Abstract
The quantum Cramér–Rao bound sets a fundamental limit on the accuracy of unbiased parameter estimation in quantum systems, relating the uncertainty in determining a parameter to the inverse of the quantum Fisher information. We experimentally demonstrate near saturation of the quantum Cramér–Rao bound in the phase estimation of a solidstate spin system, provided by a nitrogenvacancy center in diamond. This is achieved by comparing the experimental uncertainty in phase estimation with an independent measurement of the related quantum Fisher information. The latter is independently extracted from coherent dynamical responses of the system under weak parametric modulations, without performing any quantumstate tomography. While optimal parameter estimation has already been observed for quantum devices involving a limited number of degrees of freedom, our method offers a versatile and powerful experimental tool to explore the Cramér–Rao bound and the quantum Fisher information in systems of higher complexity, as relevant for quantum technologies.
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Introduction
Quantum metrology has emerged as a key quantum technological application. It allows for the improvement of sensors performance, beyond any classically achievable precision, as was demonstrated for instance in squeezedlightbased gravitational wave detectors^{1}. According to the quantum Cramér–Rao bound, the accuracy of any unbiased estimation of an unknown system parameter is limited by the inverse of the quantum Fisher information (QFI)^{2,3,4,5,6,7,8}. Importantly, the QFI only depends on the quantum state and is independent of the estimator; it is a geometric property of a quantum state in parameter space. Thus, for each parameter estimation problem, there potentially exists an optimal quantum measurement that saturates the Cramér–Rao bound. Such fully efficient estimators can be found for classical systems and for small quantum devices upon comparing to theoretical predictions^{9} or by performing fullstate tomography^{10}, which, however, becomes extremely challenging for quantum systems with higher complexity. Consequently, the identification of optimal quantum measurement schemes would highly benefit from a universal method to measure the QFI within the experimental setting. In general, this is a complicated task^{10,11,12}, which requires (by definition) a very precise determination of the “distance” (fidelity) between two quantum states. The quadratic coefficients of several fidelitylike quantities, such as Loschmidt echo^{13}, Hellinger distance^{11,12}, Euclidean distance^{14} and Bures distance^{15}, are related to the QFI. Hence, in principle, this allows for the evaluation of the QFI from the measurement of these quantities. The corresponding experiments have been demonstrated in an optical system^{14} and in BoseEinstein condensates^{11}. In experiment, these quantities are usually determined by the statistical distances of two experimental probability distributions, which are obtained by measuring two quantum states upon an infinitesimally small change of the system parameters^{4,5,16}. Considering these methods, the accurate estimation of the QFI requires precise control of system parameters and the ability to perform multiple measurements or even complete measurements^{15} on the system; this usually scales exponentially with the system size and remains challenging in manyqubit systems. Furthermore, the lower bound of the QFI can be obtained using quantum optimal control methods^{17}, variational algorithms^{18,19}, and random measurements^{20,21}, which typically require a large number of iterations or measurements.
In this work, we use a nitrogenvacancy center in diamond to perform a fully efficient phaseestimation quantum measurement by showing saturation of the Cramér–Rao bound. In contrast to a previous study^{9}, where a saturation of the bound was identified through a theoretical estimation of the QFI, we hereby demonstrate saturation through purely experimental means by independently measuring the QFI within our phaseestimation setting. This was achieved by directly probing spectroscopic responses upon weak parametric modulations, a technique which circumvents the stringent requirements of quantumstate tomography and avoids heavy experimental measurement overhead. This has the advantage of offering a more scalable approach to more complex systems. Our method is inspired by a proposal to extract the quantum metric tensor^{22,23}, which was recently implemented in NV centers^{24,25} and superconducting qubits^{26}. We demonstrate this approach in a Ramsey interferometer, which represents a standard experimental setting for the estimation of an unknown phase parameter. We determine the optimal sensitivity of the phaseparameter estimation through different resource states, and compare these results with their individual QFI. Finally, we demonstrate the applicability of our QFI measurement to the case of coupled qubits, and discuss its relation to entanglement signatures.
