Quantum thermodynamic derivation of the energy resolution limit in magnetometry

It was recently demonstrated that a multitude of realizations of several magnetic sensing technologies satisfy the energy resolution limit, which connects a quantity composed by the variance of the magnetic field estimate, the sensor volume and the measurement time, and having units of action, with $\hbar$. A first-principles derivation of this limit is still elusive. We here present such a derivation based on quantum thermodynamic arguments. We show that the energy resolution limit is a result of quantum thermodynamic work necessarily associated with quantum measurement and Landauer erasure, the work being exchanged with the magnetic field. We apply these considerations to atomic magnetometers, diamond magnetometers, and SQUIDs, spanning an energy resolution limit from $10^0\hbar$ to $10^7\hbar$. This connection between quantum thermodynamics and magnetometry can help advance quantum sensing technologies towards even more sensitive devices.