Nonunital non-Markovian dynamics induced by a spin bath and interplay of quantum Fisher information
aa r X i v : . [ qu a n t - ph ] N ov Nonunital non-Markovian dynamics induced by a spin bath andinterplay of quantum Fisher information
Xiang Hao,
1, 2, ∗ Wenjiong Wu, and Shiqun Zhu School of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou, Jiangsu 215011, People’s Republic of China Department of Physics, Renmin University of China, Beijing 100872, China School of Physical Science and Technology, Soochow University,Suzhou, Jiangsu 215006, People’s Republic of China
Abstract
We explore the nonunital non-Markovian dynamics of a qubit immersed in a spin bath. Thenonunital environmental effects on the precisions of quantum parameter estimation are investigated.The time-dependent transfer matrix and inhomogeneity vector are obtained for the description ofthe open dynamical process. The evolution of the coherent vectors for a qubit is studied so asto geometrically demonstrate the nonunital non-Markovian effects. The revivals of angles forthe fidelity between a maximal mixed state and an arbitrary trajectory state are presented inthe nonunital non-Markovian dynamics. The degree for the nonunitality is controllable with thechanges of the local magnetic field for a qubit and spin bath temperature. It is found out thatthe increase of quantum Fisher information for composite states is connected with the nonunitalnon-Markovian effects, which is helpful for the improvement of quantum metrology. ∗ Electronic address: [email protected] . INTRODUCTION There are always energy or information exchanges between open quantum systems andenvironments, owing to external measurements and inherent spontaneous decays [1, 2]. Theopen dynamical process can be regarded as Markovian dynamics in the weak coupling ap-proximation where the correlation time of the bath is much smaller than the decay time ofthe open system. During a Markovian evolution, quantum informations embodied by opensystems are gradually destroyed [3]. Indeed, actual interactions between quantum systemsand environments generally give rise to one non-Markovian dynamics [2]. The revivals of po-larization parameters [4], quantum correlations [5], quantum entanglement [6] and quantumFisher information [7, 8] can happen in the non-Markovian decoherence channel. Much moreworks have been presented in order to help people to understand non-Markovian dynami-cal processes [9–15]. As is known, quantum spin bath [16–20] has the rich non-Markovianphenomena which is useful for the solid-state quantum information processing [21, 22].Many useful measures on the non-Markovian degree for a dynamical map have been sug-gested in the recent years [23–29]. The measures based on distance fidelity [23, 24, 30, 31],entanglement [25], indivisibility [26], quantum discord [27] and Fisher information [28] areapplied to quantitatively evaluate some non-Markovian dynamics such as dephasing channeland general amplitude-damping one. Besides the non-Markovian effects, the nonunital prop-erty of the environment is necessary for the generation of quantum discord for compositestates [32]. One kind of reasonable measure is introduced to take into account the nonunitaleffects on the degree for the non-Markovianity [29].In this paper, we expect to present one spin environment with both non-Markovian effectsand nonunital ones. The general expression of a nonunital dynamical process is obtained bythe time-dependent transfer matrix and inhomogeneity vector in Sec. II. For an example, weput forward the model consisting of a qubit coupled to a spin bath with infinite number oflattices in Sec. III. To clearly understand the nonunital non-Markovian effects, we providea geometrical explanation. The actual length for Bures fidelity is utilized to measure thedegree for the non-Markovian nonunitality. As one kind of application, the impacts of thenonunital non-Markovian environment on quantum estimation are studied in Sec. IV. Themutual actions between quantum Fisher information and nonunital non-Markovianity arealso investigated. A short discussion concludes the paper.2
