\[F=Tr\sqrt{\sqrt{\rho _{\theta}}\left( \rho _{\theta}+\partial \rho _{\theta}d\theta +\partial ^2\rho _{\theta}\frac{d\theta ^2}{2} \right) \sqrt{\rho _{\theta}}} \\ \rho _{\theta}=\sum_i{p_i|i\rangle \langle i|} \\ \partial \rho _{\theta}=\sum_i{\partial p_i|i\rangle \langle i|}+\sum_i{p_i|\partial i\rangle \langle i|}+\sum_i{p_i|i\rangle \langle \partial i|} \\ \partial ^2\rho _{\theta}=\sum_i{\partial ^2p_i|i\rangle \langle i|}+\sum_i{\partial p_i|\partial i\rangle \langle i|}+\sum_i{\partial p_i|i\rangle \langle \partial i|} \\ +\sum_i{\partial p_i|\partial i\rangle \langle i|}+\sum_i{p_i|\partial ^2i\rangle \langle i|}+\sum_i{p_i|\partial i\rangle \langle \partial i|} \\ +\sum_i{\partial p_i|i\rangle \langle \partial i|}+\sum_i{p_i|\partial i\rangle \langle \partial i|}+\sum_i{p_i|i\rangle \langle \partial ^2i|} \\ =\sum_i{\partial ^2p_i|i\rangle \langle i|}+\sum_i{p_i|\partial ^2i\rangle \langle i|}+\sum_i{p_i|i\rangle \langle \partial ^2i|}+2\sum_i{\partial p_i|\partial i\rangle \langle i|}+2\sum_i{\partial p_i|i\rangle \langle \partial i|}+2\sum_i{p_i|\partial i\rangle \langle \partial i|} \\ \sqrt{\rho _{\theta}}\left( \rho _{\theta}+\partial \rho _{\theta}d\theta +\partial ^2\rho _{\theta}\frac{d\theta ^2}{2} \right) \sqrt{\rho _{\theta}} \\ \sqrt{\rho _{\theta}}\rho _{\theta}\sqrt{\rho _{\theta}}=\sum_i{{p_i}^2|i\rangle \langle i|} \\ \sqrt{\rho _{\theta}}\partial \rho _{\theta}d\theta \sqrt{\rho _{\theta}} \\ =\left( \sum_i{p_i\partial p_i|i\rangle \langle i|}+\sum_{ij}{\sqrt{p_j}p_i\sqrt{p_i}\langle j|\partial i\rangle |j\rangle \langle i|}+\sum_{ij}{\sqrt{p_i}p_i\sqrt{p_j}|i\rangle \langle \partial i|}j\rangle \right) d\theta \\ \sqrt{\rho _{\theta}}\partial ^2\rho _{\theta}\sqrt{\rho _{\theta}} \\ =\sum_i{p_i\partial ^2p_i|i\rangle \langle i|}+\sum_{ij}{\sqrt{p_j}p_i\sqrt{p_i}\langle j|\partial ^2i\rangle \langle i|}+\sum_{ij}{\sqrt{p_i}p_i\sqrt{p_j}|i\rangle \langle \partial ^2i|j\rangle}+2\sum_{ij}{\sqrt{p_j}\partial p_i\sqrt{p_i}\langle j|\partial i\rangle \langle i|}+2\sum_{ij}{\sqrt{p_i}\partial p_i\sqrt{p_j}|i\rangle \langle \partial i|j\rangle}+2\sum_{ijk}{\sqrt{p_j}p_i\sqrt{p_k}\langle j|\partial i\rangle \langle \partial i|k\rangle}\]
Unfinished
标签：distance,langle,partial,sum,sqrt,rangle,Fisher,Bures,theta From： https://www.cnblogs.com/nana22/p/16597715.html