Quantum many-body interactions can induce quantum entanglement among particles, rendering them valuable resources for quantum-enhanced sensing. In this work, we establish a link between the bound on the growth of the quantum Fisher information and the Lieb-Robinson bound, which characterizes the operator growth in locally interacting quantum many-body systems. We show that for initial separable states, despite the use of local many-body interactions, the precision cannot surpass the shot noise limit at all times. This conclusion also holds for an initial state that is the nondegenerate ground state of a local and gapped Hamiltonian. These findings strongly hint that when one can only prepare separable initial states, nonlocal and long-range interactions are essential resources for surpassing the shot noise limit. This observation is confirmed through numerical analysis on the long-range Ising model. Our results bridge the field of many-body quantum sensing and operator growth in many-body quantum systems and open the possibility to investigate the interplay between quantum sensing and control, many-body physics and information scrambling.
Quantum metrology promises sensitivity beyond classical strategies, yet it remains unsettled how quantum-enabled precision should scale with physical resources and how to interpret quantum advantage. We provide a physically grounded resource accounting that clarifies the true Heisenberg limit and resolves apparent super-Heisenberg paradoxes. We demonstrate that the Heisenberg limit is best viewed as an information-theoretic manifestation of the quantum speed limit. We illustrate these ideas with a simple, super-resolving phase-estimation protocol based on Rabi oscillations in two-level atoms driven on an
m-photon resonance. In this setting, the phase error scales as
n−m/2, where
n is the average photon number. Recasting metrological sensitivity through quantum dynamical speed limits yields operational bounds that reconcile such super-resolution strategies with the standard Heisenberg interpretation and identify the relevant resources in the norm of the generator. We also revisit the common attribution of the NOON state's
1/n scaling to quantum entanglement. We show that such an attribution is not generic and the Heisenberg
1/n scaling does not, by itself, certify entanglement as the enabling resource.
We investigate the fundamental time complexity, as constrained by Lieb-Robinson bounds, for preparing entangled states useful in quantum metrology. We relate the minimum time to the Quantum Fisher Information (
FQ) for a system of
N quantum spins on a
d-dimensional lattice with
1/rα interactions with
r being the distance between two interacting spins. We focus on states with
FQ∼N1+γ where
γ∈(0,1], i.e., scaling from the standard quantum limit to the Heisenberg limit. For short-range interactions (
α>2d+1), we prove the minimum time
t scales as
t≳Lγ, where
L∼N1/d. For long-range interactions, we find a hierarchy of possible speedups:
t≳Lγ(α−2d) for
2d<α<2d+1,
t≳logL for
(2−γ)d<α<2d, and
t may even vanish algebraically in
1/L for
α<(2−γ)d. These bounds extend to the minimum circuit depth required for state preparation, assuming two-qubit gate speeds scale as
1/rα. We further show that these bounds are saturable, up to sub-polynomial corrections, for all
α at the Heisenberg limit (
γ=1) and for
α>(2−γ)d when
γ<1. Our results establish a benchmark for the time-optimality of protocols that prepare metrologically useful quantum states.
Quantum effects in metrology can in principle enhance measurement precision from the so-called standard quantum limit to the Heisenberg limit. Further advancements in quantum metrology largely rely on innovative metrology protocols that can avoid a number of known obstacles, including the challenge of preparing entangled states with sufficient fidelity, the readout noise in measuring highly entangled states, and no-go theorems for quantum metrology under noisy environments. In this Letter, exploiting some peculiar but experimentally feasible dynamical features of a collection of spins with all-to-all time-periodic interactions, we propose a metrology protocol that can circumvent all three mentioned obstacles and yet still make good use of time as a resource for metrology. Specifically, by mapping the dynamics of such a periodically driven spin system to that of a paradigm of quantum chaos but tuned to some high-order quantum resonance, it is shown that a simple SU(2) coherent state can, after evolving to highly entangled states in the ensuing dynamics, be dynamically brought back to the same initial coherent state. The associated quantum Fisher information is found to exhibit quadratic scaling with both the number of spins and the duration of the metrology protocol. The achieved Heisenberg scaling can also largely survive in the presence of Markovian noise. Representing a previously unknown strategy for quantum metrology, the protocol proposed here can be tested on available experimental platforms.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of
N resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with
N better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources
N is strictly constrained, I show that the final measurement uncertainty is
π times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
Quantum metrology is studied in the presence of quantum correlation. The quantum correlation measure based on quantum Fisher information enables us to gain a deeper insight on how quantum correlations are instrumental in setting metrological precision. Our analysis shows that not only the entanglement but also the quantum correlation plays important roles to enhance precision in quantum metrology. Even in the absence of entanglement, quantum correlation can be exploited as the resource to beat standard quantum limit and reach Heisenberg limit in metrology. Clearly unraveling the role of quantum correlations, the tighter bounds on the metrological precision are derived.