Deterministic joint remote preparation of an arbitrary two-qubit state in noisy environments

Abstract

Using four Einstein–Podolsky–Rosen (EPR) states as the shared quantum channel, we investigate the deterministic joint remote preparation of an arbitrary two-qubit state in the presence of noisy environments through the analytical solution of the master equation in the Lindblad form. By means of unitary matrix decomposition method, quantum logic circuit for the deterministic joint remote state preparation (JRSP) protocol is first constructed. Then, we analytically derive the average fidelities of the deterministic JRSP process under the influence of Pauli noises, zero-temperature and high-temperature reservoirs acting on the four EPR pairs. It is found that the average fidelities under the action of different noises display different evolution behaviors. Moreover, for the specific noises examined in this paper, in the long-time limit, the dephasing noise and the zero-temperature environment have the relatively weak effect on their respective average fidelities, whereas the isotropic noise and the high-temperature environment have the relatively strong effect.

Appendix: Explicit expressions of the output state at Carol’s side

a) When the four EPR states are subject to the bit-flip noise, the analytical form of $\rho _{78}^{x}$ is given by

$$\begin{aligned} \rho _{78}^{x}= & {} 16e^{-16\gamma t}\left\{ \left[ \left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) g+j_{0}\zeta _{0}^{2}+j_{1}\zeta _{3}^{2}+m_{0}^{2}m_{1}^{2}\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \right] \left| 00\right\rangle \left\langle 00\right| \right. \nonumber \\&+\,\left( m_{0}+m_{1}\right) \left( \zeta _{0}\zeta _{1}m_{0}+\zeta _{2}\zeta _{3}m_{1}\right) \left[ e^{-i\varphi _{1}}m_{0}^{2}+e^{i\left( \varphi _{3}-\varphi _{2}\right) }m_{1}^{2}\right. \nonumber \\&+\left. \,m_{0}m_{1}\left( e^{i\varphi _{1}}+e^{i\left( \varphi _{2}-\varphi _{3}\right) }\right) \right] \left| 00\right\rangle \left\langle 01\right| \nonumber \\&+\,\left[ \left( m_{0}+m_{1}\right) \left( \zeta _{0}\zeta _{2}m_{0}+\zeta _{1}\zeta _{3}m_{1}\right) \left( e^{-i\varphi _{2}}m_{0}^{2}+e^{i\left( \varphi _{3}-\varphi _{1}\right) }m_{1}^{2}\right. \right. \nonumber \\&+\left. \left. \,m_{0}m_{1}e^{i\varphi _{2}}+m_{0}m_{1}e^{i\left( \varphi _{1}-\varphi _{3}\right) }\right) \right] \left| 00\right\rangle \left\langle 10\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\Big [\left( m_{1}^{2}e^{i\varphi _{3}}+m_{0}^{2}e^{-i\varphi _{3}}+m_{0}m_{1}e^{i\left( \varphi _{1}-\varphi _{2}\right) }\right. \nonumber \\&+\,m_{0}m_{1}e^{i\left( \varphi _{2}-\varphi _{1}\right) }\Big )\left( 2\zeta _{1}\zeta _{2}m_{0}m_{1}+\zeta _{0}\zeta _{3}m_{0}^{2}+\zeta _{0}\zeta _{3}m_{1}^{2}\right) \Big ]\left| 00\right\rangle \left\langle 11\right| \nonumber \\&+\,\left( m_{0}+m_{1}\right) \left( \zeta _{0}\zeta _{1}m_{0}+\zeta _{2}\zeta _{3}m_{1}\right) \left[ e^{i\varphi _{1}}m_{0}^{2}+e^{i\left( \varphi _{2}-\varphi _{3}\right) }m_{1}^{2}\right. \nonumber \\&+\left. \,m_{0}m_{1}\left( e^{-i\varphi _{1}}+e^{i\left( \varphi _{3}-\varphi _{2}\right) }\right) \right] \left| 01\right\rangle \left\langle 00\right| \nonumber \\&+\,\left[ \left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) g+j_{0}\zeta _{1}^{2}+j_{1}\zeta _{2}^{2}+m_{0}^{2}m_{1}^{2}\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \right] \left| 01\right\rangle \left\langle 01\right| \nonumber \\&+\,\left[ \left( m_{0}m_{1}e^{-i\varphi _{3}}+m_{0}m_{1}e^{i\varphi _{3}}+m_{0}^{2}e^{i\left( \varphi _{1}-\varphi _{2}\right) }\right. \right. \nonumber \\&+\left. \left. \,m_{1}^{2}e^{i\left( \varphi _{2}-\varphi _{1}\right) }\right) \left( 2\zeta _{0}\zeta _{3}m_{0}m_{1}+\zeta _{1}\zeta _{2}m_{0}^{2}+\zeta _{1}\zeta _{2}m_{1}^{2}\right) \right] \left| 01\right\rangle \left\langle 10\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\left[ \left( m_{0}+m_{1}\right) \left( \zeta _{1}\zeta _{3}m_{0}+\zeta _{0}\zeta _{2}m_{1}\right) \left( e^{i\varphi _{2}}m_{1}^{2}+e^{i\left( \varphi _{3}-\varphi _{1}\right) }m_{0}m_{1}\right. \right. \nonumber \\&+\left. \left. \,m_{0}m_{1}e^{-i\varphi _{2}}+m_{0}^{2}e^{i\left( \varphi _{1}-\varphi _{3}\right) }\right) \right] \left| 01\right\rangle \left\langle 11\right| \nonumber \\&+\,\left[ \left( m_{1}+m_{0}\right) \left( m_{0}^{2}e^{i\varphi _{2}}+m_{0}m_{1}e^{-i\varphi _{2}}+m_{0}m_{1}e^{i\left( \varphi _{3}-\varphi _{1}\right) }\right. \right. \nonumber \\&\left. \left. +\,m_{1}^{2}e^{i\left( \varphi _{1}-\varphi _{3}\right) }\right) \left( \zeta _{0}\zeta _{2}m_{0}+\zeta _{1}\zeta _{3}m_{1}\right) \right] \left| 10\right\rangle \left\langle 00\right| \nonumber \\&+\,\left[ \left( m_{0}m_{1}e^{i\varphi _{3}}+m_{0}m_{1}e^{-i\varphi _{3}}+m_{0}^{2}e^{i\left( \varphi _{2}-\varphi _{1}\right) }+m_{1}^{2}e^{i\left( \varphi _{1}-\varphi _{2}\right) }\right) \Big (2\zeta _{0}\zeta _{3}m_{0}m_{1}\right. \nonumber \\&\left. \left. +\,\zeta _{1}\zeta _{2}m_{0}^{2}+\zeta _{1}\zeta _{2}m_{1}^{2}\right) \right] \left| 10\right\rangle \left\langle 01\right| \nonumber \\&+\,\left[ \left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) g+j_{0}\zeta _{2}^{2}+j_{1}\zeta _{1}^{2}+m_{0}^{2}m_{1}^{2}\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \right] \left| 10\right\rangle \left\langle 10\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\left[ \left( m_{0}+m_{1}\right) \left( m_{1}^{2}e^{i\varphi _{1}}+m_{0}m_{1}e^{-i\varphi _{1}}+m_{0}m_{1}e^{i\left( \varphi _{3}-\varphi _{2}\right) }\right. \right. \nonumber \\&\left. \left. +~m_{0}^{2}e^{i\left( \varphi _{2}-\varphi _{3}\right) }\right) \left( \zeta _{2}\zeta _{3}m_{0}+\zeta _{0}\zeta _{1}m_{1}\right) \right] \left| 10\right\rangle \left\langle 11\right| \nonumber \\&+\,\left[ \left( m_{1}^{2}e^{-i\varphi _{3}}+m_{0}^{2}e^{i\varphi _{3}}+m_{0}m_{1}e^{i\left( \varphi _{2}-\varphi _{1}\right) }+m_{0}m_{1}e^{i\left( \varphi _{1}-\varphi _{2}\right) }\right) \Big (2\zeta _{1}\zeta _{2}m_{0}m_{1}\right. \nonumber \\&\left. \left. +\,\zeta _{0}\zeta _{3}m_{0}^{2}+\zeta _{0}\zeta _{3}m_{1}^{2}\right) \right] \left| 11\right\rangle \left\langle 00\right| \nonumber \\&+\,\left[ \left( m_{0}+m_{1}\right) \left( m_{1}^{2}e^{-i\varphi _{2}}+m_{0}m_{1}e^{i\varphi _{2}}+m_{0}^{2}e^{i\left( \varphi _{3}-\varphi _{1}\right) }\right. \right. \nonumber \\&\left. \left. +\,m_{0}m_{1}e^{i\left( \varphi _{1}-\varphi _{3}\right) }\right) \left( \zeta _{1}\zeta _{3}m_{0}+\zeta _{0}\zeta _{2}m_{1}\right) \right] \left| 11\right\rangle \left\langle 01\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\left[ \left( m_{0}+m_{1}\right) \left( m_{1}^{2}e^{-i\varphi _{1}}+m_{0}m_{1}e^{i\varphi _{1}}+m_{0}m_{1}e^{i\left( \varphi _{2}-\varphi _{3}\right) }\right. \right. \nonumber \\&\left. \left. +\,\,m_{0}^{2}e^{i\left( \varphi _{3}-\varphi _{2}\right) }\right) \left( \zeta _{2}\zeta _{3}m_{0}+\zeta _{0}\zeta _{1}m_{1}\right) \right] \left| 11\right\rangle \left\langle 10\right| \nonumber \\&\left. +\,\left[ \left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) g+j_{0}\zeta _{3}^{2}+j_{1}\zeta _{0}^{2}+m_{0}^{2}m_{1}^{2}\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \right] \left| 11\right\rangle \left\langle 11\right| \right\} , \end{aligned}$$