I. GENERAL DESCRIPTION OF NONUNITAL EVOLUTION
The dynamical physical process for an open d -dimension system is generally describedby a noisy quantum map Λ t . Without the consideration of the correlation between systemsand environments, the reduced density matrix for the open system at arbitrary time t > t ( ρ ) := Tr E h ˆ U ( t ) ρ ⊗ ρ E ˆ U † ( t ) i . (1)The initial states for the system and environment are respectively ρ and ρ E . The unitaryevolution operator is represented by ˆ U ( t ) = exp[ − i ( H S + H E + H I ) t ] where H S ( E ) denotesthe Hamiltonian for the system(environment) and the interaction Hamiltonian is expressedas H I . Assuming that ρ = P d − m,k =0 ̺ mk | m ih k | , we can also represent the expression of thereduced density matrix as Λ t ( ρ ) = P d − m,k =0 ̺ mk ˆ F mk ( t ). Here, the decoherence factors ˆ F mk ( t )are defined as, ˆ F mk ( t ) = Tr E h ˆ U ( t ) | m ih k | ⊗ ρ E ˆ U † ( t ) i . (2)To geometrically describe the dynamics of an arbitrary d -dimension system, we makeuse of a general Bloch or coherent vector [2, 33, 34] ~λ = ( λ , · · · , λ d − ) T which is basedon a ( d − × ( d −
1) time-dependent transfer matrix T (Λ t ) and inhomogeneity vector ~r (Λ t ) = ( r , · · · , r d − ) T . The general evolution expression for a coherent vector is presentedas, ~λ ( t ) := Λ t [ ~λ (0)] = T (Λ t ) · ~λ (0) + ~r (Λ t ) . (3)The density matrix ρ is also expressed as ρ = I d + P d − µ =1 λ µ ˆ O µ where I is the identity matrixand { ˆ O µ } is a complete set of Hermitian and orthonormal operators satisfying that ˆ O † µ = ˆ O µ and Tr[ ˆ O † µ ˆ O ν ] = δ µν [33, 34]. The components of the vector is calculated as λ µ = Tr[ ˆ O † µ ρ ].The nonunital non-Markovian effects from the environment can be described by the time-dependent transfer matrix and the inhomogeneity vector. These components are calculatedas T µν (Λ t ) = Tr[ ˆ O † µ Λ t ( ˆ O ν )] and r µ (Λ t ) = d Tr[ ˆ O † µ Λ t ( I )]. Furthermore, the composition ofthe quantum channels Λ A ◦ B t with respect to two non-interacting systems A , B can be easilyexpressed as Λ A ◦ B t = Λ A t · Λ B t .The transfer matrix and inhomogeneity vector are determined by the decoherence factors,3hich are also obtained as, T µν (Λ t ) = X { k,j }∈ R ν Tr[ ˆ O † µ ˆ F νkj ] ,r µ (Λ t ) = 1 d d − X m =0 Tr[ ˆ O † µ ˆ F mm ] . (4)In the above expressions, the Hermitian operator can be represented as ˆ O ν = P { k,j }∈ R ν | k ih j | . In the case of d = 2, the complete set of Hermitian operators is repre-sented by the Pauli matrix ˆ O = ˆ σ x = | ih | + | ih | , ˆ O = ˆ σ y = | ih | − i | ih | , andˆ O = ˆ σ z = | ih | − | ih | .To clearly present the nonunital effects on the dynamics, we may select any two initialorthogonal states | ψ , (0) i with the two collinear Bloch vectors of ~λ (0) + ~λ (0) = 0. Underthe condition of ~r = 0, the dynamical process is unital. From the point of view of Blochvectors, the orientations for the general Bloch vectors for any two orthogonal states alwayskeep collinear during the unital dynamics. However, for a nonunital decoherence channelwith the nonzero inhomogeneity vector ~r (Λ t ) = 0, the initial orthogonal states with collinearBloch vectors will be mapped onto others with non-collinear vectors of ~λ ( t ) + ~λ ( t ) = 0.Besides the nonunital effects, the environment is of the important non-Markovian propertywhich can give rise to the energy or information exchange between the bath and the opensystem. In the following section, we will consider a central spin coupled to a spin bath. Anonunital non-Markovian dynamical process can be obtained by using a special operatormethod without any Markovian approximation. III. ANALYSIS OF SPIN BATH MODEL
