Linear Response Theory of Entanglement Entropy

Yuan-Sheng Wang Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China

Teng Ma Beijing Academy of Information Sciences, Beijing 100193, China

Man-Hong Yung∗ Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China (Dated: May 27, 2020) Linear response theory (LRT) is a key tool in investigating the quantum matter, for quantum systems perturbed by a weak probe, it connects the dynamics of experimental with the correlation function of unprobed equilibrium states. Entanglement entropy(EE) is a measure of , it is a very important quantity of quantum physics and quantum information science. While EE is not an observable, developing the LRT of it is an interesting thing. In this work, we develop the LRT of von Neumann entropy for an open quantum system. Moreover, we found that the linear response of von Neumann entanglement entropy is determined by the linear response of an observable. Using this observable, we define the Kubo formula and susceptibility of EE, which have the same properties of its conventional counterpart. Through using the LRT of EE, we further found that the linear response of EE will be zero for maximally entangled or separable states, this is a unique feature of entanglement dynamics. A numerical verification of our analytical derivation is also given using XX chain model. The LRT of EE provides a useful tool in investigating and understanding EE.

I. INTRODUCTION however, there is still a lack of a general tool in inves- tigating how entanglement entropy (EE) response to an Entanglement provides fundamental insight into quan- external perturbation. tum physics[1,2] and quantum information science[3]. Conventional linear response theory (LRT) is a general For example, in condensed matter physics, entangle- and powerful tool in studying quantum matters[25–30], ment provides a unique tool for characterizing quantum because it connects the dynamical response of a quan- mechanical many-body phases and new kinds of quan- tum system to an external probe with the correlation tum order[4,5]. As a bridge linking quantum statisti- functions of the unperturbed equilibrium state. While cal mechanics with quantum information, entanglement the conventional LRT is applied to observable, which is has been exploited to establish efficient methods for ex- associated with a Hermitian , as EE is not an amining many-body interacting systems[6]. Deep un- observable, finding the LRT for it seems to be not only derstanding of the entanglement decoherence is also ex- interesting but also useful. pected to insights into the quantum measurement and In this paper, we develop an LRT of von Neumann en- the quantum-classical transition[7–9]. Besides the fun- tropy for an open quantum system which is subject to damental physical meaning of entanglement, it is also a external perturbation. We found that the LRT of von resource required for many quantum technologies such Neumann entropy to external perturbation is given by as quantum communication, quantum sensing[10], quan- the linear response of an observable, using this observ- tum metrology[11–13], etc. The ability to generate, ma- able we define the Kubo formula and susceptibility of EE arXiv:2005.12502v1 [quant-ph] 26 May 2020 nipulate and detect entanglement[14–17] are bases of the which have the same properties of its conventional coun- whole field of . terpart. Using the LRT of EE, we demonstrate that there The interest in the properties of entanglement has also is no linear response of von Neumann entropy for non- been extended to understanding its dynamical behaviour, degenerate systems which are initially at separable states which is a powerful perspective