How the Unruh effect affects transition between classical and quantum decoherences

We investigate how the Unruh effect affects the transition between classical and quantum decoherences for a general class of initial states and find that: $(i)$ The quantum decoherence exists while $\lambda t\leq\lambda \widetilde{t}$ (the transition time) and the classical one can also affect the system's evolution while $\lambda t\geq\lambda\widetilde{t}$ for both the bit and phase-bit flips, which are different from the cases in inertial frame; $(ii)$ The classical correlations will be different constants corresponding to different Unruh temperature and the quantum decoherence still dominates the evolution of system as $\lambda t\geq\lambda\widetilde{t}$ for the phase flip; And $(iii)$ as the Unruh temperature increases, the $\lambda\widetilde{t}$, compared with that in inertial frame, will be bigger for phase flip but smaller for bit flip. However, the $\lambda\widetilde{t}$ does not change no matter what the Unruh effect is for phase-bit flip.