Abstract We investigate the quantum Fisher information (QFI) of a qubit-qutrit system in the background of Garfinkle–Horowitz–Strominger dilation black hole. After deriving the analytical expression of the QFI, we examine its dynamics with respect to the dilation parameter D and the state parameter γ of the system. Our results show that the QFI for the estimation of γ is a fixed value, which is independent of the parameters D and γ. And the QFI for the estimation of D varies with the parameters D and γ. Additionally, we propose an effective strategy to steer the QFI by introducing weak measurement reversal. We find that the QFI can be remarkably enhanced by adjusting the appropriate reversing measurement strengths. Our findings might provide some useful insights for the study on parameter estimation of hybrid systems in the framework of relativity theory. Keywords：quantum Fisher information;Garfinkle–Horowitz–Strominger space–time;weak measurement reversal
PDF (729KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yi-Jun Lian, Jin-Ming Liu. Quantum Fisher information of a qubit-qutrit system in Garfinkle–Horowitz–Strominger dilation space–time. Communications in Theoretical Physics, 2021, 73(8): 085102- doi:10.1088/1572-9494/ac01e4
1. Introduction
Quantum metrology [1], which aims to improve the precision of parameter estimation, has aroused wide interest in recent years. As an important part of quantum metrology, quantum Fisher information (QFI) severs as an incredible indicator of parameter estimation [2, 3]. The inverse of QFI [4, 5] gives a lower bound for the estimation precision of unknown parameters based on the quantum Cramer–Rao’s theorem. Larger QFI means that the parameter can be estimated with a higher precision, and vice versa. Except for the parameter estimation, QFI was also widely applied to other fields, including quantum frequency standards [6], uncertainty relations [7, 8], and entanglement detection [9]. Until now, many schemes were proposed to examine the robustness of QFI under decoherence process [10–12].
On the other hand, the combination of quantum information science, general relativity and quantum field theory has attracted widespread attention. This combination can not only broaden the application of quantum information theory, but also deepen people’s understanding of quantum effects in the theory of relativity such as Hawking radiation and Unruh effect. Recently, the quantum effects in the background of curved space–time have been studied extensively [13–28]. For example, Martin-Martinez et al [25] analyzed the operational meaning of the residual entanglement in noninertial fermionic systems in terms of the achievable violation of the Clauser–Horne–Shimony–Holt inequality and demonstrated the quantum correlations of fermions. Shi et al [27] discussed the quantum distinguishability and geometric discord in the Schwarzschild space–time. Tavakoli et al [28] investigated the holographic entanglement entropy in the Rindler–AdS space–time to obtain an exact solution for the corresponding minimal surface. Inspired by these works, we notice that Garfinkle–Horowitz–Strominger (GHS) dilation black hole has a good approximation to the exact solution of string theory [29]. For this black hole, the dilation parameter D plays an important role on extra attractive force, and when D = 0 the black hole has an inner horizon. To investigate quantum effects in the background of GHS dilation space–time, a few studies have been carried out [30–34]. He et al [32] studied the quantum correlation for Dirac particles in the background of a GHS dilation black hole and proved that physical accessible quantum correlation decreases monotonically as the dilation parameter enhances. Zhang et al [34] analyzed the quantum-memory-assisted entropic uncertainty relation of a hybrid qubit-qutrit state for Dirac particles in the background of a GHS dilation space–time, and considered the corresponding relationship between the entropy uncertainty and the quantum entanglement.
To our knowledge, little attention has been paid to the QFI of a qubit-qutrit system in GHS space–time. In this paper, we attend to use the QFI as the indicator to study the problem of unknown parameters estimation in GHS dilation space–time. The results show that for the given initial qubit-qutrit state of system, the QFI for the estimation of the state parameter does not change with the parameters D and γ, but the QFI for the estimation of the dilation parameter varies with the values of D and γ. Moreover, the maximal QFI in the estimation of dilation parameter D is obtained in the case of D → 1 and γ = 0 or π for bipartite systems. Additionally, we propose a scheme to improve the behavior of QFI with the technique of quantum weak measurement reversal (WMR). We find that the precision of parameter estimation can be remarkably enhanced by tuning the appropriate strengths of the reversing measurement. Our results could deepen our understanding of QFI dynamics in a curved space–time and shed some new light on quantum precision measurement within the framework of relativistic theory.
This paper is organized as follows. In section 2, we briefly review the vacuum structure for Dirac particles in GHS dilation space–time. In section 3, we examine the QFI of a qubit-qutrit system in a GHS dilation black hole and propose an efficient strategy to steer the QFI through WMR. In section 4, we summarize our conclusions.