in understanding non- and maximally entangled states. To verify our analytical equilibrium quantum dynamics[18–20]. For instance, the result, we numerically solve the dynamics of von Neu- dynamics of entanglement has recently been realized as mann entropy of a subsystem in a XX spin chain model. a useful probe in studying ergodicity and its breakdown The LRT of EE provides a useful tool in understanding in quantum many-body systems[21–24]. Although great and investigating the dynamics of entanglement. progress has been made in this field, to our knowledge, The paper is organized as follows. In Sec.II, we briefly re- view the conventional LRT. In Sec.III we derive the LRT of von Neumann entropy, define the Kubo formula and ∗ [email protected] the susceptibility of EE, discuss properties and applica- 2 tions of the Kubo formula and the susceptibility of EE. B. Some properties and applications of the Kubo In Sec.IV we give a numerical verification of our analyt- formula ical results using XX spin chain model. The concluding Sec.V summarizes the findings. Technical details of Stationary. If the initial state is equilibrium state, some calculations are presented in the appendix. its then commutes with Hˆ0, inserting 0 0 iHˆ0t −iHˆ0t iHˆ0t −iHˆ0t OˆI (t) = e Oeˆ and Hˆ1,I (t) = e Hˆ1e into the expression of correlation function and using the cyclic II. REVIEW OF CONVENTIONAL LINEAR property of trace, we have RESPONSE THEORY 0 0 C(t, t ) = hOˆI (t − t )Hˆ1i (4) A. The Kubo formula that is, the equilibrium state correlation function is in- variant under time translations: C(t, t0) = C(t − t0), Consider a quantum system which is weakly coupled to which results in translation invariant Kubo formula: an external perturbation F (t)Hˆ at t = t with time 1 0 R(t, t0) = R(t − t0). dependence F (t). The Hamiltonian for such system is Causal. The system should not respond before the exter- given by nal perturbation is applied, therefore R(t) = 0 for t < 0, this is enforced by the step function in the expression Hˆ (t) = Hˆ + F (t)Hˆ (1) 0 1 of R(t). The causality will result in analyticity of χ(ω) in the upper half complex plane, we will show it in the ˆ where H0 is the Hamiltonian of unperturbed system. In following. The Fourier expansion of R(t) reads (we denote an operator or density ma- trix in interaction picture through adding ”I” to its sub- 1 Z ∞ ˆ R(t) = χ(ω)e−iωtdω (5) script), the average of any operator OI (t) over state ρI (t) 2π reads −∞ if t < 0 we can perform the integral by completing the ˆ ˆ hOI (t)i ≡ tr[ρI (t)OI (t)] (2) contour in the upper half plane, the complementary in- tegral has to be zero since the real part of the exponent ˆ ˆ † where ρI (t) = UI (t, t0)ρI (t0)UI (t, t0) with the time evo- −i(i|ωi|) · (−|t|) → −∞ along the contour in the upper R t 0 0 0 lution operator UˆI (t, t0) = T exp[−i dt F (t )Hˆ1,I (t )]. half plane, with ωi the imaginary part of ω. On the t0 If F (t) is so small such that F (t)Hˆ is much smaller other hand, according to Cauchy’s residue theorem, the 1 integral along the contour is given by the sum of the than Hˆ , we can describe hOˆ (t)i ≡ tr[ρ (t)Oˆ (t)] = 0 I I I residues inside the contour. So there is no poles of χ(ω) ˆ tr[ρ(t)O(t)] by performing an expansion in powers of for Imχ(ω) > 0, which means that χ(ω) is analytic in the ˆ F (t), working to the first order solution to UI (t) (we upper half plane. assume ~ = 1 hereafter): Kramers-Kr¨onigrelation. Because χ(ω) is analytic in the upper half complex plane, it is found that the imagi- Z ∞ 0 0 0 2 nary and real parts of χ(ω) is not independent, they are δhOˆI (t)i = dt R(t, t )F (t )) + O(F ) (3) t0 connected by the Kramers-Kr¨onigrelation[31–33]:

