states is more involving than in single-qubit cases.

We have also considered the preservation of another Bell state $\frac{1}{\sqrt{2}}[{|\uparrow\downarrow\rangle}-{|\downarrow\uparrow\rangle}]$. Following the same procedure outlined above, one finds that the required UDD control Hamiltonian should be given by $$\begin{aligned}
\tilde{H}_{c}&=& -\sum_{j=1}^{N}\frac{\pi}{4} \delta(t-T_j)(I+\sigma_{x}^{1}\sigma_{x}^{2}+\sigma_{y}^{1}\sigma_{y}^{2}+\sigma_{z}^{1}\sigma_{z}^{2}),
\label{enc2}\end{aligned}$$ which is a pulsed Heisenberg interaction Hamiltonian. Such an isotropic control Hamiltonian is consistent with the fact the singlet Bell state defining our quantum coherence measure is also isotropic.

UDD in M-level systems
----------------------

Our early consideration for two-qubit systems suggests a general strategy for establishing UDD in an arbitrary $M$-level system. Let $|0\rangle, |1\rangle, \cdots, |M-1\rangle$ be the $M$ orthogonal basis states for an $M$-level system. Their associated projectors are defined as $P_{j}\equiv |j\rangle\langle j|$, with $j=0, 1, \cdots, M-1$. Without loss of generality we consider the quantum coherence to be preserved is of the $|0\rangle$-type, as characterized by ${\cal P}_{|0\rangle}=2|0\rangle\langle 0|-I$. As learned from Sec. II-B, the important control operator is then $$\begin{aligned}
V_1 & =& {\cal P}_{|0\rangle}= 2P_{0}-I,\end{aligned}$$ with $V_1^2=I$. A UDD sequence of this control operator can be achieved by the following control Hamiltonian $$\begin{aligned}
\label{Mc}
\tilde{H}_{c}&=& \sum_{j=1}^{N}\pi \delta(t-T_j)\frac{V_1}{2}.\end{aligned}$$

In the $M$-dimensional Hilbert space, there are totally $M^2$ linearly independent Hermitian operators. We now 