2. Theory
2.1. Vacuum representation of Dirac particles in GHS dilation space–time
The metric in the background of a GHS dilation black hole is given by [29]$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -\left(\displaystyle \frac{r-2M}{r-2D}\right){\rm{d}}{t}^{2}\\ & & +\ {\left(\displaystyle \frac{r-2M}{r-2D}\right)}^{-1}{\rm{d}}{r}^{2}+r(r-2D){\rm{d}}{{\rm{\Omega }}}^{2},\end{array}\end{eqnarray}$where$\begin{eqnarray*}D={Q}^{2}/2M.\end{eqnarray*}$Here, D denotes the dilation parameter of GHS black hole. Q and M represent the charge and mass of the black hole, respectively. For simplicity, we consider G = c = ℏ = kB = 1 in this paper [30]. In a general background of curved space–time, the Dirac equation for spinor field ψ is described as [35, 36]$\begin{eqnarray}[{\gamma }^{a}{e}_{a}^{\upsilon }({\partial }_{\upsilon }+{{\rm{\Gamma }}}_{\upsilon })]\psi =0,\end{eqnarray}$where γa is the Dirac matrix, ${e}_{a}^{\upsilon }$ denotes the inverse of the tetrad ${e}_{\upsilon }^{a}$, and Γ$ \upsilon $ represents the spin connection coefficient. By solving the above Dirac equation, we can get the following positive frequency outgoing solutions for the outside and inside regions in the vicinity of the event horizon$\begin{eqnarray}{\psi }_{k}^{{\rm{I}}+}(r\gt {r}_{+})=\kappa {{\rm{e}}}^{-{\rm{i}}\omega \tau },{\psi }_{k}^{{\rm{II}}+}(r\lt {r}_{+})=\kappa {{\rm{e}}}^{{\rm{i}}\omega \tau }.\end{eqnarray}$Here, τ denotes the delay time with $\tau =t-r\,-2(M\,-D)\mathrm{ln}[(r-2M)/(2M-2D)]$, κ denotes the our-component Dirac spinor, Ω represents the monochromatic frequency of the Dirac fields, ${\psi }_{k}^{{\rm{I}}\pm }$ and ${\psi }_{k}^{{\rm{II}}\pm }$ form a completely orthogonal basis, I and II represent the exterior and interior regions near the event horizon, respectively. According to ${\psi }_{k}^{{\rm{I}}\pm }$ and ${\psi }_{k}^{{\rm{II}}\pm }$, we can get the following Dirac field formula$\begin{eqnarray}{\psi }_{\mathrm{out}}=\sum _{\eta }\int {\rm{d}}k({\zeta }_{k}^{\eta }{\psi }_{k}^{\eta +}+{\varsigma }_{k}^{\eta * }{\psi }_{k}^{\eta -}),\end{eqnarray}$where ${\zeta }_{k}^{\eta }$ and ${\varsigma }_{k}^{\eta * }$ denote the fermion annihilation and antifermion creation operators with superscript η ∈ (I, II). Based on the relationship between black hole coordinates and Kruskal coordinates, a complete basis for positive energy modes can be calculated as$\begin{eqnarray}\begin{array}{rcl}{\varpi }_{k}^{{\rm{I}}+} & = & {{\rm{e}}}^{2(M-D)\pi \omega }{\psi }_{k}^{{\rm{I}}+}+{{\rm{e}}}^{-2(M-D)\pi \omega }{\psi }_{-k}^{{\rm{II}}-},\\ {\varpi }_{k}^{{\rm{II}}+} & = & {{\rm{e}}}^{-2(M-D)\pi \omega }{\psi }_{-k}^{{\rm{I}}-}+{{\rm{e}}}^{2(M-D)\pi \omega }{\psi }_{k}^{{\rm{II}}+}.\end{array}\end{eqnarray}$Substituting the above basis into equation (4), the Dirac fields in the Kruskal space–time can be rewritten as$\begin{eqnarray}\displaystyle \begin{array}{rcl}{\psi }_{\mathrm{out}} & = & \sum _{\eta }{\rm{d}}k{\left\{2\cosh [4(M-D)\pi \omega ]\right\}}^{-1/2}\\ & & \times \ ({\alpha }_{k}^{\eta }{\varpi }_{k}^{\eta +}+{\beta }_{k}^{\eta +}{\varpi }_{k}^{\eta -}).\end{array}\end{eqnarray}$Here, ${\alpha }_{k}^{\eta }$ and ${\beta }_{k}^{\eta +}$ represent the annihilation and creation operators in the Kruskal vacuum, respectively. In terms of the Bogoliubov transformations between the creation and annihilation operators in the GHS dilation and Kruskal coordinates, we can obtain the annihilation operator$\begin{eqnarray}{\alpha }_{k}^{{\rm{I}}}=\mu {\zeta }_{k}^{{\rm{I}}}-\nu {\varsigma }_{k}^{{\rm{II}}* },\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}\mu & = & {\left[{{\rm{e}}}^{-8(M-D)\pi \omega }+1\right]}^{-1/2}\quad \mathrm{and}\\ \nu & = & {\left[{{\rm{e}}}^{8(M-D)\pi \omega }+1\right]}^{-1/2}.\end{array}\end{eqnarray}$For the sake of brevity, we set M = Ω = 1 throughout this paper. After proper calculations, the vacuum and excited states of Kruskal particle for mode K can be expressed as$\begin{eqnarray}\begin{array}{rcl}| 0{\rangle }_{K}^{+} & = & \mu | {0}_{k}{\rangle }_{{\rm{I}}}^{+}| {0}_{-k}{\rangle }_{{\rm{II}}}^{-}+\nu | {1}_{k}{\rangle }_{{\rm{I}}}^{+}| {1}_{-k}{\rangle }_{{\rm{II}}}^{-},\\ | 1{\rangle }_{K}^{+} & = & | {1}_{k}{\rangle }_{{\rm{I}}}^{+}| {0}_{-k}{\rangle }_{{\rm{II}}}^{-},\end{array}\end{eqnarray}$where $\{| {n}_{k}{\rangle }_{{\rm{I}}}^{+}\}$ and $\{| {n}_{-k}{\rangle }_{{\rm{II}}}^{-}\}$ (n = 0, 1) are the orthonormal bases for the outside and inside regions of the event horizon respectively, the superscripts {+, −} of the basis vectors denote the particle and antiparticle vacua. For simplicity, we replace $\{| {n}_{k}{\rangle }_{{\rm{I}}}^{+}\}$ and $\{| {n}_{-k}{\rangle }_{{\rm{II}}}^{-}\}$ with {∣n⟩I} and {∣n⟩II} in the rest of this paper, respectively.