Results
Experimental setting
In the experiment, we utilize a nitrogenvacancy center (NV) in diamond as the quantum sensor. The ground state of the NV center spin has three spin sublevels m_{s} = ±1, 0. By applying an external magnetic field B_{z} ≃ 510 G along the NV axis, we lift the degeneracy of the spin states m_{s} = ±1 and use the two spin sublevels m_{s} = 0, −1, with states \(\left0\right\rangle\) and \(\left1\right\rangle\), to form a quantum twolevel system with an energy gap ω_{0} = D − γ_{e}B_{z}, where the zerofield splitting is D = (2π)2.87 GHz and γ_{e} is the electronic gyromagnetic ratio [Fig. 1(c)]. We use a microwave field to coherently manipulate the NV center spin sate; see Fig. 1(d) for an illustrative Rabi oscillation.
Quantum sensing and parameter estimation have been implemented in NV centers using different approaches^{27,28}, inspired by the pioneer proposal and demonstration of magnetometry based on Ramsey spectroscopy^{29,30,31}. Building on those developments, we hereby adopt the standard protocol of a phaseparameter estimation measurement by means of Ramsey interferometry [Fig. 1(a)]. For that purpose, we first initialize the system in a coherent superposition resource state, \(\left{\psi }_{\theta }(0)\right\rangle =\cos (\theta /2)\left0\right\rangle \sin (\theta /2)\left\,1\right\rangle\), which we evolve into
according to the applied magnetic field. The phase parameter β of \(\left{\psi }_{\theta }(\beta )\right\rangle\) can be estimated by performing positiveoperator valued measurements (POVM)^{5,8}, \({{{\mathcal{M}}}}=\{{{{{\mathcal{M}}}}}_{j}\}\); as explained below, these are provided by spindependent fluorescence measurements (see Supplementary Note 2). The measurement precision is defined as the minimal change of the parameter β that can be detected from the constructed observable above the shotnoise level,
where p is the expectation value of the POVM signal, Δp is the uncertainty associated with the measurement signal. The fundamental limit of the achievable sensitivity of an unbiased estimator is given by the quantum Cramér–Rao bound^{32,33,34}
where \({{{{\mathcal{F}}}}}_{\beta }\) denotes the QFI, which for pure quantum states \(\left{\psi }_{\theta }(\beta )\right\rangle\), is given by^{4,5}
The QFI characterizes the distinguishability of adjacent quantum states over the parameter space [Fig. 1(b)]. The purity of the states in our experiment, and hence the validity of Eq. (4) to capture the QFI, is discussed below. We note that the QFI is related to the real part of the quantum geometric tensor, which can be extracted through coherent dynamical responses^{22,24}.
It is one of the central goals of this work to show the saturation of the quantum Cramér–Rao bound through an independent experimental measurement of the QFI. We extract the QFI by probing coherent dynamical responses of the quantum system upon perturbative parametric modulations^{22,24}. Our measurement protocol is shown in Fig. 2(a). The NV center spin is first initialized in the m_{s} = 0 spin state by applying a green (532 nm) laser pulse, which also polarizes the nitrogen nuclear spin associated with the NV center as we tune the magnetic field close to the excited state level anticrossing (i.e., B_{z} ≃ 510 Gauss). The subsequent microwave pulse, applied for a duration t_{θ} = (θ/Ω), rotates the NV center spin around the \(\hat{y}\) axis by an angle θ according to the Hamiltonian \({H}_{1}(t)\,=\,({\omega }_{1}/2){\sigma }_{z}+{{\Omega }}\cos ({\omega }_{1}t){\sigma }_{x}\), where ω_{1} matches the energy gap between the spin sublevels m_{s} = 0, −1 and Ω is the microwave Rabi frequency. The rotation, denoted as Y_{θ}, prepares the NV center spin into the θdependent resource state \(\left{\psi }_{\theta }(0)\right\rangle\). After the microwave pulse Y_{θ}, the system undergoes a free evolution for a time T, according to an effective Hamiltonian \({H}_{2}^{(e)}\,=\,[({\omega }_{0}{\omega }_{1})/2]{\sigma }_{z}\), which results in the final state \(\left{\psi }_{\theta }(\beta )\right\rangle\); see Eq. (1). Here, the effective Hamiltonian \({H}_{2}^{(e)}\) is defined in the interaction picture with respect to H_{0} = (ω_{1}/2)σ_{z}. The final state \(\left{\psi }_{\theta }(\beta )\right\rangle\) encodes the information about the phase parameter β = ξT to be estimated, where ξ = ω_{1} − ω_{0}.
