R1～R5 のユニタリ変換　Tong's QHE p.130あたり

page 130 of http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf


$R_3 = \frac{1}{\sqrt{2}} \left ( 1  -i +2i \Psi_2^\dagger \Psi_2 \right )$
$R_5 = \frac{1}{\sqrt{2}} \left ( 1  -i +2i \Psi_3^\dagger \Psi_3 \right )$

$R_1|0\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |0\rangle = e^{-i\pi/4}|0\rangle$,　 $R_1|1\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |1\rangle = e^{i\pi/4}|0\rangle$, 
$R_1|2\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |2\rangle = e^{-i\pi/4}|0\rangle$,　 $R_1|3\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |3\rangle = e^{i\pi/4}|0\rangle$,
$R_1|4\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |4\rangle = e^{-i\pi/4}|0\rangle $, 　$R_1|5\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |5\rangle = e^{i\pi/4}|0\rangle$, 
$R_1|6\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |6\rangle = e^{-i\pi/4}|0\rangle$, 　 $R_1|7\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |7\rangle = e^{i\pi/4}|0\rangle$, 　 

$R_1 = \frac{1}{\sqrt{2}} \left ( 1  -i +2i \Psi_1^\dagger \Psi_1 \right )$

　$ |2 \rangle=\Psi_2^\dagger |0 \rangle$, 　$|3 \rangle = \Psi_2^\dagger |1 \rangle = \Psi_2^\dagger \Psi_1^\dagger |0 \rangle$, 　$|4 \rangle = \Psi_3^\dagger |0 \rangle$, 　$|5 \rangle=\Psi_3^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |1 \rangle$, 　$|6 \rangle = \Psi_3^\dagger |2 \rangle  = \Psi_3^\dagger \Psi_2^\dagger |0 \rangle$, 　$|7 \rangle = \Psi_3^\dagger |3 \rangle = \Psi_3^\dagger \Psi_2^\dagger |1 \rangle = \Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger |0 \rangle $

$UR_1U^\dagger  = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) 
\left(\begin{array}{cccccccc} 
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} \end{array}\right) 
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  e^{i\pi/4} &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 & 0 &  e^{i\pi/4} & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & e^{-i\pi/4} &0 \\\ 
0 & e^{i\pi/4} &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  e^{-i\pi/4} & 0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  e^{-i\pi/4} & 0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &e^{i\pi/4}\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  e^{i\pi/4} &  0 & 0 & 0 & 0 &  0 & 0  \\\ 
0 &  0 &  e^{i\pi/4} & 0 & 0 &  0 & 0 & 0  \\\ 
0 &  0 &  0 & e^{-i\pi/4} & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & e^{i\pi/4} &  0 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  e^{-i\pi/4} &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0  & e^{-i\pi/4}   & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & e^{i\pi/4}\end{array}\right)$

$R_1^{(+)} =  \left(\begin{array}{cccc}
e^{-i\pi/4} &  0 &  0 & 0 \\\ 
0 &  e^{i\pi/4} &  0 & 0  \\\ 
0 &  0 &  e^{i\pi/4} & 0  \\\ 
0 &  0 &  0 & e^{-i\pi/4}   \\\ 
\end{array}\right)$, 　$R_1^{(-)} =  \left(\begin{array}{cccc}
e^{i\pi/4} &  0 &  0 & 0 \\\ 
0 &  e^{-i\pi/4} &  0 & 0 \\\ 
0 &  0  & e^{-i\pi/4}   & 0 \\\ 
0 &  0 &  0 & e^{i\pi/4}\end{array}\right)$

$R_3|0\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |0\rangle = e^{-i\pi/4}|0\rangle$,　 $R_3|1\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |1\rangle = e^{-i\pi/4}|1\rangle$, 
$R_3|2\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |2\rangle = e^{i\pi/4}|2\rangle$,　 $R_3|3\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |3\rangle = e^{i\pi/4}|3\rangle$,
$R_3|4\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |4\rangle = e^{-i\pi/4}|4\rangle $, 　$R_3|5\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |5\rangle = e^{-i\pi/4}|5\rangle$, 
$R_3|6\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |6\rangle = e^{i\pi/4}|6\rangle$, 　 $R_3|7\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |7\rangle = e^{i\pi/4}|7\rangle$, 　

$R_3 = \frac{1}{\sqrt{2}} \left ( 1  -i +2i \Psi_2^\dagger \Psi_2 \right )$

　$ |2 \rangle=\Psi_2^\dagger |0 \rangle$, 　$|3 \rangle = \Psi_2^\dagger |1 \rangle = \Psi_2^\dagger \Psi_1^\dagger |0 \rangle$, 　$|4 \rangle = \Psi_3^\dagger |0 \rangle$, 　$|5 \rangle=\Psi_3^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |1 \rangle$, 　$|6 \rangle = \Psi_3^\dagger |2 \rangle  = \Psi_3^\dagger \Psi_2^\dagger |0 \rangle$, 　$|7 \rangle = \Psi_3^\dagger |3 \rangle = \Psi_3^\dagger \Psi_2^\dagger |1 \rangle = \Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger |0 \rangle $

