Quantum Fisher information

The Quantum Fisher information measures the local sensitivity of a parametrized density matrix ${\rho }_{\theta }$. The symmetric logarithmic derivative $L_{\theta }$ satisfies ${\partial }_{\theta }\rho =(L_{\theta }\rho +\rho L_{\theta })/2$, and $F_Q=\mathrm{Tr}(\rho L_{\theta }^2)$.

For $n$ independent preparations, a locally unbiased estimator obeys $\mathrm{Var}(\hat{\theta })\ge 1/(n{F}_Q)$ under the usual regularity assumptions. For a pure family $|{\psi }_{\theta }⟩=e^{-i\theta G}|{\psi }_0⟩$, $F_Q=4\mathrm{Var}(G)$. A state with definite $G$ acquires only a global phase and has zero information about $\theta$; a superposition can have observable relative-phase sensitivity.^[ref]