(22)

where $m_{0}=\frac{1}{4}e^{4\gamma t}+\frac{1}{4}$, $m_{1}=\frac{1}{4} e^{4\gamma t}-\frac{1}{4}$, $g=m_{0}m_{1}^{3}+2m_{0}^{2}m_{1}^{2}+m_{0}^{3}m_{1}$, $j_{0}=m_{0}^{4}+2m_{0}^{3}m_{1}$, and $j_{1}=m_{1}^{4}+2m_{0}m_{1}^{3}$.

b) When the four EPR states are subject to the dephasing noise, the analytical form of $\rho _{78}^{z}$ is given by

$$\begin{aligned} \rho _{78}^{z}= & {} \zeta _{0}^{2}\left| 00\right\rangle \left\langle 00\right| +\zeta _{0}\zeta _{1}e^{-8\gamma t}e^{-i\varphi _{1}}\left| 00\right\rangle \left\langle 01\right| +\zeta _{0}\zeta _{2}e^{-8\gamma t}e^{-i\varphi _{2}}\left| 00\right\rangle \left\langle 10\right| \nonumber \\&+\,\zeta _{0}\zeta _{3}e^{-16\gamma t}e^{-i\varphi _{3}}\left| 00\right\rangle \left\langle 11\right| \nonumber \\&+\,\zeta _{0}\zeta _{1}e^{-8\gamma t}e^{i\varphi _{1}}\left| 01\right\rangle \left\langle 00\right| +\zeta _{1}^{2}\left| 01\right\rangle \left\langle 01\right| +\zeta _{1}\zeta _{2}e^{-16\gamma t}e^{i\left( \varphi _{1}-\varphi _{2}\right) }\left| 01\right\rangle \left\langle 10\right| \nonumber \\&+\,\zeta _{1}\zeta _{3}e^{-8\gamma t}e^{i\left( \varphi _{1}-\varphi _{3}\right) }\left| 01\right\rangle \left\langle 11\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\zeta _{0}\zeta _{2}e^{-8\gamma t}e^{i\varphi _{2}}\left| 10\right\rangle \left\langle 00\right| +\zeta _{1}\zeta _{2}e^{-16\gamma t}e^{i\left( \varphi _{2}-\varphi _{1}\right) }\left| 10\right\rangle \left\langle 01\right| +\zeta _{2}^{2}\left| 10\right\rangle \left\langle 10\right| \nonumber \\&+\,\zeta _{2}\zeta _{3}e^{-8\gamma t}e^{i\left( \varphi _{2}-\varphi _{3}\right) }\left| 10\right\rangle \left\langle 11\right| \nonumber \\&+\,\zeta _{0}\zeta _{3}e^{-16\gamma t}e^{i\varphi _{3}}\left| 11\right\rangle \left\langle 00\right| +\zeta _{1}\zeta _{3}e^{-8\gamma t}e^{i\left( \varphi _{3}-\varphi _{1}\right) }\left| 11\right\rangle \left\langle 01\right| \nonumber \\&+\,\zeta _{2}\zeta _{3}e^{-8\gamma t}e^{i\left( \varphi _{3}-\varphi _{2}\right) }\left| 11\right\rangle \left\langle 10\right| +\zeta _{3}^{2}\left| 11\right\rangle \left\langle 11\right| . \end{aligned}$$