2.2. Brief description of QFI
QFI is an extension of classical Fisher information under the quantum regime. As we know, classical Fisher information is used to assess how much parameter information is contained in the measured parameter. The inverse of classical Fisher information gives a lower bound of the estimation error. Generally, different positive operator valued measurement (POVM) leads to different classical Fisher information. There always exists a suitable POVM so that the optimal Fisher information is achievable, and this optimal Fisher information is called QFI. According to the Cramer–Rao inequality, the QFI is given by [37–41]$\begin{eqnarray}F(\theta )=\mathrm{Tr}[\rho (\theta ){L}^{2}],\end{eqnarray}$where L is a symmetric logarithmic derivative operator defined by$\begin{eqnarray}\displaystyle \frac{\partial \rho (\theta )}{\partial \theta }=\displaystyle \frac{L\rho (\theta )+\rho (\theta )L}{2}.\end{eqnarray}$Now let us introduce a density matrix of an N-dimensional system$\begin{eqnarray}{\rho }_{{}_{\theta }}=\sum _{i=1}^{S}{P}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| ,\end{eqnarray}$where Pi and ∣ψ⟩i are the eigenvalues and eigenvectors of ρθ respectively, S is the number of nonzero eigenvalues. Through straightforward calculations, QFI is derived as$\begin{eqnarray}\displaystyle \begin{array}{l}F(\theta )=\sum _{i=1}^{S}\displaystyle \frac{{\left({\partial }_{\theta }{P}_{i}\right)}^{2}}{{P}_{i}}+\sum _{i=1}^{S}{P}_{i}{F}_{{\psi }_{i}}(\theta )\\ \quad -\sum _{i\ne j=1}^{S}\displaystyle \frac{8{P}_{i}{P}_{j}}{{P}_{i}+{P}_{j}}| \langle {\psi }_{i}| {\partial }_{\theta }{\psi }_{j}\rangle {| }^{2},\end{array}\end{eqnarray}$where$\begin{eqnarray}{F}_{{\psi }_{i}}(\theta )=4(\langle {\partial }_{\theta }{\psi }_{i}| {\partial }_{\theta }{\psi }_{i}\rangle -| \langle {\psi }_{i}| {\partial }_{\theta }{\psi }_{i}\rangle {| }^{2}).\end{eqnarray}$For pure states, QFI described by equation (13) can be simplified to$\begin{eqnarray}F(\theta )=4(\langle {\partial }_{\theta }\psi | {\partial }_{\theta }\psi \rangle -| \langle \psi | {\partial }_{\theta }\psi \rangle {| }^{2}).\end{eqnarray}$In what follows, we will use the QFI to study the quantum metric problem in a relativistic framework.
3. QFI of a qubit-qutrit system in the GHS black hole
In this section, we concentrate on investigating the dynamic behavior of QFI in the GHS dilation black hole. Considering that the hybrid qubit-qutrit state is a nontrivial extension of qubit-qubit case, and such a state may shed some new light on quantum parameter estimation of high-dimension system. We assume that Alice and Bob, as two observers, initially share a bipartite qubit-qutrit state in the flat region of Minkowski space–time, which takes the following form$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{{AB}}} & = & \displaystyle \frac{{\cos }^{2}\gamma }{2}(| 01\rangle \langle 01| +| 01\rangle \langle 10| +| 10\rangle \langle 01| +| 10\rangle \langle 10| )\\ & & +\displaystyle \frac{{\sin }^{2}\gamma }{2}(| 20\rangle \langle 20| +| 21\rangle \langle 21| ),\end{array}\end{eqnarray}$where γ ∈ [0, π]. Note that the quantum state ${\rho }_{{}_{{AB}}}$ is maximally entangled when γ = 0 or π, and is completely separable when γ = π/2. The entanglement degree of this state decreases monotonically when γ increases from 0 to π/2, whereas the entanglement increases gradually with the growth of γ from π/2 to π.
