# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 2 Four-Party Controlled Cyclic Asymmetrical RSP Protocol of Sequentially Increasing Qubits States
### Description of Four-Party Controlled Cyclic Asymmetrical RSP Protocol
\begin{array}{l}{{}_{\
u^{\prime\prime}}\langle\Omega^{+}|_{5}\langle\Phi^{-}|_{ d^{\prime}}\langle\Omega^{\pm}|_{3}\langle\Phi^{-}\Bigm{|}\Bigm{|}\phi^{\prime} \rangle_{123456789,10,d^{\prime}b^{\prime\prime}c^{\prime\prime}}}\\ \left\{\begin{array}{l}=\frac{1}{8}[(|0000\rangle|0\rangle_{c^{\prime\prime \prime}}+|1111\rangle|1\rangle_{c^{\prime\prime\prime}})_{1234}(b_{1}e^{i\phi }|00\rangle_{67}-b_{0}e^{i\phi}|11\rangle_{67})\\ (a_{1}e^{i\phi_{1}}|0\rangle_{8}-a_{0}e^{i\phi_{0}}|1\rangle_{8})|0\rangle_{10 }+(|0011\rangle|0\rangle_{c^{\prime\prime\prime}}-|1100\rangle|1\rangle_{c^{ \prime\prime\prime}})_{1234}\\ (b_{1}e^{i\phi_{1}}|00\rangle_{67}+b_{0}e^{i\phi_{0}}|11\rangle_{67})(-a_{0}e ^{i\phi_{0}}|0\rangle_{8}-a_{1}e^{i\phi_{1}}|1\rangle_{8})|1\rangle_{10}]\\ =\frac{1}{8}[(|0000\rangle|0\rangle_{c^{\prime\prime}}+|1111\rangle|1\rangle_{ c^{\prime\prime\prime}})_{1234}(b_{1}e^{i\phi_{1}}|00\rangle_{67}-b_{0}e^{i\phi_{0}}|11 \rangle_{67})\\ (a_{1}e^{i\phi_{1}}|0\rangle_{8}+a_{0}e^{i\phi_{0}}|1\rangle_{8})|0\rangle_{10 }+(|0011\rangle|0\rangle_{c^{\prime\prime\prime}}-|1100\rangle|1\rangle_{c^{ \prime\prime\prime}})_{1234}\\ (b_{1}e^{i\phi_{1}}|00\rangle_{67}+b_{0}e^{i\phi_{0}}|11\rangle_{67})(a_{0}e ^{i\phi_{0}}|0\rangle_{8}-a_{1}e^{i\phi_{1}}|1\rangle_{8})|1\rangle_{10}]. \end{array}\right.\end{array}  \tag{17}\]  
**Step 3:** [PERSON] selects an appropriate measurement basis according to the known quantum state to perform a single qubit measurement on his qubits. After completing the measurement, [PERSON] announces his outcomes to the [PERSON] via the classical channels. [PERSON] performs a single qubit measurement on qubit 1 based on the following basis:  
\[\left\{ \begin{array}{l}|\Phi^{+}\rangle_{1}=(c_{0}|0\rangle+c_{1}|1\rangle )_{1}\\ |\Phi^{-}\rangle_{1}=(c_{1}|0\rangle-c_{0}|1\rangle)_{1}.\end{array} \right. \tag{18}\]  
Then [PERSON] performs a single qubit measurement on an auxiliary qubit \(c^{{}^{\prime\prime}}\). If [PERSON] performs a single qubit measurement on qubit 1 based on the \(|\Phi^{+}\rangle_{1}\), the basis are  
\[\left\{ \begin{array}{l}|\Psi^{+}\rangle_{c^{\prime\prime\prime}}=\frac{1}{ \sqrt{2}}(e^{-i\gamma_{1}}|0\rangle+e^{-i\gamma_{1}}|1\rangle)_{a^{\prime}}\\ |\Psi^{-}\rangle_{c^{\prime\prime\prime}}=\frac{1}{\sqrt{2}}(e^{-i\gamma_{1}} |0\rangle-e^{-i\gamma_{1}}|1\rangle)_{a^{\prime}}.\end{array} \right. \tag{19}\]  
If [PERSON] performs a single qubit measurement on qubit 1 based on the \(|\Phi^{-}\rangle_{1}\), the basis are  
\[\left\{ \begin{array}{l}|\Omega^{+}\rangle_{c^{\prime\prime\prime}}=\frac{1}{ \sqrt{2}}(e^{-i\gamma_{1}}|0\rangle+e^{-i\gamma_{0}}|1\rangle)_{a^{\prime}}\\ |\Omega^{-}\rangle_{c^{\prime\prime\prime}}=\frac{1}{\sqrt{2}}(e^{-i\gamma_{1 }}|0\rangle-e^{-i\gamma_{0}}|1\rangle)_{a^{\prime}}.\end{array} \right. \tag{20}\]  
**Step 4**: [PERSON] carries out a single qubit measurement on qubit 1 based on the following basis: \(|0\rangle\), \(|1\rangle\). With the help of the controller [PERSON], the senders [PERSON], [PERSON] and [PERSON] can exchange their desired quantum state at the same time. The final result of the controlling party [PERSON] measuring particle 10 makes the participants [PERSON], [PERSON], and [PERSON] must carry out corresponding unitary transformations, respectively, to obtain all the information of the quantum state. The controller ensures the security of the quantum communication scheme and provides robustness against complex eavesdropping attacks. **Step 5: The unitary transformations**  
If the controller, [PERSON]'s, basis is \(|0\rangle\), then the qubits 2,3,4,6,7 and 8 will collapse into the following product state \(|\varphi\rangle_{234678}=(c_{0}e^{i\gamma_{0}}|000\rangle+c_{1}e^{i\gamma_{1} }|111\rangle)\otimes(b_{0}e^{i\phi_{0}}|00\rangle+b_{1}e^{i\phi_{1}}|11\rangle )\otimes(a_{0}e^{i\phi_{0}}|0\rangle+a_{1}e^{i\phi_{1}}|1\rangle)\). [PERSON], [PERSON] and [PERSON] carry out a corresponding unitary operation \(I^{234}\otimes I^{67}\otimes I^{8}\) to obtain the desired quantum state respectively.