Quantum Fisher information measurement and verification of the quantum Cramér-Rao bound in a solid-state qubit

The study addresses how to experimentally verify and saturate the quantum Cramér-Rao bound (QCRB) for phase estimation by independently measuring the quantum Fisher information (QFI) in a solid-state qubit. Quantum metrology aims to surpass classical precision limits; the QCRB binds the variance of any unbiased estimator by the inverse QFI, a geometric property of the quantum state independent of the estimator. While optimal estimators and QCRB saturation are straightforward in small systems or via full state tomography, such approaches become intractable as system complexity grows. The authors propose and implement a tomography-free, scalable spectroscopy-based method to directly measure QFI in an NV center Ramsey interferometer and compare it against achieved estimation precision to demonstrate near-saturation of the QCRB. They further explore applicability to coupled qubits and links to entanglement.

Previous QFI evaluations rely on precise fidelity estimates between nearby quantum states or on quadratic coefficients of fidelity-like distances (Loschmidt echo, Hellinger, Euclidean, Bures), with demonstrations in optical systems and Bose-Einstein condensates. These typically demand fine parameter control and extensive measurements, often scaling exponentially with system size. Alternatives providing lower bounds use optimal control, variational algorithms, and randomized measurements, which may require many iterations or samples. Recent proposals link QFI to the quantum metric tensor accessible through coherent dynamical responses under periodic driving; implementations in NV centers and superconducting qubits have measured quantum geometry. Building on these, the present work realizes a direct, scalable QFI measurement via parametric modulation spectroscopy in a solid-state spin, enabling verification of QCRB saturation without full tomography.

Platform: A single nitrogen-vacancy (NV) center in diamond is used as a two-level system encoded in ground-state spin sublevels ms=0 and ms=−1, split by an applied magnetic field B≈510 G along the NV axis, yielding energy gap ω0 = D − γe Bz with D/(2π)=2.87 GHz. Microwave fields coherently control the spin. Resource-state preparation and phase encoding: The NV is optically initialized into |0⟩, then rotated by a microwave Yθ pulse to prepare |ψ0⟩=cos(θ/2)|0⟩−sin(θ/2)|−1⟩. Free evolution under Heff=[(ω0−ω1)/2]σz for duration T encodes a phase β=ξT with ξ=ω0−ω1, producing |ψθ(β)⟩=cos(θ/2)e^{iβ/2}|0⟩−sin(θ/2)e^{−iβ/2}|−1⟩. Direct QFI measurement by parametric modulation: An engineered effective Hamiltonian H(β) is realized via amplitude- and phase-modulated microwave driving such that |ψθ(β)⟩ is (approximately) an eigenstate of H(β). The parameter is weakly modulated as β(t)=β+αβ cos(ωt) (αβ≪1). The modulation induces coherent transitions between the two eigenstates of H(β) with Rabi frequency ν and energy gap ω. An inverse evolution sequence (two pulses Yθ and Y−θ separated by free evolution) maps populations from the instantaneous eigenbasis back to the physical |0⟩ and |−1⟩ states, allowing readout via spin-dependent fluorescence. Procedure: (1) Sweep modulation frequency to identify the resonance ω (transition peak). (2) At resonance, record coherent oscillations of the population versus modulation duration τ to extract the effective Rabi frequency νe from Po[1+cos(νe τ)]/2 fits. Relation to QFI: For a qubit, the QFI is obtained from the resonant response as Fβ = 4(νe/ω)^2 / (αβ)^2, i.e., Fβ=4 ν^2/ω^2 (generalized to multi-level as a sum over transitions). Verification of QCRB via Ramsey estimation: A Ramsey interferometer estimates β using a projective measurement Pα=|Φα⟩⟨Φα| with |Φα⟩=cos(α/2)|0⟩+sin(α/2)|−1⟩, yielding signal p(β;θ,α)=⟨ψθ(β)|Pα|ψθ(β)⟩. The working point is set near β=π/2, maximizing ∂p/∂β. Due to limited photon collection, the observable is constructed from photon counts using an estimator S with near shot-noise-limited variance; repeated runs provide Δp and slope χα=∂p/∂β to compute sensitivity δβ=Δp/χα. Two-qubit extension (NV electron + 13C nuclear spin): An effective two-qubit Hamiltonian H=½[cosβ σz + sinβ(cosφ σx + sinφ σy)] + 2Tx τx − 2Tz τz is considered. Under β(t) modulation, transitions from the ground state |Ψ1⟩ to excited states |Ψk⟩ have Rabi frequencies νk at energy gaps ωk=εk−ε1, giving Fβ=4 Σk (νk^2/ωk^2). Numerical simulations explore QFI and entanglement (concurrence) versus parameters, highlighting behavior near avoided crossings. State purity and calibration: Imperfections in interrogation may yield slight mixedness; state fidelity is reconstructed via projective measurements and exceeds 95%. Microwave modulation waveforms are synthesized and calibrated to implement H(β) with the desired amplitude/phase modulations.

