Development on quantum metrology with quantum Fisher information

Ren Zhi-Hong; Li Yan; Li Yan-Na; Li Wei-Dong

doi:10.7498/aps.68.20181965

Abstract

Quantum metrology is one of the hot topics in ultra-cold atoms physics. It is now well established that with the help of entanglement, the measurement sensitivity can be greatly improved with respect to the current generation of interferometers that are using classical sources of particles. Recently, Quantum Fisher information plays an important role in this field. In this paper, a brief introduction on Quantum metrology is presented highlighting the role of the Quantum Fisher information. And then a brief review on the recent developments for i) criteria of multi-particle entanglement and its experimental generation; ii) linear and non-linear atomic interferometers; iii) the effective statistical methods for the analysis of the experimental data.

Keywords:

Authors and contacts

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

Corresponding author: Li Wei-Dong, wdli@sxu.edu.cn

Funds: Project supported by the National Key R&D Program of China (Grant Nos. 2017YFA0304500, 2017YFA0304203), the National Natural Science Foundation of China (Grant No. 11874247), the 111 Plan of China (Grant No. D18001), the Hundred Talent Program of Shanxi Province, China (2018), and the Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices, China (Grant No. KF201703).

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图 1 相位估计的流程示意图　(1) 探测量子初态$\hat{\rho}$的制备; (2)待测相位$\theta$的编码, 量子初态演化为$\hat{\rho}(\theta)$; (3) 正定的测量算符(POVM)对末态进行测量; (4)待测相位的统计估计${\varTheta}$. 图的下部分是马赫曾德(Mach-Zehnder, MZ)干涉仪的应用实例. 图取自文献[4]

Figure 1. Here we schematically plot the elements of complete phase estimation: (1) the preparation of prob state $\hat{\rho}(0)$; (2) the encoding of phase shift $\theta$, which transform the probe state to $\hat{\rho}(\theta)$; (3) the readout measurement of the POVM and finally (4) the mapping from the measurement results to the phase provided by the estimator ${\varTheta}$. The phase sensitivity crucially depends on all these operations. The lower panel shows the application to Mach-Zehnder interferometry. Adapted from Ref. [4].

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图 2 近年来利用超冷原子体系取得的超越标准量子极限的实验进展. 对数增益(左): $10\log_{10} ({\text{Δ}}\theta_{\rm{SQL}} / {\text{Δ}}\theta)^2$ 和对应的线性增益(右): $({\text{Δ}}\theta_{\rm{SQL}} / {\text{Δ}}\theta)^2$, 其中$({\text{Δ}}\theta)^2 = {\text{ξ}}_{\rm R}^2/N$或者$({\text{Δ}}\theta)^2 = 1/F_{\rm Q}$, ${\text{ξ}}_{\rm R}^2$为自旋压缩参数, $F_{\rm Q}$为量子Fisher信息. 空心图标为净增益, 即去除测量误差的增益; 实心图标为可能增益，即实验纠缠态的可能增益; 十字交叉图标为干涉仪增益，即通过干涉仪验证的增益. $N$为实验中所用的总粒子数目(或平均数目). 图取自文献[5]

Figure 2. Gain of phase sensitivity over the standard quantum limit $\Delta\theta_{\rm{SQL}}=1/\sqrt{N}$ achieved form recent Bose-Einstein condensates experiments. The logarithmic gain is shown in left, $10\log_{10} ({\text{Δ}}\theta_{\rm{SQL}} / {\text{Δ}}\theta)^2$ and the linear gain is in right, $({\text{Δ}}\theta_{\rm{SQL}} / {\text{Δ}}\theta)^2$. The measured phase precision is obtained by spin-squeezing parameter ${\text{ξ}}_{\rm R}^2$, i.e., $({\text{Δ}}\theta)^2 = {\text{ξ}}_{\rm R}^2/N$, or from $F_{\rm Q}$, i.e., $({\text{Δ}}\theta)^2=1/F_{\rm Q}$. Open patterns are inferred values, being obtained after subtraction of detection noise; Filled patterns report witnesses of metrologically useful entanglement measured on experimentally generated states, representing potential improvement in sensitivity; Cross patterns show the measured phase sensitivity gain obtained from a full interferometer sequence. Here $N$ is the total number of particles (or mean total) used in experiments. Adapted from Ref. [5].