ˆ ˆ ˆ 0 1 Z ∞ Im[χ(ω)] where δhOI (t)i ≡ hOI (t)iF − hOI (t)iF =0 and R(t, t ) = Re[χ(ω)] = P dω0 (6) 0 0 0 −iθ(t − t ))h[OˆI (t), Hˆ1,I (t )]i0, with h·i0 ≡ tr [·ρ(t0)] an π −∞ ω − ω average with respect to the initial state ρ(t ). The real Z ∞ 0 1 Re[χ(ω)] 0 0 function R(t, t ) is the well-known Kubo formula of ob- Im[χ(ω)] = − P 0 dω . (7) π −∞ ω − ω servable Oˆ for system which is subject to a perturba- tion F (t)Hˆ1, which is also referred to as the linear re- where P denotes the Cauchy principal value. Using sponse function. It is readily to see that the Kubo for- Kramers-Kr¨onigrelation, the response function can be mula can be written in terms of two-point correlation reconstructed given just real or imaginary part. 0 ˆ ˆ 0 function C(t, t ) ≡ hOI (t)H1,I (t )i0, if ρ(t0) is an equi- Susceptibility. According to the stationary property, librium state, the correlation function provides a mea- for equilibrium initial state, the Kubo formula is time- sure of equilibrium fluctuations in the system, therefore, translation invariant: R(t, t0) = R(t−t0), Eq.(3) becomes it is revealed by Eq.(3) that although the response of an a convolution, its Fourier transform reads equilibrium system to external perturbation should move 2 the system away from equilibrium, if the perturbation is hδOˆI (ω)i = χ(ω)F (ω) + O(F ) (8) weak enough, the response is dictated by the equilibrium fluctuations. We may conclude that the knowledge of the where F (ω) is the Fourier transforms of F (t), χ(ω) is the equilibrium state is useful in predicting the behaviour of Fourier transform of Kubo formula and is oftern referred nonequilibrium process. to as generalized susceptibility. We learn from Eq.(8) 3 that the response of observable Oˆ to time-dependent per- of δρA(t) as ln ρA(t) = ln ρA(t0)| + O(δρA), then to the turbation F (t)Hˆ1 is ’local’ in frequency space, this is a first order in δρA(t), we have feature of equilibrium linear response. Thanks to this lo- cality, we can directly infer the generalized susceptibility δSA(t) = −trA [δρA(t) ln ρA(t0)] + O(δρA). (12) χ(ω) of Oˆ for a known perturbation F (t)Hˆ1 through mea- What’s interesting about this equation is that the right- suring the time evolution of hOˆ (t)i. The susceptibility I hand side of (12) has the form of the change in the expec- of some observable to certain perturbation is essential in tation value of an Hermitian operator ln ρ (t ). Similar investigating the properties of equilibrium systems (e.g. A 0 result can be seen in Ref. [35]. classifying different materials using magnetic susceptibil- Inserting Eq. (11) into Eq. (12), we obtain the time ity). Therefore the conventional LRT is an important evolution of EE under weak driving (see appendixB for tool in investigating the properties of unknown systems. detail): Fluctuation-dissipation relation. Denoting the imaginary part of generalized susceptibility χ(ω) as χ00(ω), it can Z ∞ 2 be written as δSA(t) = RE(t, τ)F (τ)dτ + O(F ) (13) t0 1 Z ∞ χ00(ω) = dteiωt[R(t) − R(−t)] (9) where RE(t, τ) is given by the following expression 2i −∞ which is not invariant under time-reversal t → −t. RE(t, τ) = −iθ(t − τ)h[ˆsA, Hˆ1,I (τ − t)]i0. (14) Hence, χ00(ω) is called the dissipative or absorptive part of the generalized susceptibility. The fluctuationdissi- wheres ˆA ≡ − ln ρA(t0), Comparing Eq. (13) and Eq. pation theorem says that the dissipation effects are de- (14) with Eq. (3), we found that to first order in F (t), termined by natural fluctuation in thermal equilibrium the change in EE is equivalent to the change in the ex- state. To be more specific, for canonical ensemble with pectation of observable hsˆA(t)i: ˆ the inverse temperature β, ρ = e−βH0 /Z, the imaginary δS (t) ≈ δhsˆ (t)i (15) part of susceptibility χ00(ω) which describe the dissipa- A A tion and (the Fourier transform of) the correlation func- Eq. (13), Eq. (14) and Eq. (15) are key results of this tion C˜(ω) are connected by the Fluctuation-dissipation work, they revealed that: for equilibrium open system A, relation: the response of EE is determined by the linear response of ˜ 00 observable sˆA = − ln ρA(t0). Eq. (14) has the same form C(ω) = −2[nB(ω) + 1]χ (ω) (10) as conventional linear response function, it describes how βω EE and the ˆA response to weak external force where nB(ω) = 1/(e − 1) is the Bose-Einstein distri- ˆ bution function. F (t)H1, we refer to RE(t, τ) as the Kubo formula of EE.

