Parameter estimation in n-dimensional massless scalar field

Ying Yang(杨颖) and Jiliang Jing(荆继良)

1Hunan Provincial Key Laboratory of Intelligent Sensors and Advanced Sensor Materials,School of Physics and Electronics,Hunan University of Science and Technology,Xiangtan 411201,China

2Department of Physics,Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education,and Synergetic Innovation Center for Quantum Effects and Applications,Hunan Normal University,Changsha 410081,China

Keywords: quantum Fisher information,parameter estimation,open quantum systems

1.Introduction

In the context of quantum measurements, the interaction between the detector and the system being measured leads to entanglement and information exchange.This results in the system being detected becoming an open quantum system.[1-3]In this context,the state of the detector is no longer described as a pure state, but rather as a mixed state.The evolution of the detector’s state is described by the master equation, typically in the Lindblad form.[4,5]During the process of quantum measurement, interaction between the detector and the measured system causes dissipation and decoherence, leading to the loss of coherence and information of the quantum system.During the process of quantum measurement, the interaction between the detector and the measured system causes dissipation and decoherence.This leads to the loss of coherence and information in the quantum system.Strategies have been developed for quantum estimation by understanding the dynamics of open quantum systems.[6-11]

Accurately estimating parameters is a challenge,and improving measurement accuracy is a key objective.There is a lot of research involved in improving parameter estimation.[12-18]Two methods have been proposed: optimizing the measurement process or optimizing the detection conditions.To enhance the measurement process, precise measurement techniques are used, such as selecting accurate and reliable instruments.Optimizing the detection conditions involves identifying the best operating conditions under which the parameter of interest can be measured with utmost precision.[19]This can be achieved by changing environmental parameters,preparing the optimal detector state,or controlling the detection time.This work focuses on the second approach,discussing the detection conditions for improving parameter estimation from a theoretical perspective.

The concept of QFI is essential when discussing the accuracy of parameter estimation.It has numerous applications in both theoretical and experimental domains.[20-23]In recent years,QFI associated with local operators has been employed to parameter estimation in open quantum systems, such as Unruh effect,[9,24]entanglement,[25,26]phase transitions,[27,28]etc.For instance, QFI has been employed to observe multipartite entanglement in Ref.[29].Besides, it has been a diagnostic for the nature of the quantum state of the system in a many-body quantum system in Ref.[30].Additionally, it has been suggested that QFI can be used as a probe for Unruh thermality in Ref.[9].Since QFI quantifies the precision of parameter estimation,several research works have focused on protecting the QFI of parameters in various ways.[31-33]

Quantum field theory commonly employs models that describe how atoms interact with quantum fields.Among these models, the Unruh-DeWitt detector is the most basic particle detector.[34]Initially proposed for studying the Unruh effect, it shows that for a uniformly accelerated observer, the vacuum of quantum fields in Minkowski spacetime is transformed into the thermal state.[35-37]The Unruh effect reveals that“vacuum”and“particle”depend on the observer.There are many applications in the model of Unruh-DeWitt detector, such as the connection of measurement uncertainty and quantum coherence for an inertial Unruh-DeWitt detector,[38]the transition rate of the Unruh-DeWitt detector in curved spacetime,[39]entanglement dynamics and entanglement harvesting for Unruh-DeWitt detector,[40,41]and more.Our research aims to explore the effect of Unruh temperature on quantum estimation.Specifically, we will consider the quantum estimation of the state parameter with an Unruh-DeWitt detector inn-dimensional Minkowski spacetime and analyze the influence of Unruh temperature on the QFI of state parameters.It is worth mentioning that the previous studies mentioned have primarily focused on state parameter estimation in 4-dimensional spacetime.[7,11,42]However, it has been observed that the response function of the detector in vacuum varies depending on the number of spacetime dimensions.[43,44]In Ref.[9], the QFI is discussed as a probe to detect the Unruh effect inn-dimensional Minkowski spacetime.Effect of spacetime dimensions on quantum entanglement has been investigated in Ref.[45].Inspired by these works, we are interested in generalizing the investigation of state parameter estimation to a more general case, namely a quantum field inn-dimensional Minkowski spacetime.

This work is organized as follows.In Section 2, we introduce the dynamical evolution of a two-level atom in the framework of open quantum system.In Section 3,we review the QFI for a single-qubit system.The dynamical evolution of a two-level atomic probe inn-dimensional Minkowski spacetime is arrived in Section 4.In Section 5, we obtain the analytical results of QFI for state parameter, and the discussion on parameter estimation is provided.The summary and conclusions are given in Section 6.The unitsc= ¯h=1 are used throughout this work.

