P}_{|0\rangle}$ at time $T$ is given by $$\begin{aligned}
  && {\rm Trace} \{ U(T) \rho_{i} U^{\dagger}(T){\cal P}_{|0\rangle}\} \nonumber \\
 &=& {\rm Trace} \{ (-iY_1)^{N}{\cal U}^{(N)}_{+} \rho_{i} \left({\cal U}^{(N)}_{+}\right)^{\dagger} (iY_1)^{N} {\cal P}_{|0\rangle}\} \nonumber  \\
&=& {\rm Trace} \{ (-iY_1)^{N}{\cal U}^{(N)}_{+} \rho_{i}{\cal P}_{|0\rangle}  \left({\cal U}^{(N)}_{-}\right)^{\dagger} (iY_1)^{N}\} \nonumber  \\
&=& {\rm Trace} \{  \left({\cal U}^{(N)}_{-}\right)^{\dagger} {\cal U}^{(N)}_{+} \rho_{i}{\cal P} _{|0\rangle}\} \nonumber \\
&=& {\rm Trace} \{ \rho_{i}{\cal P}_{|0\rangle} \} \left[1+O(T^{N+1})\right],
\label{ourmain}\end{aligned}$$ where we have used ${\cal P}_{|0\rangle}=Y_1$, $Y_1^2=I$, and the anti-commuting relation between ${\cal P}_{|0\rangle}$ and $H'$. Equation (\[ourmain\]) clearly demonstrates that, as a result of the UDD sequence of $N$ pulses, the expectation value of ${\cal P}_{|0\rangle}$ is preserved to the order of $1+O(T^{N+1})$, for an arbitrary initial state. If the initial state is set to be $|0\rangle$, i.e., ${\rm Trace} \{ \rho_{i}{\cal P}_{|0\rangle} \}=1$, then the expectation value of ${\cal P}_{|0\rangle}$ remains to be $1+O(T^{N+1})$ at time $T$, indicating that the UDD sequence has locked the system on the state $|0\rangle={|\uparrow\uparrow\rangle}$.

In our proof of the UDD applicability in preserving the coherence ${\cal P}_{|\Psi\rangle}$ associated with a non-entangled state, the first important step is to construct the control operator $Y_1={\cal P}_{|\Psi\rangle}$ and then the control Hamiltonian $H_c$. As is clear from Eq. 