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图 3 统计区分度和统计速度的示意图. 在给定$\theta_0$的条件下, 分析测量数据可得分布函数$p({\mu}|\theta_0)$(图(a)红线)和调节$\theta_0+\text{δ}\theta$后得分布函数$p({\mu}|\theta_0+\text{δ}\theta)$ (图(b)绿线). 图(c)示意由$\sqrt{p_0}=\sqrt{p({\mu}|\theta_0)}$和$\sqrt{p_{\text{δ}\theta}}=\sqrt{p({\mu}|\theta_0+\text{δ}\theta)}$定义的态矢量及其欧几里得距离(统计距离)$d_{\rm H}=2\|\sqrt{p_0}-\sqrt{p_{\text{δ}\theta}}\|$, 其中$v=v_{\rm H}$是衡量统计区分度的统计速度. 图取自文献[84]

Figure 3. Here we give the sketch of statistical distinguishability and statistical speed. The probability distribution $p({\mu}|\theta_0)$ is obtained by collecting the measurement results ${\mu}$ for different values of the parameter, here chosen to be $\theta_0$(red line) (a) and $\theta_0+\text{δ}\theta$ (green line)(b). (c) to quantify the statistical distinguishability between the two distributions we introduce unit vectors $\sqrt{p_0}=\sqrt{p({\mu}|\theta_0)}$ (red) and $\sqrt{p_{\text{δ}\theta}}=\sqrt{p({\mu}|\theta_0+\delta\theta)}$ (green), then we obtain the Euclidean distance between them: $d_{\rm H}=2\|\sqrt{p_0}-\sqrt{p_{\text{δ}\theta}}\|$. Here the $v=v_{\rm H}$ denotes the statistical speed. Adapted from Ref. [84].

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图 4 海林格距离$d_{\rm H}^2$(红线)和KL熵$2D_{\rm {KL}}$(绿线), 以及他们共同的二阶展开项$F(\theta)\text{δ}\theta^2$(蓝线)的比较. 图取自文献[84]

Figure 4. Hellinger distance, $d_{\rm H}^2$(red line), KL entropy, $2D_{\rm {KL}}$(green line), and their common low-order approximation, $F(\theta)\text{δ}\theta^2$(blue line), as a function of $\theta$. Adapted from Ref. [84].

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图 5 宇称振荡的测量结果和提取的Fisher信息　(a) GHZ($N=8$)的宇称测量图, 周期为$2\pi/8$, 取自文献[92]; (b) 基于近期实验结果得到的Fisher信息与总粒子数$N$的关系$F=V^2N^2$, 其中$V$为实验中的对比度. 上边界粗线为海森堡极限, 即$F=N^2$, 下边界黑色粗线代表标准量子极限, $F=N$. 图中其他的细线分别表示$k$粒子纠缠的边界, 即方程(60). 其中$N=10$的圆圈表明$4$粒子纠缠. 图取自文献[84]

Figure 5. Experimental results based on parity measurement and extracted Fisher information： (a) Typical parity oscillations obtained with cat states. The period is $2\pi/8$. Adapted from [92]; (b) summary of the experimental achievements, ions (circle) and photons (square). Here we show the Fisher information as a function of the number of qubits $N$, $F=V^2N^2$, obtained from the extracted experimental visibilities $V$. The upper thick line is the Heisenberg limit $F=N^2$, the lower thick line is the standard quantum limit, $F=N$. The different lines are bounds for useful $k$-particle entanglement, Eq. (60).For instance, the filled circle at $N=10$ reveals useful $4$-particle entanglement. Adapted from Ref. [84].

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图 6 有利于量子计量的$k$粒子纠缠判据. 图中蓝色的实线由方程(60)给出: $k$粒子纠缠态的量子Fisher信息. 虚线为$F_{\rm Q}/N=k$, 其中粒子数$N=100$. 该图取自文献[100]

Figure 6. Useful $k$-particle entanglement for quantum metrology. $k$-separable states have a quantum Fisher information bounded by the solid line, Eq. (60). The dashed line is $F_{\rm Q}/N=k$. Here $N=100$. Adapted from Ref. [100].

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图 7 (a) 量子相变产生双数态过程; (b) 纠缠宽度分析表明双数态纠缠宽度约为$910_{-450}^{+9900}$原子. 图取自文献[109]

Figure 7. (a) Generation of twin-Fock state by quantum phase transition; (b) analysis of entanglement breadth for the Twin-Fock state samples, and it shows the entanglement breadth is at least $910_{-450}^{+9900}$ atoms. Adapted from Ref. [109].

