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arXiv:2609.25977

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

Riccardo Pedroni, Víctor Izquierdo, Advay Burte, Paolo Abiuso, John Calsamiglia, Martí Perarnau-Llobet

quant-phcond-mat.stat-mechSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
v1 · Sep 22, 2026v2 · Sep 25, 2026
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Abstract

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.

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We consider the simultaneous estimation of multiple parameters in Hamiltonians of the form Hθ⃗=Hθ⃗P+HCH_{\vecθ}= H_{\vecθ}^P + H^CHθ​=HθP​+HC, where Hθ⃗PH_{\vecθ}^PHθP​ encodes the unknown parameters and HCH^CHC 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 HCH^CHC that saturate them. These results establish the fundamental limits of vector magnetometry for unitary dynamics, thermal equilibrium states, and ground states.
Quantum metrology aims to enhance measurement precision beyond the classical limit by leveraging quantum resources. Unlike multi-parameter dynamic quantum metrology, many questions regarding multiparameter quantum metrology at thermal equilibrium remain elusive. In particular, the ultimate precision limits achievable in this equilibrium setting are not yet well understood. In this work, we examine the fundamental limits of estimating multiple parameters with a quantum probe at thermal equilibrium. We first show that the Heisenberg limit with respect to the number of probes can be achieved, and our bound coincides with the known single-parameter bound when only one parameter is estimated. We then consider the low temperature limit, revealing a qualitatively different behavior compared to the finite temperature case. We give an example to illustrate the usage of our main results. Finally, we show the conditions under which the sensitivity bound can be attained and the optimal measurements to achieve it.
We consider the estimation of an unknown parameter θ through a quantum probe at thermal equilibrium. The probe is assumed to be in a Gibbs state according to its Hamiltonian H_{θ}, which is divided in a parameter-encoding term H_{θ}^{P} and an additional, parameter-independent control H^{C}. Given a fixed encoding, we find the maximal quantum Fisher information attainable via arbitrary H^{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 an 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.
Estimation of physical parameters encoded in a Hamiltonian is a central task in quantum sensing and learning. While the ultimate precision limit for estimating a single parameter coupled to a single generator is well established, the corresponding bound for estimating a function of multiple parameters-each coupled to distinct and possibly non-commuting generators-remains unknown in general. Here, we derive the ultimate quantum limit and present an estimation protocol for any function of parameters in a general Hamiltonian that attains this bound. We show that, although the task is fundamentally a multiparameter problem, our tight bound reduces to an optimized single-parameter quantum Cramér-Rao bound, even for arbitrary generator sets. Our result unifies and extends previous works, providing a general framework for optimal function estimation in quantum systems.
We consider the estimation of an unknown parameter θ via a many-body probe. The probe is initially prepared in a product state and many-body time-independent interactions enhance its θ sensitivity during the dynamics and/or in the steady state. We present bounds on the quantum Fisher information, and corresponding optimal interacting Hamiltonians, for two paradigmatic scenarios for encoding θ : (i) via unitary Hamiltonian dynamics (dynamical metrology), and (ii) in the Gibbs and diagonal ensembles (time-averaged dephased state), two ubiquitous steady states of many-body open dynamics. We then move to the specific problem of estimating the strength of a magnetic field via interacting spins and derive two-body interacting Hamiltonians that can approach the fundamental precision bounds. In this case, we additionally analyze the transient regime leading to the steady states and characterize trade-offs between equilibration times and measurement precision. Overall, our results provide a comprehensive picture of the potential of many-body control in quantum sensing.
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Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cramér-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.