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Journals >Acta Physica Sinica >Volume 68 >Issue 4 >Page 040601-1 > Article

Acta Physica Sinica
Vol. 68, Issue 4, 040601-1 (2019)

Development on quantum metrology with quantum Fisher information

Zhi-Hong Ren, Yan Li, Yan-Na Li, and Wei-Dong Li^*

DOI: 10.7498/aps.68.20181965 Cite this Article

Zhi-Hong Ren, Yan Li, Yan-Na Li, Wei-Dong Li. Development on quantum metrology with quantum Fisher information[J]. Acta Physica Sinica, 2019, 68(4): 040601-1 Copy Citation Text

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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

quantum entanglement quantum Fisher information quantum metrology ultra-cold atoms

$P(\varepsilon |\theta ) = {\rm{Tr}}[{{\hat E}}(\varepsilon )\hat \rho (\theta )].$

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$\hat \rho (0) = {\hat \rho ^{(1)}}(0) \otimes {\hat \rho ^{(2)}}(0) \otimes {\hat \rho ^{(3)}}(0) \otimes \ldots \otimes {\hat \rho ^{(m)}}(0).$

(2)

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$\hat E({\varepsilon }) = {\hat E^{(1)}}({\varepsilon _1}) \otimes {{\hat E}^{(2)}}({\varepsilon _2}) \otimes {\hat E^{(3)}}({\varepsilon _3}) \otimes \cdots \otimes {\hat E^{(m)}}({\varepsilon _m}).$

(3)

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$P({{\varepsilon }}|\theta ) = \prod\limits_{i = 1}^m {{P_i}} ({\varepsilon _i}|\theta ),$

(4)

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$L({{\varepsilon }}|\theta ) = \ln P({{\varepsilon }}|\theta ) = \sum\limits_{i = 1}^m {\ln } {P_i}({\varepsilon _i}|\theta ).$

(5)

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${\left\langle {\varTheta} \right\rangle _\theta } = \sum\limits_{{\varepsilon }} P ({\varepsilon }|\theta ){\varTheta}({\varepsilon }),$

(6)

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$\left( {{\text{Δ}}{\varTheta}} \right)_\theta ^2 = \sum\limits_{{\varepsilon }} P ({\varepsilon }|\theta ){\left( {{\varTheta}({\varepsilon }) - {{\left\langle {\varTheta} \right\rangle }_\theta }} \right)^2},$

(7)

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$\mathop {\lim }\limits_{m \to \infty } Pr\left( {|{\varTheta}({\varepsilon }) - \theta | > \delta } \right) = 0,$

(8)

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$\left( {{\text{Δ}}{\varTheta}} \right)_\theta ^2 \ge {\left( {{\text{Δ}}{{\varTheta}_{\rm {CR}}}} \right)^2} \equiv \dfrac{{{{\left( {\dfrac{{\partial \left\langle {\varTheta} \right\rangle }}{{\partial \theta }}} \right)}^2}}}{{F(\theta )}},$

(9)

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$F(\theta ) = {\left\langle {{{\left( {\dfrac{{ \partial L({\varepsilon }|\theta )}}{{\partial \theta }}} \right)}^2}} \right\rangle _\theta } = \sum\limits_{{{\varepsilon}}} {\dfrac{1}{{P({\varepsilon }|\theta )}}} {\left( {\dfrac{{\partial P({\varepsilon }|\theta )}}{{\partial \theta }}} \right)^2},$

(10)

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$\dfrac{\partial L({{\varepsilon}}|\theta)} {\partial \theta} = \lambda_{\theta} \left( {\varTheta}({{\varepsilon}})- \langle {\varTheta} \rangle_{\theta} \right), \forall {{\varepsilon}},$

(11)

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$F_{\rm Q} [\hat{\rho}(\theta)] = \max_{\hat{E}({{\varepsilon}})} F [ \hat{\rho}(\theta), \{ \hat{E}({{\varepsilon}}) \} ]. $

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$F_{\rm Q} [\hat{\rho}(\theta)] = {\rm{Tr}}[\hat{\rho}(\theta) \hat{L}_{\theta}^2],$

(13)

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$\dfrac{\partial \hat{\rho}(\theta)}{\partial \theta} = \dfrac{1}{2} \left( \hat{\rho}(\theta)\hat{L}_{\theta} + \hat{L}_{\theta} \hat{\rho}(\theta) \right). $

(14)

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$\left( {\text{Δ}}{\varTheta} \right )_{\theta}^2 \geqslant \left( {\text{Δ}}{\varTheta}_{\rm {CR}} \right )_{\theta}^2 \geqslant \left( {\text{Δ}}{\varTheta}_{\rm {QCR}} \right )_{\theta}^2,$

(15)

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$\Re ({\rm {Tr}}[\hat \rho (\theta ){\hat L_\theta }\hat E({\varepsilon })]) = 0,\forall {\varepsilon },$

(16)

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$\hat \rho (\theta )\left( {\mathbb{1} - {\lambda _{\theta ,{\varepsilon }}}{{\hat L}_\theta }} \right)\hat E({\varepsilon }) = 0,\forall {\varepsilon },$

(17)