Direct measurement of the QFI
Inspired by the protocol of Ref. ^{22}, we extract the QFI of the final state \(\left{\psi }_{\theta }(\beta )\right\rangle\) by monitoring coherent transitions upon parametric modulations. This probing method requires the implementation of the following Hamiltonian
such that the state \(\left{\psi }_{\theta }(\beta )\right\rangle\) approximately corresponds to an eigenstate of \({{{\mathcal{H}}}}(\beta )\). This is achieved by tuning the parameters of the microwave driving field acting on the NV center spin. The key step of our experiment then consists in generating parametric modulations^{22}. To achieve this, we synthesize and calibrate an appropriate microwave driving field with proper amplitude and phase modulations, see Supplementary Note 1 and^{22,24} using an arbitrary waveform generator as follows
such that the “probing” Hamiltonian retains the form in Eq. (5), but with a timeperiodic modulation of the parameter β, i.e., \({{{\mathcal{H}}}}(\beta )\,\to \,{{{\mathcal{H}}}}[\beta (t)]={{{\mathcal{H}}}}(\beta +{a}_{\beta }\cos (\omega t))\), where a_{β} ≪ 1 quantifies the modulation amplitude.
The parametric modulation can induce a coherent transition from the state \(\left{\psi }_{\theta }(\beta )\right\rangle\) to the other orthogonal eigenstate \(\left{\psi }_{\theta }^{\perp }(\beta )\right\rangle\) of the Hamiltonian in Eq. (5)^{22,24}. This transition can be monitored by measuring the probability that the system remains in the state \(\left{\psi }_{\theta }(\beta )\right\rangle\). In the experiment, without requiring any prior information on the parameter β, we implement an inverse evolution sequence, consisting of two pulses (Y_{π} and Y_{π−θ}) separated by a free evolution of duration T [Fig. 2(a)]. Such an inverse evolution rotates the states \(\left{\psi }_{\theta }(\beta )\right\rangle\) and \(\left{\psi }_{\theta }^{\perp }(\beta )\right\rangle\) back to the states \(\left0\right\rangle\) and \(\left1\right\rangle\), respectively, see Supplementary Note 1. We then measure the population in state \(\left0\right\rangle\), which equals to the sought population in state \(\left{\psi }_{\theta }(\beta )\right\rangle\) after the application of the parametric modulation.
The efficiency of the coherent transition induced by the modulation is optimal whenever the modulation frequency matches the energy gap between the states \(\left{\psi }_{\theta }(\beta )\right\rangle\) and \(\left{\psi }_{\theta }^{\perp }(\beta )\right\rangle\). In the experiment, we first perform the modulationinducedtransition measurement for a wide range of modulation frequencies, from which we determine the resonant modulation frequency ω ≃ A; see Fig. 2(b). We then apply the parametric modulation at the resonant frequency, and measure the population in the state \(\left{\psi }_{\theta }(\beta )\right\rangle\) as a function of the perturbation duration τ; see Fig. 2(c). This data is fitted using a function \({P}_{0}\,=\,[1+\cos ({\nu }_{\theta }t)]/2\), which defines the effective Rabi frequency ν_{θ}. From this data, we extract the θdependent QFI, \({{{{\mathcal{F}}}}}_{\beta }(\theta )\), using the relation (see Methods and Supplementary Note 1)
This experimental measurement of the QFI is displayed in Fig. 2(d), which shows excellent agreement with the theoretical prediction \({{{{\mathcal{F}}}}}_{\beta }\,=\,{\sin }^{2}\theta\). In particular, it clearly demonstrates the dependence of the QFI on the initial resource state \(\left{\psi }_{\theta }(0)\right\rangle\). The precision of our measurement relies on the accuracy of the engineered Hamiltonian \({{{\mathcal{H}}}}(\beta )\) and on the determination of the effective Rabi frequency ν_{θ}. The imperfection in the interrogation step [Fig. 2(a)] may result in a mixed state rather than a pure state \(\left{\psi }_{\theta }(\beta )\right\rangle\); this would decrease the contrast of the Rabi oscillations and affect the measurement accuracy. By reconstructing the density matrix through projective measurements, we estimate the state fidelity to be above 95% in our experiment, see Supplementary Note 2, which is evidenced by the good agreement between our results and the theoretical predictions.
