Quantum coherence controls the nature of equilibration in coupled chaotic systems

Jethin J. Pulikkottil,1 Arul Lakshminarayan,2 Shashi C. L. Srivastava,3, 4

Maximilian F. I. Kieler,5 Arnd Backer,5 and Steven Tomsovic1

1Department of Physics and Astronomy, Washington State University, Pullman, WA 99164-28142Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India

3Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064, India.4Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India

5Technische Universitat Dresden, Institut fur Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany(Dated: April 18, 2022)

A bipartite system whose subsystems are fully quantum chaotic and coupled by a perturbative interactionwith a tunable strength is a paradigmatic model for investigating how isolated quantum systems relax towardsan equilibrium. It is found that quantum coherence of the initial product states in the uncoupled eigenbasiscan be viewed as a resource for equilibration and approach to thermalization as manifested by the entangle-ment. Results are given for four distinct perturbation strength regimes, the ultra-weak, weak, intermediate, andstrong regimes. For each, three types of initially unentangled states are considered, coherent random-phasesuperpositions, random superpositions, and eigenstate products. A universal time scale is identified involvingthe interaction strength parameter. Maximally coherent initial states thermalize for any perturbation strengthin spite of the fact that in the ultra-weak perturbative regime the underlying eigenstates of the system have atensor product structure and are not at all thermal-like; though the time taken to thermalize tends to infinity asthe interaction vanishes. In contrast to the widespread linear behavior, in this regime the entanglement initiallygrows quadratically in time.

I. INTRODUCTION

Thermalization in isolated quantum many-body systemshas been an active area of research for many years [1]. Theessential question is whether the system prepared in someinitial state of interest reaches a thermal equilibrium after asufficiently long time. As proposed roughly three decadesago [2, 3], thermalization really happens at the eigenstate leveland is indicative of quantum chaotic nature of the system un-der consideration. Thus, the system relaxes to a thermal stateirrespective of the initial state and without having to do anyinitial state ensemble averaging. This thermalization is seenin the subsystem states of such isolated systems where the re-duced density matrix of the subsystem follows quantum statis-tical mechanics [3]. There are numerous contemporary stud-ies on the process of thermalization in isolated quantum many-body systems and for different classes of initial states [4–10].Whereas any lack of thermalization is often attributed to disor-der induced many-body localization [9], it may originate frommemory effects in the initial state purely from weakness of theinteractions [11].

Compelling insights pertaining to the foundations of quan-tum statistical mechanics can be gained through the study ofparadigmatic bipartite systems whose subsystems are quan-tum chaotic. By adding an interaction between the subsys-tems with a tunable strength, relaxation towards an equilib-rium in the full system of various classes of initial states canbe studied over the complete range from vanishing interac-tions to the opposite limit of extremely strong interactions.There are classes of initially unentangled (product) states ofsuch subsystems that respond quite differently to weak inter-action strengths. Some may thermalize achieving near maxi-mal entanglement as random states, others may equilibrate butwith smaller entanglement, whereas others practically developno entanglement at all.

In this context, taking advantage of the universality ofchaotic single-particle or many-body dynamics, a random ma-trix model that highlights some of the major scenarios in thecase of bipartite weakly coupled chaotic systems is exploredanalytically and numerically. In particular, three sets of un-entangled initial pure states constructed from the subsystemeigenstates in the absence of interactions are contrasted: (i)tensor products of coherent random-phase superpositions (C⊗ C type), (ii) tensor products of random superpositions (R⊗ R type), and (iii) tensor product of individual subsystemeigenstates (E ⊗ E type). In each of these cases, when in-teractions are turned on, the interest is in studying the entan-glement production and its time scale, its long-time average,and the nature of the fluctuations. Naturally the interactionstrength plays a crucial role and the full range is explored fromthe perturbative regime Λ � 1, to Λ & 1 where Λ is a scaleduniversal transition parameter identified earlier [12–14]. Inthis regime, a spectral transition is observed from the Poissonto Wigner level statistics, although the subsystem dynamics isalways fully chaotic.

The set (i) corresponds to an ensemble of initial states ofmaximal coherence [15]. Coherence in a state (represented asdensity matrix) is quantified by the off-diagonal elements ofits density matrix and is a basis dependent notion. However,fixing a preferred basis, it has been found to be useful to de-velop coherence measures in quantum information theory, andstates that are diagonal are incoherent. In parallel to the re-source theory of entanglement, a resource theory for quantumcoherence has been developed; for a review see [16]. For bothsets (i) and (ii), the infinite-time averaged entanglement canbe nearly maximal for arbitrarily small interactions, althoughthe approach to the long time limit can be arbitrarily slow. Itturns out that set (i) of random-phases with maximal coher-ence (C ⊗ C) engenders a large amount of entanglement, andalready reaches the typical thermalized entangled state value.

arX

iv:2

204.

0756

1v1

[qu

ant-

ph]

15

Apr

202

2

2

This occurs in spite of the perturbative nature of the interac-tions, i.e. Λ � 1 and Poissonian level statistics [17]. For theset (ii) of random superpositions (R ⊗ R) a smaller amount ofentanglement is obtained in comparison to set (i) states. Theset (iii) of subsystem eigenstate products (E ⊗ E) are inco-herent from this point of view as they have diagonal densitymatrices. Under perturbative interactions, their entanglementremains essentially perturbative [18]. Thus, the results sug-gest investigating quantum coherence in the uncoupled eigen-basis as a “resource” for thermalization. In addition, the uni-versal rescaled time that was identified in [18] holds true inthe current study for all initial states considered.

Previous studies have shown a linear-in-time entanglementgrowth in systems with signatures of classical chaos [19–24],and in many-body systems [25–27]. This study reveals thatthe initial entanglement growth is controlled by both the tran-sition parameter Λ and the quantum coherence in the initialstate. The former leads to a linear growth and the latter to aquadratic one where a competition between these two is ob-served and a time scale is derived depicting the transition be-tween linear and quadratic growths. In the ultra-weak regime,the linear growth is suppressed and a dominant quadraticgrowth is seen.

The structure of this paper is as follows: the next sectionbriefly presents essential background regarding thermaliza-tion, quantum coherence, quantum chaos, bipartite systems,the transition parameter, and the universal rescaled time. InSect. III, a relation is given for the infinite time averaged pu-rity, and hence linear entropy as well, along with a summaryof the four perturbation regimes. This is followed with a sec-tion on analytical and numerical results for the limiting ex-tremes of ultra-weak and ultra-strong interaction strengths.The remaining perturbation regimes, weak and intermediatestrength, are covered in Sect. V. The final section summarizesthe results of this work and gives a brief outlook.

II. BACKGROUND

It is helpful to review key background information regard-ing thermalization and equilibration in isolated quantum sys-tems, quantum coherence as a resource, bipartite systems andlinear entropy, and set up notation to be used throughout therest of the paper. Also included are the relevant random matrixtransition ensembles, and the concepts of symmetry breaking,the transition parameter, and universal rescaled time.

