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We study the behavior of quantum Fisher information for a qubit probe that is interacting with a squeezed thermal environment. We analyzed the effect of squeezing parameters on the dynamics of quantum Fisher information which affects the optimal precision of the estimation parameter. We show that the squeezed field may offer a significant role in the precise measurement of the parameter cut-off frequency which is linked to the environment correlation time. Our results may be useful in quantum metrology, communication, and quantum estimation processes.

Quantum Fisher information (QFI) is a core concept in quantum metrology and quantum parameter estimation because it establishes a lower bound for parameters estimation [

Recently, QFI has been widely used to make high precision measurements of given parameters by using quantum systems and quantum resources. Practically, the quantum systems can never be closed completely and their interaction with the environment cannot be ignored which in turn, transfers the information to the quantum system. In such situations, it is essential to study the dynamics of QFI which quantifies the quality of transmitted information. The dynamics of QFI provides information about the different parameters of the environment, i.e. its temperature, time of interaction, and cut-off frequency which is linked to the environment correlation time and indirectly to the coherence time for computation and communication [

A common methodology used to enhance the precision of the estimation parameter is to increase the QFI. The higher the QFI is, the higher will be the precision of estimation. Several techniques are used to enhance the QFI for a precise measurement of parameters. The enhancement of QFI is very meaningful because its inverse characterizes the ultimate achievable precision in parameter estimation [

Different strategies are employed for the enhancement of QFI which includes partial measurements, partially collapsing measurements, and dynamical coupling pulses [

In this work, we consider the squeezed thermal environment for the enhancement of QFI. We investigate the dynamics of QFI for a qubit probe interacting with a squeezed thermal environment. This model is well considered to describe the physical systems which include molecular oscillation, the exciton-phonon interaction, and the photosynthesis process [

We arrange the paper as follows: In Section II we present the Hamiltonian of our physical model and briefly review the concept of QFI. The analytical results obtained through numerical simulations and discussion are presented in Section III. Finally, the main results are concluded in Section IV.

We consider a single qubit probe which is interacting with a squeezed bosonic thermal environment. The Hamiltonian describing the dynamics of composite system can be expressed as

H S R = 1 2 ω 0 σ z + ∑ k ω k b k † b k + σ z ⋅ ∑ k ( g k b k † + g k * b k ) , (1)

where on the right, the parts from left to right, represent the Hamiltonian of the qubit, the reservoir, and the interaction between the two components. In the first part, ω 0 ( ℏ = 1 ) is the transition energy of the qubit and σ z is the Pauli matrix acting on the space of the qubit. In the second part, ω k is the energy corresponding to the frequency of kth mode of the reservoir, and b k ( b k † ) is the annihilation (creation) operator obeying the usual commutation relations of bosonic operators. In the third part, g k represents the coupling strength of the corresponding mode with the qubit. Since the Hamiltonian of the qubit commutes with the interaction part, the overall effect of H on the qubit’s space is to decohere it. The squeezed thermal state of the environment can be written as [

ρ R ( 0 ) = S ζ ρ T h S ζ † , (2)

where ρ T h = 1 Z e − H R / T is the thermal state, Z = T r [ e − H R / T ] represents the partition function, H R is the Hamiltonian of environment (second part in Equation (1)) and T represents the temperature of environment. The the squeezed operator S ζ for the bosonic environment can be written as

S ζ = exp [ 1 2 ζ k * b k 2 − 1 2 ζ k ( b k † ) 2 ] , ζ k = r k e i θ k . (3)

where ζ k = r k e i θ k with r ≥ 0 and θ ∈ [ 0,2 π ] that denote the squeezing and phase parameters respectively. Moving to the interaction picture, the evolved reduced density matrix of the qubit probe is written as:

ρ S ( t ) = T r R [ U I ( t ) ρ S R ( 0 ) U I † ( t ) ] , (4)

where ρ S R = ρ S ( 0 ) ⊗ ρ R ( 0 ) is the state of the combined system and is written in the tensor product because the interaction of qubit with the environment is initially considered uncorrelated whereas ρ S ( 0 ) is the initial state of the qubit.

The unitary evolution operator in the interaction picture can be written as

U I ( t ) = e σ z 2 [ α k b k † − α * b k ] (5)

with α k = 2 g k ω k ( 1 − e i ω k t ) .

Starting from the pure state of the qubit probe in the form | Ψ ( 0 ) 〉 = cos ( α 2 ) | 0 〉 + sin ( α 2 ) | 1 〉 . Along with this and ρ s ( 0 ) = | ϕ ( 0 ) 〉 〈 ϕ ( 0 ) | , the density matrix for the qubit probe after doing some algebra can be written as

ρ s ( T , t , ω c ) = cos 2 ( α 2 ) | 0 〉 〈 0 | + sin 2 ( α 2 ) | 1 〉 〈 1 | + 1 2 sin ( α ) exp [ − Γ ( T , t , ω c ) ] ( | 1 〉 〈 0 | + | 0 〉 〈 1 | ) (6)

where Γ ( T , t , ω c ) contains the information of environment which are embedded on the state of qubit and its exponent is written as

e − Γ ( T , t ) = ∑ k 〈 e ( β k b † − β * b k ) 〉 (7)

Note that the above expression for e − Γ ( T , t , ω c ) is the characteristic function of Wigner representation of ρ R ( 0 ) , which is a squeezed thermal state of all modes [

Thus we get

Γ ( T , t ) = ∑ k 1 2 | β k ( t ) | 2 coth ( ω k 2 T ) , (8)

where

β k ( t ) = α k cosh r k + α k * e i θ k sinh r k (9)

