Fitting magnetic field gradient with Heisenberg-scaling accuracy

The linear function is possibly the simplest and the most used relation appearing in various areas of our world. A linear relation can be generally determined by the least square linear fitting (LSLF) method using several measured quantities depending on variables. This happens for such as detecting the gradient of a magnetic field. Here, we propose a quantum fitting scheme to estimate the magnetic field gradient with N-atom spins preparing in W state. Our scheme combines the quantum multi-parameter estimation and the least square linear fitting method to achieve the quantum Cramér-Rao bound (QCRB). We show that the estimated quantity achieves the Heisenberg-scaling accuracy. Our scheme of quantum metrology combined with data fitting provides a new method in fast high precision measurements.

M agnetometry is important for mineral exploration and probing moving magnetic objects. High precision magnetometry 1-16 also has wide applications in modern sciences and technologies, such as in nuclear magnetic resonance (NMR) 17 , magnetic resonance imaging (MRI) 18,19 , biomedical science 20 and quantum control 21 . In some cases, the quantity interested is not the absolute strength of magnetic field but its difference and gradient. A standard measuring instrument for determining the gradient is differential atom interferometry, which utilizes two completely polarized atomic ensembles. Recently, quantum-enhanced measurements of magnetic field gradient have been proposed 22-25 . It is by now well established that quantum metrology has advantages in enhancing precision of estimation 26 which is beyond the classical method. In quantum metrology, the general framework for precision bound of estimation has been proposed and developed in Refs. 27-34, which is based on Fisher information (FI) and Cramér-Rao inequality. The precision of estimation depends on the amount of resources employed in the scheme, which might be for instance the number N of identical probes (photons, atoms) or the energy of probing field. The standard quantum limit, a consequence of the central limit theorem for statistics, shows that the precision is proportional to 1 . ffiffiffiffi N p . With quantum strategies such as entanglement and squeezing applied, one may attain better accuracy scaling as 1/N, which is the ultimate limit of precision named as Heisenberg limit. The NOON and GHZ states have been demonstrated to be able to provide a Heisenberg-limit sensitivity in some schemes [35][36][37][38][39][40][41] . Also some experiments have implemented the quantum enhanced metrology [42][43][44][45][46][47] .
In this work, we propose a quantum scheme of multi-parameter estimation to detect the gradient of magnetic field by employing N-atom spins. These atoms are initially prepared in W state, a genuine multipartite entangled state that can be generated in spin chain 48 and has been experimentally produced by trapped ions 49 . These technologies can be utilized to implement our scheme in experiment. By applying the least square linear fitting method to the quantum enhanced multi-parameter estimation, we show that our scheme saturates the QCRB with Heisenberg-scaling accuracy. Let us highlight some advantages of this scheme: (i) Our scheme does not depend on the prior assumed linear assumption for the magnetic field, we essentially apply the reliable LSLF method. We also discuss that even if the linearity of the magnetic field is prior assumed, the bound of precision is exactly the same one. (ii) This simultaneous estimation scheme is in principle faster than repeated individual estimations. (iii) This is a general quantum fitting method and can be applied to measure other physical quantities with various fitting functions.
N-atom spin chain as the probes, as shown in FIG. 1, to estimate the magnetic field gradient, where the j-th atom is located at x j 5 x 1 1 (j 2 1)a, (j~1, 2, Á Á Á , N) and the uncertainty of the location x j can be neglected. The Hamiltonian describes that each atom with two hyperfine spin states is coupled to the local magnetic field, and it takes the form,Ĥ~{ where B j andŝ j z are the magnetic field and Pauli operator of atom j, and each atom has the same gyromagnetic ratio c. The task of our scheme is to obtain optimal uncertainty bound of estimating the magnetic field gradient G that quantum mechanics permitted.
Initially, the atomic spins are prepared in a W state y 0 j i1 Because of an overall unobservable phase, it is proper to think that B 1 5 0 always holds. Thus the covariance matrix Cov(B) and Fisher information matrix F B ð Þ are size (N 2 1) 3 (N 2 1). Generalizing the expression of estimation for unitary dynamical processes 33 , the quantum Fisher information (QFI) matrix is given by , where d m,n is Kronecker's delta. Then the (N 2 1) 3 (N 2 1) sized QFI matrix and its inverse associated with the estimation of the magnetic field in our scheme is where m, n~2, 3, Á Á Á , N.
Applying the LSLF method, we have the fitting gradient of the magnetic field as, where x~X Because each atom is separated with a distance a in the x-direction, then we get Thus the gradient of magnetic field is G~X Since the uncertainties of x j are neglected, the quantum Cramér-Rao inequality gives a lower bound on the variance of the magnetic field gradient This bound is clearly a Heisengberg-scaling accuracy for large N. And the commutability of corresponding symmetric logarithmic derivatives (SLD) guarantees this bound can be saturated. Now, we turn to the problem of constructing measurement strategy that can achieve quantum advantages in multi-parameter estimation. In this scheme, we construct two von Neumann measurement strategies, labeled by a, b respectively, be performed on the atomic spin chain as the following forms, where k~1, 2, Á Á Á , N{1. Both of these two sets of quantum states are orthonormal eigenstates of the coherence operator expressed aŝ see supplementary material for detailed calculations. For strategy b, the limiting process B R 0 is equivalent to the small phases requirement ctB j =1 of local estimation theory in Method Section. For strategy a, we firstly choose the path B j 5 (j 2 1)Ga to approach the limit B R 0. It's interesting that the Fisher information matrix has the same expression if one supposes that B j 5 (j 2 1)Ga. So Eq. (10) takes the limit of B j R (j 2 1)Ga. For measurement strategy a, the Fisher information matrix is positive semi-definite and irreversible, which confirms that it is not an effective deterministic estimation. Applying Fourier transforma- Þ l j . This shows that the prerequisite for determining the magnetic field B j is knowing the module and argument of all l j . Because the probability distributions associated with experimental outcomes are p(jjB) 5 jl j j 2 /N, it is impossible to determine the argument of l j . Thus this strategy is invalid for estimating magnetic field B j . For measurement strategy b, which yields the QFI matrix, the probability of each outcome is transparently related to the magnetic field B, with p(1jB) involving only B 2 , p(2jB) involving only B 2 , B 3 , and so on 34 . Through comparing the ratio of observed measurement outcomes with the probability distributions, the estimator could sequentially determine the magnetic field B 2 ,B 3 , Á Á Á ,B N . Then the gradient can be obtained by applying the LSLF method. Based on the results of asymptotically large n independent experiments, this measurement strategy is optimal which can locally achieve quantum Cramér-Rao bound with Heisenberg-scaling accuracy. It is intriguing to explore how bad will be the degradation of this Heisenberg-scaling accuracy as some realistic imperfections kick in. Further researches are needed to conduct when one considers relevant imperfections like decoherence and particle losses.
Single parameter estimation with linear assumption. If we assume that the magnetic field satisfies the linear condition B j 5 B 1 1 G(j 2 1)a, the single parameter representing gradient G of magnetic field needs be estimated. In this case, the unitary transformation for the atomic spin chain isÛ G ð Þ~e {iĤt= h , and the QFI can be expressed as 33 It is straightforward to determine that the quantum Cramér-Rao bound s G~1 2cta ð Þ s which is exactly the same as the Heisenberg-scaling accuracy for scheme of the multi-parameter estimation. Immediately, we'll show that the previously proposed measurement strategies are optimal because they both yield the QFI and QCRB. For measurement strategy a, its probability distributions and Fisher Information are where the detailed calculations are showed in the supplementary material. The probability distribution p a (jjG) is clearly peaked around (2j/N 1 j)p/(cta) with approximate width p/(Ncta), where j is an arbitrary integer. If the condition 0 , G , p/(cta) is satisfied, one can successfully estimate G with Heisenberg-scaling accuracy. This measurement strategy is essentially a quantum Fourier algorithm for phase estimation 53,54 .
For measurement strategy b, we consider the estimation is local, i.e., the unknown parameter satisfies ctaG=1. We show in the supplementary material that its Fisher information is This implies that the Heisenberg-scaling QCRB can be reached locally via performing measurement strategy b.