A. Thermalization in isolated quantum many-body systems

An isolated many-body system prepared in an initial purestate thermalizes when evolved for a sufficiently long timeif the eigenstates of the system are quantum chaotic in na-ture, and behave according to the eigenstate thermalizationhypothesis (ETH) [2, 3]. For such systems, any generic initialstate will approach thermal equilibrium in the strong sense,meaning almost all the initial states relax to equilibrium be-yond some time, thus exhibiting thermal distributions such as

Maxwell, or Bose-Einstein, or Fermi-Dirac depending on theexchange symmetry and stationary expectation values. In con-trast to strong thermalization, weak thermalization has beenfound to exist in some types of initial product states [28, 29].Weak thermalization occurs when the observable of interestfluctuates about the thermal average and only with long-timeaveraging gives the thermal result, in contrast to the stationar-ity of strong thermalization. Numerical simulations show thata weakly interacting bipartite system may achieve an equilib-rium [30, 31] – in either the weak or strong sense – that isdifferent from a thermal one, and may be characterized sim-ilarly to that of thermal fluctuations in quantum chaotic sys-tems [32].

Given a generic initial product state |α〉 and an observableor a measure of interest, the system (of size sufficiently large[30]) may reach an equilibrium after a long time and can beidentified by looking at the infinite time average of the quan-tum expectation value of the observable or measure in the timeevolved state |α(t)〉. In this paper, the linear entropy (intro-duced ahead) serves as a suitable (entanglement) measure forthe time evolved state |α(t)〉 and is denoted by S2(t). Theinfinite time average of S2 is computed as

S2 = limτ→∞

1

τ

∫ τ

0

S2(t) dt , (1)

and the equilibrium value for an ensemble of initial states canbe taken as the initial state ensemble average of S2 denotedas 〈S2〉, where the angular brackets represent the initial stateensemble averaging.

This prompts an immediate question as to how S2 is dis-tributed across various initial states from an ensemble. For ex-ample, does the probability density of S2 behave as a power-law (indicating heavy-tails) or more localized exponentialtype? If the density contains a power-law, where the fluctu-ations can be quite conspicuous compared with 〈S2〉, then thenotion of equilibrium becomes suspect. To the extent that thevarious initial states of an ensemble generate an S2 closer andcloser to 〈S2〉, the sharper the notion of equilibrium becomes.To study fluctuations, a dimensionless normalized variance isdefined as

σ2(X) =〈(X − 〈X〉)2〉〈X〉2 (2)

for a quantity X distributed as PX(x). This fluctuation mea-sure is also used in the studies of optical and acoustic scintil-lation, or irradiance fluctuations caused by small temperaturevariations in a random medium; for example see [33].

In the context of quantifying equilibrium, the fluctuationmeasure σ2(S2) is employed, which measures the scaled vari-ance of the probability density across initial states of the linearentropy’s infinite time average and is referred to as the equilib-rium measure. If σ2(S2) ∼ 1, the equilibrium is quite weak.On the other hand, if σ2(S2) � 1, an equilibrium is possiblesince a majority of the initial states in the ensemble of interestgenerate an S2 close to 〈S2〉. Similar to the weak thermal-ization mentioned earlier, S2(t) may exhibit oscillations fromthe equilibrium, even after a long time. Performing infinite

3

time averaging removes any temporal fluctuations about theequilibrium, and thus, examining the characteristics of the S2

probability density alone is insufficient to reveal whether thesystem relaxes to an equilibrium.

To explore the relaxation to an equilibrium value, the in-finite time average of the S2(t) temporal fluctuation about〈S2(t)〉 is useful, and is given by

σ2(S2

)= limτ→∞

1

τ

∫ τ

0

dt σ2(S2(t)

), (3)

which is referred to as relaxation measure. If σ2(S2) ∼ 1 thesystem relaxes to the equilibrium in the weak sense charac-terized by glaring temporal fluctuations about the equilibriumand is referred to as weak equilibration in the spirit of weakthermalization discussed earlier. If the more stringent con-dition, σ2

(S2

)� 1, is satisfied, then the system relaxes to

equilibrium in the strong sense for almost all the initial statesin the ensemble of interest, and is referred to as strong equi-libration. Moreover, if the equilibrium value coincides withthe thermal value, which is based on random pure state Haarmeasure average of S2 [34], then the system thermalizes inthe strong sense. This is referred to as strong thermalization.

B. Quantum coherence as a resource

A formal resource theory of quantum coherence, and itsquantification was developed recently [16, 35, 36], which isnot to be confused with the concept of coherent states forbosonic many-body systems [37, 38]. Fundamentally, quan-tum coherence is a basis dependent quantity, and dependingon the problem at hand a preferred basis (denote as B) isidentified. For example, an energy eigenbasis may be prefer-able for studying coherence in thermodynamics. Coherenceresource theory stems from identifying a set of incoherentstates (denote as IB in the preferred basis), maximally coher-ent states, and incoherent operations, which will be brieflysummarized here. For more details, see the review on thistopic [16].

Consider a preferred basis B = {|m〉}m=1,...,N for a givenN -dimensional Hilbert space H. An incoherent state repre-sented by the density matrix % ∈ IB is diagonal in the basisB, i.e. % =

∑Nm=1 pm |m〉 〈m|, with {pm} being probabilities

such that∑pm = 1. For the states of the form (C-type)

|αK〉C =1√K

K∑m=1

eiϕm |m〉 (4)

involving an equal superposition, with random phases 0 6ϕm < 2π, of K basis kets, the amount of coherence in-creases with K and becomes maximal for K = N . A quan-tum operation, Φ, on a state that does not generate any co-herence, but may consume it, is regarded as an incoherentoperation. More precisely, for a quantum operation that ad-mits a set of Kraus operators {Ki} (also known as opera-tion elements) [39], i.e., %′ = Φ[%] =

∑iKi%K

†i such that

∑iK†iKi = 1 (trace-preserving), is an incoherent operation

if Ki%K†i/ tr(Ki%K†i

)∈ IB for all i.

Although there are several quantifiers of coherence, the fol-lowing measure, although lacking some desired properties, issufficient for our purposes and based on the Hilbert-Schmidtnorm. It is a valid coherence monotone for trace-preservingoperations such as unitary evolutions [36]. Furthermore, thiscoherence measure was used in recent studies [40, 41] (termedas 2-coherence) to establish a connection between quantumcoherence and either localization or quantum chaos, depend-ing upon the circumstances. Using the notation introducedin [40], this coherence measure of a quantum state ρ is givenby

c(2)B (ρ) =

∑m,m′

m6=m′

|ρmm′ |2, (5)

which for incoherent states is zero, and for a state of the formEq. (4) equals 1− 1/K.

C. Bipartite systems, linear entropy

Consider pure states |α〉 of a bipartite system whose Hilbertspace is a tensor product space,HA⊗HB with subsystem di-mensionalitiesNA andNB , respectively. Without loss of gen-erality, let NA ≤ NB . The dynamics of such a generic con-servative system could be governed by a Hamiltonian or by aunitary Floquet operator in the case of periodically driven sys-tems whose time evolution produces a quantum map. Specifi-cally, a bipartite Hamiltonian system is of the form,

Hε = HA ⊗ 1B + 1A ⊗HB + εVAB , (6)

where the non-interacting limit is ε = 0. For a quantum map,the dynamics can be described by a unitary Floquet opera-tor [13],

Uε = (UA ⊗ UB)UAB , (7)

for which the non-interacting limit is UAB → 1. Assume thatboth εVAB and UAB are entangling interaction operators forε > 0 [14].

