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arXiv:2606.26248

Optimal observables for quantum-enhanced sensing and applications in a Floquet time crystal sensor

M. A. Manya, Andrei Tsypilnikov, Fernando Iemini

quant-phcond-mat.othercond-mat.stat-mechSubmitted Jun 24, 2026• Announced Jun 26, 2026• 0 citations
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Abstract

In this work, we discuss how to determine and implement feasible optimal observables for a metrology protocol that saturates the quantum Fisher information (QFI) bound. In particular, we focus our study on a simple protocol, namely the method of moments (MoM). We first demonstrate that the symmetric logarithmic derivative (SLD) operator, a Hermitian observable, once implemented in the MoM, saturates the QFI bound. However, the SLD is generally too complex and typically non-local, rendering its direct experimental realization unfeasible. To overcome this limitation, we explore its structure in a specific sensing model - a Floquet time crystal (FTC) acting as an ac field sensor - and show that the SLD can be approximated by substantially simpler observables, such as the bare spin magnetization or a parity observable, for different relevant initial state preparations. We further corroborate our theoretical predictions in a nuclear magnetic resonance system operating as an FTC sensor, employing experimentally motivated parameters to simulate its performance in a state-of-the-art implementation. In general, our results establish a practical route toward near-optimal metrology in FTC sensors, where the inaccessible SLD operator can be replaced by simpler observables while retaining quantum-enhanced sensitivity.