$UR_3U^\dagger  = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) 
\left(\begin{array}{cccccccc} 
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} \end{array}\right) 
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  e^{i\pi/4} &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 & 0 &  e^{-i\pi/4} & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & e^{i\pi/4} &0 \\\ 
0 & e^{-i\pi/4} &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  e^{i\pi/4} & 0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  e^{-i\pi/4} & 0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &e^{i\pi/4}\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  e^{i\pi/4} &  0 & 0 & 0 & 0 &  0 & 0  \\\ 
0 &  0 &  e^{-i\pi/4} & 0 & 0 &  0 & 0 & 0  \\\ 
0 &  0 &  0 & e^{i\pi/4} & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & e^{-i\pi/4} &  0 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  e^{i\pi/4} &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0  & e^{-i\pi/4}   & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & e^{i\pi/4}\end{array}\right)$

$R_3^{(+)} =  \left(\begin{array}{cccc}
e^{-i\pi/4} &  0 &  0 & 0 \\\ 
0 &  e^{i\pi/4} &  0 & 0  \\\ 
0 &  0 &  e^{-i\pi/4} & 0  \\\ 
0 &  0 &  0 & e^{i\pi/4}   \\\ 
\end{array}\right)$, 　$R_3^{(-)} =  \left(\begin{array}{cccc}
e^{-i\pi/4} &  0 &  0 & 0 \\\ 
0 &  e^{i\pi/4} &  0 & 0 \\\ 
0 &  0  & e^{-i\pi/4}   & 0 \\\ 
0 &  0 &  0 & e^{i\pi/4}\end{array}\right)$

$R_5|0\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |0\rangle = e^{-i\pi/4}|0\rangle$,　 $R_5|1\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |1\rangle = e^{-i\pi/4}|1\rangle$, 
$R_5|2\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |2\rangle = e^{-i\pi/4}|2\rangle$,　 $R_5|3\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |3\rangle = e^{-i\pi/4}|3\rangle$,
$R_5|4\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |4\rangle = e^{i\pi/4}|4\rangle $, 　$R_5|5\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |5\rangle = e^{i\pi/4}|5\rangle$, 
$R_5|6\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |6\rangle = e^{i\pi/4}|6\rangle$, 　 $R_5|7\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |7\rangle = e^{i\pi/4}|7\rangle$, 　


$R_5 = \frac{1}{\sqrt{2}} \left ( 1  -i +2i \Psi_3^\dagger \Psi_3 \right )$

　$ |2 \rangle=\Psi_2^\dagger |0 \rangle$, 　$|3 \rangle = \Psi_2^\dagger |1 \rangle = \Psi_2^\dagger \Psi_1^\dagger |0 \rangle$, 　$|4 \rangle = \Psi_3^\dagger |0 \rangle$, 　$|5 \rangle=\Psi_3^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |1 \rangle$, 　$|6 \rangle = \Psi_3^\dagger |2 \rangle  = \Psi_3^\dagger \Psi_2^\dagger |0 \rangle$, 　$|7 \rangle = \Psi_3^\dagger |3 \rangle = \Psi_3^\dagger \Psi_2^\dagger |1 \rangle = \Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger |0 \rangle $

$UR_5U^\dagger  = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) 
\left(\begin{array}{cccccccc} 
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} \end{array}\right) 
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  e^{-i\pi/4} &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 & 0 &  e^{i\pi/4} & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & e^{i\pi/4} &0 \\\ 
0 & e^{-i\pi/4} &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  e^{-i\pi/4} & 0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  e^{i\pi/4} & 0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &e^{i\pi/4}\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  e^{-i\pi/4} &  0 & 0 & 0 & 0 &  0 & 0  \\\ 
0 &  0 &  e^{i\pi/4} & 0 & 0 &  0 & 0 & 0  \\\ 
0 &  0 &  0 & e^{i\pi/4} & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & e^{-i\pi/4} &  0 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  e^{-i\pi/4} &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0  & e^{i\pi/4}   & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & e^{i\pi/4}\end{array}\right)$

$R_5^{(+)} =  \left(\begin{array}{cccc}
e^{-i\pi/4} &  0 &  0 & 0 \\\ 
0 &  e^{-i\pi/4} &  0 & 0  \\\ 
0 &  0 &  e^{i\pi/4} & 0  \\\ 
0 &  0 &  0 & e^{i\pi/4}   \\\ 
\end{array}\right)$, 　$R_5^{(-)} =  \left(\begin{array}{cccc}
e^{-i\pi/4} &  0 &  0 & 0 \\\ 
0 &  e^{-i\pi/4} &  0 & 0 \\\ 
0 &  0  & e^{i\pi/4}   & 0 \\\ 
0 &  0 &  0 & e^{i\pi/4}\end{array}\right)$

One can find the matrix forms for $R_2$ and $R_4$ here ( https://qiita.com/KogaSense/items/9f349a18ff6ed6088802 ).

