$\{V_1,H'\}_{+}=0$. Indeed, it takes exactly the same form (with $Y_1\rightarrow V_1$) as in Eq. (\[un\]). We can then conclude that, the quantum coherence property ${\cal P}_{|\Psi\rangle}$ associated with an arbitrarily pre-selected state $|\Psi\rangle$ in an $M$-level system can be preserved with a fidelity $1+O(T^{N+1})$, with only $N$ pulses. For an $m$-qubit system, $M=2^m$. In such a multi-qubit case, our result here indicates the following: if the initial state of an $m$-qubit system is known, then by (i) setting $|\Psi\rangle$ the same as this initial state, and then (ii) setting ${\cal P}_{|\Psi\rangle}$ as the control operator, the known initial state will be efficiently locked by UDD. Certainly, realizing the required control Hamiltonian for a multi-qubit system may be experimentally challenging.

Recently, a multi-level system subject to pulsed external fields is experimentally realized in a cold-atom laboratory [@Jessen]. To motivate possible experiments of UDD using an analogous setup, in the following we consider the case of $M=3$ in detail. To gain more insights into the control operator $V_1$, here we use angular momentum operators in the $J=1$ subspace to express all the nine basis operators. Specifically, using the eigenstates of the $j_z$ operator as our representation, we have $$\begin{aligned}
j_{x}& =& \frac{1}{\sqrt{2}}\left(\begin{array}{ccc}
 