Dynamics of Quantum Fisher information in a spin-boson model
aa r X i v : . [ qu a n t - ph ] N ov Dynamics of Quantum Fisher information in a spin-boson model
Xiang Hao,
1, 2, ∗ Ning-Hua Tong, and Shiqun Zhu Department of Physics, Renmin University of China, Beijing 100872, China School of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou, Jiangsu 215011, People’s Republic of China School of Physical Science and Technology, Soochow University,Suzhou, Jiangsu 215006, People’s Republic of China
Abstract
Quantum Fisher information characterizes the phase sensitivity of qubits in the spin-boson modelwith a finite bandwidth spectrum. In contrast with Markovian reservoirs, the quantum Fisherinformation will flow from the environments to qubits after some times if the bath parameter s is larger than a critical value which is related to temperatures. The sudden change behavior willhappen during the evolution of the quantum Fisher information of the maximal entanglement statein the non-Markovian environments. The sudden change times can be varied with the change of thebath parameter s . For a very large number of entangled qubits, the sudden change behavior of themaximal quantum Fisher information can be used to characterize the existence of the entanglement.The metrology strategy based on the quantum correlated state leads to a lower phase uncertaintywhen compared to the uncorrelated product state. ∗ Electronic address: [email protected] . INTRODUCTION Fisher information is a key quantity for extracting the information about a parameterfrom a measurement-induced probability distribution [1–3]. In the classical realm, the Fisherinformation can provide a basic lower bound of the variance of any unbiased estimator due tothe Cra ´mer-Rao inequality [4]. A large value of Fisher information represents an attainablemeasurement with a high precision. The quantum Fisher information is referred to as thenatural extension of the classic Fisher information. Among many versions of quantum Fisherinformation, there is a famous definition based on the symmetric logarithmic derivatives[1, 2]. In a practical estimation, the quantum Fisher information can be used to describethe sensitivity of a quantum state with respect to a parameter such as the frequency of thesystem [5, 6], the strength of the changing external field [7, 8], and the speed of the quantumevolution [9].To improve the precision of parameter estimation, some research groups have put forwardentanglement-enhanced metrological schemes [10–17]. It is proved that the scaling with thenumber of entangled particles N can achieve the Heisenberg limit, proportional to 1 /N ,which overcomes the shot-noise limit (or called as standard quantum limit), proportional to1 / √ N . However, the practical quantum technology unavoidably induces the decoherenceand dissipation owing to the coupling of the system to the environment [18, 19]. So far,much attention has been focused on quantum metrology in the presence of Markovian ornon-Markovian noise [5, 6, 20–30]. In some Markovian channels, there is a sudden changeof the quantum Fisher information [20]. A local dephasing environment was also takeninto account in order to exactly obtain the scaling law of the parameter estimation [6]. Itwas found out that the non-Markovian dephasing environment enables quantum entangledstates to outperform the strategy based on product states. In other realistic environments,we need to study if this correlated-state metrology strategy is superior to the strategy basedon product states. Besides it, we want to investigate if the sudden change behavior of thequantum Fisher information happen in non-Markovian environments.The paper is organized as follows. We introduce the definition of quantum Fisher infor-mation in Sec. II. In Sec. III, a general non-Markovian depolarizing channel is presented bya qubit coupled to a heat bath. In, Sec. IV, the crossover between non-Markovian dynamicsand Markovian ones will be manifested by the quantum Fisher information with respect to2 phase parameter. The sensitivity of the phase estimation based on the N -particle maxi-mally entangled state is also studied in contrast with the product state. Finally, a simplediscussion is concluded in Sec. V. II. QUANTUM FISHER INFORMATION