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$P\left({{\varepsilon}} | \theta \right) = \sum\limits_{k} \gamma_k P_k \left({{\varepsilon}}| \theta \right).$

(18)

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$F(\theta) \leqslant \sum\limits_{k} \gamma_k F_k \left( \theta \right),$

(19)

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$F_{\rm Q} \left[ \sum\limits_k \gamma_k \hat{\rho}_k (\theta) \right] \leqslant \sum\limits_{k} \gamma_k F_{\rm Q} \left( \hat{\rho}_k (\theta) \right).$

(20)

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$F(\theta) = \sum\limits_{i = 1}^m F_i(\theta),$

(21)

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$F(\theta) = m F' (\theta),$

(22)

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$\left( {\text{Δ}}{\varTheta}_{\rm {CR}} \right )^2 = \dfrac{\left( \dfrac{\partial \left\langle {\vartheta} \right \rangle }{\partial \theta} \right)^2}{m F'(\theta)},$

(23)

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$F'(\theta) = \sum\limits_{\varepsilon} \dfrac{1}{P(\varepsilon |\theta)} \left(\dfrac{\partial P(\varepsilon |\theta)}{\partial \theta} \right)^2.$

(24)

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$F_{\rm Q}[\hat{\rho}(\theta)] = \sum\limits_{i = 1}^m F_{\rm Q}[\hat{\rho}^{(i)}(\theta)],$

(25)

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${F_{\rm Q}}[\hat \rho (\theta )] \!=\!\! \sum\limits_{k,k'} {{p_k}} |\langle k|{\hat L_\theta }|k'\rangle {|^2} \!=\!\! \sum\limits_{k,k'} {\dfrac{{{p_k} \!+\! {p_{k'}}}}{2}} |\langle k|{\hat L_\theta }|k'\rangle {|^2}.$

(26)

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$F_{\rm Q}[\hat{\rho}(\theta)] = \sum\limits_{k,k'} \dfrac{2}{p_k+p_{k'} } |\langle k| \partial_{\theta} \hat{\rho}(\theta) | k' \rangle|^2.$

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$\begin{split}{\partial _\theta }\hat \rho (\theta ) = & \sum\limits_k {({\partial _\theta }{p_k})|k\rangle \langle } k| + \sum\limits_k {{p_k}} |{\partial _\theta }k\rangle \langle k| \\ & + \sum\limits_k {{p_k}} |k\rangle \langle {\partial _\theta }k|,\end{split}$

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$\langle k|{\partial _\theta }\hat \rho (\theta )|k'\rangle = \left( {{\partial _\theta }{p_k}} \right){\delta _{k,k'}} + ({p_k} - {p_{k'}})\langle {\partial _\theta }k|k'\rangle .$

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$\hat{L}_{\theta} = \sum\limits_k \dfrac{\partial_{\theta} p_k} {p_k} |k\rangle \langle k | + 2 \sum\limits_{k,k'} \dfrac{p_k-p_{k'}} {p_k+p_{k'}} |k\rangle \langle \partial_{\theta} k |k' \rangle \langle k' | ,$

(28)

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$F_{\rm Q}[\hat{\rho}(\theta)] = \sum\limits_k \dfrac{\left( \partial_{\theta} p_k \right)^2} {p_k} + 2 \sum\limits_{k,k'} \dfrac{\left(p_k-p_{k'}\right)^2} {p_k+p_{k'}} | \langle \partial_{\theta} k |k' \rangle|^2.$

(29)

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$\hat{L}_{\theta} = 2 |\psi(\theta) \rangle \langle \partial_{\theta} \psi(\theta) |+ 2 |\partial_{\theta} \psi(\theta) \rangle \langle \psi(\theta) | ,$

(30)

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${F_{\rm Q}}[|\psi (\theta )\rangle ] = 4\left( {\langle {\partial _\theta }\psi (\theta )|{\partial _\theta }\psi (\theta )\rangle - |\langle {\partial _\theta }\psi (\theta )|\psi (\theta )\rangle {|^2}} \right).$

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$\hat{\rho}(\theta) = {\rm e}^{-{\rm i}\hat{H} \theta} \hat{\rho}_0 {\rm e}^{+{\rm i}\hat{H} \theta},$

(32)

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$\{ \hat{\rho}_0,\hat{L}_0 \} = 2 {\rm i} [\hat{\rho}_0,\hat{H}].$

(33)

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$F_{\rm Q}[\hat{\rho}_0,\hat{H}] = \left( {\text{Δ}}\hat{L}_0 \right)^2.$

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$\hat{L}_0 = 2 i \sum\limits_{k,k'} \dfrac{ p_k-p_{k'} }{p_k+p_{k'}} |k\rangle \langle k |\hat{H}|k'\rangle \langle k |,$

(35)

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$F_{\rm Q}[ \hat{\rho}_0, \hat{H} ] = 2 \sum\limits_{k,k'} \dfrac{\left(p_k-p_{k'} \right)^2}{p_k+p_{k'}} |\langle k |\hat{H}|k'\rangle |^2,$

(36)

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$\hat{L}_0 = 2 {\rm i} |\psi_0 \rangle \langle \psi_0 |\hat{H} - 2 {\rm i} \hat{H} |\psi_0 \rangle \langle \psi_0 |,$