B. Properties and applications of the EE Kubo III. THE LINEAR RESPONSE THEORY OF formula ENTANGLEMENT ENTROPY

A. The Kubo formula of EE The Kubo formula of EE is also the Kubo formula of an observable, this fact implies that RE(t, τ) has the same properties of conventional linear response function. Be similar to the conventional case, supposing at time Like conventional LRT, if [ρ0, Hˆ0] = 0, the EE Kubo t = t0, a time-dependent perturbation F (t)Hˆ1 is applied formula will be stationary: RE(t, τ) = RE(t − τ), then to a composite system which consists of subsystems A the Fourier transform(FT) of Eq.(13) reads and B, its Hamiltonian has the same form as (1). Solving the dynamics of this composite system using perturbative 2 δSA(ω) = χE(ω)F (ω) + O(F ), (16) method (see appendixA for detail), we have where χ (ω) and F (ω) are FT of R (t) and F (t) respec- Z t E E 2 tively. Eq. (16) has the same form as (8) except for that δρ(t) = −i dτ[Hˆ1,I (τ − t), ρ0]F (τ) + O(F ), (11) t0 SA(t) is not an expectation value of an observable. RE(t) and χ (ω) also satisfy properties such as causality, an- ˆ E where ρ0 ≡ ρ(t0), δρ(t) ≡ ρ(t) − ρ0, and H1,I (t) = alyticity of χE(ω) in the upper-half plane and there is ˆ † ˆ ˆ ˆ ˆ U0 (t, t0)H1U0(t, t0), with U0(t, t0) = exp[−iH0(t − t0)]. a Kramers-Kr¨oningrelation between real and imaginary The entanglement entropy of subsystem A is given by part of χE(ω). SA(t) = −tr[ρA(t) ln ρA(t)], with ρA(t) = trB[ρ(t)]. For Entanglement susceptibility. We refer to χE(ω) as en- infinitesimal change δρA(t) ≡ ρA(t) − ρA(t0), the time- tanglement susceptibility. According to Eq. (13), the dependent change in entanglement entropy reads[34]: time evolution of EE can be determined by measuring δSA(t) ≡ SA(t) − SA(t0) ≈ −trA [δρA(t) ln ρA(t)], fur- an observables ˆA(t). F (t) is normally known to the one thermore, we can expand the operator ln ρA(t) in terms who applies the perturbation, we thus can infer the Kubo 4 formula RE(t) from (16) and hence the entanglement sus- and B, then the Hamiltonian of this composite system ceptibility χE(ω). The Kubo formula is independent of must be an non-interacting Hamiltonian of A and B, F (t), for a given Hˆ (t), once we determine its Kubo for- 1 ˆ ˆ ˆ mula, we may predict the response for any F (t). Al- H0 = HA + HB (18) thoughs ˆA is hard to measure, for a system with few degrees of freedom, (e. g., a spin-1/2) we can evaluate it before proceeding, we assume that Hˆ0 is non-degenerate, using state tomography. then both subsystem A and B are non-degenerate, their Canonical ensemble EE linear response If the initial eigenstates and eigenvalues are given by state of subsystem A is described by canonical ensem- −βHˆA ˆ ble ρA(t0) = e /ZA with inverse temperature β = HA|fiA i = fiA |fiA i (19) 1/k T , Hˆ is the Hamiltonian of subsystem A and Z ˆ B A A HB|gjB i = gjB |gjB i (20) is the corresponding partition function, then ln ρA(t0) = ˆ −βHA − ln ZA, inserting this equation into (13), we have the eigenvectors of Hˆ0 are product states of the eigenvec- Z ∞ tors of A and B: 2 −kBT δSA(t) = (1/β)RE(t, τ)F (τ)dτ + O(F ) t0 Hˆ0 (|fi i ⊗ |gj i) = (fi + gj )|fi i ⊗ |gj i (21) (17) A B A B A B denoting |S i = |f i ⊗ |g i, it is readily to see it is readily to see that (1/β)R (t, τ) = −iθ(t − iA,jB iA jB E that if the initial state is diagonal with respect to ba- τ)h[HˆA, Hˆ1,I (τ − t)]i0, which is the Kubo formula of P sis {|Si ,j i}, that is ρ0 = Pi ,j |Si ,j ihSi ,j |, ˆ ˆ A B iA,jB A B A B A B observable HA and perturbation H1(t), δQA(t) ≡ then the reduced density matrix of subsystem A is given −kBT δSA has a physical meaning of the heat change of by subsystem A during a process with fixed temperature T . ˆ X  X  We can conclude from Eq. (17) that δQA(t) = hHA(t)i ρ (t ) = tr [ρ ] = P |f ihf | (22) to the first order in F (t), this implies that for an open A 0 B 0 iA,jB iA iA i j quantum system which is described by the canonical en- A B semble, if we know how its internal energy responds to a which is diagonal with respect to eigenbasis of Hˆ , so is known perturbation, we then know how its EE responds A ln ρ (t ). to the same perturbation. A 0 Given the intial state ρ = P P |S ihS | 0 iA,jB iA,jB iA,jB iA,jB which is diagonal, if [Hˆ0, ρ0] = 0 then an arbitrary entry C. Exceptional states ˆ of the commutator in Eq.(11) is given by hSiA,jB |[H1,I (t− ˆ τ), ρ0]|SkA,lB i = (piA,jB − pkA,lB )hSiA,jB |H1,I (t −

