Reasonable method to extract Fisher information from experimental data

Physica A 514 (2019) 606–611

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Physica A journal homepage: www.elsevier.com/locate/physa

Reasonable method to extract Fisher information from experimental data Yan Li, Weidong Li

∗

Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

highlights • An extensive investigation on FI extraction from Hellinger Distance and Kullback–Leibler entropy is presented. • Two constraints on quadratic fitting for FI in a dichotomic measurement model are provided. • Higher order fitting for reasonable FI is suggested and demonstrated to be useful with recent optical experimental data.

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Article history: Received 28 February 2018 Received in revised form 10 September 2018 Available online xxxx Keywords: Fisher information Hellinger distance Kullback–Leibler entropy Higher-order fitting

a b s t r a c t Fisher information (FI) plays a crucial role in quantum precision measurement and entanglement detection. Recently two methods have been suggested to extract it from experimental data: Hellinger distance and Kullback–Leibler entropy. In this paper, an extensive investigation is considered with the help of a dichotomic measurement model. It is found that the general quadratic √ fitting for FI with both methods has two constraints: one is the critical visibility V0 = 2/3, the other is the smallest δθ for the dichotomic measurement. To relax them, we propose the higher-order fitting (fourth order considered), by which a reasonable FI is obtained from recent optical experimental data. © 2018 Published by Elsevier B.V.

1. Introduction The degree of distinguishability between two neighboring quantum states is suggested to be characterized by statistical distance in terms of a Riemannian metric [1]. In quantum parameter estimation theory, increasing statistical distance corresponds to more reliable distinguishability, and thus to the possibility of estimating weakest signals. Therefore, the statistical distance is used to explore the sensitivity with which a parameter can be estimated [2,3]. Statistically, the relationship between the sensitivity and statistical distance is revealed by the Cramér–Rao lower bound [4], where the Fisher information (FI) is simply interpreted as the square of ‘‘statistical speed" (the rate of the change of the absolute statistical distance among two nearby states) [5]. Replacing FI with its quantum version obtained by√optimizing over all possible quantum measurements, quantum Cramér–Rao lower bound [6,7] is obtained, i.e., ∆θ ≥ 1/ FQ [ρˆ in , Jˆn⃗ ], where ∆θ is the

standard deviation on any arbitrary estimator of the parameter θ , ρˆ in is the initial state or input state and Jˆn⃗ is the Hermitian generator of the phase shift. As separable and entangled states have different statistical responses to the parameter, Quantum Fisher information (QFI) has been shown to witness entanglement [5,8], and further to quantify the quantum resource for ∗ Corresponding author. E-mail address: [email protected] (W. Li). https://doi.org/10.1016/j.physa.2018.09.118 0378-4371/© 2018 Published by Elsevier B.V.

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high-precision metrology [9]. Besides, QFI is also applied in modern quantum technologies [10], including quantum phase transitions [11–16], quantum Zeno dynamics [17], and quantum information protocols [18], etc. That is the reason why how to extract FI from a quantum system, especially from bare experimental data [19], recently became a hot task. Starting from the experimental probability distributions of some observable, the FI was firstly accessed from the curvature of Hellinger Distance (HD) [20], requiring a quadratic fitting on the data with respect to a small variation of θ . Very recently, it is also recommended to extract FI by evaluating Kullback–Leibler (KL) entropy [19]. As shown in Refs. [19– 21], there are a few advantages for both methods, such as independence on the specific shape of the probability distributions and no limitation on particle numbers. In the view of entanglement witness, the values of FI are actually expected to be less than or equal to the intrinsic value to avoid a wrong detection. Nevertheless, due to the restriction of experimental techniques, it is not easy to adjust parameter so small to meet the criteria for quadratic fitting (see Eqs. (6) and (7)). Therefore, for a relatively large δθ , the reasonable FI is hard to be extracted by quadratic fitting, where higher orders interfere the process severely. In this paper, we mainly investigate how to efficiently extract FI under available experimental conditions. With a dichotomic measurement model and recent methods (HD and KL entropy), two constraints for quadratic fitting are presented. One is on the critical visibility and the other is on the smallest δθ , i.e.,Eqs. (6) and (7). To relax them, the higher-order fitting (fourth order considered) is discussed and demonstrated to be useful with latest optical experimental data. Our paper is structured as follows. In Section 2, we present two constraints on quadratic fitting for FI in a dichotomic measurement model. In Section 3, we propose the higher-order fitting to overcome the constraints. An application of two fitting methods with the optical experimental data [22] is presented in Section 4. Finally, we present our conclusions in Section 5. 2. Constraints on quadratic fitting for FI in a dichotomic measurement model In a general unbiased parameter estimation, the FI is defined as [8], F (θ ) ≡

p(ε|θ )