For an example, the model composed of a qubit( d = 2) and spin environment is describedby the Hamiltonian as, H := H S + H E + H I , (5)where H S = 12 ǫ ˆ σ z , (6)4 is the local external magnetic field. The spin bath Hamiltonian with the Heisenberg XY interaction J between any two spin lattices, is written as, H E = JL X h ij i (ˆ τ i, + ˆ τ j, − + ˆ τ i, − ˆ τ j, + ) . (7)Here ˆ τ i, ± is the raising or lowering operator for the i -th lattice of the bath and L is the totalnumber of lattices. Meanwhile, the interaction Hamiltonian is also given as, H I = J √ L L X j =1 (ˆ σ + ˆ τ j, − + ˆ σ − ˆ τ j, + ) , (8)where J is the coupling strength between a central spin and any spin for the bath. By usingthe Holstein-Primakoff transformation [35] in the approximation of L ≫
1, we can rewritethe total spin operators ˆ S ± = P Lj =1 ˆ τ j, ± as the bosonic creation and annihilation operators,ˆ S + = √ L ˆ b † r − ˆ nL , ˆ S − = √ L r − ˆ nL ˆ b. (9)The number operator for the bosonic field is ˆ n = ˆ b † ˆ b with ˆ n | n i = n | n i and [ˆ b, ˆ b † ] = 1. Here | n i is the bosonic field number state. The mean approximation of L is valid if the number ofspin lattices in excited states n is much less than the total number L [36]. The interactionHamiltonian and bath Hamiltonian can also be simplified as, H I = J (cid:20) ˆ σ + (1 − ˆ nL )ˆ b + ˆ σ − ˆ b † (1 − ˆ nL ) (cid:21) ,H E = 2 J ˆ n (1 − ˆ nL ) . (10)Under the condition of the bath at finite temperature, the initial total state for thesystem and bath is factorized as ρ ⊗ ρ E where ρ E = Z exp( − H E /T ) with the partitionfunction Z = P Nn =0 exp[ − J n (1 − nL ) /T ]. When the integer numbers n, N ≪ L , L > TJ <
10, the dynamical results are reasonably considered as that ofthe thermodynamics limit [37]. We can obtain the analytical results in the thermodynamicslimit of L → ∞ . The effective form of the interaction Hamiltonian in Eq. (10) is verysimilar with that of a Jaynes-Cumming type which can be block-diagonalized in the productbasis of {| j ; n i} . | j = 1 , i is the spin state for the open qubit. During the dynamicalprocess, the total quantum number j + n keeps constant. By the special operator method5ntroduced in the reference [17], the unitary evolution operator acting on the open system canbe obtained analytically. In the thermodynamic limit, ˆ U ( t ) | i = ˆ A ( t ) | i + ˆ B ( t ) | i whereˆ A and ˆ B are respectively expressed by the number operator as ˆ A = exp( − i J ˆ nt ) ˆ A andˆ B = ˆ b † exp( − i J ˆ nt ) ˆ B . Similarly, ˆ U ( t ) | i = ˆ C ( t ) | i + ˆ D ( t ) | i where ˆ C = ˆ b exp( − i J ˆ nt ) ˆ C and ˆ D = exp( − i J ˆ nt ) ˆ D . Here these operators ˆ n , ˆ A , ˆ B , ˆ C and ˆ D are commuted to eachother and calculated by the following coupled differential equations like, i d ˆ A dt = ǫ A + J (ˆ n + 1) ˆ B ,i d ˆ B dt = J ˆ A − ( ǫ − J ) ˆ B ,i d ˆ C dt = J ˆ D + ( ǫ − J ) ˆ C ,i d ˆ D dt = − ǫ D + J ˆ n ˆ C . (11)After solving the above equations, we can obtain all of the decoherence factors ˆ F mk ( t ).According to Eq. (4), the time-dependent transfer matrix is expressed as, T (Λ t ) = T T T T
00 0 T , (12)where the real elements are T = −T = P ( A D ∗ + D A ∗ ) e − JnT , T = T = i P ( A D ∗ − D A ∗ ) e − JnT , and T = P [ A A ∗ − ( n + 1) B B ∗ + D D ∗ − nC C ∗ ] e − JnT .These parameters are calculated as A = e − iJt (cos Γ t + i J − ǫ sin Γ t ), B = − ie − iJt J Γ sin Γ t , C = − ie − iJt J ∆ sin ∆ t , and D = e − iJt (cos ∆ t − i J − ǫ sin ∆ t ) with Γ = q J ( n + 1) + ( J − ǫ ) and ∆ = q J n + ( J − ǫ ) . Meanwhile, the inhomogeneity vectoris also obtained as, ~r (Λ t ) = (0 , , r ) T . (13) r = P [ A A ∗ − ( n + 1) B B ∗ − D D ∗ + nC C ∗ ] e − JnT . In the thermodynamics limit, allsummation symbols are calculated as P = P ∞ n =0 .The above dynamical map is applicable in both strong and weak coupling cases. Thisresult is very different from the Markovian approximation in quantum master equation. Bymeans of the transfer matrix and inhomogeneity vector, we can further study the degree forthe nonunital non-Markovian effects from the spin bath model.6 V. NONUNITAL NON-MARKOVIANITY AND QUANTUM FISHER INFOR-MATION