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$F_{\rm Q}[ |\psi_0 \rangle, \hat{H} ] = 4 \left( {\text{Δ}}\hat{H} \right)^2.$

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$F \left( \theta \right) \geqslant \dfrac{|\dfrac{{\rm d} \langle \hat{ M} \rangle}{{\rm d}\theta} |^2}{\sum_{\mu} |c_{\mu} -f(\theta) |^2 P\left( \mu | \theta \right) },$

(39)

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$F \left( \theta \right) \geqslant \dfrac{|\langle [\hat{ M},\hat{H}] \rangle |^2}{\left( {\text{Δ}}\hat{ M} \right)^2 }.$

(40)

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$F \left( \theta \right) \geqslant | \dfrac{{\rm d} \langle \hat{ M} \rangle } {{\rm d} \theta} |^2 = | \langle [\hat{H},\hat{ M}] \rangle |^2.$

(41)

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$P({\varTheta}_{\rm {MLE}}|\theta) = \sqrt{\dfrac{m F(\theta)}{2 {\text{π}}}} {\rm e}^{-\dfrac{m F(\theta)}{2} \left( \theta- {\varTheta}_{\rm {MLE}} \right)^2}.$

(42)

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$\Delta\theta_{\rm{mom}} = \dfrac{{\text{Δ}}\mu} {\sqrt{m} | {\rm d} \bar{\mu}/{\rm d}\theta|}.$

(43)

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$P\left( \mu | \theta \right) = \dfrac{{\rm e}^{-\left (\mu-\bar{\mu}\right)^2/[2\left({\text{Δ}}\mu \right)^2]}} {\sqrt{2 {\text{π}} \left( {\text{Δ}}\mu \right)^2}},$

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$d^2_{\rm H}(P_0,P_{\theta}) = 2 \sum\limits_{\mu} \left( \sqrt{P(\mu|\theta)} -\sqrt{P(\mu|0)} \right)^2,$

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$d^2_{\rm H}(P_0,P_{\theta}) = 1-\mathcal{F}_{\rm{cl}}(P_0,P_{\theta}).$

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$d^2_{\rm H}(P_0,P_{\theta}) = \dfrac{F(0)}{8} \theta^2+ \mathcal{O}(\theta^3).$

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$D_{\rm{KL}}\equiv\underset{\mu }{{ \sum }}P(\mu|\theta_0)\ln \dfrac{P(\mu |\theta_0 )}{P(\mu |\theta_0+{\text{δ}} \theta )},$

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$d^2_{\rm H}(\hat{\rho}_0,\hat{\rho}_{\theta}) = 1-\mathcal{F}_{\rm{Q}}(\hat{\rho}_0,\hat{\rho}_{\theta}),$

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$|{{\varPsi}_{N,{\rm{cat}}}}\rangle = \dfrac{{| \uparrow {\rangle ^{ \otimes N}} + | \downarrow {\rangle ^{ \otimes N}}}}{{\sqrt 2 }}$

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$P(\pm|\theta) = \dfrac{1 \pm V \cos N \theta}{2},$

(51)

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$F(\theta) = \dfrac{V^2 N^2 \sin^2 N \theta}{1-V^2 \cos^2 N \theta}.$

(52)

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$| {\varPsi}_{\rm{sep}}\rangle = |{\varPsi}^{(1)}\rangle \otimes |{\varPsi}^{(2)}\rangle \otimes \cdots \otimes |{\varPsi}^{(N)}\rangle,$

(53)

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$\hat{\rho}_{\rm{sep}} = \sum\limits_q p_q |{\varPsi}_{\rm{ sep},q} \rangle \langle {\varPsi}_{\rm{ sep},q}|,$

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$| {\varPsi}_{k \rm{sep}}\rangle = |{\varPsi}_{N_1}\rangle \otimes |{\varPsi}_{N_2}\rangle \otimes \cdots \otimes |{\varPsi}_{N_M}\rangle,$

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$\hat{\rho}_{k \rm{sep}} = \sum\limits_q p_q |{\varPsi}_{k \rm{sep},q} \rangle \langle {\varPsi}_{k \rm{sep},q}|.$

(56)

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$F_{\rm Q}[\hat{\rho}_{\rm{sep}}, \hat{J}_n] \leqslant N,$

(57)

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${\text{Δ}}\theta_{\rm{SQL}} = \dfrac{1}{\sqrt{m N}}.$

(58)

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$F_{\rm Q}[\hat{\rho}, \hat{J}_n] > N,$

(59)

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$F_{\rm Q}[\hat{\rho}_{k \rm{sep}}, \hat{J}_n] \leqslant s k^2+ r^2,$

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$F_{\rm Q}[\hat{\rho}_{k \rm{sep}}, \hat{J}_n] \leqslant N^2.$

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${\text{Δ}}\theta_{\rm{HL}} = \dfrac{1}{ N \sqrt{m}}.$

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${\text{ζ}}_{\rm R}^2 = \dfrac{N \left( {\text{Δ}}\hat{J}_{\perp} \right)^2}{\langle \hat{J}_s \rangle^2},$