In metrology, one of basic measurements is the phase estimation with respect to a unitarytransformation of a linear interferometer. The phase φ is related to an unbiased estimatorˆ φ , i.e., h ˆ φ i = φ . The measured state is obtained by ρ φ = e iφ ˆ J ~n ρe − iφ ˆ J ~n where ρ is the inputstate. ˆ J ~n is a generator which describes the angular momentum operator along a direction ~n = ( n , n , n ). According to the quantum Cra ´mer-Rao theorem [1, 2], the precision of thephase parameter φ has a lower bound limit, which is determined by∆ ˆ φ ≥ ∆ φ QCR = 1 q νF ( ρ, ˆ J ~n ) , (1)where ν is the number of the experiments. And F ( ρ, ˆ J ~n ) denotes the quantum Fisherinformation, which can be defined as F ( ρ, ˆ J ~n ) = Tr( ρ φ L φ ) . (2) L φ is the symmetric logarithmic derivative with respect to the phase parameter φ and satisfiesthat ∂ρ φ ∂φ = ( ρ φ L φ + L φ ρ φ ). To emphasize the effects of the quantum Fisher information,we set ν = 1 in the paper. Using the above equations, we calculate the quantum Fisherinformation as F ( ρ, ˆ J ~n ) = 2 X i = j ( λ i − λ j ) λ i + λ j |h i | ˆ J ~n | j i| = ~n ˆ C~n † , (3)where | i i is the eigenvector of the density matrix ρ with the corresponding eigenvalue λ i , andthe elements of the symmetric matrix ˆ C are expressed as C kl = P i = j ( λ i − λ j ) λ i + λ j ( h i | ˆ J k | j ih j | ˆ J l | i i + h i | ˆ J l | j ih j | ˆ J k | i i ).For a pure state, F ( ρ, ˆ J ~n ) = 4(∆ ˆ J ~n ) . To maximize the quantum Fisher information, weneed to select the rotation along an optimal direction ~n o by diagonalizing the symmetricmatrix as ˆ C d = ˆ O † ˆ C ˆ O = diag( C , C , C ). ˆ O is a transformation matrix which is composedof three orthogonal and normalization eigenvectors of ˆ C . According to the result of [20], themaximum of the quantum Fisher information is written as F max = 4max( C , C , C ) . (4)3he optimal direction ~n o is determined by the eigenvector of ˆ C with the maximal eigenvalue.For the total state ρ N of N independent qubits, the maximal mean value of the quantumFisher information can be introduced as¯ F max = F max N . (5)Here, the unitary transformation related to the phase parameter φ is presented asexp( iφ P Ni =1 ˆ J i~n o ) where ˆ J i~n o is the angular momentum operator for the ith qubit along theoptimal direction ~n o . When the values ¯ F max >
1, we have ∆ φ QCR < / √ N . In this con-dition, the ultimate estimation limit is superior to the standard quantum limit. Only if¯ F max ≃ N , the Heisenberg limit of ∆ φ QCR ≃ /N can be attained. III. NON-MARKOVIAN SPIN-BOSON MODEL
A spin-boson system is a simple model that describes a effective two-level system coupledto a bosonic reservoir [19, 31]. The model has extensively been applied to noisy quantumdots [33], decoherence of qubits in quantum computation [34, 35], quantum impurities andcharge transfer in donor-acceptor systems [36]. The dissipative environment in the spin-boson model can be characterized by the structured spectral function J ( ω ) with frequencybehavior of J ( ω ) ∝ ω s ω − sc exp( − ω/ω c ). ω c is the cutoff frequency of the spectrum. Withthe change of the bath parameter s , the environments vary from sub-Ohmic ones ( s < s = 1) and super-Ohmic ones ( s > H = H + H E + H I , (6)where H = ω ˆ σ x is a local hamiltonian of the system with the tunneling frequency of ω , and H E = P k ˆ b † k ˆ b k denotes the hamiltonian of the environment including all degreesof freedom. The interaction hamiltonian between the spin and environment is expressedas H I = P k g k (ˆ σ + ˆ b k + ˆ σ − ˆ b † k ), where the strength of the couplings can be described bythe spectral function as P | g k | → R J ( ω ) δ ( ω − ω k ) dω . ˆ σ ± are the rising and loweringoperators. For the kth mode field, ˆ b k and ˆ b † k represent the annihilation and creation operatorrespectively. Using a transformation ˆ U , we can diagonalize the system hamiltonian into4 H eff = ˆ U † H ˆ U = ω ˆ¯ σ z . The spin operators are also mapped into the new ones, i.e.,ˆ¯ σ j = ˆ U † ˆ σ j ˆ U , ( j = z, ± ). We can obtain the time-convolutionless master equation of thedensity matrix of the spin in the interaction picture as dρ ( t ) dt = − i [ ¯ H eff , ρ ( t )] + ˆ L [ ρ ( t )] , (7)where ρ ( t ) is the density matrix of the system after the unitary transformation. The Lindbladsuperoperator ˆ L can be written asˆ L [ ρ ( t )] = X m = z, ± γ m ( t )[ˆ¯ σ m ρ ( t )ˆ¯ σ † m − { ˆ¯ σ † m ˆ¯ σ m , ρ ( t ) } ] . (8)We need notice that the master equation is obtained in the secular approximation [19] wherethe high-frequency oscillating terms are neglected. The secular approximation is reasonableunder the assumption of the weak coupling between the system and the environment. Thetime-dependent decaying parameters at a finite temperature T are expressed as γ z ( t, T ) = 12 Z J ( ω ) coth( ~ ω κ B T ) sin ωtω dωγ ± ( t, T ) = 12 Z J ( ω )[( n T + 1) sin( ω ± ω ) tω ± ω + n T sin( ω ∓ ω ) tω ∓ ω ] dω. (9)The mean number of the field is n T = [exp( ~ ω/κ B T ) − − . The analytical expression of ρ ( t ) can be written in the Hilbert space spanned by {| i , | i} as ( a − b ) ρ (0) + b cρ (0) c ∗ ρ (0) (1 − a ) + ( a − b ) ρ (0) , (10)where ρ ij (0) is the element of the density matrix ρ (0) at the initial time and | i denotes theeigenvector of the new operator ˆ¯ σ z with the corresponding eigenvalue ±
1. The decoherenceparameters are obtained as a = 12 [(1 + δe − Γ ) + e − Γ ] b = 12 [(1 + δe − Γ ) − e − Γ ] c = e − Λ − iω t , (11)where Γ( t ) = R t [ γ + ( t ′ ) + γ − ( t ′ )] dt ′ , Λ( t ) = Γ( t ) + 2 R t γ z ( t ′ ) dt ′ and δ ( t ) = R t e Γ( t ′ ) [ γ + ( t ′ ) − γ − ( t ′ )] dt ′ . If a + b = 1 and a − b = c , the dynamical map will be reduced to a depolarizingchannel where an initial state ρ (0) will evolve into a mixture of ρ (0) and maximally mixedstate ˆ I . 5 V. DYNAMICAL CROSSOVER AND PHASE ESTIMATION
In this general depolarizing environments, the quantum Fisher information for time-dependent mixed state ρ ( t ) can be analytically obtained as follows. We may use the Blochvector ~B ( t ) to simplify the description of the density matrix of single qubit as ρ ( t ) = ˆ I + ˆ ~σ · ~B ( t )2 , (12)where the three components of the Bloch vector is written as B ( t ) = e − Λ sin θ cos( ω t + ϕ ) ,B ( t ) = e − Λ sin θ sin( ω t + ϕ ) ,B ( t ) = e − Γ (cos θ + ϕ ) . (13)Here, the Bloch vector can also be expressed as ~B ( t ) = | ~B ( t ) | (sin α cos β, sin α sin β, cos α ).These time-dependent angle parameters satisfy the initial conditions of α (0) = θ, θ ∈ [0 , π ]and β (0) = ϕ, ϕ ∈ [0 , π ] . The maximal quantum Fisher information is obtained as F max ( t ) = e − sin θ + e − (cos θ + δ ) . (14)To obtain the maximal quantum Fisher information, we can apply the two different kindsof the optimal rotations where the directions are expressed as ~n o k = (sin β, cos β, ,~n o ⊥ = (0 , , . (15)In regard to all possible initial states, we use the average value of F max which is defined as F Amax = 14 π Z π Z π F max sin θdθdϕ. (16)Actually, the non-monotonic behaviors of the quantum Fisher information can be usedto define and quantify the non-Markovianity of quantum dynamics [32]. The flow of themaximal quantum Fisher information for a qubit at a finite temperature is shown in Fig.1(a). From Fig. 1(a), we can see that the large negative values of ∂F Amax ∂t occur if the bathparameter is less than a temperature-dependent critical value, i.e., s < s c . This also meansthat the quantum Fisher information is decreased with time. In the condition of s < s c s > s c , the non-Markovianity of the reservoirs occursbecause of the small positive values of ∂F Amax ∂t >
