# Multi-parameter estimation beyond quantum Fisher information

2022-05-18 19:31:08  阅读：32  来源： 互联网

# Eq(75)

$\left( \begin{matrix} \frac{1}{2}\left( 1+cos\theta \right)& \frac{1}{2}sin\theta e^{-i\varphi}\\ \frac{1}{2}sin\theta e^{i\varphi}& \frac{1}{2}\left( 1-cos\theta \right)\\ \end{matrix} \right) \\ \frac{1}{2}|+\rangle \langle +|_z=\frac{1}{2}\left( \begin{matrix} 1& 0\\ 0& 0\\ \end{matrix} \right) ,\frac{1}{2}|-\rangle \langle -|_z=\frac{1}{2}\left( \begin{matrix} 0& 0\\ 0& 1\\ \end{matrix} \right) \\ \frac{1}{2}|+\rangle \langle +|_y=\frac{1}{4}\left( \begin{matrix} 1& -i\\ i& 1\\ \end{matrix} \right) ,\frac{1}{2}|-\rangle \langle -|_y=\frac{1}{4}\left( \begin{matrix} 1& i\\ -i& 1\\ \end{matrix} \right) \\ p_1=\frac{1}{4}\left( 1+cos\theta \right) \\ p_2=\frac{1}{4}\left( \begin{array}{c} 1-cos\theta\\ \end{array} \right) \\ p_3=\frac{1}{8}\left[ 1+cos\theta -isin\theta e^{i\varphi}+isin\theta e^{-i\varphi}+1-cos\theta \right] =\frac{1}{4}\left( 1+sin\varphi \right) \\ p_4=\frac{1}{8}\left[ 1+cos\theta +isin\theta e^{i\varphi}-isin\theta e^{-i\varphi}+1-cos\theta \right] =\frac{1}{4}\left( 1-sin\varphi \right) \\ F=\left( \begin{matrix} \frac{1}{2}& 0\\ 0& \frac{1}{2}\\ \end{matrix} \right)$