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

Fermionic quantum field theory is the tool of choice for modeling electronic quantum sensors. To this end, we calculate the quantum Fisher information for a generic many-body fermionic system in a pure state depending on a parameter. The parameter can be imprinted in the basis states, the state coefficients, or in both. 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 sensor in the electronic ground state. The occupation of single-particle 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.