The Schmidt decomposition of a pure state [39] is given by

|α〉 =

NA∑l=1

√λl∣∣lA⟩ ∣∣lB⟩ , (8)

with Schmidt eigenvalues λl ordered such that λ1 ≥ λ2 ≥. . . ≥ λNA

and∑l λl = 1. To simplify the notation of di-

rect product states,∣∣lA⟩ ∣∣lB⟩ ≡ ∣∣lAlB⟩ notation with first en-

try for subsystem A and second for B will be adopted in thepaper, and superscripts A and B are dropped whenever it isunderstood. The state Eq. (8) is unentangled iff the largesteigenvalue λ1 = 1 (all others vanishing), and maximally en-tangled if λl = 1/NA for all l. By partial traces, it followsthat the reduced density matrices

ρA = trB(|α〉 〈α|), ρB = trA(|α〉 〈α|) (9)

4

have the property

ρA∣∣lA⟩ = λl

∣∣lA⟩ , and ρB∣∣lB⟩ = λl

∣∣lB⟩ , (10)

respectively. They are positive semi-definite, share the samenon-vanishing (Schmidt) eigenvalues λl and {

∣∣lA⟩}, and{∣∣lB⟩} form orthonormal basis sets in the respective Hilbert

spaces. For subsystem B there are NB −NA additional van-ishing eigenvalues and associated eigenvectors.

As previously mentioned, in this study the linear entropyS2 of a state |α〉 is a suitable observable and given by

S2 = 1− µ2, (11)

where µ2 is the purity defined by

µ2 = trA(ρ2A) = trB(ρ2

B) =∑l

λ2l . (12)

D. Quantum chaos, RMT and universality

Random matrix theory (RMT) can be employed to modelcomplex systems that exhibit quantum chaos, and in general,the statistical properties of a quantum chaotic system does notdepend on the system details except for the presence of fun-damental symmetries that the system may respect [42–44]. Inparticular, in this study the focus is on Floquet systems of theform Eq. (7). Since the subsystems are assumed to be quan-tum chaotic in nature, the subsystem unitary operators UAand UB can be regarded as members of the circular RMT en-sembles. Furthermore, consider systems that are time-reversalnon-invariant, hence circular unitary ensembles (CUE). Thusthe dynamics of bipartite systems (generic in the above con-text) is well captured by the random matrix transition ensem-ble given by [13, 14]

URMTε = (UCUE

A ⊗ UCUEB )UAB . (13)

The interaction operator is assumed to be of the form UAB =exp(iεV), where V is a hermitian operator. In the direct prod-uct basis of the two subsystem ensembles {|ab〉} where UAand UB are represented in Eq. (13), V is assumed to be diag-onal. That is,

Vab,a′b′ = 2πξabδab,a′b′ , (14)

where ξab is a random independent number uniformly dis-tributed in (−1/2, 1/2] for subsystem ensemble basis indexes(a, b) such that 1 ≤ a ≤ NA and similarly for b index.

Let the eigenvalues and corresponding eigenstates of theunitary operators UA and UB for the subsystems and of Uε ofthe full bipartite system Eq. (13) be

UA∣∣jA⟩ = exp

(i θAj

) ∣∣jA⟩ , j = 1, 2, 3, . . . , NA

UB∣∣kB⟩ = exp

(i θBk

) ∣∣kB⟩ , k = 1, 2, 3, . . . , NB

Uε |jk(ε)〉 = exp[i θjk(ε)] |jk(ε)〉 . (15)

To simplify the notation, the superscriptsA andB are droppedfor both eigenkets, and the eigenvalues θAj ≡ θj (θBk ≡ θk).

From here on, it is understood that the labels j and k are re-served for the subsystems A and B, respectively. Further-more, the eigenbasis {|j〉} is denoted as BA, and similarly forsubsystem B. Given the form Eq. (13) of the unitary oper-ator Uε, in the limit ε → 0 one has |jk(ε)〉 → |jk〉 whichis a product eigenstate of the unperturbed system and formsa complete basis (denoted as BAB) with spectrum θjk(0) =θj + θk mod 2π.

E. Symmetry breaking and the transition parameter

Earlier studies on spectral statistics [13] of weakly inter-acting bipartite systems, of the type considered in this study,revealed that when the interaction is turned off between thesubsystems, the system enjoys a dynamical symmetry. Thissymmetry (for ε = 0) can be viewed as having two subsys-tem total energies that are separately conserved in Eq. (6), orhaving product structure of subsystem Floquet operators forthe system Floquet operator in Eq. (7). Introducing a weakinteraction between the subsystems weakly breaks this sym-metry, and in the limit of strong interaction, the symmetry iscompletely broken. Since the subsystems are assumed quan-tum chaotic, a universal scaling parameter, the so-called tran-sition parameter – a concept originally appearing in statisticalnuclear physics [12, 45, 46] – governs the influence of thesymmetry breaking on the system’s statistical properties. Thetransition parameter is defined as

Λ =v2(ε)

D2, (16)

where D is the local mean level spacing and v2(ε) is the (lo-cal) average of off-diagonal (but close to the diagonal) inten-sities of the symmetry-breaking operator represented in thesymmetry-preserving eigenenergy basis. For unitary systemsthat are of the type considered here, the mean (quasi-energy)level spacing is uniform and equals D = 2π/(NANB) [17].

For the random matrix transition ensemble in Eq. (13) withUA and UB members of CUE [13],

Λ =N2AN

2B

4π2(NA + 1)(NB + 1)

[1− sin2(πε)

π2ε2

]≈ ε2NANB

12,

(17)

where the last result is in the limit of large NA, NB , ε � 1,and Λ ranges over 0 6 Λ 6 NANB/4π

2, where limitingcases are the fully symmetry preserving, and the fully brokensymmetry, respectively.

The transition parameter facilitates the comparison of quan-tum chaotic systems’ statistical properties regardless of sizeand kind, i.e. regardless of whether it is single particle ormany-body, fermionic or bosonic. The system dependent de-tails can be mapped onto a value for the universal transitionparameter in such a way that all quantum chaotic systemspossessing the same value of Λ possess identical statisticalproperties. It has been calculated for both weakly brokenfundamental symmetries [46–49] and dynamical symmetries[50, 51]. Calculations of Λ for weakly interacting coupled

5

kicked rotors [13] and coupled kicked tops [52] have beengiven.

For the RMT transition ensemble of Eq. (13), the off-diagonal matrix elements of the symmetry-breaking operatorV in the unperturbed product subsystem eigenbasis behave ascomplex Gaussian random variables and the diagonal ones aszero-centered Gaussian random variables. The transformedVjk,j′k′ is given by

Vjk,j′k′ =∑a,b

uA∗ja uB∗kb u

Aj′a u

Bk′b (2πξab) (18)

where uA and uB are the unitary transformation matrices forsubsystem A and B, respectively. Since the subsystems arequantum chaotic in nature, both the transformation matricesare Haar measure distributed on respective subsystem unitarygroups. For large NA and NB , the real and/or imaginaryparts (depending on diagonal element or not) are Gaussiandistributed with certain variance, where the variance is com-puted with respect to uniformly distributed ξab and the Haarmeasure on the unitary groups for both uA and uB [53]. It canbe shown explicitly that (see App. A),

〈|Vjk,j′k′ |2〉 =π2(1 + δjj′)(1 + δkk′)