To clearly show the nonunital non-Markovian effects, we choose the two initial orthogonalstates as | ψ (0) i = √ ( | i + | i ) and | ψ (0) i = √ ( | i − | i ). The time-dependent Blochvectors for the two states are given by ~λ ( t ) = ( T , T , r ) T and ~λ ( t ) = ( −T , −T , r ) T .In the Figure 1(a), the angles between the two Bloch vectors for the two states are non-monotonically decreasing in the Bloch sphere. The revivals of the angles for the two statesare presented in the non-Markovian dynamics. Moreover, the lengths of the Bloch vectors | ~λ , ( t ) | are suppressed or increased. These phenomena result from the nonunital non-Markovian effects which can lead to the backflow of the information from the environmentto the system.Compared with the spin bath model, a common Markovian amplitude damping dynamicalprocess is also shown in Figure 1(b). The angles for the two states monotonically decay.The transfer matrix is T (Λ t ) = diag( √ p, −√ p, p ) and the inhomogeneity vector ~r (Λ t ) =(0 , , p − T . The time-dependent decay parameter is defined as p ( t ) = exp( − γt ) and γ >
0. The Bloch vectors for the two states are obtained as ~λ ( t ) = ( √ p, , p − T and ~λ ( t ) = ( −√ p, , p − T . When the passing time t is very long, the Bloch vectors for thetwo states infinitely trends to the same one and the angle for the two vectors are alwaysdecreased to zero. This result means that the information of the system always loses to theMarkovian environment. It is also mentioned that the lengths of the two vectors can alsobe decreased or increased. We demonstrate that the revival behavior of the length of theBloch vector cannot exactly describe the non-Markovian effects.From the point of view of quantitative description, we can utilize the angle D B for theBures fidelity [3] between an initial state | Ψ i and the final mixed states Λ t ( | Ψ ih Ψ | ) asa measure of the degree for the nonunital non-Markovianity. The general definition of theBures angle is expressed as, D B := arccos p h Ψ | Λ t ( | Ψ ih Ψ | ) | Ψ i . (14)The Bures angle is an appropriate monotonic distance under a complete positively and tracepreserving map of the density matrix. During a Markovian dynamical map, the Bures anglebetween the two states always monotonically decreases. The non-Markovian effects canresult in the increase of the Bures angle, i.e., d D B dt > t ( I d ) = I d + P d − µ =1 r µ ˆ O µ . The non-Markovian evolvingbehavior of the inhomogeneity ~r (Λ t ) vector from the maximally mixed initial state canrepresent both the non-Markovian effects and the nonunital impacts. Therefore, similar tothe definition in [29], one measure based on the Bures angle is defined as, N := max { ρ τ } Z d D Bdt > d D B ( t, ρ τ ) dt dt, (15)where the trajectory states ρ τ = Λ τ ( I d ) , < τ < ∞ are the evolving state from the maximallymixed initial state ρ = I d and the Bures angle is D B ( t, ρ τ ) = D B [Λ t ( I d ) , Λ t ( ρ τ )].In the case of the spin bath model, we can obtain the Bures angle between the maximallymixed state and the selectable trajectory state as, D B = arccos 12 X θ = ± q (1 + θr ,t )(1 + θr ,t + θ T ,t r ,τ ) . (16)Figure. 