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We calculate the quantum Fisher information for a generic many-body fermionicsystem in a pure state depending on a parameter. We discuss the situationswhere the parameter is imprinted in the basis states, in the statecoefficients, or both. In the case where the parameter dependence ofcoefficients results from a Hamiltonian evolution, we derive a particularlysimple expression for the quantum Fisher information. We apply our findings tothe quantum Hall effect, and evaluate the quantum Fisher information associatedwith the optimal measurement of the magnetic field for a system in the groundstate of the effective Hamiltonian. The occupation of electron states with highmomentum enforced by the Pauli principle leads to a super-Heisenberg scaling ofthe sensitivity with a power law that depends on the geometry of the sensor.

Quantum parameter estimation with many-body fermionic systems and application to the Hall effect