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图 8 两空间分离的粒子的相同自由度(a)和不同自由度(b)之间的非定域关联(Bell关联); (c)产生和验证自由度间非定域关联的实验方案. Alice 和 Bob 各自制备自旋态为 $\left\vert\downarrow\right\rangle$ 的原子并使其通过混合分束器. 分束器的两个输出分别与本地和对方的分析仪器相连. 第二个混合分束器作为各自的分析仪器的一部分, 将所得信号组合, 在两个粒子的不同自由度中产生了非定域关联. 通过CHSH不等式来区别, 此时两探测器的信号同时响应. 图取自文献[122]

Figure 8. (a) Hypernonlocality represents the simultaneous presence of Bell correlations among more than one DOF of two spatially separated particles; (b) hybrid nonlocality identifies Bell correlations among the discrete DOF of one particle and the continuous DOF of another distant particle; (c)experimental scheme for the generation and verification of inter-DOF entanglement. Alice and Bob both prepare one particle in a spin-$\left\vert\downarrow\right\rangle$ state and submit it to a hybrid beam splitter. One of the output ports is sent to their local laboratory while the other is send to the opposite party. By mixing the local and the received copy using a second hybrid beam splitter, the desired correlations are established. Both parties now measure either spin or external d.o.f of their received particles, as depicted by the interchangeable measurement devices (white boxes). The recorded data from the events in which both parties receive exactly one particle violate a suitable CHSH inequality, independently of the measured DOF Adapted from Ref. [122].

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图 9 广义Bloch球上马赫-曾德干涉仪和拉姆齐干涉仪对集体自旋的操作. 图取自文献[5]

Figure 9. Representation of Mach-Zehnder and Ramsey interferometer operations as rotations of the collective spin on the generalized Bloch sphere. Adapted from Ref. [5].

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图 10 无量纲参数下的原子衍射图: KD衍射和Bragg衍射所满足的条件. 图取自文献[22]

Figure 10. Dimensionless parameter space for atom diffraction, KD labels curves corresponding to conditions that maximize Kapitza-Dirac diffraction, and Bragg indicates curves that correspond to conditions for Bragg reflection. Adapted from Ref. [22].

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图 11 (a) 马赫-曾德原子干涉仪示意图; (b) 重力计原理示意图. 图取自文献[139]

Figure 11. (a) Schematic plot of Mach-Zehnder atom interferomery; (b) schematic plot of Gravimeter configuration. Adapted from Ref. [139].

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图 12 (a) 多模式KD 干涉仪. $t=0$时, 第一束KD脉冲将谐振势中原子复制多个原子波包. $t=\pi/\omega$时, 第二束KD脉冲对这些波包进行复合, 并在$t=3\pi/2\omega$时进行检测; (b) 测量时刻不同温度下原子的密度分布. 图取自文献[140]

Figure 12. (a) Multimodes Kapitza-Dirac interferometer. The first Kapitza-Dirac pulse at $t=0$ creates several modes consisting of atomic wave packets evolving under the harmonic confinement and an external perturbing field. The second Kapitza-Dirac pulse at $t=\pi/\omega$ mixes the modes which are eventually detected in output at $t=3\pi/2\omega$; (b) density profiles of the output wave function at temperatures below (dark line), equal (blue line), or above (red line) the crossover temperature $T_0$. Adapted from Ref. [140].

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图 13 (a) 在Ising模型中, 平均自旋$\langle {\sigma}_{{n}}^{(i)}\rangle$随$\gamma$的变化情况; (b) $F_{\rm Q}/N$随$\gamma$的变化情况, 以及标准量子极限和亚散粒噪声极限的边界, 同时其也是判定有用量子纠缠态的边界[84]. 灰色区域为亚散粒噪声区域. 图取自文献[84]

Figure 13. we show the $\langle {\sigma} _{{n}}^{(i)}\rangle$ with respect to $\gamma$; (b) $F_{\rm Q}/N$ with respect to the $\gamma$ and give the boundary between standard quantum limit and sub-shot noise, which also witnesses the useful entanglement in Ref. [84]. The gray region denotes the sub-shot noise region. Adapted from Ref. [84].