Discussion
Determining the gradient of magnetic field is inherently a multiparameter estimation problem. We employ quantum enhanced multi-parameter estimation and the least square linear fitting method to achieve the Heisenberg-scaling quantum Cramér-Rao bound. Our scheme provides attainable high precision in magnetometry. This proposal is the first data fitting scheme possessing Heisenberg-scaling accuracy. This opens a new avenue for the investigations of general data fitting problems.

Methods
Here, let us introduce the method used in this work. We next will present a brief review of local estimation theory, the Fisher information and Cramér-Rao inequality [27][28][29][30][31][32][33] . Considering a curver y ð Þ characterizing dynamical process on the space of density matrix, the problem of determining the value of the parameter vector y~y 1 ,y 2 , Á Á Á ,y N ð Þ T is a fundamental problem of statistical inference based on the experimental results. Before the measurements, we know that an observable random variable j carries information about the unknown parameter vector y, which is described by the smooth probability distribution p(jjy). The normalization is ð djp j y j ð Þ~1, and j could be discrete or multivariate although it is written here as a single continuous real variable.
Then we take a random sample of size n to estimate the parameter vector y via comparing the ratio of observed measurement outcomes with the probability distribution. An essential premise of effective deterministic estimation is requiring that the smooth map p(jjy) « y is bijective. In order to avoid the periodical problems of determining the parameters y i , it is generally assumed that all components y i are small, which is called local estimation. For an effective deterministic observable random variable j, one estimates the parameter vector y via funtions y est i~y est i j 1 ,j 2 , Á Á Á ,j n ð Þ based on experimental results. The general framework of quantum parameter estimation is shown in FIG. 2. Then the expectation and covariance matrix of estimation are Taking the partial derivative of Eq.(16) with respect to y j and combining them into a bilinear quadratic form via two arbitrary real vectors a~a 1 ,a 2 , Á Á Á ,a N ð where the Fisher information (FI) matrix is defined by