0. For the temperature κ B T ~ = 0 .
01, thecritical value of the bath parameter is about s c ≈ .
4. In Fig. 1(b), the crossover betweenthe Markovian dynamics and non-Markovian ones is clearly determined by the flow of thequantum Fisher information. If the bath parameter s < s c , there is no dynamical crossover.For the non-Markovian environments of s > s c , the values of ∂F Amax ∂t > ω c τ . It means that the information can flow from the environment to thesystem after the times t > τ . This non-Markovian dynamical behavior is represented by theregion of NM. Before the time ω c τ , the values of F Amax in the region of M are monotonicallydecreased, which is referred to as the Markovian dynamics.We want to know if the metrology strategy based on the quantum correlated states willhave an advantage on improving the sensitivity of the phase estimation in the practicalenvironment. The N -particle maximally entangled state is chosen to be the input state as | Ψ i = 1 √ ⊗ Nj =1 | i j + Π ⊗ Nj =1 | i j ) . (17)We assume that N qubits are subject to the local environments independently. The totaldensity matrix is obtained as ρ N ( t ) = 12 [Π ⊗ Nj =1 ˆ ε j ( | i j h | ) + Π ⊗ Nj =1 ˆ ε j ( | i j h | )+ Π ⊗ Nj =1 ˆ ε j ( | i j h | ) + Π ⊗ Nj =1 ˆ ε j ( | i j h | )] . (18)According tho the time-dependent density matrix written by Eq. (10), the dynamical mapˆ ε j for the ith qubit can be expressed asˆ ε j ( | i j h | ) = a | i j h | + (1 − a ) | i j h | , ˆ ε j ( | i j h | ) = c | i j h | , ˆ ε j ( | i j h | ) = c ∗ | i j h | , ˆ ε j ( | i j h | ) = b | i j h | + (1 − b ) | i j h | . (19)Figure 2 shows the evolutions of the phase sensitivity described by the maximal mean val-ues of the quantum Fisher information in non-Markovian depolarizing environment. Whenthe number of the experiments ν = 1, the phase sensitivity can be given as ∆ φ ∼ √ ¯ F max | Ψ p i = Π ⊗ Nj =1 1 √ ( | i j + | i j ). In thecorrelated-state metrology strategy, there exists a critical time t c where ¯ F max = 1. Before thecritical time, we can obtain the ultimate sensitivity limit which is better than the standardquantum limit because of the high values ¯ F max >
1. However, the product-state metrologystrategy cannot achieve the standard quantum limit owing to ¯ F max <
1. The maximal valueof the variance arrives at the crossover time t = τ . When t > τ , the memory effects fromthe non-Markovian depolarizing environments can give rise to the decreasing of the phasesensitivity. After a long time, the sensitivity by using the quantum correlated state is nearlyequivalent to that by using the product state.The sudden change behavior in dynamics can be shown in the inset figure in Figure 2.Such behavior results from the competition of the two different kinds of the optimal SU (2)rotations in either the x − y plane or along z direction. Previous to the sudden changetime, the small variance of the phase parameter can be obtained by the x − y plane rotationalong ~n o k . While the optimal values of (∆ φ ) is achieved by the z direction rotation along ~n o ⊥ at the time t > t s . No sudden change behavior happens in the case of the product statebecause the value of ¯ F max is the same one with respect to the two kinds of optimal rotationdirections.On one hand, the occurrence of the sudden change behavior of the maximal quantumFisher information is determined by the optimal selection of the rotation operations. Onthe other hand, F max > t > t c , the entanglement of theopen quantum systems is nearly decreased to zero. Therefore, it is of interest to study therelationship between the critical times and sudden-changing ones. Trough the numericalcalculation, we