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$\dfrac{N}{F_{\rm Q}[\hat{\rho},\hat{J}_{ n}]} \leqslant {\text{ζ}}^2_{\rm R},$

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$V^2 > \left(1-\dfrac{1}{N} \right)^2 +\dfrac{1}{N^2}.$

(65)

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$\begin{split}\hat{H}_{\rm{SM}} =\, & [q+\lambda \left( 2 \hat{N}_0-1 \right)]\left( \hat{N}_{+1} +\hat{N}_{-1} \right) \\ & + 2 \lambda \left(\hat{a}_{-1}^{\dagger}\hat{a}_{+1}^{\dagger}\hat{a}_0\hat{a}_0 + \hat{a}_{0}^{\dagger}\hat{a}_{0}^{\dagger}\hat{a}_{-1}\hat{a}_{+1} \right),\end{split}$

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$|\downarrow, {{P}}_{\downarrow} \rangle \rightarrow \alpha |\downarrow, {{P}}_{\downarrow} \rangle + \beta |\uparrow, {{P}}_{\uparrow} \rangle,$

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$\hat{U}_{{n}}(\theta) = {\rm e}^{-{\rm i} \theta \hat{J}_{{n}}},$

(68)

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$\left( \begin{array}{c} \hat{a}_{\rm{out}} \\ \hat{b}_{\rm{out}} \\ \end{array} \right) = {{M}} \left( \begin{array}{c} \hat{a}_{\rm{in}} \\ \hat{b}_{\rm{in}} \\ \end{array} \right),$

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${{M}} = \left( \begin{array}{cc} {\rm e}^{-{\rm i} \phi_{\rm t}} \cos\dfrac{\vartheta}{2} & -{\rm e}^{-{\rm i} \phi_{\rm r}} \sin \dfrac{\vartheta}{2} \\ {\rm e}^{{\rm i} \phi_{\rm r}} \sin \dfrac{\vartheta}{2} & {\rm e}^{{\rm i} \phi_{\rm t}} \cos\dfrac{\vartheta}{2} \end{array} \right),$

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${{J}}\equiv \left( \begin{array}{c} \hat{J}_x \\ \hat{J}_y \\ \hat{J}_z \\ \end{array} \right) = \dfrac{1}{2} \left( \begin{array}{c} \hat{a}^{\dagger}\hat{b}+ \hat{b}^{\dagger}\hat{a}\\ - {\rm i}\left( \hat{a}^{\dagger}\hat{b} - \hat{b}^{\dagger}\hat{a} \right) \\ \hat{a}^{\dagger}\hat{a} - \hat{b}^{\dagger}\hat{b} \\ \end{array} \right),$

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${{J}}_{\rm{out}} = {\rm e}^{+{\rm i} \theta \hat{J_{{n}}}} {{J}}_{\rm{in}}{\rm e}^{-{\rm i} \theta \hat{J_{{n}}}},$

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$|\psi_{\rm{out}}\rangle = {\rm e}^{- {\rm i} \theta \hat{J_{{n}}}} | \psi_{\rm{in}}\rangle.$

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${{M}}_{\rm{BS}} = \left( \begin{array}{cc} \cos\dfrac{\theta}{2} & -{\rm i} \sin \dfrac{\theta}{2} \\ -{\rm i} \sin \dfrac{\theta}{2} & \cos\dfrac{\theta}{2}\end{array} \right).$

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$\left( \begin{array}{c} \hat{J}_x \\ \hat{J}_y \\ \hat{J}_z \\ \end{array} \right)_{\rm{out}} = \left( \begin{array}{ccc} 1 & 0 &0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \\ \end{array} \right) \left( \begin{array}{c} \hat{J}_x \\ \hat{J}_y \\ \hat{J}_z \\ \end{array} \right)_{\rm{in}}.$

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${{M}}_{\rm{MZ}} = \left( \begin{array}{cc} \cos\dfrac{\theta}{2} & - \sin \dfrac{\theta}{2} \\ \sin \dfrac{\theta}{2} & \cos\dfrac{\theta}{2} \\ \end{array} \right).$

(76)

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$\left( \begin{array}{c} \hat{J}_x \\ \hat{J}_y \\ \hat{J}_z \\ \end{array} \right)_{\rm{out}} = \left( \begin{array}{ccc} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \\ \end{array} \right) \left( \begin{array}{c} \hat{J}_x \\ \hat{J}_y \\ \hat{J}_z \\ \end{array} \right)_{\rm{in}}. $

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$\hat{U}_{\rm{MZ}} = {\rm e}^{{\rm i} \dfrac{{\text{π}}}{2} \hat{J}_x} {\rm e}^{- {\rm i} \theta \hat{J}_z} {\rm e}^{-{\rm i} \dfrac{{\text{π}}}{2} \hat{J}_x} = {\rm e}^{- {\rm i} \theta \hat{J}_y}. $

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$U({{x}}) = \dfrac{ \hbar {\varOmega}_1^2} { 4 {\varDelta}+ {\rm i} 2 {\varGamma}} \propto \dfrac{I({{x}})}{ 2 {\varDelta}+ {\rm i} {\varGamma}},$