Let us suppose that Alice remains at the asymptotically flat region, Bob freely falls towards a GHS dilation black hole and locates near the event horizon at a constant acceleration. According to equation (9), we can rewrite the initial qubit-qutrit state in terms of Minkowski modes for Alice and black hole modes for Bob as$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{{{AB}}_{{\rm{I}}}{B}_{{\rm{II}}}}} & = & \displaystyle \frac{{\cos }^{2}\gamma }{2}({\mu }^{2}| 100\rangle \langle 100| +\mu \nu | 100\rangle \langle 111| \\ & & +\mu \nu | 111\rangle \langle 100| +{\nu }^{2}| 111\rangle \langle 111| +\mu | 100\rangle \langle 010| \\ & & +\nu | 111\rangle \langle 010| +\mu | 010\rangle \langle 100| +\nu | 010\rangle \langle 111| \\ & & +| 010\rangle \langle 010| )+\displaystyle \frac{{\sin }^{2}\gamma }{2}({\mu }^{2}| 200\rangle \langle 200| \\ & & +\mu \nu | 200\rangle \langle 211| +\mu \nu | 211\rangle \langle 200| \\ & & +{\nu }^{2}| 211\rangle \langle 211| +| 210\rangle \langle 210| ).\end{array}\end{eqnarray}$Due to the fact that the exterior region is causally disconnected from the interior region of the event horizon, we can trace over the modes of the interior region to obtain the physically accessible density matrix as follows$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{{{AB}}_{{\rm{I}}}}} & = & \displaystyle \frac{1}{2}[{\cos }^{2}\gamma | 01\rangle \langle 01| +\mu {\cos }^{2}\gamma | 01\rangle \langle 10| \\ & & +\mu {\cos }^{2}\gamma | 10\rangle \langle 01| +{\mu }^{2}{\cos }^{2}\gamma | 10\rangle \langle 10| \\ & & +{\nu }^{2}{\cos }^{2}\gamma | 11\rangle \langle 11| +{\mu }^{2}{\sin }^{2}\gamma | 20\rangle \langle 20| \\ & & +(1+{\nu }^{2}){\sin }^{2}\gamma | 21\rangle \langle 21| ].\end{array}\end{eqnarray}$Similarly, by tracing over the modes of the exterior region and the system A respectively, the physically inaccessible density matrix can be derived as$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{{{AB}}_{{\rm{II}}}}} & = & \displaystyle \frac{1}{2}[{\cos }^{2}\gamma | 00\rangle \langle 00| +\nu {\cos }^{2}\gamma | 00\rangle \langle 11| \\ & & +{\mu }^{2}{\cos }^{2}\gamma | 10\rangle \langle 10| +\nu {\cos }^{2}\gamma | 11\rangle \langle 00| \\ & & +{\nu }^{2}{\cos }^{2}\gamma | 11\rangle \langle 11| +(1+{\mu }^{2}){\sin }^{2}\gamma | 20\rangle \langle 20| \\ & & +{\nu }^{2}{\sin }^{2}\gamma | 21\rangle \langle 21| ],\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{{B}_{{\rm{I}}}{B}_{{\rm{II}}}}} & = & \displaystyle \frac{1}{2}[{\mu }^{2}| 00\rangle \langle 00| +\mu \nu | 00\rangle \langle 11| +| 10\rangle \langle 10| \\ & & +\mu \nu | 11\rangle \langle 00| +{\nu }^{2}| 11\rangle \langle 11| ].\end{array}\end{eqnarray}$
3.1. Quantum estimation of initial state parameter γ and dilation parameter D
