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The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information. The quantum Fisher information F Q {\displaystyle F_{\rm {Q}}} of a state ϱ {\displaystyle \varrho } with respect to the observable A {\displaystyle A} is defined as
where λ k {\displaystyle \lambda _{k}} and | k ⟩ {\displaystyle \vert k\rangle } are the eigenvalues and eigenvectors of the density matrix ϱ , {\displaystyle \varrho ,} respectively, and the summation goes over all k {\displaystyle k} and l {\displaystyle l} such that λ k + λ l > 0 {\displaystyle \lambda _{k}+\lambda _{l}>0}.
When the observable generates a unitary transformation of the system with a parameter θ {\displaystyle \theta } from initial state ϱ 0 {\displaystyle \varrho _{0}} ,
the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter θ {\displaystyle \theta } via the quantum Cramér–Rao bound as