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]

For a differentiable family of density matrices ${\rho }_{\theta }$, the symmetric logarithmic derivative is a Hermitian operator $L_{\theta }$ satisfyingIf ${\rho }_{\theta }={\sum }_j{p}_j|j⟩⟨j|$, its determined matrix entries are $(L_{\theta }{)}_{jk}=2⟨j|{\partial }_{\theta }{\rho }_{\theta }|k⟩/(p_j+p_k)$ whenever $p_j+p_k>0$. Entries wholly in the kernel are undetermined when a solution exists and do not affect $\mathrm{Tr}({\rho }_{\theta }{L}_{\theta }^2)$. Rank-changing boundary points need extra care.

For ${\rho }_{\theta }=\mathrm{diag}(\theta ,1-\theta )$, $0<\theta <1$, one gets $L_{\theta }=\mathrm{diag}(1/\theta ,-1/(1-\theta ))$ and Quantum Fisher information $F_Q=1/[\theta (1-\theta )]$. This recovers ordinary Fisher information for a binary observation. In noncommuting families the symmetrized equation supplies the operator version needed to quantify local sensitivity.^[ref]