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${\text{Δ}}\theta \!=\! \dfrac{1}{\sqrt{2} V}\! \dfrac{1}{|\sin (\theta/2)|}\! \dfrac{1} {\sqrt{m N}} \simeq \!\dfrac{\sqrt{2}}{|\sin (\theta/2)|} \dfrac{1}{M\!-\!1}\! \dfrac{1} {\sqrt{m N}},$

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$\dfrac{{\text{Δ}}g}{g} \simeq \dfrac{\sqrt{2}}{8 {\text{π}}} \dfrac{\omega^2 \lambda}{g} \dfrac{1}{M-1} \dfrac{1} {\sqrt{m N}},$

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$\hat{H} = \hat{H}_0 + \hat{H}_{\rm{int}} = \sum\limits_{i = 1}^{N} \dfrac{\alpha _{i}}{2} { {\sigma}} _{{m}}^{(i)}+\sum\limits_{\substack{ i,j = 1}}^{N}\dfrac{V_{ij}}{4}{ {\sigma}} _{{n}}^{(i)}{ {\sigma}} _{{n}}^{(j)},$

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$F_{\rm Q}[| \psi _{\rm{sep}}\rangle,\hat{H}] = f_{0}^{2}+f_{1}^{2}+f_{2}^{2},$

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$f_{0}^{2} = 4\left( {\text{Δ}}\hat{H}_{0}\right) ^{2} = \sum\limits_{i = 1}^{N}\alpha _{i}^{2} \big(1-\langle { {\sigma}} _{{m}}^{(i)}\rangle ^{2} \big),$

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$\begin{split} f_{1}^{2} =\, & 4\left( {\text{Δ}}\hat{H}_{1}\right) ^{2} = \sum\limits_{\substack{ i,j = 1 \\ i\neq j}}^{N}\dfrac{V_{ij}^{2}}{2}\big[ 1-\langle { {\sigma}} _{{n}}^{(i)}\rangle ^{2}\langle { {\sigma}} _{{n} }^{(j)}\rangle ^{2}\big] \\ & +\sum\limits_{\substack{ i,j,l = 1 \\ i\neq j\neq l}} ^{N}V_{ij}V_{il}\big[1-\langle { {\sigma}} _{{n}}^{(i)}\rangle ^{2}\big] \langle { {\sigma}} _{{n}}^{(j)}\rangle \langle \sigma _{\bf{n} }^{(l)}\rangle,\end{split}$

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$\begin{split} f_{2}^{2} & = 4\big( \left\langle \{\hat{H}_{0},\hat{H}_{1}\}\right\rangle -2\left\langle \hat{H}_{0}\right\rangle \left\langle \hat{H}_{1}\right\rangle \big) \\ & = 2\sum\limits_{\substack{i,j = 1 \\ i\neq j}}^{N}V_{ij}\alpha _{i}\big[ {{n}}\cdot {{m}}-\langle { {\sigma}} _{{n}}^{(i)}\rangle \langle { {\sigma}} _{{m}}^{(i)}\rangle \big]\langle { {\sigma}} _{{n} }^{(j)}\rangle.\end{split}$

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$\left\vert \psi_{\rm sep} \right\rangle \!=\! \bigotimes_{i = 1}^{N} \left(\!\!\sqrt{\dfrac{1\!+\! \langle \sigma _{{n}}^{(i)} \rangle }{2}}\left\vert \uparrow \right\rangle _{i}\!+\!{\rm e}^{-{\rm i}\varphi _{i}}\sqrt{\dfrac{1\!-\!\langle \sigma _{{n}}^{(i)} \rangle }{2}} \left\vert \downarrow \right\rangle _{i}\!\right)\!,$

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$F_{\rm Q}[|\psi_{\rm{sep}}\rangle,\hat{H}]\leqslant N + \gamma \dfrac{N}{2} + \gamma^2 \dfrac{N}{2}.$

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$\begin{split}\dfrac{F_{\rm Q}}{N} = \, & (1-a^2) + 2 \tilde{\gamma} a (1-a^2) \\ & + \tilde{\gamma}^2 \dfrac{1+2(N-2)a^2-(2N-3)a^4}{2(N-1)}, \end{split}$

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$\dfrac{F_{\rm Q}}{N} = 1+\dfrac{2N-1}{2(N-1)} \tilde{\gamma}^2 + O(\tilde{\gamma}^4),$

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$\dfrac{F_{\rm Q}}{N} = 2\tilde{\gamma} \dfrac{N-1}{2N-3} \sqrt{\dfrac{N-2}{2N-3}}+ \tilde{\gamma}^2 \dfrac{(N-1)}{2 (2N-3)}.$

(90)

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$\dfrac{F_{\rm Q}}{N} \leqslant 1 + \dfrac{4}{3\sqrt{3}} \tilde{\gamma} +\dfrac{(N-1)}{2(2N-3)} \tilde{\gamma}^2,$

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$\big(\Delta^2\theta_{\rm {est}}\big)_{\mu|\theta_0} = \sum\limits_{{{\mu}}}\big(\theta_{\rm {est}}({{\mu}})-\langle \theta_{\rm {est}} \rangle_{{{\mu}}|\theta_0} \big)^2 P({{\mu}}|\theta_0),$