In this subsection, we give the analytical expression of QFI for bipartite systems and single-particle system, and then examine the influence of parameters γ and D on the QFI. For the bipartite physically accessible system ${\rho }_{{}_{{{AB}}_{{\rm{I}}}}}$, based on equation (18), the nonzero eigenvalues λi and the corresponding eigenvectors ψi are obtained as$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{1} & = & \displaystyle \frac{(1+{\mu }^{2}){\cos }^{2}\gamma }{2},\quad {\psi }_{1}\\ & = & \left\{0,\sqrt{\displaystyle \frac{1}{1+{u}^{2}}},\sqrt{\displaystyle \frac{{u}^{2}}{1+{u}^{2}}},0,0,0\right\},\\ {\lambda }_{2} & = & \displaystyle \frac{{\nu }^{2}{\cos }^{2}\gamma }{2},{\psi }_{2}=\left\{0,0,0,1,0,0\right\},\\ {\lambda }_{3} & = & \displaystyle \frac{{\mu }^{2}{\sin }^{2}\gamma }{2},{\psi }_{3}=\left\{0,0,0,0,1,0\right\},\\ {\lambda }_{4} & = & \displaystyle \frac{(1+{\nu }^{2}){\sin }^{2}\gamma }{2},{\psi }_{4}=\left\{0,0,0,0,0,1\right\}.\end{array}\end{eqnarray}$Substituting equation (21) into equation (13), the QFI for γ can be calculated as$\begin{eqnarray}{F}_{{{AB}}_{{\rm{I}}}}(\gamma )=\displaystyle \frac{{\left({\partial }_{\gamma }{\lambda }_{1}\right)}^{2}}{{\lambda }_{1}}+\displaystyle \frac{{\left({\partial }_{\gamma }{\lambda }_{2}\right)}^{2}}{{\lambda }_{2}}+\displaystyle \frac{{\left({\partial }_{\gamma }{\lambda }_{3}\right)}^{2}}{{\lambda }_{3}}+\displaystyle \frac{{\left({\partial }_{\gamma }{\lambda }_{4}\right)}^{2}}{{\lambda }_{4}}=4.\end{eqnarray}$Clearly, ${F}_{{{AB}}_{{\rm{I}}}}(\gamma )$ is independent of the values of dilation parameter D and state parameter γ, which means that D and γ will not cause any disturbance on ${F}_{{{AB}}_{{\rm{I}}}}(\gamma )$. Likewise, the QFI for D can be derived as$\begin{eqnarray}{F}_{{{AB}}_{{\rm{I}}}}(D)=2\mu {{\prime} }^{2}+2\nu {{\prime} }^{2}{\cos }^{2}\gamma +\displaystyle \frac{2{\nu }^{2}\nu {{\prime} }^{2}{\sin }^{2}\gamma }{{\nu }^{2}+1},\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}\mu ^{\prime} & = & \displaystyle \frac{\partial \mu }{\partial D}=-4\pi \omega {\mu }^{3}{{\rm{e}}}^{-8(M-D)\pi \omega }\quad \mathrm{and}\\ \nu ^{\prime} & = & \displaystyle \frac{\partial \nu }{\partial D}=4\pi \omega {\nu }^{3}{{\rm{e}}}^{8(M-D)\pi \omega }.\end{array}\end{eqnarray*}$
In figure 1, we plot ${F}_{{{AB}}_{{\rm{I}}}}(D)$ as functions of γ and D. For clarity, the corresponding two-dimensional figures are exhibited in figures 1(b) and (c). From figure 1(b), we can see that for the fixed γ, ${F}_{{{AB}}_{{\rm{I}}}}(D)$ enhances as D increases, and in the limit of D→1, i.e. the Hawking temperature T = 1/[8π(M − D)] → ∞ , ${F}_{{{AB}}_{{\rm{I}}}}(D)$ tends to its maximum value. That is to say, the parameter estimation precision about dilation parameter can reach a maximum when the black hole approximates to evaporate completely. Figure 1(c) shows that ${F}_{{{AB}}_{{\rm{I}}}}(D)$ decreases first and then enhances with γ increasing for the fixed D. This implies that the higher precision in dilation parameter estimation can be attained for an appropriate value of parameter γ.
Figure 1.
New window|Download| PPT slide Figure 1.(a) The QFI ${F}_{{{AB}}_{{\rm{I}}}}(D)$ versus dilation parameter D and state parameter γ. (b) ${F}_{{{AB}}_{{\rm{I}}}}(D)$ as a function of D for different γ. (c) ${F}_{{{AB}}_{{\rm{I}}}}(D)$ as a function of γ for different D.