2.The framework of open quantum system

A general parameter estimation consists of four processes:the preparation of the probe,the evolution of the probe state in the measured system,the measurement of the evolved probe state, and the estimation of the measured parameters from the measurement results.Here we use an Unruh-DeWitt detector as a probe which is modeled by a two-level atom.The detector is regarded as an open quantum system,which is coupled to a massless scalar field inn-dimensional Minkowski spacetime,and the Hamiltonian of the combined system reads

where the atomic HamiltonianHatom=(1/2)ω0σ3,Hfieldis the Hamiltonian of the scalar field.The interaction between the detector and the field is represented by the interaction Hamiltonian

withσ+,σ-,andω0being the atomic raising,lowering operators,and the energy level spacing of the atom respectively.

The dynamic map of the whole system is introduced in the following.The state of atom-field combined system is approximated asρtot(0)=ρ(0)atom⊗ρfieldat the initial timeτ=0,whereρ(0)atomis the atomic initial state andρfieldrepresents the state of quantum field.Then the total density matrixρtotis described by the von Neumann equation in the interaction picture as

The density matrix of the detector is then governed by a master equation in Lindblad form in Eq.(4), which represents a dissipative evolution due to the interaction between the detector and the quantum fields,andCi jis the Kossakowski matrix.Before giving the expression ofCi j,the Wightman function of scalar field should be introduced at

which is the Hilbert transform of Wightman functions.

After resolving the master equation(4)with a general initial state

where the state of this single qubit is determined by the parameterθandφ.Considering the effect of quantum field on quantum state,the density matrixρ(τ)evolving over time can be expressed as

For a two-level atomic detector,the density matrix can be expressed in a Bloch form as

For a general initial state expressed in Eq.(11), the initial Bloch vector isω= (sinθcosφ,sinθsinφ,cosθ).Assuming the atom is considered as a closed system, the Bloch vector of the state with atomic proper timeτbecomesω=(sinθcos(φ+ω0τ),sinθsin(φ+ω0τ),cosθ).Due to the coupling of the two-level atomic system with the quantum field,the atom cannot be regarded as a closed system but an open quantum system, thus influence of the quantum fields will be encoded in the atomic state, and Bloch vector evolutes with time has an exponential decay factor due to decoherence in Eq.(14).

3.Quantum Fisher information for a singlequbit system

One of the basic characteristics of QFI is that we can get its Lower bound on the achievable mean-square error of the estimated parameter.The unbiased estimator for the parameterϖis called quantum Cramer-Rao(QCR)theorem,and the QCR bound is given in the following inequality:[47-49]

With the expression of the Bloch vector in Eq.(13),the explicit form of QFI for a single-qubit system can be further expressed as[50]

For the mixed state, we calculate the QFI of parameterϖby the first line expression in Eq.(17), while for pure states, we use the second line expression in the above equation.Due to the interaction between the quantum system and the environment,the quantum state generally takes a mixed state after evolution,thus we will use the expression of QFI for the mixed state in the following parameter estimation.

4.Dynamical evolution of an Unruh–DeWitt detector in n-dimensional Minkowski spacetime

In order to arrive the dynamic evolution of a two-level atom, we need to discuss specific trajectories.In this section we will talk about the following uniformly accelerated trajectory:

where

in Eqs.(21), (23), and (24) represents the conventional gamma function.From Euler’s reflection formula,[51]we have|Γ(ix)|2=π/xsinh(πx),|Γ(1/2+ix)|2=π/cosh(πx), and recurrence relation Γ(z+1) =zΓ(z).Then we obtain the Bloch vectorsω=(ω1,ω2,ω3)Tinn-dimensional Minkowski spacetime as

The information about the evolution of quantum states is encoded in Eq.(25), thus we can calculate QFI by using the above Bloch vectors.

5.Discussion on quantum Fisher information

In this section,we will explore the behavior of QFI with different dimensions,and analysis how spacetime dimension,Unruh temperature and evolution time affect QFI, and then analyze the estimated accuracy of the initial parameter estimation via QFI.

5.1.Quantum Fisher information of parameter θ

By substituting Eq.(25)into Eq.(17),we obtain the analytical results of QFI for state parameterθas follows:

It is interesting to find thatFθis independent of initial phase parameterφfrom Eq.(26).To analyze how spacetime dimension influences parameter estimation,we discuss several cases with the dimensionsn=4,5,6,7.In the following context,we would like to discuss the variation of QFI with several parameters, such as evolution time, Unruh temperature, and initial state parameter.