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In this work, we discuss how to determine and implement feasible optimal observables for a metrology protocol that saturates the quantum Fisher information (QFI) bound. In particular, we focus our study on a simple protocol, namely the method of moments (MoM). We first demonstrate that the symmetric logarithmic derivative (SLD) operator, a Hermitian observable, once implemented in the MoM, saturates the QFI bound. However, the SLD is generally too complex and typically non-local, rendering its direct experimental realization unfeasible. To overcome this limitation, we explore its structure in a specific sensing model - a Floquet time crystal (FTC) acting as an ac field sensor - and show that the SLD can be approximated by substantially simpler observables, such as the bare spin magnetization or a parity observable, for different relevant initial state preparations. We further corroborate our theoretical predictions in a nuclear magnetic resonance system operating as an FTC sensor, employing experimentally motivated parameters to simulate its performance in a state-of-the-art implementation. In general, our results establish a practical route toward near-optimal metrology in FTC sensors, where the inaccessible SLD operator can be replaced by simpler observables while retaining quantum-enhanced sensitivity.
Quantum Fisher information (QFI) determines the ultimate precision of a quantum sensor after optimization over all allowed measurements, but it does not quantify how difficult the optimal readout is to implement. We introduce a moment Fisher hierarchy FM(k)F_M^{(k)}FM(k)​, which measures the fraction of QFI accessible to collective spin moments up to degree kkk, together with a readout-complexity scale k90k_{90}k90​ defined by the minimum degree required to recover 90%90\%90% of the QFI. Applying this framework to a nonlinear collective-spin sensor, we show that many-body dynamics can preserve large QFI while transferring the optimal readout into high-rank SU(2)SU(2)SU(2) tensor sectors inaccessible to low-order collective observables. For system sizes up to N=100N=100N=100, first moments access only ∼10−2\sim 10^{-2}∼10−2 of the available QFI, while the degree required to recover 90%90\%90% of it grows to k90≃28k_{90}\simeq 28k90​≃28. A rank-resolved analysis of the symmetric logarithmic derivative (SLD) under weak damping shows that the nonlinear protocol pushes the optimal readout toward the highest available tensor ranks while strongly depleting its low-rank weight. These results identify nonlinear tangent-space shear as the mechanism behind a finite-size readout-complexity bottleneck in many-body quantum metrology.
Studies of quantum metrology have shown that the use of many-body entangled states can lead to an enhancement in sensitivity when compared with unentangled states. In this paper, we quantify the metrological advantage of entanglement in a setting where the measured quantity is a linear function of parameters individually coupled to each qubit. We first generalize the Heisenberg limit to the measurement of nonlocal observables in a quantum network, deriving a bound based on the multiparameter quantum Fisher information. We then propose measurement protocols that can make use of Greenberger-Horne-Zeilinger (GHZ) states or spin-squeezed states and show that in the case of GHZ states the protocol is optimal, i.e., it saturates our bound. We also identify nanoscale magnetic resonance imaging as a promising setting for this technology.
We develop a quantum metrological framework for resonant nanophotonic sensors based on subwavelength Fabry--Perot slit cavities. Building on classical Fisher-information analyses of resonant transmission sensors, we model parameter encoding as a phase-and-loss quantum channel embedded in one arm of a Mach-Zehnder interferometer. We derive the quantum Fisher information (QFI) for coherent and Gaussian probe states under linear loss and show that, even at the quantum limit, optimal estimation precision is governed by the generator of parameter-dependent phase shifts rather than by the cavity quality factor. Consequently, the operating point that maximizes the QFI does not generally coincide with the maximum-Q resonance. Quantum resources enhance sensitivity but do not redefine the optimal geometry. Our results provide physically transparent design principles for quantum-enhanced nanophotonic sensing.
Quantum-enhanced sensing with atomic ensembles has predominantly focused on qubit-based protocols, despite the growing ability of many experimental platforms to coherently control and entangle multi-level systems. Here, we investigate quantum-enhanced sensing with qutrit ensembles by introducing three experimentally feasible qutrit generalisations of the one-axis twisting (OAT) model that involve entangling operations only between two levels, while the third level primarily acts as a spectator. We characterize the metrological utility of the dynamically generated states using the quantum Fisher information toolbox. We find numerically that all three models offer considerable freedom in encoding direction for quantum-enhanced sensing, with up to 6 out of 8 possible directions exhibit near-Heisenberg scaling after a short evolution time. We discuss experimental access to this enhanced metrological precision via effective time-reversal protocols. Furthermore, we examine the practical issues of estimation ambiguity and local dissipation, and show that they can be largely overcome by optimizing the sensor operating point. Finally, we show that the measurement incompatibility in estimating multiple parameters simultaneously encoded in different directions with near-Heisenberg scaling of precision is suppressed at the zero operating point as the system size increases. In the process, we find that one of the models enables near-Heisenberg scaling metrology with a pair of commuting generators, a possibility that arises from the su(3)su(3)su(3) algebra and is thus absent in qubit-ensemble based sensors of collective SU(2)SU(2)SU(2) rotations.
We investigate the use of a boundary time crystals (BTCs) as quantum sensors of AC fields. Boundary time crystals are non-equilibrium phases of matter in contact to an environment, for which a macroscopic fraction of the many-body system breaks the time translation symmetry. We find an enhanced sensitivity of the BTC when its spins are resonant with the applied AC field, as quantified by the quantum Fisher information (QFI). The QFI dynamics in this regime is shown to be captured by a relatively simple Ansatz consisting of an initial power-law growth and late-time exponential decay. We study the scaling of the Ansatz parameters with resources (encoding time and number of spins) and identify a moderate quantum enhancement in the sensor performance through comparison with classical QFI bounds. Investigating the precise source of this performance, we find that despite of its long coherence time and multipartite correlations (advantageous properties for quantum metrology), the entropic cost of the BTC (which grows indefinitely in the thermodynamic limit) hinders an optimal decoding of the AC field information. This result has implications for future candidates of quantum sensors in open system and we hope it will encourage future study into the role of entropy in quantum metrology.
Quantum-enhanced sensing is commonly benchmarked using the quantum Fisher information (QFI), often interpreted as a direct indicator of achievable precision. However, such an interpreta- tion is only justified within a fully specified inference framework that consistently incorporates state preparation, measurement design, resource accounting, estimator construction, prior information, and finite data effects. Here we develop an operational, end-to-end framework for quantum sensing under finite resources and identify general principles required for meaningful performance assess- ment. A central conceptual point is that the relevant unit of estimation is not a single detection event, but the complete data set required to construct a consistent estimator. Rather than relying on abstract statistical formalism, we analyze a set of paradigmatic and experimentally relevant ex- amples that expose common pitfalls in widely discussed quantum sensing strategies. These include quantum interferometry, sensing with squeezed states, and criticality-inspired protocols. Our results clarify when large Fisher information or nonclassical resources translate into genuine metrological advantage, and when they do not. They further provide a practical methodology for designing and evaluating quantum sensing protocols under realistic experimental constraints.