We revisit well-known protocols in quantum metrology using collective spins and propose a unifying picture for optimal state preparation based on a semiclassical description in phase space.We show how this framework allows for quantitative predictions of the timescales required to prepare various metrologically useful states, and that these predictions remain accurate even for moderate system sizes, surprisingly far from the classical limit.Furthermore, this framework allows us to build a geometric picture that relates optimal (exponentially fast) entangled probe preparation to the existence of separatrices connecting saddle points in phase space.We illustrate our results with the paradigmatic examples of the two-axis countertwisting and twisting-and-turning Hamiltonians, where we provide analytical expressions for all the relevant optimal timescales.Finally, we propose a generalization of these models to include p-body collective interaction (or p-order twisting), beyond the usual case of p = 2. Using our geometric framework, we prove a no-go theorem for the local optimality of these models for p > 2.
We revisit well-known protocols in quantum metrology using collective spins and propose a unifying picture for optimal state preparation based on a semiclassical description in phase space.We show how this framework allows for quantitative predictions of the timescales required to prepare various metrologically useful states, and that these predictions remain accurate even for moderate system sizes, surprisingly far from the classical limit.Furthermore, this framework allows us to build a geometric picture that relates optimal (exponentially fast) entangled probe preparation to the existence of separatrices connecting saddle points in phase space.We illustrate our results with the paradigmatic examples of the two-axis countertwisting and twisting-and-turning Hamiltonians, where we provide analytical expressions for all the relevant optimal timescales.Finally, we propose a generalization of these models to include p-body collective interaction (or p-order twisting), beyond the usual case of p = 2. Using our geometric framework, we prove a no-go theorem for the local optimality of these models for p > 2.
Interference provides a fundamental mechanism for generating and manipulating entanglement in many-body quantum systems. Here, we develop an interference framework in which the nonlinear dynamics of collective spin-21 ensembles are mapped onto phase accumulation and self-interference in phase space, providing a direct and physically transparent description of entanglement formation. Within this framework, one-axis twisting produces Greenberger-Horne-Zeilinger (GHZ) states, while two-axis twisting generates multi-component GHZ superpositions relevant for multiparameter quantum metrology. Building on this interference-based description, we analyze metrological performance under realistic Markovian noise, including local and collective emission, pumping, and dephasing, and examine the role of Hamiltonian control based on linear and nonlinear interactions. We show that the optimal control enhances sensitivity in both single- and multiparameter estimation across noise-dependent regimes. These results establish interference as a unifying principle linking nonlinear dynamics, entanglement generation, and metrological performance. This framework offers a broadly applicable route to robust quantum-enhanced sensing in noisy many-body systems.
Nonlinear effects are widely utilized in quantum metrology to enhance measurement precision by leveraging complex dynamical behaviors. Here, we propose a quantum scrambling-integrated scheme that optimizes the selection of initial states and parameter encoding by exploiting the nonlinear dynamical trajectories and their geometric properties in phase space. We consider a two-mode Bose-Einstein condensate as a typical example. Within a certain parameter range, using uncorrelated spin coherent states located at saddle points and separatrices enables the measurement precision to reach the Heisenberg scaling. An analytical expression for the quantum Fisher information under long-time evolution is provided, and our analysis further reveals that the enhancement mechanism is governed by the sensitivity of the energy spectrum to parameter variations. Compared to the criticality-enhanced protocol that requires parameters to approach the critical point, our scheme offers a distinct advantage in its applicability over a broader range of parameters. Moreover, this scheme can also be applied to models that include one- and two-axis countertwisting interactions and chaotic dynamics. This Letter offers a new perspective for advancing the application of nonlinear dynamics to enhance the parameter estimation precision in quantum systems.
Entanglement can enhance precision in phase estimation and in frequency estimation with atomic clocks, but it remains a central challenge to identify useful states and measurements under realistic noise processes. Here, we study quantum metrology with an ensemble of atoms subject to spontaneous emission, which limits frequency estimation by constraining the useful interrogation time. We identify spin grid states (SGSs) as the relevant near-optimal probes beyond a crossover at an ensemble size of 51, where GHZ-like states cease to be optimal. SGSs display a periodic grid on the Bloch sphere and can be generated with two one-axis-twisting (OAT) operations separated by a collective rotation. By optimizing the quantum Fisher information over permutationally symmetric states, we find that SGSs perform close to the globally optimal probes for intermediate and large ensembles, which turn out to be spin GKP states. We further present a sequential readout strategy that nearly saturates the corresponding quantum Fisher information. This strategy uses an OAT echo to map spontaneous-emission events onto collective rotations, making the corresponding jump sectors distinguishable and allowing the measurement basis to be conditioned on the number of decay events. Together, these results show that SGSs provide a practical route to quantum-enhanced phase and frequency metrology in atomic ensembles limited by spontaneous emission.
Quantum metrology and sensing seek advantages in estimating an unknown parameter of some quantum state or channel, using entanglement such as spin squeezing produced by one-axis twists or other quantum resources. In particular, qubit phase estimation, or rotation sensing, appears as a ubiquitous problem with applications to electric field sensing, magnetometry, atomic clocks, and gyroscopes. By adopting the Bayesian formalism to the phase estimation problem to account for limited initial knowledge about the value of the phase, we formulate variational metrology and treat the state preparation (or encoding) and measurement (or decoding) procedures as parameterized quantum circuits. It is important to understand how effective various parametrized protocols are as well as how robust they are to the effects of complex noise such as spatially correlated noise. First, we propose a family of parametrized encoding and decoding protocols called arbitrary-axis twist ansatzes, and show that it can lead to a substantial reduction in the number of one-axis twists needed to achieve a target estimation error. Furthermore, we demonstrate that the estimation error associated with these strategies decreases with system size in a faster manner than classical (or no-twist) protocols, even in the less-explored regimes where the prior information is limited. Last, using a polynomial-sized tensor network algorithm, we numerically analyze practical variational metrology beyond the symmetric subspace of a collective spin and find that quantum advantage persists for the arbitrary-axis twist ansatzes with a few one-axis twists and smaller total twisting angles for practically relevant noise levels. Published by the American Physical Society 2024
Quantum metrology offers a quadratic advantage over classical approaches to parameter estimation problems by utilising entanglement and nonclassicality. However, the hurdle of actually implementing the necessary quantum probe states and measurements, which vary drastically for different metrological scenarios, is usually not taken into account. We show that for a wide range of tasks in metrology, 2D cluster states (a particular family of states useful for measurement-based quantum computation) can serve as flexible resources that allow one to efficiently prepare any required state for sensing, and perform appropriate (entangled) measurements using only single qubit operations. Crucially, the overhead in the number of qubits is less than quadratic, thus preserving the quantum scaling advantage. This is ensured by using a compression to a logarithmically sized space that contains all relevant information for sensing. We specifically demonstrate how our method can be used to obtain optimal scaling for phase and frequency estimation in local estimation problems, as well as for the Bayesian equivalents with Gaussian priors of varying widths. Furthermore, we show that in the paradigmatic case of local phase estimation 1D cluster states are sufficient for optimal state preparation and measurement.
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.