As is shown in Fig.1, the QFI exhibits a monotonically decreasing behavior from 1 to 0 over time.For fixed Unruh temperatureTU=0.1,1,5, we find that the time it takes for the QFI to decay from 1 to 0 varies significantly at different Unruh temperatures.For example, forn=5 the decay timeτis around 100 for QFI to decay from 1 to 0 withTU=0.1,while the decay timeτis around 6 for QFI to decay from 1 to 0 withTU=1, and the decay timeτis around 0.3 for QFI to decay from 1 to 0 withTU=5, which present that for different Unruh temperatures,the decay time differs by 3 orders of magnitude.Hence the behavior of the QFI with evolution time is strongly influenced by the Unruh temperature.We find that forTU=0.1, in the case of high dimensions, the QFI of state parameters is larger.However, with the increase of Unruh temperature, the measurement advantage of high dimensions gradually disappears, and the measurement accuracy in the case of 4-dimensional spacetime is higher than other dimensions.When the Unruh temperatureTUand initial state parameterθare the same, we find that the attenuation rate of QFI corresponding to different dimensions over time is not consistent,and even there is a big difference.Therefore,may be we can distinguish different spacetime dimensions via QFI.

Fig.1.The Fθ as a function of the evolution time τ with fixed values of the initial state parameter θ =0.We take ω0=1.From the top panel to the bottom panel,as indicated in the figure,we take the Unruh temperature TU =0.1,1,and 5 respectively.

Fig.2. Fθ as a function of the Unruh temperature TU with fixed values of the initial state parameter θ =0.We take ω0=1.From the top panel to the bottom panel, as indicated in the figure, we take τ =1,3, and 5 respectively.

Fig.3.The Fθ as a function of the initial state parameter θ with fxied values of Unruh temperature TU =1.We take ω0=1.From the left panel to the right panel,as indicated in the fgiure,we take τ =1,3,and 5 respectively.

5.2.Quantum Fisher information of parameter φ

The analytic expression of parameterφis obtained after calculation as

From the above equation, we find thatFφdecays from 1 to 0 with increasing proper time forθ=π/2, and the decay rates depend on the function 4A+,n.Besides,we obtain thatFφ=0 whenθ=0,π,and hereθ=0,πcorrespond to the initial excited state.Since the variation of QFI for parametersφwith evolution timeτand Unruh temperatureTUis similar to that of the case of parametersθwhich we have discussed above,then we will not analyze the variation of phase parameters with time and Unruh temperature in detail.Here we just gives the figures ofFφas a function of initial parameterθ.

Fig.4. Fφ as a function of initial state parameter θ with fxied values of the Unruh temperature TU =1.We take ω0 =1.From the left panel to the right panel,as indicated in the fgiure,we take τ =1,3,and 5 respectively.

As is shown in Fig.4, it is obviously to see that the value ofFφis the maximum forθ=π/2, and it is the minimum forθ=0,π.When other parameters (Unruh temperature,initial parameter,and spacetime dimension)are the same,only the evolution time is different, we find that the QFI forτ=1 is about one order of magnitude higher than the case ofτ=5.For Unruh temperatureTU=1,the initial state parameterθ=π/2 and the evolution timeτ=1,3,5,we obtain that

6.Conclusions

We conducted a study on parameter estimation using local quantum estimation.Our focus was on an Unruh-DeWitt detector as an open quantum system that interacts with a massless scalar background inn-dimensional spacetime.The detector’s dynamics are described by a Lindblad master equation that governs the evolution of its density matrix.We discovered that the QFI of state parametersθandφdepends on various factors,including evolution time,Unruh temperature,and scalar field dimensionality.By studying the QFI’s behavior under different parameters,we aimed to improve the accuracy of parameter estimation.Our results show that the QFI exhibits a monotonically decreasing behavior over time, decaying from 1 to 0 at a varying rate in different Unruh temperatures.It is observed that the QFI of the state parameter depends onθ, andFθis the maximum forθ=0 orθ=π,Fφis the maximum forθ=π/2.We also found that the attenuation rate of QFI corresponding to different dimensions over time is not consistent.This indicates that we may be able to distinguish different spacetime dimensions using QFI.As the Unruh temperature increases,the QFI value first decreases,then gradually trends to 0.We attribute this trend to the thermal fluctuations caused by the increasing Unruh temperature,which makes the system more disordered,thus decreasing the QFI of the state parameter.We also observed that the QFI for small evolution time is about several orders of magnitude higher than that of long evolution time.

Acknowledgments

Project supported by the National Natural Science Foundation of China(Grant Nos.12105097 and 12035005)and the Science Research Fund of the Education Department of Hunan Province,China(Grant No.23B0480).