Reaching the quantum Cramér–Rao bound
The QFI measurement enables us to experimentally show that our phaseparameter estimator exhibits optimal performance by saturating the quantum Cramér–Rao bound in Eq. (3). In order to analyze the relation between the measurement precision and the QFI, we now determine the measurement sensitivity for the estimation of the parameter β within our Ramsey interferometry experiment. To do so, we first apply the rotation Y_{θ} on the NV center spin qubit to prepare the initial state \(\left{\psi }_{\theta }(0)\right\rangle\); the system then evolves freely for a time T = β/ξ. To build an estimator of the parameter β, we apply a rotation Y_{α}, which is equivalent to a projective measurement \({P}_{\alpha }\,=\,\left{\phi }_{\alpha }\right\rangle \left\langle {\phi }_{\alpha }\right\) on the final state \(\left{\psi }_{\theta }(\beta )\right\rangle\), where \(\left{\phi }_{\alpha }\right\rangle =\cos (\alpha /2)\left0\right\rangle +\sin (\alpha /2)\left1\right\rangle\) [Fig. 1(a)]. The observable of interest is then provided by the function \(p(\beta ;\theta ,\alpha )\,=\,\left\langle {\psi }_{\theta }(\beta )\right{P}_{\alpha }\left{\psi }_{\theta }(\beta )\right\rangle\), from which we aim to estimate the parameter β with optimal accuracy [Eq. (3)]. We tune the free evolution time such that the parameter β = ξT is close to the working point where the best sensitivity occurs, i.e., β ≃ π/2 where the slope ∂p/∂β is maximal [Fig. 3(a)].
Ramsey parameter estimation can, in principle, achieve optimal efficiency. However, in practice, this would require an ideal projective measurement of the sensor upon reaching the shotnoise limit. Such an ideal measurement cannot be perfectly performed, due to a limited collection efficiency or other types of measurement noise (e.g., Gaussian fluctuations in the photon number). To overcome this limitation, one may adopt the technique of singleshot readout^{35,36,37,38}, which consists in setting a threshold n_{s} of photon number to distinguish the state \(\left{m}_{s}=1\right\rangle\) and \(\left{m}_{s}=0\right\rangle\) and assign a value s = 0 or 1 depending on whether n_{j} > n_{s} or n_{j} < n_{s}.
In our experiment, the observable \(p(\beta ;\theta ,\alpha )\,=\,\left\langle {\psi }_{\theta }(\beta )\right{P}_{\alpha }\left{\psi }_{\theta }(\beta )\right\rangle\) is estimated from the collected photons of a fluorescence signal (see Methods). Due to the limited collection efficiency, the signal photons are accumulated over many sweeps of an experimental sequence, which constitutes one experimental run of our measurement. In the jth run, based on the photon number n_{j} detected from the rotated spin state \({Y}_{\alpha }\left{\psi }_{\theta }(\beta )\right\rangle\), we define the ratio p_{j} = (n_{j} − n_{1})/(n_{0} − n_{1}) where n_{0} and n_{1} are the average photon numbers obtained from the bare spin states m_{s} = 0 and m_{s} = −1, respectively. We proceed to assign a measurement value s_{j} = k + 1 or k according to the probabilities \({p}_{j}^{(k)}={p}_{j}k\) and \(1{p}_{j}^{(k)}\) for ⌊p_{j}⌋ = k, see Methods. This allows us to introduce a quantity \(S\,=\,(1/N)\mathop{\sum }\nolimits_{j = 1}^{N}\,{s}_{j}\), whose expectation value yields the desired function \(\left\langle S\right\rangle \,=\,p(\beta ;\theta ,\alpha )\). Using this quantity, we can construct an estimator for the parameter β, and find that the influence of measurement noise on S is eliminated to a large extent (apart from the shotnoise), which also provides a data analysis alternative for the spin readout techniques of NV centers^{35,36,37,38}, see Methods. The data obtained from repeated measurements [Fig. 3(a)] allows us to determine the slope of the signal, which is defined as \({\chi }_{\alpha }\,=\,\partial p/\partial \beta \,=\,\left[p(\beta +d\beta )p(\beta )\right]/d\beta\). From the experimental data, we can also extract the measurement uncertainty Δp associated with the observable S; see Fig. 3(b). We note that the uncertainty scales with the number of repetitions N as \({{\Delta }}p\,=\,{{{\Delta }}}_{0}/\sqrt{N}+{\xi }_{0}\), see Methods. The first term arises from the shotnoise with Δ_{0} = [p(1−p)]^{1/2}, while the second term ξ_{0} represents the contribution from the measurement fluctuation that cannot be averaged out. We remark that other advanced readout techniques, such as the singleshot measurement based on spin to charge conversion^{39}, can further reduce such measurement noise (see Eq. ((18), (21)) in Methods) and enhance the sensitivity.