3(NA + 1)(NB + 1), (19)

for any given pairs of indexes jk and j′k′. Furthermore, theoff-diagonal elements can be rewritten as

|Vjk,j′k′ |2 = 〈|Vjk,j′k′ |2〉wjk,j′k′ , (20)

where wjk,j′k′ is distributed as an exponential Pw(x) =exp(−x). Note that the off-diagonal elements that are close tothe diagonal that enter into the Λ definition given in Eq. (16)has index pair such that j 6= j′ and k 6= k′, whereas off-diagonal elements with either j = j′ or k = k′ are muchfurther away from the diagonal but can be related to Λ viaEq. (19). Thus the off-diagonal absolute squared matrix ele-ments can be rescaled as

ε2|Vjk,j′k′ |2 = ΛD2(1 + δjj′)(1 + δkk′)wjk,j′k′ . (21)

Notice that, from Eq. (19), the variance of diagonal matrixelements is four times that of the off-diagonal ones with j 6= j′

and k 6= k′. Scaling the diagonal matrix elements with itsstandard deviation gives

Vjk,jk = xjk

√〈|Vjk,jk|2〉, (22)

in which xjk follows a zero-centered Gaussian distributionwith unit variance. It is worth mentioning that the variousdiagonal elements are correlated to each other and the covari-ance between any two diagonal elements can be calculatedsimilar to Eq. (19) mentioned earlier (see App. A), which interms of rescaled variables is given by

〈xjk xj′k′〉 =1

4(1 + δjj′)(1 + δkk′). (23)

Moreover, the unperturbed spectrum {θjk(0)} is an uncorre-lated spectrum and behaves as Poissonian (for large NA, and

NB), so adding in the first order perturbation corrections (i.e.εVjk,jk) that are random will not change the statistical natureof the spectrum.

With these in mind, it is useful to cast the theory in termsof universal parameters, namely, the transition parameter andrescaled time (introduced ahead in Subsec. II F). Let

sjk,j′k′(ε) =θjk(ε)− θj′k′(ε)

D(24)

be the unfolded level spacing of the perturbed spectrum whose(local) mean level spacing is unity [54]. Define, sjk,j′k′ =sjk,j′k′(0). Then, in terms of Λ and other rescaled quantities,the standard perturbation expression for Eq. (24) is given by

sjk,j′k′(Λ) ≈ sjk,j′k′ + 2√

Λ (xjk − xj′k′)

+2Λwjk,j′k′

sjk,j′k′, (25)

where the approximation in Eq. (25) is obtained by consider-ing up to O(ε2) corrections. Furthermore, the second ordercorrections due to levels other than jk and j′k′ are ignored,since the main effect of ignored levels is just shifting levelsjk and j′k′ back and forth, and will mostly cancel out. How-ever, the terms involving just the levels jk and j′k′ push themaway from each other and contribute to opening the gap, andis more pronounced when they are nearest neighbors due tothe small energy denominator.

The approximation in Eq. (25) fails when the energy lev-els become too close resulting in divergences, and for a Pois-sonian spectrum such close lying levels occur far more oftenthan in the case of a CUE spectrum. This, however, can be reg-ularized using degenerate perturbation theory giving [13, 46]

|sjk,j′k′(Λ)| ≈√[sjk,j′k′ + 2

√Λ (xjk − xj′k′)

]2+ 4Λwjk,j′k′ (26)

and the sign is given by sgn[sjk,j′k′ + 2√

Λ (xjk − xj′k′)].

F. Universal rescaled time

In a recent study of the average entanglement productionof initially unperturbed eigenstates {|jk〉} for bipartite sys-tems (same arrangement as described here – weakly interact-ing chaotic subsystems) [18], a universal rescaled time, t, wasidentified as

t = nD√

Λ, (27)

where n is the number of iterations of a unitary operator gen-erating the dynamics. Independent of system details, any twosystems possessing the same value of Λ have the same en-tropy production curve in terms of this time scale. Thus, themean level spacing times the square root of the transition pa-rameter identifies the time scale of relaxation towards equili-bration. Naturally, as the interaction strength gets weaker, this

6

time scale gets longer, tending to infinity as ε → 0. Further-more, if normalized by the infinite time saturation value, inthe perturbative regime, i.e. Λ . 10−2, all entropy produc-tion curves collapse onto the same curve as a function of time.Even beyond the perturbative regime, this universal curve isonly slightly altered as Λ grows.

It turns out that this same rescaled time extends to the timeevolution of a generic pure state as follows. Consider an arbi-trary initial state |α(0)〉 whose density operator evolves aftern iterations as

|α(n; ε)〉 〈α(n; ε)| = Unε |α(0)〉 〈α(0)|(U†ε)n

=∑jk,j′k′

exp(in[θj′k′(ε)− θjk(ε)])

× 〈j′k′(ε)|α(0)〉 〈α(0)|jk(ε)〉× |j′k′(ε)〉 〈jk(ε)| . (28)

Applying the rescalings introduced in the previous subsection,and relabeling the eigenstates by Λ instead of ε gives

|α(t; Λ)〉 〈α(t; Λ)| =∑jk,j′k′

exp

(it√Λsj′k′,jk(Λ)

)× 〈j′k′(Λ)|α(0)〉 〈α(0)|jk(Λ)〉× |j′k′(Λ)〉 〈jk(Λ)| , (29)

where it is understood that |α(t; Λ)〉 ≡ |α(n; ε)〉 with appro-priate variable changes. For the rest of the paper, the rescaledparameters Λ and t are used instead of ε and n. As shownahead in Subsect. IV A, the universal rescaled time emergesnaturally for ultra-weak perturbation strengths for any kindof pure state, in the case where only the lowest order cor-rection to the eigenphase θjk(ε) is relevant and no rotationto the eigenstate |jk(ε)〉 is considered. This generalizes theuniversal nature of the rescaled time beyond its relevance tothe time evolution of initial unperturbed product eigenstates{|jk〉} presented in [18].

III. EQUILIBRATION AND THERMALIZATION -GENERALITIES

The central question of interest is to what extent does quan-tum coherence in the initial state play a role in the entangle-ment generated at long times, and thus, the thermalization ofthe system with an eye on whether it happens in the weakor strong sense. Various ensembles of initially unentangledstates, based on the amount of coherence present are consid-ered. To begin though, an exact expression for the infinite timeaverage of S2 is calculated valid for a generic initial state andfor a given interaction strength characterized by the transitionparameter Λ.

Consider a generic initial pure state |α(0)〉 whose densityoperator evolution (for Floquet systems) is given by Eq. (29).The time-dependent reduced density matrix ρA(t; Λ) of sub-

system A can be expressed as

ρA(t; Λ) =∑j,j′,k

|j〉 〈j′| 〈jk|α(t; Λ)〉 〈α(t; Λ)|j′k〉

= ρA(Λ) + δρA(t; Λ), (30)

where, ρA(Λ) is the infinite time average of ρA(t; Λ) given by

ρA(Λ) =∑jk

ρA,jk| 〈jk(Λ)|α(0)〉 |2, (31)

in which ρA,jk is the reduced density matrix of subsystem Afor the state |jk(Λ)〉, and let µ2,jk(Λ) be the correspondingpurity of the eigenstate. The matrix elements of δρA(t; Λ) are

(δρA)jj′ =∑

j′′k′′ 6=j′′′k′′′k

〈jk|j′′k′′(Λ)〉 〈j′′′k′′′(Λ)|j′k〉

× 〈j′′k′′(Λ)|α(0)〉 〈α(0)|j′′′k′′′(Λ)〉

× exp

(it√Λsj′′k′′,j′′′k′′′(Λ)

