Fundamental limits of metrology at thermal equilibrium

We consider the estimation of an unknown parameter $\theta$ through a quantum probe at thermal equilibrium. The probe is assumed to be in a Gibbs state according to its Hamiltonian $H_\theta$, which is divided in a parameter-encoding term $H^{\rm P}_\theta$ and an additional, parameter-independent, control $H^{\rm C}$. Given a fixed encoding, we find the maximal Quantum Fisher Information attainable via arbitrary $H^{\rm C}$, which provides a fundamental bound on the measurement precision. We elucidate the role of quantum coherence between encoding and control in different temperature regimes, which include ground state metrology as a limiting case. In the case of locally-encoded parameters, the optimal sensitivity presents a $N^2$-scaling in terms of the number of particles of the probe, which can be reached, at finite temperature, with local measurements and no entanglement. 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.