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.
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.
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.
We investigate the physical requirements to reach the ultimate limits of ground-state metrology -- i.e., the estimation of an unknown parameter θ of an N-body Hamiltonian via measurements on its ground state -- using many-body probes featuring either short-range or long-range interactions. We characterize the conditions to saturate two fundamental limits of the quantum Fisher information Fθ: the static ground-state bound Fθ≲N2/Δ2, where Δ is the spectral gap, and the dynamical Heisenberg limit Fθ≲N2τ2, where τ is the total protocol duration. To saturate the static limit, we argue that short-range interacting systems require a spectral gap that closes with N. At second-order quantum critical points this leads to a universal condition on the critical exponents, ν(d+z)=1, which can be approached arbitrarily closely, as we show for the XXZ chain; cat-like ground states at first-order transitions provide an alternative route to static optimality. In contrast, we find that all-to-all interacting models can achieve optimality even in gapped regimes. Furthermore, reaching the dynamical bound is tightly related to the ground-state preparation time; we demonstrate that its optimal, inverse-gap profile can be obtained via local adiabatic protocols. By analyzing paradigmatic systems -- the transverse-field Ising chain, the XXZ chain, a squeezing Hamiltonian, and the two-mode Bose-Hubbard model -- we identify specific many-body probes capable of approaching both limits. Overall, our results provide a theoretical basis for the design of (local) many-body ground-state sensors operating at the ultimate limits of quantum precision.
We consider the simultaneous estimation of multiple parameters in Hamiltonians of the form Hθ=HθP+HC, where HθP encodes the unknown parameters and HC 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 HC that saturate them. These results establish the fundamental limits of vector magnetometry for unitary dynamics, thermal equilibrium states, and ground states.
Measuring stochastic signals ("noise metrology") constitutes a central task in quantum sensing and the characterization of open quantum systems. Here we establish ultimate precision bounds for multiparameter estimation of stochastic signals encoded through Markovian Lindblad dynamics, allowing for arbitrary quantum control and noiseless ancillae. Although Markovianity enforces standard-quantum-limit scaling with sensing time T, our bounds reveal Heisenberg-type scaling in the number of dissipative channels, R: when the stochastic signal exhibits high-rank correlations across the R channels and the probe is entangled, the average variance (per parameter) scales no better than Ω(1/(TR2)). For collective k-body dissipation, R=Θ(Nk), signifying super-Heisenberg scaling with the system size N. We further show that, when the unknown parameters enter through the dissipative eigenrates, a Rapid Prepare-and-Measure (RPM) protocol that tracks many distinct quantum jumps in parallel attains these limits. In this regime, the estimation problem reduces to a multi-Poisson counting model, providing a conceptually clean route to optimal quantum noise metrology. We illustrate the breadth of the framework with applications to networked noise metrology, collective many-body dissipation, learning Pauli noise, and subdiffraction quantum imaging.
We develop a variational principle to determine the quantum controls and initial state that optimizes the quantum Fisher information, the quantity characterizing the precision in quantum metrology. When the set of available controls is limited, the exact optimal initial state and the optimal controls are, in general, dependent on the probe time, a feature missing in the unrestricted case. Yet, for time-independent Hamiltonians with restricted controls, the problem can be approximately reduced to the unconstrained case via Floquet engineering. In particular, we find for magnetometry with a time-independent spin chain containing three-body interactions, even when the controls are restricted to one- and two-body interaction, that the Heisenberg scaling can still be approximately achieved. Our results open the door to investigate quantum metrology under a limited set of available controls, of relevance to many-body quantum metrology in realistic scenarios.
We analyze the scaling of quantum Fisher information with the number of system particles in the limit of large number of particles, as a function of the number of parties interacting with each other, for encoding Hamiltonians having arbitrary-body interactions. We find that estimation of coupling strength of such arbitrary-body encoding Hamiltonians provide a super-Heisenberg scaling that increases monotonically with an increase in the number of interacting particles, in the limit of large number of system particles. Moreover, we also find that the optimal probes corresponding to Hamiltonians that contain even-body interaction terms, may be entangled, but certainly not so in all bipartitions, and particularly, it is possible to attain optimal precision using asymmetric probes. Thereby we find a complementarity in the requirement of asymmetry and genuine entanglement in optimal probes for estimating strength of odd- and even-body interactions respectively. Additionally, we provide an upper bound on the number of parties up to which one can always obtain an asymmetric product state that gives the best metrological precision for even-body interactions. En route, we find the quantum Fisher information in closed form for two- and three-body interactions for arbitrary number of parties. We also provide an analysis of the case when the Hamiltonian contains local fields and up to k-body interaction terms, where the strength of interaction gradually decreases with an increase in the number of parties interacting with each other. Interestingly, we find a similar dichotomy in the nature of the optimal probe in this case as well. Further, we identify conditions on the local component of the Hamiltonian, for which this dichotomy is still shown to exist for two- and three-body encoding Hamiltonians with arbitrary local dimensions.