). (32)

The infinite time average of(δρA

)jj′

vanishes assuming nodegeneracy in the spectrum {θjk(Λ)}. Furthermore, assumethat all possible level spacings, sjk,j′k′(Λ), are unique. Theseare reasonable assumptions to make because the spectrum{θjk(Λ)} is a result of superposition of two uncorrelated spec-tra that are quantum chaotic in nature. This gives the infinitetime average of the purity µ2(t; Λ) = trA[ρ2

A(t; Λ)] as

µ2(t; Λ) =∑jk

µ2,jk| 〈α(0)|jk(Λ)〉 |4

+∑

jk 6=j′k′[trA(ρA,jkρA,j′k′) + trB(ρB,jkρB,j′k′)]

× | 〈α(0)|jk(Λ)〉 |2| 〈α(0)|j′k′(Λ)〉 |2,(33)

where the infinite time average of trA[δρ2A],

trA[δρ2A(t; Λ)] =

∑jk 6=j′k′

trB(ρB,jkρB,j′k′)

× | 〈α(0)|jk(Λ)〉 |2| 〈α(0)|j′k′(Λ)〉 |2,(34)

is used to derive Eq. (33), and ρB,jk is the equivalent ofρA,jk but for subsystem B. Thus, the infinite time averageof S2(t; Λ) can be obtained using Eq. (33) in Eq. (11).

To gain greater insight into the range of possible behaviors,there are various strength of interaction regimes to consider.First, there are two limiting regimes, an ultra-weak perturba-tion strength regime denoted as Λ → 0+, and a strong in-teraction regime denoted as Λ � 1. In the former, no rota-tion of the unperturbed eigenbasis describing the initial stateneeds to be taken into account, i.e., |jk(Λ)〉 ≈ |jk〉. In thestudy of the irreversibility in quantum theory by Peres [55],precisely such a regime was analysed. There the quantity ofinterest was the squared overlap of two time evolved states

7

via an unperturbed and its perturbed Hamiltonian, or the so-called fidelity. The decay law of ensemble averaged fidelityfor a chaotic Hamiltonian was found to follow a Gaussianbehavior in time. In [56, 57], the fidelity of a chaotic sys-tem was also studied where similar Gaussian behavior wasderived, and moreover associated the width of the Gaussian(also related to a transition parameter) to the phase space vol-ume of the system and the classical action diffusion coeffi-cient. For the scenario presented in this paper, there is a phasemixing between the unperturbed eigenstate components of theinitial state whereas their respective intensities remain nearlythe same. This causes the time evolved state to depart fromthe product structure and generate entanglement, which sat-urates at a common rescaled time tsat. Note however, sinceΛ → 0+, the actual (nonrescaled) saturation time nsat → ∞by virtue of Eq. (27). As a consequence, performing the infi-nite time average of S2 first and then taking the limit Λ → 0gives very different results to the reverse order of the limits,which gives S2 = 0. In the latter limiting regime, the full sys-tem eigenstate components follow a Haar measure behavioralong with orthonormalization constraints. This leads to theexpected known results given ahead.

There are two further regimes, first a weak perturbationregime (0+ < Λ < 10−2), which can be characterized asthe regime in which an eigenstate |jk(Λ)〉 remains Schmidtdecomposed in the unperturbed eigenbasis {|jk〉} [14, 58].Consequently, the time evolution of an unperturbed eigen-state, which is incoherent, will also remain Schmidt decom-posed in the unperturbed eigenbasis [18]. Furthermore, it wasshown that for this regime, the majority of the contribution(∼ O(

√Λ) on average) to |jk(Λ)〉 and its S2 is due to the

first two largest Schmidt eigenvalues. The rest of the Schmidteigenvalues contribute at a higher order (∼ O(Λ ln Λ) on av-erage) that can be neglected.

Finally, an intermediate perturbation regime (10−2 . Λ '1) occurs for interaction strengths in which an eigenstate|jk(Λ)〉 is not Schmidt decomposed in the unperturbed eigen-basis. This regime controls the transition in behaviors be-tween weak perturbation regime and the strong interactionlimit, but is the most difficult to treat analytically as the eigen-states possess neither a Schmidt decomposed nor Haar mea-sure form.

To summarize the various regimes are:

• ultra-weak perturbation regime: Λ→ 0+,

• strong interaction regime: Λ� 1,

• weak perturbation regime: 0+ < Λ < 10−2,

• intermediate regime: 10−2 . Λ ' 1.

IV. LIMITING REGIMES

In [18], a theory for the entropy production of direct prod-ucts of subsystem eigenstates was given, which has a vanish-ing coherence measure. Here, much more general classes ofinitially unentangled states are considered with non-vanishing

products of coherence measures, such as product states havingthe form of Eq. (4), and product states which are randomizedwithin some subspace of the subsystem eigenstates.

A. Ultra-weak perturbation limit

1. Entanglement production

Let B′A ⊆ BA be a subset of eigenstates of subsystemA containing KA elements, and similarly for B. Now con-sider an arbitrary initial product state whose components areformed in these subspaces

|α(0)〉 =

( ∑|j〉∈B′

A

zA,j |j〉)⊗( ∑|k〉∈B′

B

zB,k |k〉), (35)

where the {zA,j} are a particular set of complex numbers withno constraints other than satisfying the normalization con-dition

∑j |zA,j |2 = 1, and likewise for B. Using the ap-

proximation mentioned earlier that defines this regime, i.e. aperturbed eigenstate remains close enough to the correspond-ing unperturbed eigenstate so that it is sufficient to consider|jk(Λ)〉 ≈ |jk〉, gives the reduced density matrix, ρA,jk ≈|j〉 〈j| and similarly for subsystem B. In this case, the infinitetime average of the purity in Eq. (33) for an initial state of theform Eq. (35) becomes

µ2(t; Λ) ≈∑jk

|zA,j |4|zB,k|4

+∑

j′k′ 6=jk

[δjj′(1− δkk′) + δkk′(1− δjj′)

]× |zA,j |2|zA,j′ |2|zB,k|2|zB,k′ |2

= 1− c(2)A c

(2)B . (36)

The initial coherence measure of subsystem A (and similarlyfor subsystem B) in the BA basis is given by

c(2)A =

∑j 6=j′|zA,j |2|zA,j′ |2 = 1−

∑j

|zA,j |4 (37)

and is used in the last step in Eq. (36). Thus, the infinite timeaverage of S2 for an initial product state |α(0)〉 in Eq. (35) isgiven by

S2 ≈ c(2)A c

(2)B , (38)

which is just the product of coherence measures of subsystemA and B in their preferred eigenbasis. Note that for initialpure states, either of whose coherence measure of the subsys-tems vanishes, higher order corrections must be incorporatedin order to find a non-vanishing infinite time average of S2

leading to some function of the transition parameter Λ. Thus,such systems saturate at values that depend on Λ, unlike sys-tems for which the right hand side of Eq. (38) vanishes.

