Dynamical, thermal, and ground-state multiparameter quantum metrology: Fundamental limits and their attainability in magnetometry
We consider the simultaneous estimation of multiple parameters in Hamiltonians of the form $H_{\vecθ}= H_{\vecθ}^P + H^C$, where $H_{\vecθ}^P$ encodes the unknown parameters and $H^C$ is a control term. We consider three basic paradigmatic frameworks for metrology: (i) dynamical, where the parameters are encoded unitarily, (ii) thermal, and (iii) ground state. We generalize previous bounds for single-parameter estimation to simultaneous multiparameter estimation for all three metrological settings and establish the conditions required for their saturability. We then focus on vector magnetometry, deriving tight bounds for the simultaneous estimation of magnetic-field components and identifying optimal many-body control Hamiltonians $H^C$ that saturate them. These results establish the fundamental limits of vector magnetometry for unitary dynamics, thermal equilibrium states, and ground states.