(

∑ ε

∂ log p(ε|θ ) ∂θ

)2

,

(1)

ˆ ε ] denotes the conditional probability depending on the final states ρˆ θ and positive-operator valued where p(ε|θ ) = Tr[ρˆ θ M ˆ ε . Usually, it is not easy to have an analytical expression for p(ε|θ ), especially in some cases [23] we cannot measure (POVM) M obtain it any more. However, from the experimental point of view, it is not hard to have the information about p(ε|θ ) by repeated measurements with adjusting parameter θ . As pointed in [19], with this and the direct connection between FI and HD (or KL entropy) in the limit of δθ → 0, FI was successfully obtained by quadratic fitting [20,21]. More clearly, for given two neighboring probability distributions p(ε|θ ) and p(ε|θ + δθ ), HD is defined as DHD ≡ 1 −

∑√ ε

p(ε|θ )p (ε|θ + δθ ),

(2)

and the corresponding Kullback–Leibler entropy [24] (also called relative entropy [25]), a well-established informationtheoretic measure for the discrepancy of two probability distributions, is given by DKL ≡

p(ε|θ ) ln

∑ ε

p(ε|θ ) p(ε|θ + δθ )

.

(3)

It is easy to check that the coefficients of the second order (δθ )2 in series expansion of Eqs. (2) and (3) around any given θ contain Eq. (1) but with different ratio factors [26,27], DHD =

C2HD

DKL = C2KL

(

(

1 8

1 2

)

F (θ ) δθ 2 + C3HD δθ 3 + C4HD δθ 4 + O(δθ 5 ),

(4)

)

F (θ ) δθ 2 + C3KL δθ 3 + C4KL δθ 4 + O(δθ 5 ).

(5)

As such, it is suggested to extract FI by quadratic fitting: expanding DHD (DKL ) to the second order. Obviously, the smaller the δθ , the better the evaluation for FI. However, once δθ is not so small, it may induce a false value, larger or smaller than the true FI [28]. Therefore, the study of the criteria for quadratic fitting in extracting FI is necessary. To explore this we consider a dichotomic measurement model, parity measurement on the multi-ion GHZ state [29], where the conditional probability can be expressed as p(±|θ ) = (1±V cos(N θ ))/2 with the total ion number N and the visibility V from the oscillating fringe acquired by measurement εi ∈ {+, −} on the state. Here, ‘‘+ (−)" denotes the state with an even (odd) number ( of excitations. )With this conditional probability the optimal FI can be obtained by maximizing F (θ ) = V 2 N 2 sin2 (N θ )/ 1 − V 2 cos2 (N θ) over all possible θ , i.e., Fopt = V 2 N 2 at θ0 = π/(2N) + nπ/N, where n = 0, 1, 2, . . .. In Fig. 1, we plot the DHD as black curves, DKL as red curves, as well as the fitting curves for quadratic fitting (blue curves) and higher-order fitting (dashed lines) at θ0 with two different visibility V = 0.787 (Fig. 1(a, b)) and V = 0.950 (Fig. 1(c, d)). As mentioned before, for small enough δθ we find no difference in extracting FI by comparing all of the curves. However, a

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Fig. 1. (Color online) Hellinger distance (black solid lines) in Eq. (2) and Kullback–Leibler entropy (red solid lines) in Eq. (3) for ideal trapped ions. The green vertical dashed lines show the critical values δθcHD (δθcKL ), over which a clear divergence between the fitting curves and black (red) lines is shown. Here two different values of visibility are considered, V = 0.787 for (a) and (b), V = 0.950 for (c) and (d). The total ion number is N = 8. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