Ensembles of either the C ⊗ C type (coherent randomphase) or the R ⊗ R type (random superpositions) can be cre-ated by defining the appropriate probability densities for the

8

0.0 0.5 1.0 1.5 2.0 2.5 3.0t

0.0

0.2

0.4

0.6

0.8

1.0〈S

2(t

;Λ)〉

(a) C ⊗ C

K = 2

K = 6

K = 10

K = 50

0.0 0.5 1.0 1.5 2.0 2.5 3.0t

0.0

0.2

0.4

0.6

0.8

1.0

〈S2(t

;Λ)〉

(b) R ⊗ R

K = 2

K = 6

K = 10

K = 50

FIG. 1. Ensemble averaged linear entropy 〈S2(t,Λ)〉 versusrescaled time for: (a) initial states of C ⊗ C type for the RMT tran-sition ensemble defined in Eq. (13), and (b) R ⊗ R type. Both useNA = NB = 50, Λ = 10−6, and various K values. The black solidlines show the corresponding theory of Eq. (41), using Eqs. (43) and(45), respectively.

values of the {zA,j} and {zB,k} in Eq. (35), respectively. Foreither ensemble, an approximate expression for the ensembleaveraged time evolution curve of S2 can be derived beginningfrom Eq. (29). This gives

S2(t; Λ) ≈ c(2)A c

(2)B −

∑j 6=j′k 6=k′

|zA,j |2|zA,j′ |2|zB,k|2|zB,k′ |2

× exp

(i t√Λ

[sjk,j′k(Λ) + sj′k′,jk′(Λ)]

), (39)

where only the first order (in ε) correction to the quasi-eigenenergies is to be included. The second-order correctionto the quasi-eigenenergies is due to the rotation of eigenstatesand are omitted in this regime. After ensemble averaging, S2

is given by

〈S2(t; Λ)〉 ≈ 〈c(2)A 〉〈c

(2)B 〉[1−

⟨exp(2 i t x)

⟩], (40)

where x = xjk−xj′k+xj′k′−xjk′ is a sum of four (rescaled)diagonal matrix elements, which behaves like a zero-centered

Gaussian random variable of unit variance (shown usingEq. (23)) giving 〈exp(2 i t x)〉 = exp

(−2 t2

). Thus, the C

⊗ C type or R ⊗ R type ensemble-averaged S2 follows thevery simple behavior given by

〈S2(t; Λ)〉 ≈ 〈c(2)A 〉〈c

(2)B 〉[1− exp

(−2 t2

)]. (41)

Surprisingly, the initial entanglement generation is quadraticin time as opposed to the generally expected linear in-crease [19–24], and this is linked to the full system eigenstatesretaining their product nature in this regime. Note that if ei-therKA = 1 orKB = 1, the coherence measure vanishes andit is necessary to calculate the Λ dependent functional formfollowing [18].

Consider C ⊗ C type initial states of the form of Eq. (35)where all |zA,j | = 1/

√KA with random, independently cho-

sen phases, and likewise for subsystem B, i.e.

|α(0)〉 = |αKA〉C ⊗ |αKB

〉C (42)

where |αK〉C is of the form Eq. (4). The long time limitingevolution for a C ⊗ C initial state is statistically equivalent toan entangled random phase state with equal intensities. ForK = KA = KB , the singular values and various entropies ofentangled random phase states are studied in [59]. For theseinitial states, the ensemble average of the coherence measureis same as the individual coherence measures (no fluctuationsin coherence measures within the ensemble), and is given by

〈S2〉 = 〈c(2)A 〉〈c

(2)B 〉 =

(1− 1

KA

)(1− 1

KB

). (43)

The ensemble averaged time evolution of S2(t; Λ) for C⊗ C type initial states is shown in Fig. 1 (a) for K =2, , 6, 10, 50 compared with the combined results of Eq. (41)and Eq. (43). The agreement is quite good considering thatthere should be finite NA, NB and Λ > 0+ corrections. Allthe necessary details about the numerical calculations shownin Fig. 1 and the other figures are provided in App. B. ForKA,KB comparable to the respective subsystem dimension-ality, i.e. initial states that are tensor product of (nearly) max-imally coherent states of the subsystems, the system time evo-lution generates entanglement 〈S2〉 ≈ 1 − 1/NA − 1/NB .This is close to the well-known result derived in [34], shownin Eq. (53) ahead. Given that the eigenstates are not thermaland have the product structure, this is a remarkable result inthe sense that for an arbitrarily small interaction between thesubsystems, a near-maximal entanglement is achieved after along time by the virtue of maximal coherence in the initialproduct state.

Next consider R ⊗ R type initial product states of the formof Eq. (35), where the {zA,j} and {zB,k} are random Haarmeasure complex coefficients where the only constraint is theunit normalization, i.e.

|α(0)〉 = |αKA〉R ⊗ |αKB

〉R . (44)

The ket |αKA〉R is a random pure state in a given subspace

of subsystem A Hilbert space spanned by B′A (R type) andsimilarly for B. Performing initial state ensemble averaging

9

of the coherence measure of subsystems, the equilibrium S2

value can be computed as

〈S2〉 = 〈c(2)A 〉〈c

(2)B 〉 =

(KA − 1

KA + 1

)(KB − 1

KB + 1

), (45)

where the Haar measure average 〈|zA,j |4〉 = 2/KA(KA + 1)is used to calculate the above equilibrium value. Figure 1(b) shows the ensemble averaged S2(t; Λ) for R ⊗ R typeinitial states, and an excellent agreement between the theoryand numerics is found. For large KA, KB close to subsys-tem dimensionality, the equilibrium value is given by 〈S2〉 ≈1− 2/NA − 2/NB which is slightly less than that of the C ⊗C type initial states. This can be attributed to the fluctuationsin the intensities in the initial states, and is discussed ahead.

Remarkably, dividing Eq. (41) by the equilibrium value〈S2〉makes all (ensemble-averaged) entanglement productioncurves for both C ⊗ C and R ⊗ R types and various KA andKB fall on to one universal curve as illustrated in Fig. 2. Itmust be emphasized that for obtaining this universal behaviorit is crucial to use the rescaled time given by Eq. (27).

2. Equilibrium and relaxation

The nature of the equilibrium onto which the system even-tually settles can be investigated by examining the varia-tions of S2 in time and across the ensemble. As discussedin Sect. II A, two different useful fluctuation measures aregiven by σ2(S2) (equilibrium measure) and σ2(S2) (relax-ation measure). An equilibrium can be inferred from the for-mer measure if a majority of the initial states evolve to anequilibrium value 〈S2〉 to within small or negligible residualfluctuations. In the Λ → 0+ regime, the equilibrium measureσ2(S2) defined in Eq. (2), can be calculated via Eqs. (38) and(37) in a straightforward way for various initial state ensem-bles. The relaxation measure σ2(S2) defined in Eq. (3) can becalculated using Eq. (39) in S2

2(t; Λ) followed by finding itsinfinite-time average. This gives

〈S22(t; Λ)〉 = 〈(c(2)

A )2〉〈(c(2)B )2〉+ 2〈c(4)

A 〉〈c(4)B 〉, (46)

where a higher order coherence measure c(4)A is introduced

based on the l4-norm [16] and is defined as

c(4)A =

∑j 6=j′|zA,j |4|zA,j′ |4, (47)

and similarly for subsystem B. The relaxation measure canthen be written as

σ2(S2) = σ2(S2) + 2〈c(4)A 〉〈c

(4)B 〉

〈c(2)A 〉2〈c

(2)B 〉2

. (48)

It must be emphasized that fluctuation measures derived herefor the Λ → 0+ regime are not valid for initial states eitherof whose subsystem coherence measures are vanishing. Suchcases need special treatment due to the rotation of eigenstates,regardless of how infinitesimal Λ is, and are discussed in the

0.0 0.5 1.0 1.5 2.0 2.5 3.0t

0.0

0.2

0.4

0.6

0.8

1.0

〈S2(t,Λ)〉〈S2〉

FIG. 2. Ensemble averaged linear entropy 〈S2(t,Λ)〉 divided by〈S2〉. Curves are shown for the RMT transition ensemble of Eq. (13)with NA = NB = 50, Λ = 10−6, and for various K values. Both C⊗ C type (blue markers) and R ⊗ R type (red markers) initial statesare indicated. The black solid line shows corresponding theory curvebased on Eq. (41).

next section. It is worth mentioning that for initial unperturbedeigenstates, the S2 probability density behaves similarly toheavy-tailed type densities spanning S2 values from O(Λ) toO(1) [18].