Hence the decay factor can be written as

Γ ( T , t ) = ∑ k 4 | g k | 2 ( 1 − cos ω k t ω k 2 ) [ cosh 2 r k − cos ( θ − ω k t ) sinh 2 r k ] coth ( ω k 2 T ) . (10)

For continuous reservoir modes, the summation over | g k | 2 changes into integral and introducing the ohmic spectral density J ( ω ) that is ∑ k | g k | 2 → ∫ J ( ω ) d ω . Applying continuous mode approximations, the decay function can explicitly be written as

Γ ( T , t ) = ∫ J ( ω ) ( 1 − cos ω t ω 2 ) [ cosh 2 r − cos ( θ − ω t ) sinh 2 r ] coth ( w 2 T ) . (11)

The above decoherence function depends on temperature and also on the spectral density J ( ω , ω c ) of the coupling frequencies of reservoir, so for ohmic family, it is given by

J ( ω , ω c ) = ω s ω c s − 1 exp { − ω ω c } (12)

where ω c is the cut off frequency which describes a natural boundary in frequency response of the system and s is the true positive number that controls the behavior of spectral density at low frequencies and is called ohmicity parameter. Various values of the ohmicity parameter usually correspond to radically different types of dynamics, and therefore, it would be greatly recommendable to have an estimation scheme for their precise characterization. This dimensionless parameter classifies the environment into three classes, i.e. ohmic (s = 1), sub-ohmic (s = 0.5) and super-ohmic (s = 3).

QFI has different useful versions but in this paper we will consider symmetric logarithmic derivative based QFI [

I θ = T r [ ρ L θ 2 ] (13)

where L θ is found by ∂ ρ ∂ θ = 1 2 [ ρ L θ + L θ ρ ] .

The QFI with respect to a parameter β , for a spectrally decomposed density matrix ρ ( β ) = ∑ i λ i ( β ) | ϕ j ( β ) 〉 〈 ϕ i ( β ) | matrix can be expressed as [

I ( ω c ) = ∑ s ( ∂ ω c λ s ) 2 λ s + 2 ∑ l ≠ m ( λ l − λ m ) 2 λ l + λ m | 〈 ϕ l | ∂ ω c ϕ m 〉 | 2 , (14)

where the first part depends only on the eigenvalues and corresponds to classical Fisher information. The second part, in addition to eigenvalues, also depends on eigenvectors and is thus quantum mechanical in nature. A measurement on the final density matrix is said to be optimal for which the QFI reduces to classical Fisher Information. Moreover, being used as an estimation tool for different purposes [

λ ± = 1 2 ( 1 ± exp [ Γ ( T , t , ω c ) ] ) κ ( Γ , α ) (15)

and

Φ ± ( T , t , ω c ) = [ exp { Γ ( T , t , ω c ) } cot θ ± csc θ κ ( Γ , α ) ] | 0 〉 + | 1 〉 , (16)

with function κ ( Γ , α ) = [ exp { 2 Γ ( T , t , ω c ) } cos 2 α + sin 2 α ] 1 / 2 After substituting Equations (15) and (16) in (14), expression for the QFI reads to

I ( T , t , ω c ) = sin 2 ( α ) [ ∂ ω c Γ ( T , t , ω c ) ] 2 e 2 Γ ( T , t , ω c ) − 1 (17)

The QFI is maximum when α = π / 2 and the corresponding initial pure state of qubit probe is | + 〉 = 1 2 ( | 0 〉 + | 1 〉 ) . By considering this initial state of the qubit, the function κ ( Γ , α ) = 1 and eigenvectors become temperature independent and the second part of Equation (14) vanishes.

In this section, we will present the results by using the numerical simulations of Equation (17) due to a complex form of Γ ( T , t , ω c ) . Using these results, we will investigate the effects of different parameters of the environment on the dynamics of QFI. For simplicity, the interaction time t has been scaled out, i.e., one unit of time is given by the inverse of the coupling constant g.

towards the origin which shows that maximum information about the parameter cut-off frequency is encoded onto the state of qubit at these peaks. This type of behavior of QFI is very advantageous in various estimation procedures.

For different values of phase parameter, θ (

Finally we study the dynamics of QFI for a super-ohmic regime ( s = 3 ) by considering the effects of squeezing parameters on QFI. In

We have theoretically studied the dynamics of quantum Fisher information for a qubit probe that is interacting with a squeezed thermal environment. We have examined the behavior of QFI through numerical simulations and found that how QFI is maximized through the initial state of the qubit probe. Furthermore, we have described the effects of squeezing parameters on the QFI to extract maximum information about ω c that is imprinted onto the state of the qubit. We extended our results for three types of environments, i.e. sub-Ohmic, Ohmic, and super-Ohmic environments. It was found that for all three cases squeezing enhances the QFI which means that the precision of the estimation parameter is improved. When QFI is investigated with respect to time and cut off frequency, it is found that we have more time required to obtain optimal precision of the parameter, i.e. cut off frequency in sub-Ohmic and Ohmic case. However, in the super-Ohmic case, the QFI decays rapidly which shortens the interaction time for optimal measurements due to decoherence effects. Our results show that more squeezed fields lead to interesting and nontrivial effects on the optimal precision of cut off frequency. These results can play a significant role in quantum metrology, communication, and quantum estimation processes.

The authors declare no conflicts of interest regarding the publication of this paper.

Ullah, A. and Khan, K. (2021) Squeezing Effects on the Estimation Precision of Cut off Frequency. Journal of Quantum Information Science, 11, 13-23. https://doi.org/10.4236/jqis.2021.111002