# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 4 Four-Party Controlled Cyclic Asymmetrical RSP Protocol in Noisy Environments  
In this section, we discuss our scheme in four noisy environments, including amplitude-damping, phase-damping noise, bit-flip noise, and phase-flip noise. The content shared by the participants in advance is constructed by an organization called the QDC (Qubits Distribution Center). After preparing the quantum channel, QDC allocates the qubits \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\) to [PERSON], the qubits \(B_{1}\), \(B_{2}\) to [PERSON], and sends the qubits \(C_{1},C_{2},C_{3}\) to [PERSON], sends the qubits \(D_{1}\) to [PERSON]. If the quantum channel is in an ideal environment without noise influence, it is a pure state, and its density matrix can be expressed as \(\rho=|\zeta\rangle\langle\xi|\). In practical applications, when affected by a noisy environment [38],the pure state may be converted to a mixed state. The corresponding density matrix can be written as follows:  
\[ \begin{array}{l}\Lambda^{q}(\rho)=\sum\limits_{m}(E_{j}^{q})_{C_{1}}(E_{j}^{q}) _{A_{1}}(E_{j}^{q})_{A_{2}}(E_{j}^{q})_{A_{3}}(E_{j}^{q})_{B_{1}}(E_{j}^{q})_{C_ {2}}(E_{j}^{q})_{C_{3}}(E_{j}^{q})_{B_{2}}\\ \times(E_{j}^{q})_{A_{4}}(E_{j}^{q})_{D_{1}}\rho(E_{j}^{q})_{C_{1}}^{\dagger}(E_{ j}^{q})_{A_{1}}^{\dagger}(E_{j}^{q})_{A_{2}}^{\dagger}(E_{j}^{q})_{A_{3}}(E_{j}^{q})_{ B_{1}}^{\dagger}(E_{j}^{q})_{C_{2}}^{\dagger}\\ \times(E_{j}^{q})_{C_{3}}^{\dagger}(E_{j}^{q})_{B_{2}}(E_{j}^{q})_{A_{4}}^{ \dagger}(E_{j}^{q})_{D_{1}}^{\dagger},\end{array}  \tag{23}\]  
where \(j\) is a label, and different noise channels have different values; the superscript indicates that the operator E acts on different noise environment and \(q\in\{D,A,P,W,B,S\}\). If \(q=D\), for depolarized noise, then \(j=0,1,2,3\); if \(q=A\), for amplitude damping noise, then \(j=0,1\); if \(q=P\), for phase damping noise, then \(j=0,1\) ; if \(q=S\), for bit-phase flip noise, then \(j=0,1\). For reconstructing the desired state, three participants and the controller have to perform single-qubit measurements based on appropriate measurement bases.The desired states were then reconstructed via unitary transformations. If the seven measurement bases are \(\ket{\Phi^{+}}_{9},\ket{\Psi^{+}}_{a^{\prime}},\ket{\Phi^{+}}_{5},\ket{\Psi^{+ }}_{b^{\prime}},\ket{\Phi^{+}}_{1},\ket{\Psi^{+}}_{c^{\prime\prime}},\ket{0}_ {10},\) then the corresponding unitary transformations would be \(I_{A_{2}}\otimes I_{A_{3}}\otimes I_{A_{4}}\otimes I_{B_{2}}\otimes I_{B_{3}} \otimes I_{C_{2}}\) to recover the desired states. We obtain the reconstructed desired state in an ideal environment as follows,  
\[\ket{\zeta}=\left\{ \begin{array}{l}\left[c_{0}e^{i\tau_{0}}\ket{000}+c_{1}e^ {i\tau_{1}}\ket{111}\right]\right\}_{A_{1}A_{2}A_{3}}\otimes\left[b_{0}e^{i \phi_{0}}\ket{00}+b_{1}e^{i\phi_{1}}\ket{11}\right]_{C_{2}C_{3}}\\ \otimes\left[a_{0}e^{i\phi_{0}}\ket{0}+a_{1}e^{i\theta_{1}}\ket{1}\right]_{B_{ 2}}\ket{0}\right]_{D_{1}}\\ +\left[c_{0}e^{i\tau_{0}}\ket{011}-c_{1}e^{i\tau_{1}}\ket{100}\right]_{A_{1}A_ {2}A_{3}}\otimes\left[b_{0}e^{i\phi_{0}}\ket{11}-b_{1}e^{i\phi_{1}}\ket{00} \right]_{C_{2}C_{3}}\\ \otimes\left[a_{1}e^{i\theta_{1}}\ket{0}-a_{0}e^{i\theta_{1}}\ket{1}\right]_{B_ {2}}\left[\ket{1}\right]_{D_{1}}.\end{array} \right. \tag{24}\]  
For any two quantum states, including pure state and mixed state, the fidelity is \(F(\rho,\sigma)=Tr\sqrt{\rho^{\frac{1}{2}}\sigma\rho^{\frac{1}{2}}}\), which is defined as a formula composed of density operators in two states. Then the fidelity between two pure states \(\ket{\psi}\) and \(\ket{\varphi}\) is the modulus of their inner product, which is \(F{=}|\bra{\psi}\ket{\varphi}|\). As in this article, when the ideal output state is a pure state and the observed output state is a mixed state, the fidelity formula defined by the original fidelity is simplified to \(F=\langle\xi|\rho_{out}|\zeta\rangle\).