# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 4 Four-Party Controlled Cyclic Asymmetrical RSP Protocol in Noisy Environments
### Analysis of the Effect of the Scheme in Four Noisy Environments  
From the above calculation results of fidelity, it can be found that in the environment of depolarization, amplitude damping and phase damping noise, the fidelity of the output state depends on the coefficient of the desired state and the decoherence rate. However, in a bit-phase flip noise environment, the fidelity of the output state is only decided by the decoherence rate.  
As can be seen from the Figure 6, in the depolarization noise, amplitude damped noise, and bit-phase flip noise channels, as the probability of depolarization increases, the fidelity of the output state gradually tends to 0; while in phase damping noise channel, the fidelity of the output state decreases with the increase of decoherence rate and then increases when P \(>\) 0.8.  
When the decoherence rate of bit-phase flip noise and phase damped noise channel is 0.4, the fidelity of the output state drops to 0, and the fidelity of the phase damping noise is gradually increasing after the decoherence rate is 0.8. When the decoherence rate of the depolarized noise channel is 0.6, the fidelity of the output state drops to 0; and the amplitude damping noise is less affected by the decoherence rate than the other three noises.  
When simultaneously considering the influence of the coefficient of the desired quantum state and the decoherence rate, it is assumed that the coefficients of the desired quantum state are all equal. There are the variations of fidelity in different noises as the decoherence rate and the coefficient of desired quantum state varied simultaneously is shown in the figure below:  
From Figure 7, it can be observed that the Fidelity(D) and the Fidelity(S) decrease with the P(D) and P(S). The Fidelity(D) and the Fidelity(S) independent of the coefficient of the desired quantum state. In the amplitude damped noise, when the coefficient of the desired state is greater than 0.7, the Fidelity(A) is positively correlated with the transformation of P(A). In the phase damping noise channel, when the coefficient of the desired state is 1 and the P(p) is higher than 0.5, the Fidelity(p) increases as the P(p) increases. When the coefficient of the desired state is less than 1, the Fidelity(p) decreases with the increasing of P(p).  
Figure 6: The trend of the fidelity of the output state with the change of the decoherence rate(P) in four types of noisy environments, where different types of lines represent different noise environments. Suppose the coefficient of the desired state \(a_{0}=a_{1}=b_{0}=b_{1}=c_{0}=c_{1}=\frac{1}{\sqrt{2}}\).  
Figure 7: The variations of the fidelity in different noise with different coefficients of the initiated state and decoherence rate, where P(D) is short for the decoherence rate of the depolarized noise, P(A) is short for the decoherence rate of the Amplitude damping noise, P(p) is short for the decoherence rate of the phase damping noise, P(S) is short for the decoherence rate of the bit-phase flip noise. Suppose the coefficient of the desired state is \(a_{0}=a_{1}=b_{0}=b_{1}=c_{0}=c_{1}\in[0,1]\).