HD KL HD KL Fig. 2. (Color √ online) Contributions from 4C4 and C4 . The black (red) curves denote 4C4 (C4 ) with respect to the visibility V for N = 8. The black dot (V0 = 2/3) shows the boundary for KL entropy or HD method, suggested for good estimation on FI with quadratic fitting. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

clear divergence can be found when δθ is larger than a critical value, which also ∑can be found from Eqs. (4) and (5). We find that the zeroth order and first order vanish due to the normalization condition ε p(ε|θ ) = 1 [20]. Moreover, it is interesting to see that all odd orders are equal to zero since that they all have a multiplier sin(2N θ ). Thus all even orders become the workhorse for our work. The coefficient C2 of the second order can be used to extract FI. Naturally the fourth order turns into the most influential one in FI extraction. √ Firstly, we note that the sign of C4 (fourth order coefficient) for KL entropy can be positive when V is larger than V0 = 2/3, while for HD it keeps negative, as shown in Fig. 2. This can also be read from Fig. 1(b) and 1(d), where quadratic fitting (blue solid curve) may be tighter (or more relaxed) than KL entropy (red solid curve), hence it may induce a wrong determination for FI because a relative large value may be extracted when C4 > 0 [28]. Meanwhile, we also note that |C4KL |< |4C4HD | when V < V0 in Fig. 2. To obtain a good estimation on FI, we suggest to extract

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Table 1 Experimental data for two-photon state.

θ/π

θ0 − 3δθ

θ0 − 2δθ

θ0 − δθ

θ0

θ0 + δθ

θ0 + 2δθ

θ0 + 3δθ

⟨Mθ⊗2 ⟩ σ (error)

0.705 0.039

0.477 0.034

0.233 0.034

−0.033

−0.273

−0.511

−0.711

0.034

0.034

0.044

0.044

Table 2 Experimental data for eight-photon state.

θ/π ⟨ ⊗8 ⟩

Mθ σ (error)

θ0 − 2δθ

θ0 − δθ

θ0

θ0 + δθ

θ0 + 2δθ

0.563 0.050

0.418 0.058

0.039 0.050

−0.357

−0.525

0.062

0.050

FI by fitting HD (Eq. (2)) for V > V0 , while by fitting KL entropy (Eq. (3)) for V ⩽ V0 . Secondly, when the value of C4 δθ 4 and C2 δθ 2 become comparable (here the ratio between them is 1/100), we find that the criteria on how small δθ is required to have a good estimation,

δθ < δθcHD =

√

48/(16 − 15V 2 )/(10N)

(6)

for HD (0.011π in Fig. 1(a) and 0.018π in Fig. 1(c)).

δθ < δθcKL =

√

6/|3V 2 − 2|/(10N)

(7)

for KL entropy (0.026π in Fig. 1(b) and 0.012π in Fig. 1(d)). We have shown those critical values by green vertical dashed lines in Fig. 1. Once the visibility and the spacing between two measurements (e.g., δθ ) satisfy those requirements, FI from HD has no difference with it from KL entropy. 3. Higher-order fitting for FI However, it is well known that the spacing is not so easy to adjust to fulfill those criteria, especially in the experiment with large scale quantum states. Therefore, we suggest to consider one more general fitting method involving higher orders, and then extract the coefficient of second order as FI. As shown in Fig. 1, we have included the contribution of fourth order for fitting, and a good agreement with KL entropy and HD has been found. Even better consistence can be seen for KL entropy. In the following, we will show some features about the higher-order fitting (the fourth order considered) in two aspects: (i) weakened or reduced constraint on visibility; (ii) much more tolerance on the spacing between two independent measurements. 4. Applications For an intuitive comparison, we apply the quadratic fitting and higher-order GHZ state ⟨ fitting on a recent ten-photon ⟩

experiment [22], wherein the expected values of the observable Mθ⊗N

⟨

⟩

=

( )⊗N σx cos θ + σy sin θ with relative phase

θ are reported in Fig. 3 in [22], with to its eigenvalues ±1. A nice oscillation behavior has shown a possible N⟨ respect ⟩ photon GHZ state [22,30,31], i.e., Mθ⊗N = V cos N θ , then the experimental conditional probabilities can be written as p(±1|θ ) = (1 ± V cos N θ) /2. This is the reason why we choose these ⟨experimental data to check our suggestions. ⟩