(23)

c) When the four EPR states are subject to the isotropic noise, the analytical form of $\rho _{78}^{iso}$ is given by

$$\begin{aligned} \rho _{78}^{iso}= & {} e^{-\frac{32}{\sqrt{3}}\gamma t}\left\{ \left[ 16\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \left( n_{0}^{3}n_{1}+n_{0}n_{1}^{3}+2n_{0}^{2}n_{1}^{2}\right) +16\zeta _{0}^{2}(n_{0}^{4}+2n_{0}^{3}n_{1}) \nonumber \right. \right. \\&\left. +\,16\zeta _{3}^{2}(n_{1}^{4}+2n_{0}n_{1}^{3})+16n_{0}^{2}n_{1}^{2}\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \right] \left| 00\right\rangle \left\langle 00\right| +4(\zeta _{0}\zeta _{1}n_{0}+\zeta _{2}\zeta _{3}n_{1})\nonumber \\&\,\qquad \left( e^{-i\varphi _{1}}n_{0}+e^{i\left( \varphi _{2}-\varphi _{3}\right) }n_{1}\right) \left| 00\right\rangle \left\langle 01\right| \nonumber \\&+\,\,4\left( \zeta _{0}\zeta _{2}n_{0}+\zeta _{1}\zeta _{3}n_{1}\right) \left( e^{-i\varphi _{2}}n_{0}+e^{i(\varphi _{1}-\varphi _{3})}n_{1}\right) \left| 00\right\rangle \left\langle 10\right| +\zeta _{0}\zeta _{3}e^{-i\varphi _{3}}\left| 00\right\rangle \left\langle 11\right| \nonumber \\&+\,\,4\left( \zeta _{0}\zeta _{1}n_{0}+\zeta _{2}\zeta _{3}n_{1}\right) \left( e^{i\varphi _{1}}n_{0}+e^{i\left( \varphi _{3}-\varphi _{2}\right) }n_{1}\right) \left| 01\right\rangle \left\langle 00\right| \nonumber \\&+\,\left[ 16\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \left( n_{0}^{3}n_{1}+n_{0}n_{1}^{3}+2n_{0}^{2}n_{1}^{2}\right) \right. \nonumber \\&+\left. \,16\zeta _{1}^{2}\left( n_{0}^{4}+2n_{0}^{3}n_{1}\right) +16\zeta _{2}^{2}(n_{1}^{4}+2n_{0}n_{1}^{3})+16n_{0}^{2}n_{1}^{2}\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \right] \left| 01\right\rangle \left\langle 01\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\zeta _{1}\zeta _{2}e^{i\left( \varphi _{1}-\varphi _{2}\right) }\left| 01\right\rangle \left\langle 10\right| +4\left( \zeta _{0}\zeta _{2}n_{1}+\zeta _{1}\zeta _{3}n_{0}\right) \left( e^{-i\varphi _{2}}n_{1}+e^{i\left( \varphi _{1}-\varphi _{3}\right) }n_{0}\right) \left| 01\right\rangle \left\langle 11\right| \nonumber \\&+~4\left( \zeta _{0}\zeta _{2}n_{0}+\zeta _{1}\zeta _{3}n_{1}\right) \left( e^{i\varphi _{2}}n_{0}+e^{i\left( \varphi _{3}-\varphi _{1}\right) }n_{1}\right) \left| 10\right\rangle \left\langle 00\right| +\zeta _{1}\zeta _{2}e^{i\left( \varphi _{2}-\varphi _{1}\right) }\left| 10\right\rangle \left\langle 01\right| \nonumber \\&+\,\left[ 16\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \left( n_{0}^{3}n_{1}+n_{0}n_{1}^{3}+2n_{0}^{2}n_{1}^{2}\right) +16\zeta _{1}^{2}\left( n_{1}^{4}+2n_{0}n_{1}^{3}\right) \right. \nonumber \\&+\,16\zeta _{2}^{2}\left( n_{0}^{4}+2n_{0}^{3}n_{1}\right) \nonumber \\&+\left. \,16n_{0}^{2}n_{1}^{2}\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \right] \left| 10\right\rangle \left\langle 10\right| +4\left( \zeta _{0}\zeta _{1}n_{1}+\zeta _{2}\zeta _{3}n_{0}\right) \nonumber \\&\times \left( e^{-i\varphi _{1}}n_{1}+e^{i\left( \varphi _{2}-\varphi _{3}\right) }n_{0}\right) \left| 10\right\rangle \left\langle 11\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\zeta _{0}\zeta _{3}e^{i\varphi _{3}}\left| 11\right\rangle \left\langle 00\right| +4\left( \zeta _{0}\zeta _{2}n_{1}+\zeta _{1}\zeta _{3}n_{0}\right) \left( e^{i\varphi _{2}}n_{1}+e^{i\left( \varphi _{3}-\varphi _{1}\right) }n_{0}\right) \left| 11\right\rangle \left\langle 01\right| \nonumber \\&+\,\,4\left( \zeta _{0}\zeta _{1}n_{1}+\zeta _{2}\zeta _{3}n_{0}\right) \left( e^{i\varphi _{1}}n_{1}+e^{i\left( \varphi _{3}-\varphi _{2}\right) }n_{0}\right) \left| 11\right\rangle \left\langle 10\right| \nonumber \\&+\,\left[ 16\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \left( n_{0}^{3}n_{1}+n_{0}n_{1}^{3}+2n_{0}^{2}n_{1}^{2}\right) \right. \nonumber \\&\left. \left. +\,16\zeta _{0}^{2}\left( n_{1}^{4}+2n_{0}n_{1}^{3}\right) +16\zeta _{3}^{2}\left( n_{0}^{4}+2n_{0}^{3}n_{1}\right) +16n_{0}^{2}n_{1}^{2}\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \right] \left| 11\right\rangle \left\langle 11\right| \right\} ,\nonumber \\ \end{aligned}$$

