Quantum Fisher information
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 formula_1 of a state formula_2 with respect to the observable formula_3 is defined as formula_4 where formula_5 and formula_6 are the eigenvalues and eigenvectors of the density matrix formula_7 respectively. When the observable generates a unitary transformation of the system with a parameter formula_8 from initial state formula_9, formula_10 the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter formula_8 via the quantum Cramér–Rao bound as formula_12  where formula_13 is the number of independent repetitions. It is often desirable to estimate the magnitude of an unknown parameter formula_14 that controls the strength of a system's Hamiltonian formula_15 with respect to a known observable formula_3 during a known dynamical time formula_17. In this case, defining formula_18, so that formula_19, means estimates of formula_8 can be directly translated into estimates of formula_14. Relation to the Symmetric Logarithmic Derivative. The quantum Fisher information equals the expectation value of formula_22, where formula_23 is the Symmetric Logarithmic Derivative. Convexity properties. The quantum Fisher information equals four times the variance for pure  states formula_24. For mixed states, when the probabilities are parameter independent, i.e., when formula_25, the quantum Fisher information is convex: formula_26 The quantum Fisher information is the largest function that is convex and that equals four times the variance for pure states. That is, it equals four times the convex roof of the variance formula_27 where the infimum is over all decompositions of the density matrix formula_28 Note that formula_29 are not necessarily orthogonal to each other. When the probabilities are formula_30-dependent, an extended-convexity relation has been proved: formula_31 where formula_32 is the classical Fisher information associated to  the probabilities contributing to the convex decomposition. The first term, in the right hand side of the above inequality, can be considered as the average quantum Fisher information of the density matrices in the convex decomposition. Inequalities for composite systems. We need to understand the behavior of quantum Fisher information in composite system in order to study quantum metrology of many-particle systems. For product states, formula_33 holds. For the reduced state, we have formula_34 where formula_35. Relation to entanglement. There are strong links between quantum metrology and quantum information science. For a multiparticle system of formula_36 spin-1/2 particles formula_37  holds for separable states, where formula_38 and formula_39 is a single particle angular momentum component. The maximum for general quantum states is given by formula_40 Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology. Moreover, for quantum states with an entanglement depth formula_41, formula_42 holds, where formula_43 is the largest integer smaller than or equal to formula_44 and formula_45 is the remainder from dividing formula_36 by formula_41. Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation. It is possible to obtain a weaker but  simpler bound formula_48 Hence, a lower bound on the entanglement depth is obtained as formula_49 Similar quantities. The Wigner–Yanase skew information is defined as formula_50 It follows that formula_51 is convex in formula_52 For the quantum Fisher information and the Wigner–Yanase skew information, the inequality formula_53 holds, where there is an equality for pure states.