Following the process of [19], we extract the experimental data of Mθ⊗N shown in Tables 1 and 2 from Fig. 3 in [22] for two cases, N = 2 and N = 8, respectively. In both cases, we make θ0 ∼ π/(2N) as requirement for the FI. Since the same conditional probabilities as ion case [29], we have a simple answer for the value of FI, Fopt = V 2 N 2 defined in previous section. From [22], we get V ∼ 0.9305 ± 0.0003 to evaluate Fopt ∼ 3.463 ± 0.002 for two-photon state, and get V ∼ 0.538 ± 0.029 to evaluate Fopt ∼ 18.524 ± 1.997 for eight-photon state. To have the values of HD and KL entropy around θ0 , i.e., Eqs. (2) and (3), we need the group values of p(±|θ0 ± nδθ ), where n = 1, 2, 3 for two-photon state and n = 1, 2 for eight-photon state (the limited experimental data points). Since we do not have enough experimental data only from [22], we resort ⟨ ⟩ to Monte Carlo simulation to produce them with the required average values Mθ⊗N and standard variance σ in Tables 1 and 2. For each group of the simulated data, we obtain the HD (green dots) and KL entropy (black dots) firstly, as shown in HD KL Fig. 3. Secondly the quadratic fitting is executed and corresponding coefficient C2HD (C2KL ) is obtained, therefore, 2 Fopt (2 Fopt ) is HD KL acquired. Similarly, we obtain 4 Fopt (4 Fopt ) for higher-order fitting. Finally, after 1000 times simulation we take the averaged values as the final results and present them in Tables 3 and 4. √ In the two-photon state, the experimental visibility is V ∼ 0.9305 and larger than the critical value V0 = 2/3 defined in Fig. 2. And the experimental spacing is around δθ ∼ 0.045π , which meets the requirement on the δθ for HD and KL entropy, where δθcHD ∼ 0.064π and δθcKL ∼ 0.050π . Based on the criteria in Section 2, it is suggested to have a better FI with HD method. This can also be read from Table 3 and Fig. 3(a, b), where with quadratic fitting a clear difference can be found by comparing with Fopt (light blue lines). Specifically speaking, a tight curve (red line) in Fig. 3(b) may indicate a false

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Fig. 3. (Color online) FI estimation for two- and eight-photon entanglement experiments [22]. Two-photon state with HD method (a) and KL entropy method (b). Eight-photon state with HD method (c) and KL entropy method (d). The light and dark gray regions denote the error of quadratic fitting and Higher-order fitting. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 3 FI from KL entropy and HD for two-photon state. Fitting methods

Quadratic fitting

Higher-order fitting

HD Fopt KL Fopt

3.420 ± 0.302 3.893 ± 0.405

3.356 ± 0.698 3.317 ± 0.798

Table 4 FI from KL entropy and HD for eight-photon state. Fitting methods

Quadratic fitting

Higher-order fitting

HD Fopt KL Fopt

9.178 ± 1.125 9.724 ± 1.359

19.627 ± 5.133 20.265 ± 5.359

estimation on FI. However, this risk can be avoided by including higher orders, such as the results (red dashed line) in Fig. 3(b) and Table 3. Therefore, with the help of higher-order fitting we can reduce the constraint of visibility in quadratic fitting. In the √ eight-photon state, the visibility dramatically decreases to V ∼ 0.538, which is smaller than the critical value V0 = 2/3. And the experimental spacing is around δθ ∼ 0.031π , which is much larger than the requirement on δθ for both cases, δθcHD = 0.008π and δθcKL = 0.009π . According to the criteria this actually means that the quadratic fitting will fail to present any helpful information on FI, which can also be read from Table 4 and black (red) solid curve in Fig. 3(c(d)). While, the higher-order fitting makes a great improvement on the estimation of the values of FI, as shown in Table 4 and black (red) dashed lines in Fig. 3(c(d)). Compared with the value of Fopt obtained by the visibility, a slight bigger value for FI can be found from higher-order fitting. We believe that with more data or even higher-order fitting (more than fourth order) a reasonable value for FI can be obtained, even for larger δθ and bad visibility. 5. Conclusions This work studies how to efficiently extract FI from experimental data based on the Hellinger Distance and Kullback– Leibler entropy. Considering a dichotomic measurement model, the general quadratic fitting for FI with respect to δθ is

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√

thoroughly studied and shows two constraints. The first constraint is about the critical visibility V0 = 2/3. In the case of V > V0 , HD gives the positive result than it from KL entropy, while for V < V0 KL entropy shows the better outcome. The second constraint is about the smallest δθ , which is required to be smaller than δθcHD (δθcKL ). The higher-order fitting, here the fourth order considered, may help to overcome them and gives a reasonable value for FI. With recent experimental reported data the extractions of FI by quadratic fitting and higher-order fitting are compared, which demonstrates the higher-order fitting is useful and meaningful. We believe our work will contribute more for future research in the field of quantumenhanced phase estimation. Acknowledgments We thank A. Smerzi for helpful discussions. This work was supported by the National Key R & D Program of China (No. 2017YFA0304500 and No. 2017YFA0304203), the National Natural Science Foundation of China (Grant No. 11874247), the 111 plan of China (No. D18001), the Hundred Talent Program of the Shanxi Province (2018), and the Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices of China (No. KF201703). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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