Given an LRT of EE, a natural question arises: the sign of τ)|SkA,lB i, and response is determined by the sign of driving, or the sign of α, consider a separable initial state, then S (t ) = 0, Z t A 0 ˆ we can always choose an appropriate sign of α such that hδρ(t)im,m = −i dτhSm|[H1,I (t − τ), ρ0]|SmiF (τ) t0 the linear correction of EE is negative, that is δSA(t) < 0, + O(F 2) then SA(t) = SA(t0) + δSA(t) < 0, this is, of course, not true. We may have a similar question for maximally en- = O(F 2) (23) tangled states. Viewed from another perspective, the EE of product states and maximally entangled states must where hδρ(t)im,m ≡ hρ(t) − ρ0im,m. The above equation be extreme values, therefore δSA = 0 for these states. To implies that after applying perturbation, any diagonal el- address these problems, we propose Theorem1. ement of linear correction to initial state is zero, does this Theorem 1 For any product state or maximally entan- means that any diagonal element of δρA(t) is zero? Con- gled state of subsystems A and B which is an eigenstate sider an arbitary off-diagonal element |SiA,jB ihSkA,lB | of of composite system A∪B, if a time-dependent perturba- the initial state, tion F (t)Hˆ1 (with F (t)  1) is applied to the composite system, then the linear response of EE of system A or B trB(|SiA,jB ihSkA,lB |) = δjB ,lB |fiA ihfkA | (24) 2 must be zero. Moreover, δSA/B(t) = O(α ). Proof. We here give the proof of the statement about which does not contribute to the diagonal element of sub- separable states in Theorem1, the proof about maxi- system A with respect to basis |fiA i. As any diagonal mally states is similar and we show it in appendixC. element of ρ0 is zero, we may conclude that the δρA(t) is The basic idea of this proof is to prove that ln ρA(t0) is hollow, that is the diagonal entries are all zero. diagonal while any diagonal element of δρA(t) is zero, In summary, if the initial eigenstate is a separable state then trA[δρA(t) ln ρA(t0)] = 0. of A and B, moreover, if HˆA + HˆB is non-degenerate, For a composite system consists of two subsystem A and then ln ρA(t0) is diagonal, while δρA(t) is hollow. Thus 2 B, if its eigenstates are separable or product states of A the linear response of EE δSA(t) = O(F ). 5

External driving reads (we assume ~ = 1 hereafter)

L L−1 driving λ X J X Hˆ = σˆz + (ˆσxσˆx +σ ˆyσˆy ) (25) 0 2 j 2 j j+1 j j+1 j=1 j=1

k whereσ ˆj (k = x, y, z) are the Pauli operators with j labeling the spins in the chain; λ denotes the longitudinal magnetic field exerted homogeneously on all the spins; J …… is the coupling strengths between the nearest-neighbor spins of the chain. Set the spin at the first site as subsys- tem A, a time-dependent perturbation of the following FIG. 1. XX spin chain with external driving on the first spin. form is applied on this subsystem, σˆz F (t)Hˆ (t) = F (t) 1 (26) 0.001 1 2

then the total Hamiltonian is given by Hˆ (t) = Hˆ0 + 10-4 Hˆsource(t). Diagonalizing Hˆ0 in the single-excitation sub- space, its eigenstates read