- Direct, tomography-free measurement of the QFI in a solid-state qubit: The experimentally extracted QFI from parametric modulation spectroscopy matches the theoretical prediction F=sin^2θ across resource states, demonstrating the dependence on the initial superposition angle θ. - Near-saturation of the quantum Cramér-Rao bound: By independently measuring QFI and the Ramsey estimation sensitivity, the product δβ × √Fβ is found to be 1.041 ± 0.036, indicating near-optimal performance consistent with the QCRB. - Optimal measurement basis: The best sensitivity is achieved for projective measurements with α=π/2, as predicted, and sensitivity improves as θ→π/2 (maximally coherent resource state). - Noise and scaling: The measurement uncertainty Δp extracted from repeated runs shows near shot-noise scaling with repetitions N (approximately ∝1/√N), with an additional small offset attributed to measurement fluctuations; a data-processing estimator S mitigates readout noise from limited photon collection. - State preparation quality: Reconstructed state fidelity exceeds 95%, supporting the applicability of pure-state QFI expressions. - Two-qubit generalization: Numerical simulations for an NV–13C coupled system show that QFI of the ground state peaks near avoided level crossings, coinciding with enhanced entanglement (concurrence), indicating that large QFI correlates with strong entanglement in coupled qubits.

The work demonstrates, in a fully experimental manner, that a Ramsey phase-estimation protocol with an NV center can reach the quantum Cramér-Rao limit by independently measuring the QFI without state tomography. The spectroscopy-based QFI extraction via weak parametric modulation provides a practical and scalable tool for quantifying information content of quantum states directly in the experimental platform. The agreement between measured QFI and theoretical F=sin^2θ, together with the optimal measurement basis and observed δβ × √Fβ ≈ 1, confirms that the implemented estimator is fully efficient. Extending the approach conceptually to coupled qubits shows that QFI enhancement near avoided crossings correlates with entanglement growth, suggesting a route to assess and exploit entanglement-enhanced metrology. Overall, the method enables identification and validation of optimal measurements in more complex quantum sensors where conventional tomography is prohibitive.

The authors introduced and implemented a direct, tomography-free measurement of quantum Fisher information in a solid-state spin using parametric modulation spectroscopy and verified near-saturation of the quantum Cramér-Rao bound in Ramsey phase estimation. The method’s scalability and independence from detailed theoretical modeling make it a universal tool for optimizing quantum sensors. Simulations in a coupled-qubit setting indicate that high QFI aligns with strong entanglement near avoided crossings, opening avenues to engineer entanglement-enhanced sensing. Future work includes experimental application to multi-qubit and many-body systems, leveraging the protocol to map quantum geometry, identify optimal measurement strategies, probe multipartite entanglement, and explore quantum speed limits and optimal control in complex platforms.

- Measurement noise and finite detection efficiency: Limited photon collection introduces additional fluctuations beyond shot noise; while mitigated by the estimator S and could be further reduced by advanced single-shot readout or spin-to-charge conversion, residual noise contributes to slight deviations from ideal scaling and symmetry. - Hamiltonian engineering accuracy: The QFI extraction relies on precise implementation of the effective Hamiltonian H(β) and accurate determination of resonant frequencies and Rabi rates; imperfections (e.g., off-resonant effects, detuning) can affect measurement accuracy. - State purity: Imperfections in interrogation can lead to slight mixedness (state fidelity >95%), potentially reducing Rabi contrast and impacting QFI estimation. - Weak-modulation regime: The approach presumes αβ≪1 for perturbative validity; larger modulations could introduce systematic errors. - Multi-qubit demonstration: The two-qubit results presented are based on numerical simulations of a realistic NV–13C system; an experimental multi-qubit verification was not reported here. - Resource requirements: Accumulation over many experimental sweeps per run is needed due to limited collection efficiency, which may limit real-time sensing applications.