2 shows the effects of the local magnetic field ǫ and the bath temperature T on thedynamics of the Bures angle. It is found out that the non-Markovian revivals of the Buresangle are mostly pronounced at the low bath temperature under the resonant condition of ǫ = 2 J . On one hand, the non-Markovian behavior is suppressed when the detuning caseof ǫ > J is considered. With the increasing of detuning, the interactions between thecentral qubit and bath are weaker than the transition energy of the qubit applied by thelocal field. Therefore, the impacts of the environment on the dynamics of the qubit becomefragile. On the other hand, the high bath temperature dynamical process can also reducethe non-Markovian behavior of the Bures angle. This is the reason that the bath at hightemperature appears to be in a very disorderly mixed state. The information of the qubitvery quickly loses to the bath and very little information can return to the system from theenvironment.As one efficient application of the nonunital non-Markovian dynamics, it is of value toinvestigate the nonunital non-Markovian effects on the precision of quantum estimation.Quantum Fisher information [38, 39] is a key quantity for describing the sensitivity of aquantum state with respect to a parameter χ . Quantum Fisher information can providea lower bound of the variance of any unbiased estimator due to the quantum Cra ´mer-Rao inequality [31]. A large value of quantum Fisher information represents an attainablemeasurement with a high precision. Among many versions of quantum Fisher information,8here is a famous definition [38, 39] as, F χ := Tr( ∂ρ χ ∂χ L χ ) = Tr( ρ χ L χ ) . (17)The symmetric logarithmic derivatives L χ is determined by ∂ρ χ ∂χ = { ρ χ , L χ } . By diagonal-izing the density matrix for the quantum state as ρ χ = ̺ m | ψ m ih ψ m | with P m ̺ m = 1, wecan transform the expression of the quantum Fisher information into, F χ := X m ̺ m ( ∂̺ m ∂χ ) + 2 X m = k ( ̺ m − ̺ k ) ̺ m + ̺ k (cid:12)(cid:12) h ψ m | ∂∂χ ψ k i (cid:12)(cid:12) . (18)The amplitudes ϑ for quantum states are changing because of the nonunital non-Markovian decocherence. We will evaluate the precision of quantum estimation about theamplitudes. We may assume that the initial composite state is chosen as the product state | Ψ ϑ (0) i = Π ⊗ j =A , B cos ϑ | j i + sin ϑ | j i with ϑ ∈ [0 , π ). According to Eq. (18), the quantumFisher information with the parameter ϑ = π is calculated as, F Prod ϑ = π = T + r T − r − T − T . (19)If the initial entangled state is given as | Ψ ϑ (0) i = cos ϑ | i AB + sin ϑ | i AB , the quan-tum Fisher information F QCϑ with the parameter ϑ can also be numerically obtained byEq. (18). The density matrix for the composite states can be diagonalized as ρ A ◦ B ( t ) = P m =1 ̺ m ( t ) | ψ m ih ψ m | with the eigenvalues and corresponding eigenvectors as, ̺ , = cos ϑ αξ + sin ϑ βδ,̺ , = 12 cos ϑ α + ξ ) + 12 sin ϑ β + δ )+ 12 s cos ϑ (cid:20) ( α − ξ ) + sin ϑ β − δ ) (cid:21) + sin ϑ (cid:12)(cid:12) ( T + i T ) (cid:12)(cid:12) , | ψ , i = | i AB ( | i AB ) , | ψ , i = x , | i AB + y , | i AB . (20)Here, α = P ∞ n =0 A A ∗ e − JnT , β = P ∞ n =0 nC C ∗ e − JnT and ξ = α − T − r , δ = β + T − r . The coefficients of the eigenvectors for j = 3 , x j /y j = sin ϑ ( T + i T ) / [2( ̺ j − cos ϑ α − sin ϑ β )] and | x j | + | y j | = 1.The dynamical behavior of the quantum Fisher information for composite states withrespect to the amplitude are shown in Figure 3. It is clearly seen that the oscillation of9 QCϑ is markedly strong in the resonant case of ǫ = 2 J . With the increasing of the bathtemperature T , the revivals of F QCϑ are reduced. Under the condition of the large detuning ǫ = 6 J , the values of the quantum Fisher information weakly oscillate in the vicinity of acertain high value. Besides it, the quantum Fisher information for composite entangled states F QCϑ is always superior to that for product state F P rodϑ . It is found out that the nonunitalnon-Markovian effects can enhance the precision of quantum parameter estimation.To furthermore demonstrate the relations of quantum Fisher information and nonunitalnon-Markovianity, we study the derivatives of them with respect to time. The positivederivatives of quantum Fisher information denote the backflow of the information from thebath to system, which is regarded as non-Markovian behavior. Meanwhile, we use the Buresangle to evaluate the degree for nonunital non-Markovianity. In Figure 4, the dynamics of d F ϑ dt is synchronous with that of d D B dt . That is, the increase of the quantum Fisher informationalso represents the existence of the nonunital non-Markovian effects. V. DISCUSSION