The S2 probability density for C ⊗ C type initial states isshown in Fig. 3 (a) for various K values. It is evident fromFig. 3 that for K = 2 the density is broad and reminiscent ofthe heavy-tailed nature of the S2 probability density of E ⊗ Etype initial state ensemble discussed in the section ahead. AsK increases, more unperturbed eigenstates participate result-ing in an increasingly sharper S2 density.

Now consider the two fluctuation measures, σ2 (S2) andσ2(S2) for C ⊗ C type initial states. The relaxation measurecan be calculated via Eq. (48) giving

σ2(S2) =2

KA(KA − 1)KB(KB − 1)∼ 2

K2AK

2B

, (49)

where the last expression above is valid for largeKA andKB .Excellent agreement is found between the theory and numer-ical calculations as illustrated in Fig. 3 (b). This shows thatC ⊗ C type initial states evolve to an equilibrium state in thestrong sense as K increases. Also in Fig. 3 (b) is the compar-ison of σ2(S2) for Λ = 10−8 and 10−6, which illustrates itsΛ-dependent nature and that they lie below σ2(S2). It turnsout that the leading term of σ2(S2) vanishes for this ensem-ble, and the effect of eigenstate rotation cannot be neglectedfor small K, leading to this Λ-dependent behavior.

For R⊗ R type initial states, the probability density of S2 isshown for various K-values in Fig. 4 (a). Compared to the C⊗ C type densities, the shapes are quite different and the widthof the R ⊗ R type densities are wider for any given K. ForK = 2, the heavy-tail behavior (straight line in a log-log plot)due to the eigenstate rotation is quite prominent. This is notvery surprising since the probability density for random stateshas inverse square root singularities in the region in which oneof the two states is the dominant contribution and the other is

10

0.2 0.4 0.6 0.8x

10−2

10−1

100

101

102

103PS

2(x

)(a) K = 2

K = 4

K = 6

K = 8

0.5 1.0 1.5log10(K)

−8

−6

−4

−2

0

log

10(v

aria

nce

)

(b)

FIG. 3. (a) Probability density of S2 values, and (b) the twofluctuation measures, σ2 (S2) and σ2

(S2

)for C ⊗ C type initial

states. Various K values are shown for NA = NB = 50. In (a)Λ = 10−6. In (b) the black solid line is from Eq. (49) for σ2 (S2).Compare this prediction to calculated σ2 (S2) values for Λ = 10−8

(×) and Λ = 10−6 (#). In addition, calculated values for σ2(S2)

(which should lie below σ2 (S2) ) are plotted for Λ = 10−8 (+) andΛ = 10−6 (♦) to illustrate its Λ-dependence.

very small. This leads the K = 2 case to being much closerto the unperturbed eigenstate case, which does have the heavytail (discussed in the next section). As the K-value increases,the probability densities become more narrow and the fluc-tuations about the mean are mainly due to the Haar measureprobability density of the components {zA,j} and {zB,k}. Us-ing Eq. (38) and various Haar measure moment averages (seeApp. C), the fluctuation measure can be computed as

σ2(S2) =4 [K3

A +K3B + 4(K2

A +K2B) +KA +KB − 8]

(K3A + 4K2

A +KA − 6)(K3B + 4K2

B +KB − 6)

∼ 4

K3A

+4

K3B

, (50)

where the last line is good for large enough KA, KB . Simi-larly, it can be shown that

〈c(4)A 〉 =

4(KA − 1)

(KA + 1)(KA + 2)(KA + 3)∼ 4

K2A

, (51)

10−5 10−4 10−3 10−2 10−1 100

x

10−2

10−1

100

101

102

PS

2(x

)

(a)

K = 2

K = 4

K = 6

K = 8

0.5 1.0 1.5log10(K)

−4

−3

−2

−1

0

log

10(v

aria

nce

)

(b)

FIG. 4. (a) Probability density of S2 values, and (b) the two fluc-tuation measures, σ2

(S2

)and σ2 (S2) for R ⊗ R type initial states.

Various K values are shown for NA = NB = 50. In (a) and (b)Λ = 10−6 for the calculations. In (b) the solid black line is thetheoretical prediction for σ2 (S2) and the black dashed line is forσ2

(S2

). The # are the calculated values for σ2 (S2) and the ♦ are

the values for σ2(S2

).

and along with Eq. (48) an expression for σ2(S2) can be foundin a straightforward way. In contrast to the C ⊗ C type case,where the two fluctuation measures are significantly different,for R ⊗ R type initial states they are identical to leading orderin large-KA, KB and

σ2(S2) ≈ σ2(S2) . (52)

This is shown in Fig. 4 (b) where a good agreement is ob-served between the theory and numerical values. Thus, bothmeasures are dominated by their variations about the infinitetime average, and the temporal fluctuations are lower order inK.

For this regime, the equilibrium measure for various initialstate ensembles with non-vanishing coherence implies equi-librium, which becomes sharper as the coherence increases.This is consistent with a transition from weak to strong equi-libration as the initial state coherence increases. For initialproduct states with maximal coherence, the entanglement sat-urates to that of thermalized states, and they exhibit a strong

11

relaxation (strong thermalization), although the non-scaled re-laxation time lengthens to infinity as Λ→ 0.

B. Strong interaction regime

In this interaction regime, the eigenstates {|jk(Λ)〉} es-sentially behave just like that of NANB - dimensional CUEmatrices, and the time evolution of the system shows van-ishingly small initial state dependence. Using eigenvectorstatistics of unitary ensembles for the full system space, thelimiting behavior of 〈S2〉 can be derived. The complex co-efficients 〈α(0)|jk(Λ)〉 in this limit behave the same as theeigenvector components of an NANB-dimensional CUE. For|jk(Λ)〉 =

∑j′k′ zjk |j′k′〉, the average purity of the eigen-

state is given by [34]

〈µ2,jk〉 =NA +NBNANB + 1

≈ 1

NA+

1

NB. (53)

The average cross-term trace of the reduced density matricesin Eq. (33) can be calculated as

trA(ρA,jkρA,j′k′) =⟨ ∑jk,j′k′

zjkz∗j′kz

′j′k′z

′∗jk′⟩

=⟨ ∑j,k,k′

|zjk|2|z′jk′ |2⟩

≈∑j,k,k′

( 1

NANB

)2

=1

NA(54)

for large NA, NB , and similarly for the subsystem B traceterm. Putting all these together, it can be shown that

