\ast }$, $ -i a _{f} ^{ \prime \prime \ast} ) B _{ {\bf k} } \propto ( a _{f} ^{ \prime \ast}$, $ -i a _{f} ^{ \prime \prime \ast} )$ are normalized such that $\langle \varphi _{f} | \varphi _{f} \rangle =1$. We may exchange the components in Eq. (5) to obtain $$\begin{aligned}
T | \varphi _{ f } \rangle =  \left[
\begin{array}{c} 
\varphi ^{\prime \prime} _{ f }  \\   \varphi  ^{\prime} _{ f }
\end{array}
\right],\end{aligned}$$ which also represents the ket $\varphi _{ f }$, by the operator $$\begin{aligned}
T =  \left[
\begin{array}{c} 
0 \text{ \ \ }  I  \\  I  \text{ \ \ }  0  
\end{array}
\right]. \end{aligned}$$ The extended AF density matrix $ \sum  ( | \varphi _{ f } \rangle  \langle \varphi _{ f } | + T  | \varphi _{ f } \rangle  \langle \varphi _{ f } | T ^{\dag} ) $ is of the form of $\rho _{ea}$ given by Eq. (2) because it commutes with $T$. We can decompose $\rho _{ea}$ as $$\begin{aligned}
\rho _{ea}  = \rho ^{(1)} \otimes e _{1} e _{1} ^{\dag} + \rho ^{(2)} \otimes e _{2} e _{2} ^{\dag}. \end{aligned}$$ such that $\rho ^{(1)}  = \rho 