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$\rm{MSE}(\theta_{\rm{est}})_{{{\mu}}|\theta_0} = \sum\limits_{{{\mu}}} \big( \theta_{\rm{est}}({{\mu}}) - \theta_0 \big)^2 P({{\mu}} \vert \theta_0),$

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$\rm{MSE}(\theta_{\rm{est}})_{{{\mu}}|\theta_0} =( {\varDelta}^2\theta_{\rm est})_{{{\mu}} \vert \theta_0} + \left(\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_0}-\theta_0\right)^2.$

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$\begin{split} & \left( \varDelta^2 \theta_{\rm{est}} \right)_{{{\mu}}|\theta_0} \geqslant \varDelta^2 \theta_{\rm BB} \\ & \quad \equiv \sup_{\theta_i,a_i,n} \dfrac{\left\{\sum_{i = 1}^{n}a_i[ \langle \theta_{\rm{est}} \rangle_{{{\mu}} \vert \theta_i} - \langle \theta_{\rm{est}} \rangle_{{{\mu}} \vert \theta_0} ]\right\}^2}{ \sum_{{{\mu}}} \left[\sum_{i = 1}^{n}a_i \mathcal{L}({{\mu}} \vert \theta_i,\theta_0)\right]^2 P({{\mu}}|\theta_0)},\end{split}$

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$\begin{split} & \left( \varDelta^2 \theta_{\rm{est}} \right)_{{{\mu}}|\theta_0} \geqslant \varDelta^2 \theta_{\rm BB}^{\rm{ub}} \\ &\quad \equiv \sup_{\theta_i,a_i,n} \dfrac{\left\{\sum_{i = 1}^{n}a_i[\theta_i - \theta_0 ]\right\}^2}{ \sum_{{{\mu}}} \left[\sum_{i = 1}^{n}a_i \mathcal{L}({{\mu}} \vert \theta_i,\theta_0)\right]^2 P({{\mu}}|\theta_0)}.\end{split}$

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$\begin{split} & \left( \varDelta^2 \theta_{\rm{est}} \right)_{{{\mu}}|\theta_0} \geqslant \varDelta^2 \theta_{\rm BB} \geqslant \varDelta^2 \theta_{\rm EChRB} \\ &\quad \geqslant \varDelta^2 \theta_{\rm ChRB} \geqslant \varDelta^2 \theta_{\rm CRB},\end{split}$

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$\centering P_{\rm{post}}(\theta|{{\mu}}) = \dfrac{P({{\mu}}|\theta) P_{\rm{pri}}(\theta)}{P_{\rm mar}({{\mu}})}.$

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$\left(\varDelta^{2}\theta_{\rm BL}({{\mu}})\right)_{\theta|{{\mu}}} = \int_{a}^{b} {\rm d}\theta \, P_{\rm{post}}(\theta|{{\mu}}) \big( \theta - \theta_{\rm BL}({{\mu}}) \big)^{2}.$

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$\varDelta^2 \theta_{\rm GB}({{\mu}}) = \dfrac{(f\left( {{\mu}},a,b\right) -1)^{2}}{\int_{a}^{b} {\rm d} \theta \, \dfrac{1}{P_{\rm{post}}(\theta|{{\mu}})} \left( \dfrac{{\rm d} P_{\rm{post}}(\theta|{{\mu}} )}{{\rm d} \theta}\right) ^{2} },$

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$\theta-\theta_{\rm BL}({{\mu}}) = \lambda_{{{\mu}}} \dfrac{{\rm d} \log P(\theta \vert {{\mu}})}{{\rm d} \theta},$

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$\begin{split}\left( \varDelta^{2}\theta_{\rm BL} \right)_{{{\mu}},\theta \vert \theta_0} & = \sum\limits_{{{\mu}}} \big( \varDelta^2 \theta_{\rm BL}({{\mu}}) \big)_{\theta \vert {{\mu}}} \, P({{\mu}} \vert \theta_0) \\ & = \sum\limits_{{{\mu}}} \int_{a}^{b} {\rm d} \theta \, P(\theta, {{\mu}} |\theta_0)\big( \theta-\theta_{\rm BL}({{\mu}})\big) ^{2}.\end{split}$

(102)

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$\varDelta^2 \theta_{\rm aGB} = \sum\limits_{{{\mu}}} \dfrac{(f\left( {{\mu}},a,b\right) -1)^{2}}{\int_{a}^{b} {\rm d} \theta \, \dfrac{1}{P_{\rm{post}}(\theta|{{\mu}})} \big( \dfrac{\partial P_{\rm{post}}(\theta|{{\mu}})}{\partial\theta} \big)^{2}} \, P({{\mu}}|\theta_{0}).$

(103)

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$\begin{split} P({{\mu}}|\theta_0) & = \prod_{i = 1}^{m}P(\mu_i|\theta_0)\\ &= \left(\dfrac{1+\cos(N \theta_0)}{2} \right)^{m_{+}} \left(\dfrac{1-\cos(N \theta_0)}{2} \right)^{m_{-}},\end{split}$

(104)

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$\theta_{\rm{MLE}}({{\mu}}) \equiv \rm{\arg \max_{\theta_0}} \{ P({{\mu}}|\theta_0) \}.$

