= Quantum Fisher information
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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 $F_Q=\operatorname{Tr}(\rho L_\theta^2)$.
For $n$ independent preparations, a locally unbiased estimator obeys $\operatorname{Var}(\hat\theta)\geq1/(nF_Q)$ under the usual regularity assumptions. For a pure family $|\psi_\theta\rangle=e^{-i\theta G}|\psi_0\rangle$, $F_Q=4\operatorname{Var}(G)$. A state with definite $G$ acquires only a global phase and has zero information about $\theta$; a superposition can have observable relative-phase sensitivity.<Quantum estimation for quantum technology by Matteo G. A. Paris>[Section 2, logarithmic derivative and quantum Cramér-Rao bound]{ref}
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