Eq(75)
\[\left( \begin{matrix}
\frac{1}{2}\left( 1+cos\theta \right)& \frac{1}{2}sin\theta e^{-i\varphi}\\
\frac{1}{2}sin\theta e^{i\varphi}& \frac{1}{2}\left( 1-cos\theta \right)\\
\end{matrix} \right)
\\
\frac{1}{2}|+\rangle \langle +|_z=\frac{1}{2}\left( \begin{matrix}
1& 0\\
0& 0\\
\end{matrix} \right) ,\frac{1}{2}|-\rangle \langle -|_z=\frac{1}{2}\left( \begin{matrix}
0& 0\\
0& 1\\
\end{matrix} \right)
\\
\frac{1}{2}|+\rangle \langle +|_y=\frac{1}{4}\left( \begin{matrix}
1& -i\\
i& 1\\
\end{matrix} \right) ,\frac{1}{2}|-\rangle \langle -|_y=\frac{1}{4}\left( \begin{matrix}
1& i\\
-i& 1\\
\end{matrix} \right)
\\
p_1=\frac{1}{4}\left( 1+cos\theta \right)
\\
p_2=\frac{1}{4}\left( \begin{array}{c}
1-cos\theta\\
\end{array} \right)
\\
p_3=\frac{1}{8}\left[ 1+cos\theta -isin\theta e^{i\varphi}+isin\theta e^{-i\varphi}+1-cos\theta \right] =\frac{1}{4}\left( 1+sin\varphi \right)
\\
p_4=\frac{1}{8}\left[ 1+cos\theta +isin\theta e^{i\varphi}-isin\theta e^{-i\varphi}+1-cos\theta \right] =\frac{1}{4}\left( 1-sin\varphi \right)
\\
F=\left( \begin{matrix}
\frac{1}{2}& 0\\
0& \frac{1}{2}\\
\end{matrix} \right) \]
Eq.(80) - (82)
\[Z\left[ X \right] =Tr\left( \rho _{\vec{\theta}}XX^T \right) =\left( \begin{matrix}
1-r^2+{\alpha _r}^2& -i\alpha _r+\alpha _r\alpha _{\theta}\\
i\alpha _r+\alpha _r\alpha _{\theta}& \frac{1}{r^2}+{\alpha _{\theta}}^2\\
\end{matrix} \right)
\\
Tr\left( CZ\left[ X \right] \right) =c\left( r \right) \left( 1-r^2+{\alpha _r}^2 \right) +r^2\left( \frac{1}{r^2}+{\alpha _{\theta}}^2 \right)
\\
so\,\,we\,\,can\,\,see\,\,that\,\,Tr\left( CZ\left[ X \right] \right) \,\,get\,\,minimal\,\,when\,\,\alpha _r=\alpha _{\theta}=0
\\
so\,\,we\,\,have\,\,C^{SLD}=c\left( r \right) \left( 1-r^2 \right) +1
\\
and\,\,since\,\,C^H\,\,differ\,\,from\,\,C^{SLD}\,\,merely\,\,by\,\,Tr\left( CV \right) |V\ge Z\left[ X \right]
\\
so\,\,we\,\,have\,\,that\,\,minTr\left( CV \right) |V\ge Z\left[ X \right] \,\,should\,\,be\,\,when\,\,\alpha _r=\alpha _{\theta}=0, in\,\,which\,\,V=Z\left[ X \right]
\\
so\,\,we\,\,have\,\,C^{SLD}=C^H=c\left( r \right) \left( 1-r^2 \right) +1 this\,\,time\]