  We calculate the quantum Fisher information for a generic many-body fermionic
system in a pure state depending on a parameter. We discuss the situations
where the parameter is imprinted in the basis states, in the state
coefficients, or both. In the case where the parameter dependence of
coefficients results from a Hamiltonian evolution, we derive a particularly
simple expression for the quantum Fisher information. We apply our findings to
the quantum Hall effect, and evaluate the quantum Fisher information associated
with the optimal measurement of the magnetic field for a system in the ground
state of the effective Hamiltonian. The occupation of electron states with high
momentum enforced by the Pauli principle leads to a super-Heisenberg scaling of
the sensitivity with a power law that depends on the geometry of the sensor.
