Dynamical, thermal, and ground-state multiparameter quantum metrology: Fundamental limits and their attainability in magnetometry

Quantum metrology aims to enhance parameter estimation beyond classical limits. While single-parameter estimation is well-understood, many practical applications—such as vector field sensing and quantum imaging—require the simultaneous estimation of multiple parameters. This paper investigates the fundamental limits of multiparameter quantum metrology and identifies the conditions under which these limits can be reached using different physical encodings.

The authors analyze three paradigmatic metrological frameworks: dynamical (unitary evolution), thermal equilibrium (Gibbs states), and ground-state metrology. They define the total Hamiltonian as the sum of a parameter-encoding term and a control term, $H_{\vec{\theta}} = H_{\vec{\theta}}^P + H^C$. By generalizing the Quantum Fisher Information Matrix (QFIM) and the Quantum Cramér-Rao Bound (QCRB) to the multiparameter regime, the authors derive lower bounds on the weighted mean square error for these three settings. They specifically apply these results to vector magnetometry, identifying the optimal control Hamiltonians $H^C$ that allow for the saturation of these bounds.

The study establishes that the saturability of the QCRB in multiparameter settings is constrained by the weak commutativity condition (WCC). The authors provide tight analytical bounds for the simultaneous estimation of two and three magnetic field components. Notably, they find that thermal states are uniquely suited for estimating a large number of parameters due to their high-rank QFIM, whereas ground states and unitarily evolved pure states are better suited for distributed private sensing tasks where a lower rank is desired. The paper also identifies specific many-body control Hamiltonians that allow these systems to achieve the fundamental precision limits, effectively bridging the gap between theoretical bounds and potential experimental implementations.