Quantum metrology aims to improve sensor performance beyond classical limits, as demonstrated in gravitational wave detectors using squeezed light. The quantum Cramér-Rao bound dictates that the accuracy of unbiased parameter estimation is limited by the inverse of the quantum Fisher information (QFI). The QFI, a geometric property of a quantum state in parameter space, is independent of the estimator. Finding optimal measurements that saturate this bound is crucial, achievable in small quantum systems through comparisons with theoretical predictions or full-state tomography. However, this becomes challenging in complex systems. Therefore, a universal method to measure QFI experimentally is needed. Existing methods, which often rely on precise parameter control and multiple measurements, including those using Loschmidt echo, Hellinger distance, Euclidean distance, and Bures distance, face scalability issues in many-qubit systems. Alternative approaches using quantum optimal control, variational algorithms, and random measurements require extensive iterations or measurements. This paper addresses these challenges by experimentally demonstrating near saturation of the Cramér-Rao bound in phase estimation using a nitrogen-vacancy (NV) center in diamond. Unlike previous studies relying on theoretical QFI estimations, this work uses purely experimental methods to measure QFI through spectroscopic responses under weak parametric modulations, circumventing the need for quantum-state tomography and offering a more scalable approach to complex systems. The experiment utilizes a Ramsey interferometer, a standard setting for phase parameter estimation, to determine optimal sensitivity using different resource states and comparing these to their individual QFI. The method is further extended to coupled qubits, exploring the relation between QFI and entanglement signatures.

The authors review previous work in quantum metrology, highlighting the quantum Cramér-Rao bound and the importance of quantum Fisher information (QFI) in determining the precision limits of parameter estimation. They discuss existing experimental methods for measuring QFI, such as those based on statistical distances between quantum states (e.g., using Loschmidt echo, Hellinger distance, Euclidean distance, and Bures distance), and computational methods like quantum optimal control and variational algorithms. These methods, however, face limitations in terms of scalability and experimental complexity, especially for many-qubit systems. The authors contrast their work with previous studies on parameter estimation in NV centers and other quantum systems, emphasizing the novelty of their approach in achieving a purely experimental determination of the QFI without resorting to quantum state tomography.

The experiment uses a nitrogen-vacancy (NV) center in diamond as a quantum sensor. The two-level system is encoded in the |0⟩ and |-1⟩ ground states of the NV center spin, manipulated using microwave pulses. A Ramsey interferometry protocol is implemented for phase parameter estimation. The system is initialized in a coherent superposition resource state |ψ₀⟩ = cos(θ/2)|0⟩ - sin(θ/2)|-1⟩, which evolves into |ψθ(β)⟩ = cos(θ/2)e^(iβ/2)|0⟩ - sin(θ/2)e^(-iβ/2)|-1⟩, encoding the phase parameter β. The measurement precision (δβ)ₘ is determined from the minimal detectable change in β, given by (δβ)ₘ = Δp/∂p, where p is the expectation value of the positive-operator valued measurement (POVM) signal, and Δp is the associated uncertainty. The quantum Cramér-Rao bound (1/δβ ≥ √Fβ) provides the fundamental limit to sensitivity, with Fβ representing the QFI. For pure states, Fβ is given by equation (4) in the paper. The QFI is measured directly by probing the coherent dynamical responses to weak parametric modulations. The NV center spin is initialized in the |0⟩ state using a laser pulse, prepared in the resource state via a microwave pulse, and then undergoes free evolution for time T, resulting in state |ψθ(β)⟩. Parametric modulation is applied via an amplitude and phase-modulated microwave field. This induces coherent transitions, monitored by measuring the probability of the system remaining in state |ψθ(β)⟩ using an inverse evolution sequence that returns the states to |0⟩ and |-1⟩ for readout. The parametric modulation efficiency is maximized when the modulation frequency matches the energy gap between |ψθ(β)⟩ and its orthogonal state. The QFI is extracted from the measured Rabi frequency using equation (7) in the paper. For parameter estimation, a rotation Yα is applied (equivalent to a projective measurement), yielding an observable p(β; θ, α). The parameter β is estimated from this observable at a working point (β ≈ π/2) where the slope ∂p/∂β is maximal. To account for measurement noise, due to limited collection efficiency, the number of signal photons is accumulated over multiple experimental runs. A quantity S is defined and an estimator for β is constructed, reducing the effects of measurement noise. The measurement uncertainty Δp is determined from repeated measurements. The optimal measurement sensitivity is found by varying the angle α of the projective measurement and the angle θ of the resource state. Finally, the methodology is extended to a two-qubit system (NV center and a nearby 13C nuclear spin) where the QFI is related to Rabi frequencies of induced transitions.