(24)

where $n_{0}=\frac{1}{4}e^{\frac{8}{\sqrt{3}}\gamma t}+\frac{1}{4}$ and $n_{1}=\frac{1}{4}e^{\frac{8}{\sqrt{3}}\gamma t}-\frac{1}{4}.$

d) When the four EPR states are subject to high-temperature environment, the analytical form of $\rho _{78}^{high}$ is given by

$$\begin{aligned} \rho _{78}^{high}= & {} 16e^{-16\gamma t}\left[ \zeta _{0}^{2}\left( f_{0}^{4}+2f_{1}f_{0}^{3}+f_{1}^{2}f_{0}^{2}\right) +\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \left( f_{0}f_{1}^{3}+f_{1}f_{0}^{3}+2f_{1}^{2}f_{0}^{2}\right) \right. \nonumber \\&\left. +\,\zeta _{3}^{2}\left( f_{1}^{4}+2f_{0}f_{1}^{3}+f_{1}^{2}f_{0}^{2}\right) \right] \left| 00\right\rangle \left\langle 00\right| \nonumber \\&+\,\left[ 4e^{-12\gamma t}\left( f_{0}\zeta _{0}\zeta _{1}+f_{1}\zeta _{2}\zeta _{3}\right) \left( f_{0}e^{-i\varphi _{1}}+f_{1}e^{i\left( \varphi _{2}-\varphi _{3}\right) }\right) \right] \left| 00\right\rangle \left\langle 01\right| \nonumber \\&+\,\left[ 4e^{-12\gamma t}\left( f_{0}\zeta _{0}\zeta _{2}+f_{1}\zeta _{1}\zeta _{3}\right) \left( f_{0}e^{-i\varphi _{2}}+f_{1}e^{i\left( \varphi _{1}-\varphi _{3}\right) }\right) \right] \left| 00\right\rangle \left\langle 10\right| \nonumber \\&+\,\zeta _{0}\zeta _{3}e^{-i\varphi _{3}}e^{-8\gamma t}\left| 00\right\rangle \left\langle 11\right| \nonumber \\&+\,\left[ 4e^{-12\gamma t}\left( f_{0}\zeta _{0}\zeta _{1}+f_{1}\zeta _{2}\zeta _{3}\right) \left( f_{0}e^{i\varphi _{1}}+f_{1}e^{i\left( \varphi _{3}-\varphi _{2}\right) }\right) \right] \left| 01\right\rangle \left\langle 00\right| \nonumber \\&+\,16e^{-16\gamma t}\left[ \zeta _{1}^{2}\left( f_{0}^{4}+2f_{1}f_{0}^{3}+f_{1}^{2}f_{0}^{2}\right) \right. \nonumber \\&\left. +\,\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \left( f_{0}f_{1}^{3}+f_{1}f_{0}^{3}+2f_{1}^{2}f_{0}^{2}\right) +\zeta _{2}^{2}\left( f_{1}^{4}+2f_{0}f_{1}^{3}+f_{1}^{2}f_{0}^{2}\right) \right] \left| 01\right\rangle \left\langle 01\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\zeta _{1}\zeta _{2}e^{i\left( \varphi _{1}-\varphi _{2}\right) }e^{-8\gamma t}\left| 01\right\rangle \left\langle 10\right| \nonumber \\&+\,\left[ 4e^{-12\gamma t}\left( f_{1}\zeta _{0}\zeta _{2}+f_{0}\zeta _{1}\zeta _{3}\right) \left( f_{1}e^{-i\varphi _{2}}+f_{0}e^{i\left( \varphi _{1}-\varphi _{3}\right) }\right) \right] \left| 01\right\rangle \left\langle 11\right| \nonumber \\&+\,\left[ 4e^{-12\gamma t}\left( f_{0}\zeta _{0}\zeta _{2}+f_{1}\zeta _{1}\zeta _{3}\right) \left( f_{0}e^{i\varphi _{2}}+f_{1}e^{i\left( \varphi _{3}-\varphi _{1}\right) }\right) \right] \left| 10\right\rangle \left\langle 00\right| \nonumber \\&+\,\zeta _{1}\zeta _{2}e^{i\left( \varphi _{2}-\varphi _{1}\right) }e^{-8\gamma t}\left| 10\right\rangle \left\langle 01\right| \nonumber \\&+\,16e^{-16\gamma t}\left[ \zeta _{2}^{2}\left( f_{0}^{4}+2f_{1}f_{0}^{3}+f_{1}^{2}f_{0}^{2}\right) +\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \left( f_{0}f_{1}^{3}+f_{1}f_{0}^{3}+2f_{1}^{2}f_{0}^{2}\right) \right. \nonumber \\&\left. +\,\zeta _{1}^{2}\left( f_{1}^{4}+2f_{0}f_{1}^{3}+f_{1}^{2}f_{0}^{2}\right) \right] \left| 10\right\rangle \left\langle 10\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,\left[ 4e^{-12\gamma t}\left( f_{1}\zeta _{0}\zeta _{1}+f_{0}\zeta _{2}\zeta _{3}\right) \left( f_{0}e^{-i\varphi _{1}}+f_{1}e^{i\left( \varphi _{2}-\varphi _{3}\right) }\right) \right] \left| 10\right\rangle \left\langle 11\right| \nonumber \\&+\,\zeta _{0}\zeta _{3}e^{i\varphi _{3}}e^{-8\gamma t}\left| 11\right\rangle \left\langle 00\right| \nonumber \\&+\,\left[ 4e^{-12\gamma t}\left( f_{1}\zeta _{0}\zeta _{2}+f_{0}\zeta _{1}\zeta _{3}\right) \left( f_{1}e^{i\varphi _{2}}+f_{0}e^{i\left( \varphi _{3}-\varphi _{1}\right) }\right) \right] \left| 11\right\rangle \left\langle 01\right| \nonumber \\&+\,\left[ 4e^{-12\gamma t}\left( f_{1}\zeta _{0}\zeta _{1}+f_{0}\zeta _{2}\zeta _{3}\right) \left( f_{0}e^{i\varphi _{1}}+f_{1}e^{i\left( \varphi _{3}-\varphi _{2}\right) }\right) \right] \left| 11\right\rangle \left\langle 10\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,16e^{-16\gamma t}\left[ \zeta _{3}^{2}\left( f_{0}^{4}+2f_{1}f_{0}^{3}+f_{1}^{2}f_{0}^{2}\right) \nonumber \right. \\&\left. +\,\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \left( f_{0}f_{1}^{3}+f_{1}f_{0}^{3}+2f_{1}^{2}f_{0}^{2}\right) +\zeta _{0}^{2}\left( f_{1}^{4}+2f_{0}f_{1}^{3}+f_{1}^{2}f_{0}^{2}\right) \right] \left| 11\right\rangle \left\langle 11\right| ,\nonumber \\ \end{aligned}$$

