos \eta|j\rangle_{E}|r\rangle_{F}+\sin\eta|r\rangle_{E}|j\rangle_{F})\otimes|0 \rangle_{G}\\ +\beta_{r}|rr\rangle_{CD}\otimes|00\rangle_{EF}\otimes|1\rangle_ {G}.\end{split}