Fundamental limits of metrology at thermal equilibrium

We consider the estimation of an unknown parameter through a quantum probe at thermal equilibrium. The probe is assumed to be in a Gibbs state according to its Hamiltonian H_, which is divided in a parameter-encoding term HP_ and an additional, parameter-independent, control HC. Given a fixed encoding, we find the maximal Quantum Fisher Information attainable via arbitrary HC, which provides a fundamental bound on the measurement precision. Our bounds show that: (i) assuming full control of HC, quantum non-commutativity does not offer any fundamental advantage in the estimation of ; (ii) an exponential quantum advantage arises at low temperatures if HC is constrained to have a spectral gap; (iii) in the case of locally-encoded parameters, the optimal sensitivity presents a Heisenberg-like N²-scaling in terms of the number of particles of the probe, which can be reached with local measurements. We apply our results to paradigmatic spin chain models, showing that these fundamental limits can be approached using local two-body interactions. Our results set the fundamental limits and optimal control for metrology with thermal and ground state probes, including probes at the verge of criticality.