operators. We further assume that the qubit-qubit interaction is given by $H_{12}=\sum_{k,l=\{x,y,z\}}c_{kl}\sigma_{k}^{1}\sigma_{l}^{2}$, where the coefficients $c_{kl}$ may also depend on arbitrary bath operators. A general total Hamiltonian describing a two-qubit system in a bath hence becomes $$\begin{aligned}
H & = & H_{E}+H_{1E}+H_{2E}+H_{12} \nonumber \\
 & = & c_0+\sigma_{x}^{1}c_{x,1}+\sigma_{y}^{1}c_{y,1}+\sigma_{z}^{1}c_{z,1}
 + \sigma_{x}^{2}c_{x,2} \nonumber \\
 && +\ \sigma_{y}^{2}c_{y,2}+\sigma_{z}^{2}c_{z,2} +\sigma_{x}^{1}\sigma_{x}^{2}c_{xx}+\sigma_{x}^{1}\sigma_{y}^{2}c_{xy}\nonumber \\
& & +\ \sigma_{x}^{1}\sigma_{z}^{2}c_{xz}+\sigma_{y}^{1}\sigma_{x}^{2}c_{yx}+\sigma_{y}^{1}\sigma_{y}^{2}c_{yy}
 +\sigma_{y}^{1}\sigma_{z}^{2}c_{yz}\nonumber\\
 && +\ \sigma_{z}^{1}\sigma_{x}^{2}c_{zx}+\sigma_{z}^{1}\sigma_{y}^{2}c_{zy}
 +\sigma_{z}^{1}\sigma_{z}^{2}c_{zz}.
\label{eq:2 Qubits with bath-total Hamiltonian}\end{aligned}$$ For convenience each term in the above total Hamiltonian is assumed to be time independent (this assumption will be lifted in the end).

Focusing on the two-qubit subspace, the above total Hamiltonian is seen to consist of 16 linearly-independent terms that span a natural set of basis operators for all possible Hermitian operators acting on the two-qubit system. This set of basis operators can be summarized as $$\begin{aligned}
\{X_{i}\}_{i=1,2,\cdots,16}=\{\sigma_{k}\otimes\sigma_{l}\}, \\end{aligned}$$ where $
\sigma_{k},\sigma_{l}\in\{I,\sigma_{x},\sigma_{y},\sigma_{z}\}$, with the orthogonality condition $\mathrm{Trace}(X_{j}X_{k})=4\delta_{jk}$. But this choice of basis operators is rather arbitrary. We find that this operator basis set should be changed to new ones to facilitate operator manipulations. In the following we examine the suppression of two types of coherence, one is associated with non-entangled states and the other is associated with a 