We first compare the sensitivity δβ = Δp/χ_{α} obtained by projective measurements over different bases P_{α}. The experimental results shown in the inset of Fig. 3(c) demonstrate that the optimal measurement sensitivity is obtained when α = π/2, which agrees with the theoretical prediction (see Supplementary Note 2), \({(\delta \beta )}^{2}\,=\,[1{(\cos \beta \sin \theta )}^{2}]/ \sin \beta \sin \theta { }^{2}\). The slight deviation arises from other sources (apart from shot noise). The measurement precision also depends on the angle θ of the resource state \(\left{\psi }_{\theta }(0)\right\rangle\), which accounts for the QFI of the final state \(\left{\psi }_{\theta }(\beta )\right\rangle\): we proceed by determining the optimal measurement sensitivity with different resource states \(\left{\psi }_{\theta }(0)\right\rangle\) in view of testing the quantum Cramér–Rao bound in Eq. (3). It can be seen from the results shown in Fig. 3(c) that the optimal measurement sensitivity improves as the angle θ approaches π/2, i.e., when the resource state \(\left{\psi }_{\theta }(0)\right\rangle\) becomes a maximally coherent superposition state. We remark that the result in the inset of Fig. 3(c) is skewed as the pulse Y_{α} is offresonant; the influence of the corresponding detuning is the asymmetry observed around α = π/2. Moreover, the optimal measurement sensitivity verifies the quantum Cramér–Rao bound [Eq. (3)], as we finally demonstrated in Fig. 3(d).
Generalization to entangled qubits
Single NV centers in diamond allow to perform quantum sensing with unprecedented spatial resolution^{40}. In this context, the saturation of the Cramér–Rao bound is of particular importance as it may allow quantum sensing with unparalleled accuracy. Still, it is a natural question whether our QFI measurement can also be extended to the multiqubit case, where quantum entanglement can provide a further key factor to increase the performance of a quantum sensor.
For that purpose, we now demonstrate the applicability of our parametric modulation scheme in view of measuring the QFI in a realistic twoqubit correlated system^{24}, which consists of an NV center and a nearby strongly coupled ^{13}C nuclear spin via the hyperfine interaction. The effective Hamiltonian of the system is given by (see Supplementary Note 3)
where σ and τ denote the Pauli matrices of the NV center and of the ^{13}C nuclear spin, respectively. We denote the four eigenstates of this Hamiltonian as \(\left{{{\Psi }}}_{1}\right\rangle\), \(\left{{{\Psi }}}_{3}\right\rangle\), \(\left{{{\Psi }}}_{3}\right\rangle\) and \(\left{{{\Psi }}}_{4}\right\rangle\), with their associated eigenvalues ϵ_{1} < ϵ_{2} < ϵ_{3} < ϵ_{4}. Similarly to the single qubit case treated above, we are interested in the quantumparameterestimation problem associated with the parameter β, and in particular, to the related QFI. Without loss of generality, we focus our study on the QFI contained in the lowestenergy eigenstate \(\left{{{\Psi }}}_{1}\right\rangle\).
Considering the parametric modulation \(\beta (t)=\beta +a\cos (\omega t)\), the QFI can be related to the three Rabi frequencies ν_{k} associated with the induced transitions between the ground state \(\left{{{\Psi }}}_{1}\right\rangle\) and the other three eigenstates \(\left{{{\Psi }}}_{k}\right\rangle\) according to
where ω_{k} = ϵ_{k} − ϵ_{1}. We have performed a numerical simulation of this setting and we present the results in Fig. 4(a). We find that the QFI of the ground state reaches its peak value when the energy of the corresponding eigenstate becomes very close to another eigenenergy, in the form of an avoided crossing (see Supplementary Note 3). In this situation, a small variation of the parameter (i.e., a perturbation) would indeed result in a significant change of the ground state. Importantly, this increase of the QFI is accompanied with a significant growth of entanglement, as quantified by the concurrence^{41}, as we demonstrate in Fig. 4(b). The connection between the QFI and the entanglement of such a coupledqubit setting (see Supplementary Note 3) is known to arise from the level anticrossing^{42,43}, which represents a general feature in systems beyond the singlequbit context. These results suggest that a large QFI is linked to strong entanglement upon measuring the QFI based on parametric modulations as introduced here. We remark that the proposed protocol can be extended to experimentally determine the QFI of manybody quantum systems by measuring the excitation rate under parametric modulation following the idea as presented in Ref. ^{22}. The approach does not require full state tomography, which is an experimentally demanding task for a multiqubit system. The present technique which allows us to estimate the QFI, and hence the quantum Cramér–Rao bound, will be helpful in resolving the challenging task of determining the optimal measurement for manybody ground states that can reach the bound.
