# Quantum Multi-User Broadcast Protocol for the \"Platform as a Service\" Model
## 3 Multi-User Broadcast Protocol Based on _Qcm_
### Multicast Date Packet Communication on Quantum Cloning Process  
We apply the phase-covariant cloning machine to multicast communication (in Figure 3). For each input single photons, we use a unitary transformation matrix \(A\) to prepare the input state on the equator of the Bloch sphere:  
\[A=\frac{1}{\sqrt{2}} \begin{pmatrix}1&e^{-i\varphi}\\ e^{i\varphi}&-1\end{pmatrix} . \tag{6}\]  
The quantum multicast router encodes these strings as a block of \((4n+\delta)\), and sends \(\ket{\Psi_{0}}=\sum_{i=0}^{4n+\delta}\ket{\psi_{i}}\) to \(F_{i}(i=1,2,\cdots,M)\) by quantum cloning via the quantum communication channel. For simplicity, we assume that the quantum channel is noiseless. The quantum multicast router transmits \(\ket{\Psi_{0}}\) through quantum channel to construct \(QCM\). The quantum multicast router acts on an input state \(\ket{\psi}\) as follows:  
\[ \begin{split}\sum_{i=0}^{4n+\delta}U_{1,M}\ket{\psi_{i}}\otimes \ket{R}&=\sum_{i=0}^{4n+\delta}\sum_{j=0}^{M-1}\alpha_{M-1-j}|(M-1-j) \psi_{i},(j+1)\psi_{i}^{\perp}\rangle\otimes|R_{j}\left(\psi_{i}\right)\\ &=\sum_{i=0}^{4n+\delta}\sum_{j=0}^{M-1}\alpha_{M-1-j}|(M-1-j) \psi_{i},(j+1)\psi_{i}^{\perp}\rangle\otimes|(M-1-j)\psi_{i}^{*},(j+1)\left( \psi_{i}^{*}\right)^{\perp}\rangle,\end{split}  \tag{7}\]  
where \(\ket{R_{j}\left(\psi_{i}\right)}\) represents the internal state of \(QCM\) with \(\ket{R_{j}\left(\psi_{i}\right)}\bot\ket{R_{k}\left(\psi_{i}\right)}\) for all \(j\
eq k\), and the equatorial qubits take the forms  
\[ \begin{split}\ket{\psi_{i}}&=\frac{1}{\sqrt{2}}( \ket{0}+e^{i\varphi}\ket{1}),\\ \ket{\psi_{i}^{*}}&=\frac{1}{\sqrt{2}}(\ket{0}+e^{-i \varphi}\ket{1}),\\ \ket{\psi_{i}^{\perp}}&=\frac{1}{\sqrt{2}}(e^{-i \varphi}\ket{0}-\ket{1}),\\ \ket{(\psi_{i}^{*})^{\perp}}&=\frac{1}{\sqrt{2}}(e^{ i\varphi}\ket{0}-\ket{1}).\end{split}  \tag{8}\]  
Quantum correlation is the key concept for the quantum computation, quantum processing, and quantum information. Entanglement is a special case of quantum correlation. It is a property of correlations between two or more quantum systems. This nonlocal nature of entanglement has also been identified as an essential resource for many novel tasks. The preparation of entangled states in different physical systems has been widely studied and constitutes an essential step in many quantum  
Figure 3: Multicast quantum cloning process. information processing and transmission tasks [21; 22; 23]. For [PERSON] and \(M\) [PERSON] brothers, we assume that they all share a multi-particle entangled state \(\ket{\Omega}\), and a choice of \(\ket{\Omega}\) with these properties in the following \(M\)-qubit state:  
\[\ket{\Omega}=\frac{1}{\sqrt{2}}(\ket{0}\otimes\ket{\phi_{0}}_{AC}+\ket{1} \otimes\ket{\phi_{1}}_{AC}), \tag{9}\]  
where \(\ket{\phi_{0}}_{AC}\) and \(\ket{\phi_{1}}_{AC}\) are the optimal cloning states given by Equation (4), and \(\ket{\Omega}\) turns into  
\[\ket{\Omega}=\tfrac{1}{\sqrt{2}}(\ket{0}\sum_{j=0}^{M-1}\alpha_{j} \ket{(M-j)0,j1}\otimes\ket{(M-1-j)0,j1}+\ket{1}\sum_{j=0}^{M-1}\alpha_{j}\ket{ 0,(M-j)1}\otimes\ket{j0,(M-1-j)1}). \tag{10}\]  
The tenser product of \(\ket{\Omega}\) with the equator qubits \(\ket{\Psi_{0}}\) held by the \((2M+4n+\delta+1)\)-qubit state of [PERSON]. [PERSON] performs a joint measurement of the system \(\ket{\Psi_{0}}\otimes\ket{\Omega}\) by using Bell measurement [24], where the four Bell states are defined as usual as the following  
\[\ket{\phi_{\pm}} =\frac{1}{\sqrt{2}}\left(\ket{00}\pm\ket{11}\right), \tag{11}\] \[\ket{\psi_{\pm}} =\frac{1}{\sqrt{2}}\left(\ket{01}\pm\ket{10}\right).\]  
Once one of the Bell states is obtained, we can recover the correct state by exploiting the symmetries of states \(\ket{\phi_{0}}_{AC}\) and \(\ket{\phi_{1}}_{AC}\) under the unitary transformation. Consider three Pauli matrices; the unitary transformation can be expressed as  
\[\sigma_{0}=\left( \begin{array}{cc}1&0\\ 0&1\end{array} \right),\sigma_{x}=\left( \begin{array}{cc}0&1\\ 1&0\end{array} \right),\sigma_{z}=\left( \begin{array}{cc}1&0\\ 0&-1\end{array} \right).