# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 4 Four-Party Controlled Cyclic Asymmetrical RSP Protocol in Noisy Environments
### Bit-Phase Flip Noise  
The bit-phase flip noise is a combination of a phase flip and a bit flip, and its influence to the quantum channel can be expressed by the Kraus operators given in Formula (34).  
\[E_{0}^{S}=\sqrt{1-s_{P}}\begin{bmatrix}1&0\\ 0&1\end{bmatrix},E_{1}^{S}=\sqrt{s_{P}}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}, \tag{34}\]  
where \(p_{S}\in[0,1]\) is the decoherence rates for bit-phase flip noise. \(E_{0}^{S}\) means that the quantum state \(|0\rangle\) is not changed, but the amplitude of quantum state \(|1\rangle\) is reduced; \(E_{1}^{S}\) means that quantum state \(|0\rangle\) transitions to i\(|1\rangle\), quantum state \(|1\rangle\) transitions to -i\(|0\rangle\).  
If the scheme is implemented in bit-phase flip noisy environment, then the concrete density matrices of the output states can be obtained as follows.  
\[(\rho_{out}^{S})=\left\{\begin{array}{l}(1-p_{S})^{10}\{\{c_{0}e^{i\gamma_{0} }|000\rangle+c_{1}e^{i\gamma_{1}}|111\rangle\}_{A_{1}A_{2}A_{3}}\times\{b_{0}e^ {i\phi_{0}}|00\rangle+b_{1}e^{i\phi_{1}}|11\rangle\}_{C_{2}C_{3}}\\ \times\{a_{0}e^{i\theta_{0}}|0\rangle+a_{1}e^{i\theta_{1}}|1\rangle\}_{B_{1}} \{0\}_{D_{1}}+\{c_{0}e^{i\gamma_{0}}|011\rangle-c_{1}e^{i\gamma_{1}}|100 \rangle\}_{A_{1}A_{2}A_{3}}\\ \times\{b_{0}e^{i\phi_{0}}|11\rangle-b_{1}e^{i\phi_{1}}|00\rangle\}_{C_{2}C_{3 }}\times\{a_{1}e^{i\theta_{1}}|0\rangle-a_{0}e^{i\theta_{0}}|1\rangle\}_{B_{2 }}\{1\}_{D_{1}}\}\\ \times\{c_{0}e^{-i\gamma_{0}}(000|+c_{1}e^{-i\gamma_{1}}(111|)_{A_{1}A_{2}A_{ 3}}\times\{b_{0}e^{-i\phi_{0}}\langle 00|+b_{1}e^{-i\phi_{1}}\langle 11|\}_{C_{2}C_{3}} \\ \times\{a_{0}e^{-i\theta_{0}}\langle 0|+a_{1}e^{-i\theta_{1}}\langle 1|\}_{B_{2}}\{ 0\}_{D_{1}}+\{c_{0}e^{-i\gamma_{0}}\langle 011|-c_{1}e^{-i\gamma_{1}} \langle 100|\}_{A_{1}A_{2}A_{3}}\\ \times\{b_{0}e^{-i\phi_{0}}\langle 11|-b_{1}e^{-i\phi_{0}}\langle 00|\}_{C_{2}C_{3 }}\times\{a_{1}e^{-i\theta_{1}}\langle 0|-a_{0}e^{-i\theta_{0}}\langle 1|\}_{B_{2 }}\{1|_{D_{1}}\}\\ +\{c_{0}e^{i\gamma_{1}}\langle 11|\}_{A_{1}A_{2}A_{3}}\times\{[i_{0}e^{i \phi_{0}}|00\rangle-b_{1}e^{i\phi_{1}}|11\rangle\}_{C_{2}C_{3}}\\ \times\{-a_{0}e^{i\theta_{0}}|0\rangle-a_{1}e^{i\theta_{1}}|1\rangle\}_{B_{2 }}\{i|1\}_{D_{1}}+\{c_{1}e^{i\gamma_{1}}|100\rangle-c_{0}e^{i\gamma_{0}}|011 \rangle\}_{A_{1}A_{2}A_{3}}\\ \times\{-i_{0}e^{i\phi_{0}}|11\rangle-ib_{1}e^{i\phi_{1}}|00\rangle\}_{C_{2}C_{ 3}}\times\{a_{1}e^{i\theta_{1}}|1\rangle-a_{0}e^{i\theta_{0}}|0\rangle\}_{B_{2 }}\{-i|0\rangle\}_{D_{1}}\\ \times\{c_{0}e^{-i\gamma_{0}}(000|+c_{1}e^{-i\gamma_{1}}\langle 111|)_{A_{1}A_{2}A_{ 3}}\times\{[i_{0}e^{-i\phi_{0}}\langle 00|+b_{1}e^{-i\phi_{1}}\langle 11|]\}_{C_{2}C_{3}} \\ \times\{-a_{0}e^{-i\theta_{0}}\langle 0|-a_{1}e^{-i\theta_{1}}\langle 1|\}_{B_{2}}\{i|1\}_{D_{1}}+ \{c_{1}e^{-i\gamma_{1}}\langle 100|-c_{0}e^{-i\gamma_{0}}\langle 011|\}_{A_{1}A_{2}A_{3}} \\ \times\{-i(b_{0}e^{-i\phi_{0}}\langle 11|-b_{1}e^{-i\phi_{1}}\langle 00|)\}_{C_{2}C_{3}}\times\{a_{1}e^{-i \theta_{1}}\langle 1|-a_{0}e^{-i\theta_{0}}\langle 0|\}_{B_{2}}\{-i\langle 0|\}_{D_{1}} \}.\end{array} \tag{35}\]  
Then, the fidelity of the output state in bit-phase flip noisy environment can be calculated as:  
\[\mathrm{F}^{S}\!\!=\!\!\langle\zeta|\rho_{out}^{S}|\zeta\rangle\!\!=(1\!-\!p_{S })^{10}. \tag{36}\]  
It can be seen from the above formula that the fidelity of the output quantum state is only related to the decoherence rate in bit-phase flip noise.