10-5 L i(2π/L)kj X e + |ϕki = √ σˆ |{↓j}i (27) L j 10-6 j=1

-4 -2 0 2 4 which is a spin wave with wave vector k, and its cor- responding eigenenergies are Ek = λ + 2J cos k. Here |{↓j}} is the ferromagnetic state of the chain with all its FIG. 2. Spectrum of linear response (yellow) and numerical + x y spin pointing to the −eˆz direction andσ ˆj = (ˆσj +iσˆj )/2. results(blue) for driving amplitude α = 0.1, other parameters: Any state of the system can be expressed as the super- L = 100, J = 2.0, T = 0.5π. position of these eigenstates

L X |ψ(t)i = c (t)|ϕ i (28) IV. LINEAR RESPONSE OF EE IN XX SPIN k k CHAIN k=1

where ck(t) = hϕk|ψ(t)i. In numerical calculation, we choose ck(t) = δ1,k i.e. |ψ(t0)i = |ϕ1i, and F (t) = To verify our analytical result, we numerically investigate −t2/2 the dynamics of a one-dimension spin chain, which is α cos(2πt/T )e with α a small prefactor and JT = π. composed of L spin-1/2 particles coupled via XX-type Through numerically solving the dynamics of XX spin interaction. Hamiltonian of this model in the absence of chain in single-excitation subspace, we obtain δSA(t), its spectrum δSA(ω) is given by the discrete Fourier trans- form of numerical data. In Fig.2 and Fig.3, we show δSA(ω) of the numerical result (blue line) and analyti- 0.010 cal one which only consider linear correction of dynamics (yellow line). There are two characteristic peaks in ana- lytical result, to explain where do these peaks come from, 0.001 we show the spectrum of RE(t) and F (t) in Fig.4, it is readily to see that the one near position 2ω/J ≈ ±1.27 10-4 result from the spectrum of response function χ(ω); the other peaks near 2ω/J ≈ ±0.66 result from the spectrum of driving F (ω). Comparing results with different α, it 10-5 is found that when the driving is weak, i.e. α is small -4 -2 0 2 4 enough, numerical results match the linear response quite well, while α became large enough, there is a fairly obvi- ous deviation between numerical spectrum and analytical FIG. 3. Spectrum of linear response (yellow) and numerical one, especially at a position near 2ω/J = 0, 2ω/J = 2 results(blue) for driving amplitude α = 2.5. Other parameters and position near 2ω/J = 2.5, these deviations result are the same as Fig.2. from the emergent of non-linear response. 6

0.08 No.U1801661 and No.11905100), the Economy, Trade and Information Commission of Shenzhen Municipality 0.06 (Grant No.201901161512), the Guangdong Provincial 0.04 Key Laboratory(Grant No.2019B121203002).

0.02 Appendix A: Derivation of Eq. (11) 0.00

0.20 In interaction picture, we formally integral the Liouville 0.15 equation and obtain 0.10 Z t ρ (t) = ρ − i dτ[F (τ)Hˆ (τ), ρ (τ)] (A1) 0.05 I 0 1,I I t0 0.00 -3 -2 -1 0 1 2 3 ˆ ˆ † ˆ ˆ where ρ0 ≡ ρ(t0), H1,I (t) = U0 (t, t0)H1U0(t, t0) and ˆ † ˆ ρI (t) = U0 (t, t0)ρ(t)U0(t, t0), this expression is accurate. Assuming the value of F (t) is very small, we can expand FIG. 4. The spectrum χ(ω) and F (ω), which is given by (0) (1) ρI (t) in powers of F (t): ρI (t) = ρI (τ) + F ρI (τ) + the discrete Fourier transform of response function RE (t) and O(F 2), inserting the expansion into (A1), we obtain driving F (t). Z t 2 ρI (t) = ρ0 − i dτ[F (τ)Hˆ1,I (τ), ρ0] + O(F ), (A2) t0 V. CONCLUSION in the derivation of the above equation, we have used (0) ρI (t) = ρ(t0) ≡ ρ0. we further assume the initial state An LRT of von Neumann entropy is developed for an ˆ ˆ open quantum system A which is subject to weak per- ρ0 commutes with H0, that is [ρ0, H0] = 0. After trans- turbation, we assumed A is a subsystem of composite forming the density matrix in the above equation into system A ∪ B, and the composite system is initially at Schr¨odingerpicture, we have a state which density matrix is commuted with the un- Z t ˆ 2 perturbed Hamiltonian. We found that the LRT of von δρ(t) = −i dτ[H1,I (τ − t), ρ0]F (τ) + O(F ) Neumann entropy to external perturbation is given by t0 the linear response of an observable, using this observable where δρ(t) = ρ(t) − ρ(t0), we thus get Eq. (11). we define the Kubo formula of EE which has the same properties of its conventional counterpart; using the LRT of EE, we demonstrate that there is no linear response of Appendix B: Derivation of Eq. (13) von Neumann entropy for a non-degenerate system which is initially at separable states and maximally entangled Working in the interaction picture, we insert Eq. (11) states. To verify our analytical results, we numerically into Eq. (12), to the first order in F (t), we obtain the solve the dynamics of EE for a subsystem of a XX spin following equation chain. Numerical results show that when the perturba- δS (t) tion is weak enough, the dynamics of EE will approach A the result given by LRT. The LRT of EE provides a use- Z t n  o ≈ − i dτF (τ)tr sˆ tr Hˆ (τ − t), ρ  ful tool in understanding and investigating the dynamics A A B 1,I 0 t0 of entanglement. Z t n  o ≈ − i dτF (τ)tr sˆA Hˆ1,I (τ − t), ρ0 (B1) t0