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图 14 (a) 在LMG模型中, 平均自旋$\langle {\sigma}_{{n}}^{(i)}\rangle$随$\gamma$的变化情况; (b) $F_Q/N$随$\gamma$的变化情况, 以及标准量子极限和亚散粒噪声极限的边界, 同时其也是判定有用量子纠缠态的边界[84]. 图取自文献[84]

Figure 14. (a) We show the $\langle \rm {\sigma} _{{n}}^{(i)}\rangle$ with respect to $\gamma$ in LMG model; (b) $F_{\rm Q}/N$ with respect to the $\gamma$ and give the boundary between standard quantum limit and sub-shot noise, which also witnesses the useful entanglement in Ref. [84]. The gray region denotes the sub-shot noise region. Adapted from Ref. [84].

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图 15 (a) 最大似然估计偏差(绿色圆点)随独立测量次数$m$的变化, 误差为$(\Delta\theta_{\rm{MLE}})_{{\mu} \vert \theta_0}$; 红色曲线为$\pm \Delta \theta_{\rm CRB} = $$\pm |{\rm d} \langle \theta_{\rm MLE} \rangle_{{\mu} \vert \theta_0} / {\rm d}\theta_0|/\sqrt{mF(\theta_0)}$; (b) $ m F(\theta_0) (\varDelta^2\theta_{\rm{MLE}})_{{\mu} \vert \theta_0}$ (空心红圈)随$m$的变化. 红色实线为$({\rm d} \langle \theta_{\rm MLE} \rangle_{{\mu} \vert \theta_0} / {\rm d}\theta_0)^2$. 图取自文献[164]

Figure 15. (a) Bias $\langle \theta_{\rm MLE} \rangle_{{\mu} \vert \theta_0} - \theta_0$ (green dots) as function of $m$ with error bars $(\Delta\theta_{\rm{MLE}})_{{\mu} \vert \theta_0}$. The red lines are $\pm \Delta \theta_{\rm CRB} = $$\pm |{\rm d} \langle \theta_{\rm MLE} \rangle_{{\mu} \vert \theta_0} / {\rm d}\theta_0|/\sqrt{mF(\theta_0)}$; (b) variance of the maximum likelihood estimator multiplied by the Fisher information, $ m F(\theta_0) (\varDelta^2\theta_{\rm{MLE}})_{{\mu} \vert \theta_0}$ (red circles), as a function of the sample size $m$. It is compared to the bias $({\rm d} \langle \theta_{\rm MLE} \rangle_{{\mu} \vert \theta_0} / {\rm d}\theta_0)^2$ (red line). We recall that $\theta_0=\pi/4$ and $F(\theta_0)=4$ here. Adapted from Ref. [164].

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图 16 不同先验概率下, 频率论方法得到相位估计值的方差($m (\varDelta^2 \theta_{\rm BL})_{{\mu} \vert \theta_0}$, 红色圆圈)及其边界CRB($m\varDelta^2 \theta_{\rm CRB}$, 红色虚线), 和贝叶斯相位估计的方差($m (\Delta^2 \theta_{\rm BL})_{{\mu},\theta \vert \theta_0}$, 蓝色圆圈)及其边界($m\varDelta^2 \theta_{\rm aGB}$, 蓝色实线)随样本$m$的变化 (a) $\alpha=-100$; (b) $\alpha=-10$; (c) $\alpha=1$; (d) $\alpha=10$. 每张图中内嵌的图是先验概率$p_{\rm pri}(\theta)$的分布图. 图取自文献[164]

Figure 16. Comparisons of phase estimation variance as a function of the sample size for Bayesian and frequentist data analysis under different prior distributions: (a) $\alpha=-100$; (b) $\alpha=-10$; (c) $\alpha=1$; (d) $\alpha=10$. In all figures, Red circles (frequentist) are $m (\varDelta^2 \theta_{\rm BL})_{{\mu} \vert \theta_0}$, the red dashed line is the CRB. $m\varDelta^2 \theta_{\rm CRB}$, Eq. (124). Blue circles (Bayesian) are $m (\varDelta^2 \theta_{\rm BL})_{{\mu},\theta \vert \theta_0}$, the blue solid line is the likelihood-averaged Ghosh bound $m\varDelta^2 \theta_{\rm aGB}$, Eq. (103). The inset in each panel is $p_{\rm pri}(\theta)$. Adapted from Ref. [164].

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