(105)

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$\begin{split} P(\theta_{\rm{MLE}}|\theta_{0}) = \; &\sqrt{\dfrac{mF\left( \theta_{0}\right) }{2{\text{π}}} }{\rm e}^{-\frac{mF\left( \theta_{0}\right) }{2}\left( \theta_{0}-\theta _{\rm{MLE}}\right) ^2 }\\ & (m\gg 1).\end{split}$

(106)

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$P_{\rm pri}(\theta) = \dfrac{2}{{\text{π}}} \dfrac{{\rm e}^{\alpha\sin(2 \theta)^2}-1}{{\rm e}^{\alpha/2} I_0(\alpha/2)-1},$

(107)

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$P_{\rm{post}}(\theta|{{\mu}}) = \sqrt{\dfrac{m F(\theta_0)}{2 {\text{π}}}} {\rm e}^{-\textstyle\frac{m F(\theta_0)}{2} (\theta-\theta_0)^2} \quad (m \gg 1),$

(108)

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${F_{\rm Q}}[\hat \rho ,\hat q({{g}})] = {{{g}}^{\rm T}}{{ \varOmega}^{\rm T}}{{ \varOmega}}_{\hat \rho }^{ - 1}{{ \varOmega}{g}},$

(109)

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${F_{\rm Q}}[{\hat \rho _{{\text{Λ}}}},\hat q({{g}})] \leqslant 4{{{g}}^{\rm T}}{{ \varOmega}_{{{ \varPi}_{{\text{Λ}}}}({{\hat \rho }_{{\text{Λ}}}})}}{{g}},$

(110)

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${\text{ζ}}_{{\text{Λ}}}^2(\hat{\rho}): = \min_{{g}}4({{g}}^{\rm T}{ \varOmega}^{\rm T}{ \varOmega}_{{ \varPi}_{{\text{Λ}}}(\hat{\rho})}{ \varOmega}{{g}})({{g}}^{\rm T}{ \varOmega}_{\hat{\rho}}{{g}}).$

(111)

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$ \begin{align} {\text{ζ}}_{{\text{Λ}}}^{-2}(\hat{\rho}_{\rm{sep}})\leqslant 1. \end{align} $

(112)

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${{B}} ({\theta}) \geqslant \dfrac{[{{F}}({\theta})]^{-1}}{m} \geqslant \dfrac{[{{F}}_{\rm Q}({\theta})]^{-1}}{m},$

(113)

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$[{{F}}({\theta})]_{l,m} = \sum\limits_{\mu}\dfrac{\partial_l P({mu}|{\theta}) \partial_m P({mu}|{\theta})}{P({mu}|{\theta})},$

(114)

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$\begin{split}\dfrac{\partial \langle {\varTheta} \rangle }{\partial\theta} & = \dfrac{\partial }{\partial\theta} \sum\limits_{{{{\varepsilon}}}} P({{\varepsilon}}|\theta) {\varTheta} ({{{\varepsilon}}}) = \sum\limits_{{{{\varepsilon}}}} P({{\varepsilon}}|\theta){\varTheta} ({{{\varepsilon}}}) \dfrac{\partial L({{{\varepsilon}}} \vert \theta)}{\partial \theta} \\ & \equiv \left \langle {\varTheta} \dfrac{\partial L({{{\varepsilon}}} \vert \theta)}{\partial \theta} \right \rangle.\end{split}\tag{A1}$

(115)

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$\dfrac{\partial }{\partial\theta} \sum\limits_{{{{\varepsilon}}}} P({{\varepsilon}}|\theta) \!=\! \sum\limits_{{{{\varepsilon}}}} P({{\varepsilon}}|\theta) \dfrac{\partial L({{{\varepsilon}}} \vert \theta)}{\partial \theta} \!=\! \left \langle \dfrac{\partial L({{{\varepsilon}}} \vert \theta)}{\partial \theta} \right \rangle \!=\! 0 .\tag{A2}$

(116)

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$\bigg( \dfrac{\partial \langle {\varTheta} \rangle }{\partial\theta} \bigg)^2 = \left\langle \big( {\varTheta} ({{{\varepsilon}}}) -\langle {\varTheta} \rangle \big) \dfrac{\partial L({{{\varepsilon}}} \vert \theta)}{\partial \theta} \right\rangle^2.\tag{A3}$

(117)

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$\left\langle({\varTheta} - \langle {\varTheta} \rangle)^2 \right\rangle \left\langle \left(\dfrac{\partial L({{{\varepsilon}}} \vert \theta)}{\partial \theta}\right)^2 \right\rangle \geqslant \left(\dfrac{\partial \langle {\varTheta} \rangle}{\partial \theta}\right)^2,\tag{A4}$

(118)

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$F[\hat \rho (\theta ),\hat { E}({\epsilon})] = \sum\limits_{\epsilon} {\dfrac{{{\rm{Tr}}{{\left[ {{{\hat {\rm E}}}({\epsilon}){\partial _\theta }\hat \rho (\theta )} \right]}^{\rm{2}}}}}{{{\rm{Tr}}\left[ {{{\hat { E}}}({\epsilon})\hat \rho (\theta )} \right]}}} ,\tag{A5}$