(25)

where $f_{0}=\frac{1}{4}e^{4\gamma t}+\frac{1}{4}$ and $f_{1}=\frac{1}{4} e^{4\gamma t}-\frac{1}{4}$.

e) When the four EPR states are subject to zero-temperature environment, the analytical form of $\rho _{78}^{zero}$ is given by

$$\begin{aligned} \rho _{78}^{zero}= & {} e^{-8\gamma t}[\zeta _{0}^{2}\epsilon +\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \tau +\zeta _{3}^{2}\omega ]\left| 00\right\rangle \left\langle 00\right| +\zeta _{0}\zeta _{3}e^{-4\gamma t}e^{-i\varphi _{3}}\left| 00\right\rangle \left\langle 11\right| \nonumber \\&+\,e^{-6\gamma t}\left\{ e^{-i\varphi _{1}}\left[ \zeta _{0}\zeta _{1}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{2}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\&\left. +\,e^{i\left( \varphi _{2}-\varphi _{3}\right) }\left[ \zeta _{0}\zeta _{1}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{2}\zeta _{3}s_{1}^{2}\right] \right\} \left| 00\right\rangle \left\langle 01\right| \nonumber \\&+\,e^{-6\gamma t}\left. \{e^{-i\varphi _{2}}\left[ \zeta _{0}\zeta _{2}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{1}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\&\left. +\,e^{i\left( \varphi _{1}-\varphi _{3}\right) }\left[ \zeta _{0}\zeta _{2}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{1}\zeta _{3}s_{1}^{2}\right] \right\} \left| 00\right\rangle \left\langle 10\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,e^{-6\gamma t}\left\{ e^{i\varphi _{1}}\left[ \zeta _{0}\zeta _{1}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{2}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\&\left. +\,e^{i\left( \varphi _{3}-\varphi _{2}\right) }\left[ \zeta _{0}\zeta _{1}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{2}\zeta _{3}s_{1}^{2}\right] \right\} \left| 01\right\rangle \left\langle 00\right| \nonumber \\&+e^{-8\gamma t}\left[ \zeta _{1}^{2}\epsilon +\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \tau +\zeta _{2}^{2}\omega \right] \left| 01\right\rangle \left\langle 01\right| +\zeta _{1}\zeta _{2}e^{-4\gamma t}e^{i\left( \varphi _{1}-\varphi _{2}\right) }\left| 01\right\rangle \left\langle 10\right| \nonumber \\&+\,e^{-6\gamma t}\left\{ e^{i\left( \varphi _{1}-\varphi _{3}\right) }\left[ \zeta _{1}\zeta _{3}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{0}\zeta _{2}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\&\left. +\,e^{-i\varphi _{2}}\left[ \zeta _{1}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{0}\zeta _{2}s_{1}^{2}\right] \right\} \left| 01\right\rangle \left\langle 11\right| \nonumber \\&+\,e^{-6\gamma t}\left\{ e^{i\varphi _{2}}\left[ \zeta _{0}\zeta _{2}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{1}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\ \end{aligned}$$

