# Quantum Multi-User Broadcast Protocol for the \"Platform as a Service\" Model
## 2 Overview
### Phase-Covariant Quantum Cloning  
The no-cloning theorem [15] states that it is impossible to build a quantum copying machine that would perfectly copy arbitrary quantum states. However, we can try to clone a quantum state  
Figure 1: The simple multicast communication network.  
approximately with the optimal fidelity, or instead, we can try to clone it perfectly with the largest probability. Thus, various quantum cloning machines have been designed for different quantum information protocols. Some well-known quantum cloning machines include universal quantum cloning machine, phase-covariant cloning machine, the asymmetric quantum cloning machine, and the probabilistic quantum cloning machine. For instance, a cloning machine that achieves equal fidelity for every state is called a universal quantum cloning machine (\(UQCM\)). This problem is equivalent to distributing information to different receivers, and it is natural to require the performance is the same for every input state, since we do not have any specific information about the input state ahead. According to no-cloning theorem, it is expected that the original input state will be destroyed and become as one of the output copies. Because of the different types of copies, there are symmetric and asymmetric \(UQCMs\). In the past years, much progress has been made in studying quantum cloning machines and their applications and implementations, both theoretically and experimentally. More details about quantum cloning are proposed in [16; 17; 18].  
Here, we mainly discuss the phase-covariant quantum cloning, which has developed on the basis of universal quantum cloning and can produce equally good copies for all input states that lie on the equator of the Bloch sphere. The quantum state  
\[\ket{\psi}=\frac{1}{\sqrt{2}}\left(\ket{0}+e^{i\varphi}\ket{1}\right), \tag{1}\]  
where \(\varphi\in[0,2\pi)\) is an arbitrary phase parameter, is often used as the input qubit of the phase-covariant \(QCM\). For instance, the optimal \(1\to 2\) phase-covariant cloning transformation takes the form [19]  
\[\ket{0}_{A}\ket{0}_{B}\ket{0}_{C}\rightarrow\sqrt{\frac{1}{2}}\left(\ket{0}_ {B}\ket{00}_{AC}+\ket{1}_{B}\ket{\chi}_{AC}\right)=\ket{\phi_{0}}_{ABC},\]  
\[\ket{1}_{A}\ket{0}_{B}\ket{0}_{C}\rightarrow\sqrt{\frac{1}{2}}\left(\ket{1}_ {B}\ket{11}_{AC}+\ket{0}_{B}\ket{\chi}_{AC}\right)=\ket{\phi_{1}}_{ABC}, \tag{2}\]  
where \(\ket{\chi}=1/\sqrt{2}(\ket{10}+\ket{01})\), A is the initial state of the cloning machine, B is an ancilla state of the system, and C is the blank state. The optimal fidelity of phase-covariant \(QCM\) is \(F=1/2+1/\sqrt{8}\approx 0.85\), which is higher than the fidelity (\(F\approx 0.83\)) of \(UQCM\).  
Then, considering the optimal \(1\to M\) phase-covariant \(QCM\), one of the cloning transformations is  
\[\ket{\psi}\otimes\left(\ket{R}^{\otimes M-1}\right)\otimes\ket{M}\overset{U_ {1,M}}{\rightarrow}\ket{\psi}^{\otimes M}\otimes\ket{M(\psi)}, \tag{3}\]  
where \(\ket{\psi}\) is the state of Hilbert space \(H\), \(\ket{R}\) is a blank state, and \(\ket{M}\) is the state of auxiliary system (ancilla). The \(U_{1,M}\) is described by the following unitary operator[20]:  
\[\ket{\phi_{0}}_{AC} =U_{1,M}\ket{0}\otimes\ket{R}=\sum_{j=0}^{M-1}\alpha_{j}|(M-j)0,j 1\rangle\otimes\ket{R_{j}}, \tag{4}\] \[\ket{\phi_{1}}_{AC} =U_{1,M}\ket{1}\otimes\ket{R}=\sum_{j=0}^{M-1}\alpha_{M-1-j}|(M-1 -j)0,(j+1)1\rangle\otimes\ket{R_{j}},\]  
in which \(\alpha_{j}=\sqrt{2(M-j)/M(M+1)}\), \(\ket{R}\) is the initial state of the copy machine and the \(M-1\) blank copies, and \(\ket{R_{j}}\equiv|(M-1-j)0,1j\rangle\) are orthogonal normalized internal states of the \(QCM\). With the help of an ancilla qubit, the optimal fidelity of \(1\to M\) phase-covariant \(QCM\) for equatorial qubits takes the form[18]  
\[F_{1,M}=\begin{cases}1/2+\sqrt{M(M+2)}/4M,&\text{$M$ is even},\\ 1/2+(M+1)/4M,&\text{$M$ is odd},\end{cases} \tag{5}\]  
which is the decreasing function for \(M\) and is better than the fidelity of \(UQCM\) via numerical computation.