(119)

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${\rm Tr}\left[\hat{ E}({{\varepsilon}})\partial_{\theta}\hat{\rho}(\theta)\right] = \Re\big({\rm Tr}\left[\hat{\rho}(\theta)\hat{L}_{\theta}\hat{ E}({{\varepsilon}})\right]\big),\tag{A6}$

(120)

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$\begin{split} &\Re\big({\rm Tr}\left[\hat{\rho}(\theta)\hat{L}_{\theta}\hat{ E}({{\varepsilon}})\right]\big)\leqslant \big|{\rm Tr}\left[\hat{\rho}(\theta)\hat{L}_{\theta}\hat{ E}({{\varepsilon}})\right]\big|^2\\ &\quad \leqslant {\rm Tr}\big[\hat{\rho}(\theta)\hat{ E}({{\varepsilon}})\big]{\rm Tr}\big[\hat{ E}({{\varepsilon}})\hat{L}_{\theta}\hat{\rho}(\theta)\hat{L}_{\theta}\big],\end{split}\tag{A7}$

(121)

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$\dfrac{\big({\rm Tr}\left[\hat{ E}({{\varepsilon}})\partial_{\theta}\hat{\rho}(\theta)\right]\big)^2}{{\rm Tr}\left[\hat{\rho}(\theta)\hat{ E}({{\varepsilon}})\right]}\leqslant {\rm Tr}\big[\hat{ E}({{\varepsilon}})\hat{L}_{\theta}\hat{\rho}(\theta)\hat{L}_{\theta}\big],\tag{A8}$

(122)

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$\begin{split} & F[ \hat{\rho}(\theta), \hat{ E}({{\varepsilon}}) ] \leqslant \sum\limits_{{{\varepsilon}}} {\rm Tr}\big[\hat{ E}({{\varepsilon}})\hat{L}_{\theta}\hat{\rho}(\theta)\hat{L}_{\theta}\big] \\ = & {\rm Tr}\big[\hat{\rho}(\theta)\hat{L}_{\theta}^2\big] = F_{\rm Q}[\hat{\rho}(\theta)],\end{split}\tag{A9}$

(123)

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$\left( {\text{Δ}}{\varTheta}_{\rm QCR} \right )^2 = \dfrac{\left( \dfrac{\partial \left\langle {\varTheta} \right \rangle }{\partial \theta} \right)^2}{F_{\rm Q}[\hat{\rho}(\theta)]},\tag{A10}$

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$\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta} = \sum\limits_{{\mu}}\theta_{\rm{est}}({\mu})P({\mu}|\theta).\tag{A11}$

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$\mathcal{L}({\mu}|\theta_i,\theta_0) = \dfrac{P({\mu}|\theta_i)}{P({\mu}|\theta_0)},\tag{A12}$

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$\sum\limits_{{\mu}}\theta_{\rm{est}}({\mu})L({\mu}|\theta_i,\theta_0)P({\mu}|\theta_0) = \langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_i},\tag{A13}$

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$\sum\limits_{{\mu}}\mathcal{L}({\mu}|\theta_i,\theta_0)P({\mu}|\theta_0) = \sum\limits_{{\mu}}P({\mu}|\theta_i) = 1,\tag{A14}$

(128)

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$\begin{split} & \sum\limits_{{\mu}}\left(\theta_{\rm{est}}({\mu})-\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_0}\right) \mathcal{L}({\mu}|\theta_i,\theta_0)P({\mu}|\theta_0) \\= & \langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_i}-\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_0},\end{split}\tag{A15}$

(129)

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$\begin{split} & \sum\limits_{{\mu}}\left(\theta_{\rm{est}}({\mu})-\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_0}\right)\left(\sum\limits_{i = 1}^na_i\mathcal{L}({\mu}|\theta_i,\theta_0)\right) P({\mu}|\theta_0) \\ = & \sum\limits_{i = 1}^na_i\left(\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_i}-\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_0}\right),\end{split}\tag{A16}$

(130)

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$\begin{split} & \left(\sum\limits_{i = 1}^na_i\left(\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_i}-\langle\theta_{\rm{est}}\rangle_{{\mu}|\theta_0}\right)\right)^2 \\ & \leqslant \left(\varDelta^2\theta_{\rm{est}}\right)_{{\mu}|\theta_0} \left(\sum\limits_{{\mu}}\bigg(\sum\limits_{i = 1}^na_i \mathcal{L}({\mu}|\theta_i,\theta_0)\bigg)^2P({\mu}|\theta_0)\right),\end{split}\tag{A17}$

(131)

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$\left(\varDelta^2\theta_{\rm{est}}\right)_{{\mu}|\theta_0} = \sum\limits_{{\mu}}\left(\theta_{\rm{est}}({\mu})- \langle \theta_{\rm{est}}\rangle_{{\mu}|\theta_0}\right)^2P({\mu}|\theta_0),\tag{A18}$

(132)

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Zhi-Hong Ren, Yan Li, Yan-Na Li, Wei-Dong Li. Development on quantum metrology with quantum Fisher information[J]. Acta Physica Sinica, 2019, 68(4): 040601-1

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