$$\begin{aligned}&\left. +\,e^{i\left( \varphi _{3}-\varphi _{1}\right) }\left[ \zeta _{0}\zeta _{2}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{1}\zeta _{3}s_{1}^{2}\right] \right\} \left| 10\right\rangle \left\langle 00\right| \nonumber \\&+\,\zeta _{1}\zeta _{2}e^{-4\gamma t}e^{i\left( \varphi _{2}-\varphi _{1}\right) }\left| 10\right\rangle \left\langle 01\right| +e^{-8\gamma t}\left[ \zeta _{2}^{2}\varepsilon +\left( \zeta _{0}^{2}+\zeta _{3}^{2}\right) \tau +\zeta _{2}^{2}\omega \right] \left| 10\right\rangle \left\langle 10\right| \nonumber \\&+\,e^{-6\gamma t}\left\{ e^{i\left( \varphi _{2}-\varphi _{3}\right) }\left[ \zeta _{2}\zeta _{3}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{0}\zeta _{1}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\&\left. +\,e^{-i\varphi _{1}}\left[ \zeta _{2}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{0}\zeta _{1}s_{1}^{2}\right] \right\} \left| 10\right\rangle \left\langle 11\right| \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\,e^{-6\gamma t}\left\{ e^{i\left( \varphi _{3}-\varphi _{1}\right) }\left[ \zeta _{1}\zeta _{3}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{0}\zeta _{2}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\&\left. +\,e^{i\varphi _{2}}\left[ \zeta _{1}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{0}\zeta _{2}s_{1}^{2}\right] \right\} \left| 11\right\rangle \left\langle 01\right| \nonumber \\&+\,e^{-6\gamma t}\left\{ e^{i\left( \varphi _{3}-\varphi _{2}\right) }\left[ \zeta _{2}\zeta _{3}\left( 1/4+s_{0}+s_{0}^{2}\right) +\zeta _{0}\zeta _{1}\left( s_{1}+2s_{0}s_{1}\right) \right] \right. \nonumber \\&\left. +\,e^{i\varphi _{1}}\left[ \zeta _{2}\zeta _{3}\left( s_{1}+2s_{0}s_{1}\right) +4\zeta _{0}\zeta _{1}s_{1}^{2}\right] \right\} \left| 11\right\rangle \left\langle 10\right| \nonumber \\&+\,\zeta _{0}\zeta _{3}e^{-4\gamma t}e^{i\varphi _{3}}\left| 11\right\rangle \left\langle 00\right| +e^{-8\gamma t}\left[ \zeta _{3}^{2}\epsilon +\left( \zeta _{1}^{2}+\zeta _{2}^{2}\right) \tau +\zeta _{0}^{2}\omega \right] \left| 11\right\rangle \left\langle 11\right| ,\nonumber \\ \end{aligned}$$

(26)

where $s_{0}=e^{2\gamma t}-e^{\gamma t}+\frac{1}{2}$, $s_{1}=\frac{1}{2} e^{\gamma t}-\frac{1}{2}$, $\epsilon =\frac{1}{16}+\frac{1}{2} s_{1}+s_{1}^{2}+\frac{1}{2}s_{0}+\frac{3}{2} s_{0}^{2}+2s_{0}^{3}+s_{0}^{4}+3s_{0}s_{1}+4s_{0}s_{1}^{2}+6s_{0}^{2}s_{1}+4s_{0}^{2}s_{1}^{2}+4s_{0}^{3}s_{1}$, $\tau =\frac{1}{4} s_{1}+2s_{1}^{2}+4s_{1}^{3}+3s_{0}s_{1}+8s_{0}s_{1}^{2}+8s_{0}s_{1}^{3}+3s_{0}^{2}s_{1}+8s_{0}^{2}s_{1}^{2}+2s_{0}^{3}s_{1}$, and $\omega =s_{1}^{2}+8s_{1}^{3}+16s_{1}^{4}+4s_{0}s_{1}^{2}+16s_{0}s_{1}^{3}+4s_{0}^{2}$.