$UR_2U^\dagger  = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) 
\frac{1}{\sqrt{2}}\left( \begin{array}{cccccccc}
1 & 0 & 0 & i & 0 & 0 & 0 &0 \\\ 
0 & 1 & -i & 0 & 0 & 0 & 0 &0 \\\ 
0 & -i & 1 & 0 & 0 & 0 & 0 &0 \\\ 
i & 0 & 0 & 1 & 0 & 0 & 0 &0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 &i \\\ 
0 & 0 & 0 & 0 & 0 & 1 & -i &0 \\\ 
0 & 0 & 0 & 0 & 0 & -i & 1 & 0 \\\ 
0 & 0 & 0 & 0 & i & 0 & 0 &1\end{array} \right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 & 0 &  0 &  i &  0 &  0 & 0 &0 \\\ 
i & 0 &  0 &  1 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 & 0 &  1 & -i &0 \\\ 
0 & 0 &  0 &  0 &  0 &  -i & 1 &0 \\\ 
0 & 1 &  -i &  0 &  0 &  0 & 0 &0 \\\ 
0 & -i &  1 & 0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  1 & 0 & 0 &i \\\ 
0 & 0 &  0 &  0 &  i &  0 & 0 &1 \end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 &  i &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
i &  1 &  0 & 0 & 0 & 0 &  0 & 0  \\\ 
0 &  0 &  1 & -i & 0 &  0 & 0 & 0  \\\ 
0 &  0 & -i & 1 & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 1 &  -i &  0 & 0 \\\ 
0 &  0 &  0 & 0 & -i &  1 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0  & 1 & i \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  i & 1 \end{array}\right)$

$R_2^{(+)} =  \frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
1 &  i &  0 & 0 \\\ 
i &  1 &  0 & 0  \\\ 
0 &  0 &  1 & -i  \\\ 
0 &  0 &  -i & 1  \\\ 
\end{array}\right)$, 　$R_2^{(-)} =  \left(\begin{array}{cccc}
1 &  -i &  0 & 0 \\\ 
-i &  1 &  0 & 0 \\\ 
0 &  0  & 1 & i \\\ 
0 &  0 &  i & 1\end{array}\right)$


 
$UR_4U^\dagger  = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) 
\frac{1}{\sqrt{2}}\left( \begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & i &0 \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 &i \\\ 
0 & 0 & 1 & 0 & -i & 0 & 0 &0 \\\ 
0 & 0 & 0 & 1 & 0 & -i & 0 &0 \\\ 
0 & 0 & -i & 0 & 1 & 0 & 0 &0 \\\ 
0 & 0 & 0 & -i & 0 & 1 & 0 &0 \\\ 
i & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & i & 0 & 0 & 0 & 0 & 0 &1\end{array} \right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 & 0 &  0 &  0 &  0 &  0 & i &0 \\\ 
0 & 0 &  0 &  1 &  0 &  -i & 0 &0 \\\ 
0 & 0 &  0 & -i &  0 &  1 & 0 &0 \\\ 
i & 0 &  0 &  0 &  0 &  0 & 1 &0 \\\ 
0 & 1 &  0 &  0 &  0 &  0 & 0 &i \\\ 
0 & 0 &  1 & 0 &  -i &  0 & 0 &0 \\\ 
0 & 0 & -i &  0 &  1 & 0 & 0 & 0 \\\ 
0 & i &  0 &  0 &  0 &  0 & 0 &1 \end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 &  0 &  0 & i & 0 &  0 &  0 & 0 \\\ 
0 &  1 &  -i & 0 & 0 & 0 &  0 & 0  \\\ 
0 &  -i &  1 & 0 & 0 &  0 & 0 & 0  \\\ 
i &  0 & 0 & 1 & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 1 &  0 &  0 & i \\\ 
0 &  0 &  0 & 0 & 0 &  1 & -i & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  -i  & 1 & 0 \\\ 
0 &  0 &  0 & 0 & i &  0 &  0 & 1 \end{array}\right)$

$R_4^{(+)} =  \frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
1 &  0 &  0 & i \\\ 
0 &  1 &  -i & 0  \\\ 
0 &  -i &  1 & 0  \\\ 
i &  0 &  0 & 1  \\\ 
\end{array}\right)$, 　$R_4^{(-)} =  \left(\begin{array}{cccc}
1 &  0 &  0 & i \\\ 
0 &  1 &  -i & 0 \\\ 
0 &  -i  & 1 & 0 \\\ 
i &  0 &  0 & 1\end{array}\right)$