a different coherence property as defined by our generalized polarization operator ${\cal P}_{|\Psi\rangle}$, with $|\Psi\rangle$ taken as a Bell state $$\begin{aligned}
\label{bell}
|\tilde{0}\rangle=\frac{1}{\sqrt{2}}[{|\uparrow\downarrow\rangle}+{|\downarrow\uparrow\rangle}].\end{aligned}$$ The other three orthogonal basis states for the two-qubit system are now denoted as $|\tilde{1}\rangle$, $|\tilde{2}\rangle$, $|\tilde{3}\rangle$. For example, they can be assumed to be ${|\tilde{1}\rangle} = \frac{1}{\sqrt{2}}[|\uparrow\uparrow\rangle +
|\downarrow\downarrow\rangle]$, ${|\tilde{2}\rangle} =
\frac{1}{\sqrt{2}}[{|\uparrow\uparrow\rangle} -
{|\downarrow\downarrow\rangle}]$, and ${|\tilde{3}\rangle} =
\frac{1}{\sqrt{2}}[{|\uparrow\downarrow\rangle} -
{|\downarrow\uparrow\rangle}]$. To preserve such a new type of coherence, we follow our early procedure to first construct a control operator $\tilde{Y}_1$ and then a new set of basis operators. In particular, we require $$\begin{aligned}
\tilde{Y}_1&=& {\cal P}_{|\tilde{0}\rangle} = 2|\tilde{0}\rangle\langle \tilde{0}|- I \nonumber \\
&=& \frac{1}{2}(-I+\sigma_{x}^{1}\sigma_{x}^{2}+\sigma_{y}^{1}\sigma_{y}^{2}-\sigma_{z}^{1}\sigma_{z}^{2}).\end{aligned}$$ We then construct other 9 basis operators that all commute with $\tilde{Y}_1$, e.g., $$\begin{aligned}
\tilde{Y}_{2} & =& \frac{1}{2}(I+\sigma_{x}^{1}\sigma_{x}^{2}),\nonumber \\
\tilde{Y}_{3} & =& \frac{1}{2}(I-\sigma_{x}^{1}\sigma_{x}^{2}+2\sigma_{y}^{1}\sigma_{y}^{2}),\nonumber \\
\tilde{Y}_{4} & =& \frac{1}{2}(I-\sigma_{x}^{1}\sigma_{x}^{2}-\sigma_{y}^{1}\sigma_{y}^{2}-3\sigma_{z}^{1}\sigma_{z}^{2}),\nonumber \\
\tilde{Y}_{5} & =& \frac{1}{2}(\sigma_{z}^{1}\sigma_{x}^{2}-\sigma_{x}^{1}\sigma_{z}^{2}),\nonumber \\
\tilde{Y}_{6} & =& \frac{1}{2}(\sigma_{y}^{2}-\sigma_{y}^{1}),\nonumber \\
\tilde{Y}_{7} & =& \frac{1}{2}(\sigma_{x}^{2}-\sigma_{x}^{1}),\nonumber \\
\tilde{Y}_{8} & =& -\frac{1}{2}(\sigma_{y}^{1}\sigma_{z}^{2}-\sigma_{z}^{1}\sigma_{y}^{2}),\nonumber \\
\tilde{Y}_{9} & =&\frac{1}{2}(\sigma_{z}^{1}+\sigma_{z}^{2}),\nonumber \\
\tilde{Y}_{10} & =& -\frac{1}{2}(\sigma_{x}^{1}\sigma_{y}^{2}+\sigma_{y}^{1}\sigma_{x}^{2}).\end{aligned}$$ The remaining 6 linearly independent basis operators are found to be anti-commuting with $\tilde{Y}_1$. They can be written as $$\begin{aligned}
\tilde{Y}_{11} & =&\frac{1}{2}(\sigma_{x}^{1}+\sigma_{x}^{2}),\nonumber \\
\tilde{Y}_{12} & =&-\frac{1}{2}(\sigma_{y}^{1}\sigma_{z}^{2}+\sigma_{z}^{1}\sigma_{y}^{2}),\nonumber \\
\tilde{Y}_{13} & =&\frac{1}{2}(\sigma_{x}^{1}\sigma_{z}^{2}+\sigma_{z}^{1}\sigma_{x}^{2}),\nonumber \\
\tilde{Y}_{14} & =&-\frac{1}{2}(\sigma_{y}^{1}+\sigma_{y}^{2}),\nonumber 