Next, we focus on the QFI of the other bipartite divisions, i.e. ${\rho }_{{}_{{{AB}}_{{\rm{II}}}}}$ and ${\rho }_{{}_{{B}_{{\rm{I}}}{B}_{{\rm{II}}}}}$, which are physically inaccessible. Similarly, we derive the nonzero eigenvalues and the corresponding eigenvectors of ${\rho }_{{}_{{{AB}}_{{\rm{II}}}}}$ as$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{1}^{{\prime} } & = & \displaystyle \frac{(1+{\nu }^{2}){\cos }^{2}\gamma }{2},\\ {\psi }_{1}^{{\prime} } & = & \left\{\sqrt{\displaystyle \frac{1}{1+{\nu }^{2}}},0,0,\sqrt{\displaystyle \frac{{\nu }^{2}}{1+{\nu }^{2}}},0,0\right\},\\ {\lambda }_{2}^{{\prime} } & = & \displaystyle \frac{{\mu }^{2}{\cos }^{2}\gamma }{2},{\psi }_{2}^{{\prime} }=\left\{0,0,1,0,0,0\right\},\\ {\lambda }_{3}^{{\prime} } & = & \displaystyle \frac{(1+{\mu }^{2}){\sin }^{2}\gamma }{2},{\psi }_{3}^{{\prime} }=\left\{0,0,0,0,1,0\right\},\\ {\lambda }_{4}^{{\prime} } & = & \displaystyle \frac{{\nu }^{2}{\sin }^{2}\gamma }{2},{\psi }_{4}^{{\prime} }=\left\{0,0,0,0,0,1\right\}.\end{array}\end{eqnarray}$
The nonzero eigenvalues and the corresponding eigenvectors of ${\rho }_{{B}_{{\rm{I}}}{B}_{{\rm{II}}}}$ are$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{1}^{{\prime\prime} } & = & \displaystyle \frac{1}{2},{\psi }_{1}^{{\prime\prime} }=\left\{\mu ,0,0,\nu \right\},\\ {\lambda }_{2}^{{\prime\prime} } & = & \displaystyle \frac{1}{2},{\psi }_{2}^{{\prime\prime} }=\left\{0,0,1,0\right\}.\end{array}\end{eqnarray}$Based on equation (13), we have$\begin{eqnarray}{F}_{{{AB}}_{{\rm{II}}}}(\gamma )=4,{F}_{{B}_{{\rm{I}}}{B}_{{\rm{II}}}}(\gamma )=0,\end{eqnarray}$and$\begin{eqnarray}{F}_{{{AB}}_{{\rm{II}}}}(D)=2\nu {{\prime} }^{2}+2\mu {{\prime} }^{2}{\cos }^{2}\gamma +\displaystyle \frac{2{\mu }^{2}\mu {{\prime} }^{2}{\sin }^{2}\gamma }{{\mu }^{2}+1},\end{eqnarray}$$\begin{eqnarray}{F}_{{B}_{{\rm{I}}}{B}_{{\rm{II}}}}(D)=2(\mu {{\prime} }^{2}+\nu {{\prime} }^{2}).\end{eqnarray}$It is obvious that the estimation for γ remains unchanged in the bipartite physically inaccessible region. Meanwhile, we plot the behavior of ${F}_{{{AB}}_{{\rm{II}}}}(D)$ as functions of D and γ in figure 2(a). We can see that ${F}_{{{AB}}_{{\rm{II}}}}(D)$ always has a maximum at a certain D for a given γ. Notably, for the fixed value of D, the maximum of the QFI is obtained when γ = 0 or π, which implies that the parameter γ of the initial state is vital to enhance the precision of quantum estimation. Furthermore, it can be seen from figure 2(b) that the value of ${F}_{{B}_{{\rm{I}}}{B}_{{\rm{II}}}}(D)$ becomes large gradually with the increase of D.
Figure 2.
New window|Download| PPT slide Figure 2.(a) The QFI ${F}_{{{AB}}_{{\rm{II}}}}(D)$ versus dilation parameter D and state parameter γ. (b) ${F}_{{B}_{{\rm{I}}}{B}_{{\rm{II}}}}(D)$ as a function of D.
Finally, we calculate the QFIs of single-particle system for individual modes A, BI and BII, which are given by$\begin{eqnarray}\begin{array}{rcl}{F}_{A}(\gamma ) & = & 4,{F}_{A}(D)=0,\\ {F}_{{B}_{{\rm{I}}}}(\gamma ) & = & 0,{F}_{{B}_{{\rm{I}}}}(D)=2\mu {{\prime} }^{2}+\displaystyle \frac{2{\nu }^{2}\nu {{\prime} }^{2}}{{\nu }^{2}+1},\\ {F}_{{B}_{{\rm{II}}}}(\gamma ) & = & 0,{F}_{{B}_{{\rm{II}}}}(D)=2\nu {{\prime} }^{2}+\displaystyle \frac{2{\mu }^{2}\mu {{\prime} }^{2}}{{\mu }^{2}+1}.\end{array}\end{eqnarray}$In figure 3, the QFI of subsystems BI and BII with respect to D is illustrated. We can find that the value of ${F}_{{B}_{{\rm{I}}}}(D)$ remains near zero for the small D, then increases rapidly after D = 0.8, and reaches the maximum value for D → 1. However, ${F}_{{B}_{{\rm{II}}}}(D)$ increases first and then decreases with the growth of D.
Figure 3.
New window|Download| PPT slide Figure 3.The QFI ${F}_{{B}_{{\rm{I}}}}(D)$ (a) and ${F}_{{B}_{{\rm{II}}}}(D)$ (b) as a function of dilation parameter D.
3.2. Steering the QFI by WMR
According to the above discussion, it is quite clear that QFI is affected by the dilation parameter of the black hole and the parameter of the initial state. As we know, weak measurement has less disturbance on quantum systems than the traditional von Neumann orthogonal measurement. After a weak measurement is performed on a state of a quantum system, the quantum state may be restored to its initial status through appropriate measurement reversal operations, which is beneficial to protect the quantum state under decoherence [42, 43]. In the following, we will investigate whether the WMR can be adopted to improve the QFI of the qubit-qutrit system or not.
