# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 4 Four-Party Controlled Cyclic Asymmetrical RSP Protocol in Noisy Environments
### Depolarized Noise  
For a single qubit, supposing that the qubit is depolarized with probability p and remains unchanged with probability 1-p. The effect of the depolarization noise on the Bloch sphere is the entire sphere shrinks uniformly as a function of p. Then the state of the quantum system in depolarized noise is \(\varepsilon(\rho)=\frac{p! }{2}+(1-p)\rho\), and supposing there is \(\frac{I}{2}=\frac{\rho+X\rho X+Y\rho Y+Z\rho Z}{4}\), and then get \(\varepsilon(\rho)=(1-p)\rho+\frac{p}{3}(X\rho X+Y\rho Y+Z\rho Z)\). The depolarized noise can be expressed by Kraus operators given in Formula (25). \[E_{0}^{D}{=}\sqrt{1-\frac{3}{4}p_{D}},E_{1}^{D}=\sqrt{\frac{p_{D}}{4}} \begin{bmatrix} 0&1\\ 1&0\end{bmatrix} ,E_{2}^{D}=\sqrt{\frac{p_{D}}{4}} \begin{bmatrix}0&-i\\ i&0\end{bmatrix} ,E_{3}^{D}=\sqrt{\frac{p_{D}}{4}} \begin{bmatrix}1&0\\ 0&-1,\end{bmatrix}  \tag{25}\]  
where \(p_{D}\in[0,1]\) is the decoherence rate for depolarized noise. \(E_{0}^{D}\) means that the quantum state \(\ket{0}\), \(\ket{1}\) is not changed, but the amplitude of the quantum state is reduced; \(E_{1}^{D}\) means the quantum state \(\ket{0}\) and \(\ket{1}\) transition with probability \(\sqrt{\frac{p_{D}}{4}}\); \(E_{2}^{D}\) means the quantum state \(\ket{0}\) transitions to \(i\ket{1}\), the quantum state \(\ket{1}\) transitions to \(-i\ket{0}\); \(E_{2}^{D}\) means the quantum state \(\ket{1}\) transition to \(\ket{-1}\). If the scheme is implemented in depolarized noisy environment, then the concrete density matrices of the output states can be obtained as follows. If the scheme is implemented in amplitude damping noisy environment, then the concrete density matrices of the output states can be obtained as follows. \[(\rho_{out}^{A})=\left\{ \begin{array}{l}\{\left[c_{0}e^{-i\gamma_{0}}\langle 00 0|+(1-P_{A})^{2}c_{1}e^{i\gamma_{1}}|11\rangle\right]_{A_{1}A_{2}A_{3}}\\ \times\{\left[b_{0}e^{i\phi_{0}}|00\rangle+(\sqrt{1-P_{A}})^{3}b_{1}e^{i\phi_{ 1}}|11\rangle\right]_{C_{2}C_{3}}\\ \times\{a_{0}e^{i\phi_{0}}|0\rangle+(1-P_{A})a_{1}e^{i\phi_{1}}|1\rangle\}_{ B_{2}}[|0\rangle]_{D_{1}}\\ +\left[(1-P_{A})(c_{0}e^{i\gamma_{0}}|011\rangle-c_{1}e^{i\gamma_{1}}|100 \rangle)\right]_{A_{1}A_{2}A_{3}}\\ \times\left[((1-P_{A})b_{0}e^{i\phi_{0}}|11\rangle-\sqrt{1-P_{A}}b_{1}e^{i\phi _{1}}|00\rangle)\right]_{C_{2}C_{3}}\\ \times\{\left[\sqrt{1-P_{A}}(a_{1}e^{i\phi_{1}}|0\rangle-a_{0}e^{i\phi_{0}}|1 \rangle)\right]_{B_{2}}[\sqrt{1-P_{A}}|1\rangle\}_{D_{1}}\}\\ \times\{\left[c_{0}e^{-i\gamma_{0}}\langle 000|+(1-P_{A})^{2}c_{1}e^{-i\gamma_ {1}}\langle 111|\right]_{A_{1}A_{2}A_{3}}\\ \times\left[b_{0}e^{-i\phi_{0}}\langle 00|+(\sqrt{1-P_{A}})^{3}b_{1}e^{-i\phi_{ 1}}\langle 11|\right]_{C_{2}C_{3}}\\ \times\left[a_{0}e^{-i\phi_{0}}\langle 01|+(1-P_{A})a_{1}e^{-i\phi_{1}}\langle 1 |\right]_{B_{2}}[\langle 0|\right]_{D_{1}}\\ +\left[(1-P_{A})(c_{0}e^{-i\gamma_{0}}\langle 11|-c_{1}e^{-i\gamma_{1}}\langle 1 00|)\right]_{A_{1}A_{2}A_{3}}\\ \times\left[(1-P_{A})(b_{0}e^{-i\phi_{0}}\langle 11|-b_{1}e^{-i\phi_{1}}\langle 0 0|)\right]_{C_{2}C_{3}}\\ \times\left[\sqrt{1-P_{A}}(a_{1}e^{-i\phi_{1}}\langle 0|-a_{0}e^{-i\phi_{0}} \langle 11|)\right]_{B_{2}}[\sqrt{1-P_{A}}\langle 1|\right]_{D_{1}}\}\\ +P_{A}^{10}c_{0}^{2}b_{0}^{2}a_{0}^{2}(|0000000\rangle+|0000000\rangle)_{A_{1}A _{2}A_{3}C_{2}C_{3}B_{2}D_{1}}\\ \times(\langle 000000|+\langle 0000000|\rangle_{A_{1}A_{2}A_{3}C_{2}C_{3}B_{ 2}D_{1}}.\end{array} \right. \tag{29}\]  
Then, the fidelity of the output state in amplitude damping noisy environment can be calculated as:  
\[\begin{array}{l}\mathrm{F}^{A}\!\!=\!\!\langle\zeta|\rho_{out}^{A}|\zeta \rangle\\ =\!\!\left\{\begin{array}{l}\frac{1}{2}\{\left\{\left[c_{0}^{2}+(1-P_{A})^{2 }c_{1}^{2}\right]\right.\right.\right. }\left.\left.\left.\left.\left.\left[ \zeta_{0}^{2}+(1-P_{A})^{2}c_{1}^{2}\right]\right.\right.\right.\right.\left. \left.\left.\left.\left.\left.\left.\left.\left[1-P_{A})(c_{0}^{2}+c_{1}^{2} \right]\right]\right.\right.\right.\right.\left.\left.\left.\left.\left.\left.