find that the vanishing of the entanglement is always more early than thehappening of the sudden change behavior. The difference between the sudden-changing timeand the critical time is plotted with the increasing of the number of qubits in Figure 3. It isfound out that the scaling property can be obtained as ω c ( t s − t c ) ∝ N . When N → ∞ , thesudden-changing times are infinitely close to the critical times. For a very large number ofentangled qubits, the sudden change behavior of the maximal quantum Fisher informationcan be used to characterize the existence of the entanglement. With respect to the maximal8uantum Fisher information of large- N entangled qubits, the selection of the optimal SU (2)rotation along ~n o k in the x − y plane denotes the existence of the entanglement of opensystems.The impacts of the reservoirs on the sudden change dynamics of the phase sensitivity areshown in Figure 4. It is clearly seen that the sudden change behavior can occur in bothMarkovian ( s < s c ) environments and non-Markovian ( s > s c ) ones. For the cases of thesmaller values of bath parameter s , the lower variance of phase estimation will be obtained.The values of the sudden-changing time t s are decreased with the increasing of the bathparameter. When t < t s , the better resolution limit is achieved in the sub-Ohmic heat bathwhere the higher values of the maximal quantum Fisher information can be kept. V. DISCUSSION
We employ the spin-boson system to construct the practical non-Markovian depolarizingchannel. The evolution of quantum Fisher information is applied to discriminate the Marko-vian dynamics and non-Markovian ones. There exists the temperature-dependent criticalbath parameter s c . Only if s > s c , the crossover between the non-Markovian decoherenceand Markovian one will exist. The sensitivity of the phase estimation was studied in bothcorrelated-state metrology strategy and product-state metrology strategy. It is found outthat the quantum correlated states can be used to improve the phase sensitivity. During theevolution, there are the critical time and sudden change time. Before the critical time, theresolution limit is intermediatly between the standard quantum limit (or shot-noise limit)and Heisenberg limit. The sudden change behavior will occur because of the competitionof two different optimal rotations. The change of the reservoir parameter s can lead to thevariation of the sudden changing times. Moreover, for the large- N cases, the sudden changetimes approximately equal to the critical times. For large- N entangled states, we can obtainthe maximal quantum Fisher information using the optimal SU (2) rotation in the x − y plane, which denotes the existence of the entanglement of open systems.9 I. ACKNOWLEDGEMENT
This work is supported by the National Natural Science Foundation of China under GrantNo. 11074184 and No. 11174114. X. H. is financially supported from the China PostdoctoralScience Foundation funded project No. 2012M520494, the Basic Research Funds in RenminUniversity of China from the central government project No. 13XNLF03 and ResearchProject (No. 03040813) of Natural Science in Nantong University. [1] C. W. Helstrom,
Quantum Detection and Estimation Theory (Academic Press, New York,1976).[2] A. S. Holevo,
Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Ams-terdam, 1982).[3] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics , 222(2011).[4] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. , 3439(1994).[5] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Phys.Rev. Lett. , 3865(1997).[6] A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. , 233601(2012).[7] C. Invernizzi, M. Korbman, L. C. Venuti, and M. G. A. Paris, Phys. Rev. A , 042106(2008).[8] Z. Sun, J. Ma, X. M. Lu, and X. Wang, Phys. Rev. A , 022306(2010).[9] F. Fr¨owis, Phys. Rev. A , 052127(2012).[10] M. H¨ubner, Phys. Lett. A , 239(1992).[11] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A , R4649(1996).[12] J. P. Dowling, Phys. Rev. A , 4736(1998).[13] ˇC. Brukner, V. Vedral, and A. Zeilinger, Phys. Rev. A , 012110(2006).[14] S. Y. Cho and R. H. McKenzie, Phys. Rev. A , 012109(2006).[15] G. R. Jin and S. W. Kim, Phys. Rev. Lett. , 170405(2007).[16] J. Est`eve, C. Gross, A. Weller, S. Giovannazzi, and M. K. Oberthaler, Nature(London) ,1216(2008).[17] L. Pezz´e and A. Smerzi, Phys. Rev. Lett. , 100401(2009).[18] U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1999)
19] H.-P. Breuer and F. Petruccione,
The Theory of Open Quantum Systems (Oxford UniversityPress, Oxford, 2001)[20] J. Ma, Y. X. Huang, X. Wang, and C. P. Sun, Phys. Rev. A , 022302(2011).[21] D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A , 052106(2001).[22] A. Shaji and C. M. Caves, Phys. Rev. A , 032111(2007).[23] A. Monras and M. G. A. Paris, Phys. Rev. Lett. , 160401(2007).[24] Y. Li, Y. Castin, and A. Sinatra, Phys. Rev. Lett. , 210401(2008).[25] R. Demkowicz-Dobrza´nski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Ba-naszek, and I. A. Walmsley, Phys. Rev. A , 013825(2009).[26] T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. Min, D. B. Uskov, C. F.Wildfeuer, G. Veronis, and J. P. Dowling, Phys. Rev. A , 063803(2009).[27] Y. Watanabe, T. Sagawa, and M. Ueda, Phys. Rev. Lett. , 020401(2010).[28] P. Hyllus, L. Pezz´e, and A.Smerzi, Phys. Rev. Lett. , 120501(2010).[29] J. Kacprowicz, R. Demkowicz-Dobrza´nski, W. Wasilewski, K. Banaszek, and I. A. Walmsley,Nature Photonics , 357(2010).[30] B. M. Escher, R. L. de MatosFilho, and L. Davidovich, Nature Physics , 406(2011).[31] X. Hao, X. Wang, C. Liu, and S. Zhu, J. Phys. B: At. Mol. Opt. Phys. , 025502(2013).[32] X. M. Lu, X. Wang and C. P. Sun, Phys. Rev. A , 042103(2010).[33] K. Le Hur, Phys. Rev. Lett. , 196804(2004).[34] M. Thorwart and P. H¨anggi, Phys. Rev. A , 012309(2001).[35] T. A. Costi and R. H. McKenzie, Phys. Rev. A , 034301(2003).[36] S. Tornow, N. H. Tong, and R. Bulla, Europhys. Lett. , 913(2006).[37] L. Pezz´eand and A. Smerzi, Phys. Rev. Lett. , 100401(2009). igure Captions Fig. 1 (a). The flow of the average of the maximal quantum Fisher information is plotted asfunctions of the bath parameter s and scaled time ω c t if κ B T ~ = 0 .
01 and ω c = 10 ω ; (b).The dynamical crossover related to the bath parameter s is shown. The flow of the quantumFisher information ∂F Amax ∂t > ∂F Amax ∂t < Fig. 2
The dynamics of the variance of the phase estimation (∆ φ ) is plotted in both correlated-state metrology strategy and product-state metrology strategy if κ B T ~ = 0 . ω c = 10 ω , s = 3 and the number of qubits is N = 5. The solid line represents the case of the maximallyentangled state and dash-dot line denotes the case of the uncorrelated product state. Fig. 3
The scaling property of the critical time t c and sudden change time t s is plotted by thesquares if κ B T ~ = 0 . ω c = 10 ω and s = 3. The dashed line represents the fit result. Fig. 4
For the strategy based on the maximally entangled state, the sudden change behaviorcan be shown with the change of the bath parameter s if κ B T ~ = 0 . ω c = 10 ω and N = 5.12 NMM (b) ω c t s (a) Q F I f l o w s ω c t .0 0.5 1.0 1.5 2.0 2.50714212835 ( ∆ φ ) ω c t
10 15 200.0080.0120.0160.0200.024 ω c ( t s - t c ) N .81.21.6 2.0 2.4 2.80123456 0.00.20.40.60.8 ( ∆ φ ) ω c tt