Now, we introduce a tripartite WMR operator taking the following form [44]$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal M }}_{r} & = & \sqrt{(1-p)(1-q)}| 0\rangle \langle 0| +\sqrt{1-q}| 1\rangle \langle 1| \\ & & +\ \sqrt{1-p}| 2\rangle \langle 2| ,\end{array}\end{eqnarray}$where p and q denote the strengths of the reversing measurements with p, q ∈ [0, 1]. We assume that the WMR acts on Alice’s side, and the qubit-qutrit state ρAB after the reversal operator is given by$\begin{eqnarray}\begin{array}{l}{\widetilde{\rho }}_{{AB}}=\displaystyle \frac{({{ \mathcal M }}_{r}\otimes {\mathbb{I}}){\rho }_{{AB}}{\left({{ \mathcal M }}_{r}\otimes {\mathbb{I}}\right)}^{\dagger }}{\mathrm{Tr}[({{ \mathcal M }}_{r}\otimes {\mathbb{I}}){\rho }_{{AB}}{\left({{ \mathcal M }}_{r}\otimes {\mathbb{I}}\right)}^{\dagger }]}\\ =\ N[{\cos }^{2}\gamma (1-p)(1-q)| 01\rangle \langle 01| +{\cos }^{2}\gamma (1-q)\sqrt{1-p}| 01\rangle \langle 10| \\ \ \ +\ {\cos }^{2}\gamma (1-q)\sqrt{1-p}| 10\rangle \langle 01| +{\cos }^{2}\gamma (1-q)| 10\rangle \langle 10| \\ \ \ +\ {\sin }^{2}\gamma (1-p)| 20\rangle \langle 20| +{\sin }^{2}\gamma (1-p)| 21\rangle \langle 21| ],\end{array}\end{eqnarray}$where$\begin{eqnarray*}N=\displaystyle \frac{1}{{\cos }^{2}\gamma (2-p)(1-q)+2{\sin }^{2}\gamma (1-p)}.\end{eqnarray*}$Meanwhile, according to equation (9), the bipartite systems of the qubit-qutrit state in the GHS space–time can be reduced to the following forms after a series of trace operation$\begin{eqnarray}\begin{array}{rcl}{\widetilde{\rho }}_{{}_{{{AB}}_{{\rm{I}}}}} & = & N[{\cos }^{2}\gamma (1-p)(1-q)| 01\rangle \langle 01| \\ & & +\mu {\cos }^{2}\gamma (1-q)\sqrt{1-p}| 01\rangle \langle 10| \\ & & +\mu {\cos }^{2}\gamma (1-q)\sqrt{1-p}| 10\rangle \langle 01| \\ & & +{\mu }^{2}{\cos }^{2}\gamma (1-q)| 10\rangle \langle 10| \\ & & +{\nu }^{2}{\cos }^{2}\gamma (1-q)| 11\rangle \langle 11| \\ & & +{\mu }^{2}{\sin }^{2}\gamma (1-p)| 20\rangle \langle 20| \\ & & +(1+{\nu }^{2}){\sin }^{2}\gamma (1-p)| 21\rangle \langle 21| ],\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\widetilde{\rho }}_{{}_{{{AB}}_{{\rm{II}}}}} & = & N[{\cos }^{2}\gamma (1-p)(1-q)| 00\rangle \langle 00| \\ & & +\nu {\cos }^{2}\gamma (1-q)\sqrt{1-p}| 00\rangle \langle 11| \\ & & +{\mu }^{2}{\cos }^{2}\gamma (1-q)| 10\rangle \langle 10| \\ & & +\nu {\cos }^{2}\gamma (1-q)\sqrt{1-p}| 11\rangle \langle 00| \\ & & +{\nu }^{2}{\cos }^{2}\gamma (1-q)| 11\rangle \langle 11| \\ & & +(1+{\mu }^{2}){\sin }^{2}\gamma (1-p)| 20\rangle \langle 20| \\ & & +{\nu }^{2}{\sin }^{2}\gamma (1-p)| 21\rangle \langle 21| ].\end{array}\end{eqnarray}$
Through numerical calculations based on the equation (13), the QFI for the dilation parameter D corresponding to the bipartite states ${\widetilde{\rho }}_{{}_{{{AB}}_{{\rm{I}}}}}$ and ${\widetilde{\rho }}_{{}_{{{AB}}_{{\rm{II}}}}}$ can be obtained as follows$\begin{eqnarray}\begin{array}{rcl}{\widetilde{F}}_{{{AB}}_{{\rm{I}}}}(D) & = & 4a(\mu {{\prime} }^{2}+\nu {{\prime} }^{2})(1-q)+4b\mu {{\prime} }^{2}(1-p)\\ & & +\displaystyle \frac{4b\nu {{\prime} }^{2}{\nu }^{2}(1-p)}{1+{\nu }^{2}},\\ {\widetilde{F}}_{{{AB}}_{{\rm{II}}}}(D) & = & 4a(\mu {{\prime} }^{2}+\nu {{\prime} }^{2})(1-q)+4b\nu {{\prime} }^{2}(1-p)\\ & & +\displaystyle \frac{4b\mu {{\prime} }^{2}{\mu }^{2}(1-p)}{1+{\mu }^{2}},\end{array}\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}a & = & \displaystyle \frac{{\cos }^{2}\gamma }{{\cos }^{2}\gamma (2-p)(1-q)+2{\sin }^{2}\gamma (1-p)},\\ b & = & \displaystyle \frac{{\sin }^{2}\gamma }{{\cos }^{2}\gamma (2-p)(1-q)+2{\sin }^{2}\gamma (1-p)}.\end{array}\end{eqnarray*}$