Discussion
In this work, we have introduced an experimental technique to measure the QFI in a solidstate spin system based on spectroscopic responses. Importantly, this approach does not require full state tomography, and it can therefore be potentially applied to more complex systems. We have shown that this technique offers a genuine experimental probe of the quantum Cramér–Rao bound saturation, which does not rely on any theoretical knowledge, hence providing a universal tool to identify fully efficient estimators. The presented technique provides a versatile tool to explore the fundamental role of the QFI in various physical scenarios, including quantum metrology, but also entanglement properties of manybody quantum systems^{23,44} and the quantum speed limit in the context of optimal control^{45,46,47,48,49}.
Methods
QFI and parametric modulationinduced transition
At first we can rewrite the QFI (4) of \(\left{\psi }_{n}(\beta )\right\rangle\) which is the nth eigenstate of Hamiltonian \({{{\mathcal{H}}}}(\beta )\), i.e., \({{{\mathcal{H}}}}(\beta )\left{\psi }_{n}(\beta )\right\rangle ={E}_{n}(\beta )\left{\psi }_{n}(\beta )\right\rangle\)
with the following identity
Now, we consider the weak parametric modulation i.e., a_{β} ≪ 1, the timeperiodic Hamiltonian can be expanded as
According to timedependent perturbation theory, if the system is initialized in the eigenstate state \(\left{\psi }_{n}(\beta )\right\rangle\), the second term will excite the system to another eigenstate \(\left{\psi }_{k}(\beta )\right\rangle\) under a resonate condition ω = ω_{kn} = E_{k}(β) − E_{n}(β). In the subspace spanned by \(\{\left{\psi }_{k}(\beta )\right\rangle ,\left{\psi }_{n}(\beta )\right\rangle \}\), the parametric modulation Hamiltonian can be written as
with \({{{\Omega }}}_{kn}={a}_{\beta }\left\langle {\psi }_{k}(\beta )\right{\partial }_{\beta }{{{\mathcal{H}}}}(\beta )\left{\psi }_{n}(\beta )\right\rangle\). This Hamiltonian induces a Rabi oscillation between \(\left{\psi }_{n}(\beta )\right\rangle\) and \(\left{\psi }_{k}(\beta )\right\rangle\) with the corresponding Rabi frequency \({\nu }_{k}= {{{\Omega }}}_{kn} ={a}_{\beta } \left\langle {\psi }_{n}(\beta )\right{\partial }_{\beta }{{{\mathcal{H}}}}(\beta )\left{\psi }_{k}(\beta )\right\rangle \), and the QFI becomes^{22,24}
which gives the two specific forms ((7), (9)) in the qubit system.
Quantum parameter estimation protocol
The sensitivity of quantum parameter estimation is dependent on the measurement protocol. In the experiment, we perform projective measurement on the NV center spin that is described by the operator \({\hat{P}}_{\alpha }=\left{\phi }_{\alpha }\right\rangle \left\langle {\phi }_{\alpha }\right\) with the basis state \(\left{\phi }_{\alpha }\right\rangle =\cos (\alpha /2)\left0\right\rangle +\sin (\alpha /2)\left1\right\rangle\). We count the number of photons in the first 300 ns of the laser pulse as the signal photons. Due to the limit of collection efficiency, the signal photons are accumulated over a number of sweeps of an experimental measurement sequence, which constitutes one experiment run of measurement. We denote the averaged photon number obtained from the bare spin state m_{s} = 0 and m_{s} = −1 as n_{0} and n_{1} respectively. We introduce a variable s = 1/0 to represent the spin state m_{s} = 0/m_{s} = −1. For the NV center spin system, the signal photons are spindependent, namely (n_{0} − n_{1})/n_{0} ≃ 30%, see Fig. 5(a). For a quantum state ρ with the state \(\left0\right\rangle\) population \(p=\left\langle 0\right\rho \left0\right\rangle\), the number of photons n_{j} collected in the jth experiment run fluctuates and follows the distribution \({n}_{j} \sim p{{{\mathcal{N}}}}({n}_{0},{\sigma }_{0}^{2})+(1p){{{\mathcal{N}}}}({n}_{1},{\sigma }_{1}^{2})\), see an example shown in Fig. 5(b). According to the properties of the normal distribution, the random variable p_{j} = (n_{j} − n_{1})/(n_{0} − n_{1}) follows the probability distribution Q(p_{j})
where Δn = n_{0} − n_{1} and \({\tilde{\sigma }}_{m}={\sigma }_{m}/{{\Delta }}n,\,m=0,1\). Q(p_{j}) is shown in Fig. 6 and is divided into a series of intervals by the integers k = ⌊p_{j}⌋.
