\frac{i}{2}(j_{-}^{2}-j_{+}^{2}).\end{aligned}$$ Some linear combinations of these operators will be required to construct the control Hamiltonian to preserve the coherence associated with other states.

Simple Numerical Experiments
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To further confirm the UDD control sequences we explicitly constructed above, we have performed some simple numerical experiments. We first consider a model of a two-spin system coupled to a bath of three spins. The total Hamiltonian in dimensionless units is hence given by $$\begin{aligned}
H & = & \sum_{m=3}^{5}\sum_{j=\{x,y,z\}}b_{j,m}\sigma_{j}^{m}\nonumber \\
 &  & + \sum_{n=1}^{5}\sum_{k=\{x,y,z\}}\sum_{m>n}^{5}\sum_{j=\{x,y,z\}}c_{jk}\sigma_{j}^{m}\sigma_{k}^{n}\nonumber \\
 &  & +\ H_{c},\end{aligned}$$ where the first two spins constitute the two-qubit system in the absence of any external field, $H_{c}$ represents the UDD control Hamiltonian, and the coefficients $b_{j,m}$ and $c_{jk}$ take randomly chosen values in $[0,1]$ in dimensionless units. In addition, to be more realistic, we replace the instantaneous $\delta(t-T_{j})$ function in our control Hamiltonians by a Gaussian pulse, i.e., $(1/c\sqrt{\pi})e^{-[(t-T_{j})^{2}/c^{2}]}$, with $c=T/100$ unless specified otherwise. Further, we set $T=0.1$, because this scale is comparable to the decoherence time scale.

Figure 1 depicts the time dependence of the expectation value of the coherence measure ${\cal P}_{|\Psi\rangle}$, denoted $F(t)$, with $|\Psi\rangle$ being the non-entangled state ${|\uparrow\uparrow\rangle}$ of the two-qubit system. The initial state of the system 