The researchers experimentally demonstrated near saturation of the quantum Cramér-Rao bound in phase estimation using a solid-state qubit (NV center in diamond). They achieved this through an independent experimental measurement of the Quantum Fisher Information (QFI) without the need for quantum state tomography. The QFI was extracted by probing coherent dynamical responses of the system to weak parametric modulations using a Ramsey interferometry setup. Their results showed excellent agreement between the experimentally measured QFI and theoretical predictions, clearly demonstrating the dependence of the QFI on the initial resource state. This validates their approach as a reliable method for experimentally determining the QFI. Specifically, the experimental measurement of the sensitivity (δβ) in phase estimation showed a linear relationship with the inverse square root of the experimentally measured QFI (δβ ∝ 1/√Fβ), confirming the near saturation of the quantum Cramér-Rao bound. The proportionality factor obtained was 1.041 ± 0.036, very close to the theoretical value of 1. Optimal measurement sensitivity was achieved by adjusting the projective measurement basis and the initial resource state, highlighting the ability to control the estimation precision. The methodology was further extended to a coupled two-qubit system (NV center and a nearby 13C nuclear spin), numerically demonstrating a correlation between the QFI of the ground state and entanglement, indicated by the concurrence, especially around avoided crossings in the energy spectrum. This suggests that their method of QFI measurement, through parametric modulations, might also provide insights into entanglement signatures in multi-qubit systems.

This work provides a significant advancement in quantum metrology by offering a scalable experimental approach to measure the quantum Fisher information (QFI) and verify the quantum Cramér-Rao bound. The ability to experimentally determine the QFI without quantum state tomography is a major improvement, making this technique applicable to complex quantum systems. The near saturation of the Cramér-Rao bound demonstrates the practical potential for achieving optimal quantum sensing. The extension of the technique to a two-qubit system opens up possibilities for studying entanglement properties and their relationship to metrological precision. The results have implications for developing more advanced quantum sensors and for fundamental studies of quantum systems, bridging the gap between theoretical predictions and experimental validation. The accurate determination of QFI is crucial for optimizing quantum sensors and identifying optimal measurement strategies in various applications.

The study successfully demonstrated a novel experimental method for measuring quantum Fisher information (QFI) and verified the quantum Cramér-Rao bound in a solid-state qubit. This method, based on spectroscopic responses to weak parametric modulations, avoids the need for quantum state tomography, improving scalability and applicability to complex systems. The near-saturation of the Cramér-Rao bound was experimentally confirmed, highlighting the potential for optimal quantum sensing. Further research could focus on applying this technique to larger-scale quantum systems, investigating the relationship between QFI and entanglement in more detail, and exploring applications in other areas like quantum speed limit studies and optimal control.

While the experiment achieved near-saturation of the Cramér-Rao bound, some limitations exist. The accuracy of the QFI measurement relies on the precise control of the engineered Hamiltonian and the accurate determination of the effective Rabi frequency. Imperfections in the experimental setup, such as limited collection efficiency in photon counting, could introduce noise and affect the accuracy of the measurement. The study focused on phase estimation; further investigation is needed to explore the applicability of the method to other types of parameter estimation. The extension to the two-qubit system was numerical, and experimental verification is necessary. Future work could explore mitigating the effects of noise and improving the accuracy of QFI measurements, expanding the scope to more complex scenarios.