VI. ACKNOWLEDGMENTS wheres ˆA ≡ − ln ρA(t0). Using the cyclic property of trace tr(XY ) = tr(YX), we rewrite the above equation We acknowledge helpful discussions with Han-Tao Lu as and Chong Chen. This work was supported by the Nat- Z t  ˆ  2 ural Science Foundation of Guangdong Province (Grant δSA(t) = −i h sˆA, H1,I (τ − t) i0F (τ)dτ + O(F ) t0 No.2017B030308003), the Key R&D Program of Guang- (B2) dong province (Grant No. 2018B030326001), the Science, Technology and Innovation Commission of Shenzhen denoting RE(t, τ) = −iθ(t − τ)h[ˆsA, Hˆ1,I (τ − t)]i0, the Municipality (Grant No.JCYJ20170412152620376 above equation can be simplified into and No.JCYJ20170817105046702 and Z ∞ 2 No.KYTDPT20181011104202253), the National Natural δSA(t) ≈ RE(t, τ)F (τ)dτ + O(F ) (B3) Science Foundation of China (Grant No.11875160, t0 7 we thus get Eq. (13). inserting the above equation into δSA(t) = −trA[δρA(t) ln ρA(t)], we have

δSA(t) It is worth noting that the Kubo formula RE(t, τ) is Z t n   o invariant under time translations, that is, RE(t, τ) = 2 = −i dτtrA trB [Hˆ1,I (τ − t), ρ0] sˆA + O(F ). RE(t − τ). t0 (C2)

Consider a quantum system which consists of two subsys- tems A and B, the degree of freedom of A and B are both d, assuming initially the total system is at maximally en- 1 Pd tangled state: ρ0 = d i,j=1 |iAiBihjAjB|, where {|iAi} and {|jBi} are complete sets of orthonormal basis of sub- system A and B respectively, we then have

1 X A ρ (t ) = tr (ρ ) = |i ihi | = I Appendix C: No linear response for maximally A 0 B 0 d A A d entangled states i

then ln ρA(t0) = −IA ln d, which is diagonal. On the other hand,   trB [Hˆ1,I (τ − t), ρ0] Starting from Eq. (11), and assume [ρ0, Hˆ0] = 0, we may obtain the change in the time evolution state of subsys- d tem A: X  ˆ (jB iB ) ˆ (iB jB )  = H1,I (τ − t)|iAihjA| − H1,I (τ − t)|jAihiA| i,j=1 (C3)

ˆ (jB iB ) ˆ δρA(t) = trB[δρ(t)] where H1,I (τ − t) ≡ hjB|H1,I (τ − t)|iBi. It is not Z t hard to see that the above operator is hollow with re-   2 = −i dτtrB [Hˆ1,I (τ − t), ρ0] F (τ) + O(F ) spect to basis {|iAi}, therefore according to Eq.(C3), we t0 conclude that the response of EE to first order in α is (C1) zero: δSA(t) ≈ 0.

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