Based on the distribution Q(p_{j}), we proceed to assign a measurement value s_{j} = k + 1 or k according to the probabilities \({p}_{j}^{(k)}={p}_{j}k\) and \(1{p}_{j}^{(k)}\) in the k^{th} interval. This allows us to construct a quantity as \(S=(1/N)\mathop{\sum }\nolimits_{j = 1}^{N}{s}_{j}\), the expectation value of which is
The variance of the quantity S is given by
The first term can be calculated as
We define p_{j} = ⌊p_{j}⌋ + δ_{j} = k + δ_{j} with δ_{j} ∈ [0, 1), and F_{k} can be write as
It can be seen that for all k ∈ Z, F_{k} ≥ 0, and if and only if k = 0, F_{k} = 0. Therefore, Eq. (18) satisfies
and the variance of the quantity S is bounded by the shot noise
If the distribution Q(p_{j}) is strictly localized in the zeroth (k = 0) interval, i.e the black areas in Fig. 6 are negligible, all the components F_{k} ≃ 0. Therefore, the variance of the observable S achieves the shot noise
In our experiment, the measurements are performed at the working points β = π/2, which makes p ≃ 1/2 and
with \(\sigma =(1/2)\sqrt{{\tilde{\sigma }}_{0}^{2}+{\tilde{\sigma }}_{1}^{2}}\). The distribution of p_{j} obtained in our experiment satisfies \(\sqrt{{\tilde{\sigma }}_{0}^{2}+{\tilde{\sigma }}_{1}^{2}}\simeq 1/2\), which guarantees a more than 95% confidence interval of k = 0 (see Fig. 6).
Furthermore, we note that
thus we can construct the following estimator for the parameter β as
where N_{0} and N_{1} represents the number of s_{j} = 0 and 1 respectively. With α = π/2, the estimator becomes
The precision can be written as
which gives the optimal sensitivity with α = π/2 satisfying the quantum Cramér–Rao bound at the working point β = π/2.
Data availability
The data generated during this work are available from the corresponding author J.M.C upon reasonable request.
Code availability
The code used for generating the plots are available from the corresponding author J.M.C. upon reasonable request.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (11874024, 11690032), the National Key R&D Program of China (Grant No. 2018YFA0306600), the Open Project Program of Wuhan National Laboratory for Optoelectronics (No. 2019WNLOKF002). T.O. is supported by JSPS KAKENHI Grant Number JP18H05857, JST PRESTO Grant Number JPMJPR19L2, JST CREST Grant Number JPMJCR19T1, and the Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) at RIKEN. N.G. is supported by the ERC Starting Grant TopoCold and the Fonds De La Recherche Scientifique (FRSFNRS) (Belgium). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 853443), and M.H. further acknowledges support by the Deutsche Forschungsgemeinschaft via the Gottfried Wilhelm Leibniz Prize program.
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M.Y. and Y.L. contributed equally to this work. M.Y., P.C.Y., M.S.G., Q.Y.C., J.M.C. performed the experiments, Y.L., S.L.Z., H.B.L., M.H., T.O., N.G., J.M.C. performed theoretical analysis and simulation, Y.L., M.H., T.O., N.G., J.M.C. wrote the manuscript. All authors discussed the results, and agreed with the conclusions.
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Yu, M., Liu, Y., Yang, P. et al. Quantum Fisher information measurement and verification of the quantum Cramér–Rao bound in a solidstate qubit. npj Quantum Inf 8, 56 (2022). https://doi.org/10.1038/s4153402200547x
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DOI: https://doi.org/10.1038/s4153402200547x
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