To unveil the relation among the reversal strengths (p and q) and the QFI in GHS space–time, we depict the QFI of ${\widetilde{\rho }}_{{}_{{{AB}}_{{\rm{I}}}}}$ with respect to p and q in figure 4. Without loss of generality, we set the parameter of the initial state γ = 0.3. From figure 4(a), we find that for the fixed D∈(0, 1), the QFI enhances monotonously with p increasing, but with the enhancement of q, the QFI will diminish invariably. This implies that the WMR is capable of enhancing the precision of parameter estimation by increasing parameter p and reducing parameter q. To elaborate further, we plot a two-dimensional figure of the QFI as a function of p for different D under the fixed q. From figure 4(b), we can see that the value of ${\widetilde{F}}_{{{AB}}_{{\rm{I}}}}(D)$ increases with growing p no matter what the value of D is. Next, we investigate the effect of the WMR on the QFI of ${\widetilde{\rho }}_{{}_{{{AB}}_{{\rm{II}}}}}$. As depicted in figure 5, the QFI increases monotonously as p grows, which is similar to the situation of figure 4.
Figure 4.
New window|Download| PPT slide Figure 4.(a) Contour plot of ${\widetilde{F}}_{{{AB}}_{{\rm{I}}}}(D)$ versus the reversal strengths p and q with D = 0.8. (b) ${\widetilde{F}}_{{{AB}}_{{\rm{I}}}}(D)$ as a function of p for different D with q = 0.1.
Figure 5.
New window|Download| PPT slide Figure 5.(a) Contour plot of ${\widetilde{F}}_{{{AB}}_{{\rm{II}}}}(D)$ versus the reversal strengths p and q with D = 0.8. (b) ${\widetilde{F}}_{{{AB}}_{{\rm{II}}}}(D)$ as a function of p for different D with q = 0.1.
As we know, purity of the initial state is a critical element on the realization of optimal parameter estimation. To further understand the relationship among QFI and reversal strengths p and q, now we explore the impact of WMR on the state ${\widetilde{\rho }}_{{}_{{AB}}}$ by analyzing its purity ξ of such a state. Assume that $\xi =\mathrm{Tr}{\widetilde{\rho }}_{{}_{{AB}}}^{2}$, clearly if ξ = 1 (≠ 1), ${\widetilde{\rho }}_{{}_{{AB}}}$ is a pure state (mixed state). From figure 6, one can easily see that the reversal strength p has an incremental impact on ξ while ξ has a negative correlation with q. As a result, enlarging p and reducing q are helpful in raising the purity of ${\widetilde{\rho }}_{{AB}}$, which contributes to achieving better parameter estimation.
Figure 6.
New window|Download| PPT slide Figure 6.Purity ξ for the state ${\widetilde{\rho }}_{{AB}}$ versus reversal strengths p and q.
4. Conclusions
In summary, by considering that a qubit locates near the event horizon of the black hole and a qutrit stays at the asymptotically flat region, we have studied the influence of state parameter γ and dilation parameter D on the QFI of a hybrid qubit-qutrit system in the GHS space–time. In particular, we had derived the analytical forms of QFIs for the hybrid system and revealed their dynamic evolution in the framework of relativity. The results show that the F(γ) is a fixed value but the F(D) varies with the values of D and γ. Notably, we find that the high precision of dilation parameter estimation can be attained with respect to an appropriate value of γ for systems ${\rho }_{{}_{{{AB}}_{{\rm{I}}}}}$ and ${\rho }_{{}_{{{AB}}_{{\rm{II}}}}}$. The maximal QFI for the estimation of D is obtained in the limit of D → 1 and γ = 0 or π. Furthermore, we have proposed a strategy to steer the behavior of the QFI with the technique of WMR. Remarkably, the evolution of QFI with respect to D shows a distinct enhancement after the measurement reversal operation. This indicates that WMR has a significant influence on the precision of estimation for D. We believe that our findings might provide some useful insights for quantum precision measurement and parameter estimation of hybrid systems in the curved space–time.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant Nos. 91950112 and 11174081, and the National Key Research and Development Program of China under Grant No. 2016YFB0501601.
,State Key Laboratory of Precision Spectroscopy, 
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