The nonunital non-Markovian effects from the environment can be studied by the time-dependent transfer matrix and inhomogeneity vector which are determined by the decoher-ence factors. The nonunital non-Markovian dynamics of the open system in a spin bath isanalytically obtained in the thermodynamics limit by using the special operator method.We may select any two orthogonal initial states with collinear Bloch vectors. The nonuni-tal non-Markovian environment leads to the two evolving states with non-collinear vectors.The revivals and suppressing of the angles between the two Bloch vectors happen in thenonunital non-Markovain dynamics, which is different from the monotonic decrease of theangles in the Markovian dynamics.We also use the Bures angle to measure the degree for the nonunital non-Markovianity.In the resonant case, the nonunital non-Markovain effects are prominent at the low bathtemperature. As one possible application, the nonunital non-Markovian bath can give riseto the enhancement of the precision of quantum parameter estimation. The increase ofthe quantum Fisher information for composite states with respect to the amplitude is inaccordance with that of the Bures angle. This also provides another efficient way to thequantitative description of the nonunital non-Markovian effects.10
I. ACKNOWLEDGMENT
This work is supported by the National Natural Science Foundation of China under GrantNo. 11074184, 11174363 and No. 11174114. X. H. is financially supported from the ChinaPostdoctoral Science Foundation funded project No. 2012M520494, the Basic ResearchFunds in Renmin University of China from the central government project No. 13XNLF03. [1] U. Weiss,
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Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Ams-terdam, 1982). igure Captions Fig. 1 (a). The dynamical behavior of the two orthogonal with the collinear Bloch vectorsduring the time interval
J t ∈ [0 ,
3] are plotted in the Bloch sphere when ǫ = 2 J , J = J and T = J ; (b). The dynamics of these vectors during the time interval γt ∈ [0 ,
3] is also shownin the nonunital Markovian map when the decaying parameter is p ( t ) = exp( − γt ) with thepositive value of γ >
0. The black lines and blue ones denote the Bloch vectors for theinitial state | ψ (0) i = √ ( | i + | i ) and | ψ (0) i = √ ( | i − | i ) respectively. The black solidarrows are the Bloch vectors at t = 0 and blue dotted arrows represent the Bloch vectorsafter the time interval. Fig. 2
The dynamical processes of the Bures angle between the Bures angle between the max-imally mixed state and the selectable trajectory state are plotted in order to measure thedegree for the nonunital non-Markovianity under the condition of τ = 2 and J = J . Fig. 3
The nonunital non-Markovian effects of the bath on the quantum Fisher information forcomposite states with the parameter ϑ = π/ T and the local magnetic field ǫ are changed. In the case of ǫ = 6 J , T = J and J = J , theblue solid line represents F QCϑ for the initial entangled state and green line denotes F P rodϑ forthe product coherent state. The black dotted line is the one for F QCϑ when ǫ = 2 J , T = J and J = J . And the red dashed line is the one for F QCϑ when ǫ = 2 J , T = 6 J and J = J . Fig. 4
The time-dependent derivatives of the Bures angle (red line) and quantum Fisher infor-mation (black line) for entangled initial states with ϑ = π/ ǫ = 6 J , T = J and J = J . 13 a) (b) D B ( t, ) Jt /J=2, T/J=1 /J=6, T/J=1/J=2, T/J=6 F Jt Jt J0/J=1, T/J=1, /J=6;QC,dF/dt(black), dD B /dt(red), zero(blue) d D B / d t, d F / d tt