〈S∞2 〉 = limΛ→∞

〈S2(t; Λ)〉 ∼ 1− 1

NA− 1

NB, (55)

as expected. Furthermore, the expression for average linearentropy derived in [18] in the non-perturbative regime for aninitial state ensemble of product eigenstates (shown ahead inEq. (82)) can be used to describe the situation here with anapproximation to the function C(2; t) (defined in Eq. (57) andgiven by Eq. (D3)) that appears in the expression. Since thesaturation happens quickly in this regime, the small t approx-imation of C(2; t) shown in Eq. (58) will suffice giving

〈S2(t; Λ)〉 ≈ 〈S∞2 〉[

1− exp

(−4π t

√Λ

〈S∞2 〉

)]. (56)

This is in good agreement with numerical simulations whoseinitial states are of R ⊗ R type with K = 2, 50 as illus-trated in Fig. 5 (a). No initial state dependence is seen andit saturates as expected to 〈S∞2 〉. As Λ → ∞, where thetransition becomes complete, the time evolution curve ap-proaches a Heavyside step function scaled by 〈S∞2 〉, saturat-ing almost instantly. The expression in Eq. (56), originallyderived with a regularized perturbation theory, extends to thenon-perturbative regime using an ‘embedding technique’ de-veloped in [14].

0.0 0.1 0.2 0.3 0.4 0.5t

0.0

0.2

0.4

0.6

0.8

1.0

〈S2(t

;Λ)〉

(a)

K = 2

K = 50

0.5 1.0 1.5log10(K)

−7.50

−7.25

−7.00

−6.75

−6.50

−6.25

log

10(v

aria

nce

)

(b)

FIG. 5. Ensemble averaged linear entropy 〈S2(t; Λ)〉 in (a) and thetwo variance measures, σ2(S2) and σ2(S2) in (b) for R ⊗ R typeinitial states and Λ = 10, NA = NB = 50. In (a), K = 2, 50 areshown along with theory curve (of product eigenstates) in Eq. (82).The red dotted line shows the saturation value 〈S∞2 〉. In (b), σ2(S2)

(♦) and σ2(S2) (�) are shown as a function of K.

Quite remarkably, the expression agrees with a recent studybased on the entangling power of sequentially applied ran-dom diagonal nonlocal operators interlaced with random lo-cal unitary operations [60]. Note that entangling power ep(U)is defined as the average entanglement (here linear entropy)produced by the action of a nonlocal gate U on product statessampled from the Haar measure on the subspaces. For U sam-pled from the diagonal ensemble UAB defined in this paperaround Eq. (13) with ε � 1, the entangling power is quitesmall ep(U) � 1. For large enough subsystem dimensions(and hence satisfying Λ � 1), Eq. (55) in [60] representingep(U) averaged over local unitaries is same as Eq. (56) herewith the identification 4πt

√Λ = 2π2ε2n/3 where n is the

actual time. Note that the average of ep(U) over a Haar distri-bution of U denoted as ep in [60] is approximately the sameas 〈S∞2 〉. It may be noted that the entangling power approachinvolves the local unitaries UA and UB to be different at eachtime step and this leads to a decorrelation that gives rise to theinitial linear growth, also evident in Eq. (56) here. Though UAand UB are the same at each time step here in this paper, in

12

the strong interaction regime the memory from previous timesteps is essentially washed out justifying this connection tothe entangling power approach.

The equilibrium and relaxation measures are negli-gibly small compared to the mean value 〈S2〉, i.e.σ2(S2), σ2(S2) � 1, as evident from Fig. 5 (b), and theyshow a lack of initial state dependence. Thus the initial statecoherence does not play any significant role in the entangle-ment production in this regime and saturates to the thermalvalue rapidly. Furthermore, as revealed by the fluctuationmeasures, the system possesses a very sharp equilibrium andthermalizes in the strong sense where almost all the initialstates regardless of their coherence relaxes rapidly to the ther-mal value.

V. WEAK AND INTERMEDIATE PERTURBATIONREGIMES

A. Weak perturbation regime

1. E ⊗ E type initial states

Consider the initial state ensemble of unperturbed eigen-states {|jk〉}. The coherence measures of both the subsystemsare vanishing in the respective preferred subsystem eigenba-sis. As mentioned earlier, the entanglement produced for thistype of initial states is entirely due to the rotation of eigen-states instigated by the interaction. For sufficiently weak Λ,the leading order effects arise from the rotation of the initialstate |jk〉 with its energetically nearest neighbor. Since theunperturbed spectrum is a direct product of two independentspectra, the level statistics are Poissonian [17]. There is anabsence of level repulsion and two-level near degeneracies arequite frequent. Depending on the interaction strength betweenthe initial state |jk〉 and its nearest neighbor, the rotation of thepair to perturbed eigenstates can range from little to complete(π/4 rotation angle).

For an ensemble of eigenstates {|jk(Λ)〉}, it was shownin [58] that the linear entropy ranges over the full possible in-terval 0 ≤ S2 ≤ 1/2. Its probability density displays a heavy-tailed behavior towards large values. In addition, the near de-generacies increased the order of the average eigenstate en-tanglement from O(Λ) to O(

√Λ). In [18], it was shown that

the mean entanglement production rate increased as

〈S2(t; Λ)〉 ≈ C(2; t)√

Λ , (57)

where C(2; t) is a function of rescaled time and is given inEq. (D3). The short-time behavior of 〈S2(t; Λ)〉 is linear-in-time,

〈S2(t; Λ)〉 ≈ 4π t√

Λ , (58)

and after long time it saturates to

〈S2〉 ≈5

8π3/2

√Λ . (59)

In this Λ regime, the time evolution of such initial states willremain Schmidt decomposed in the unperturbed eigenstate ba-sis BAB to O(N−1

A ) (up to some phase) and is given by

|jk(t; Λ)〉 ≈NA∑l=1

√λl,jk(t; Λ) |(jk)l〉 , (60)

where {|(jk)l〉} for l > 1 are unperturbed eigenstates that areenergetically close to the initial state |(jk)1〉 = |jk〉. The ex-pression for Schmidt eigenvalues λl,jk(t; Λ) derived in [18],is given by

λl,jk(t; Λ) ≈ 4Λwjk,(jk)l

s2jk,(jk)l

+ 4Λwjk,(jk)l

× sin2( t

2√

Λ

√s2jk,(jk)l

+ 4Λwjk,(jk)l

)(61)

for l > 1 and λ1,jk = 1−∑l>1 λl,jk. This expression is de-rived using a degenerate perturbation theory where the diver-gences due to near two-level degeneracies are regularized in aself-consistent manner. The expression for Schmidt eigenval-ues in Eq. (61) shows non-self-averaging and oscillatory be-havior (quite similar to Rabi oscillations). The linear entropycan be computed using

S2(t; Λ) ≈ 2∑l>1

(λl,jk(t; Λ)− λ2

l,jk(t; Λ)). (62)

The cross terms λl,jk(t; Λ)λl′,jk(t; Λ) are neglected in theabove since on average they contribute to higher order,O(Λ) [18].

Following the approach in [58], an approximate expressionfor the probability density of S2 can be derived in this regimewhere only the closest unperturbed eigenstate to the initialstate |jk〉 is relevant. In this limit, the infinite time averageof Eq. (62) can be calculated as

S2 ≈ 4u(1− 3u), (63)

where the variable u is given by

u =(

4 +s2

Λw

)−1

. (64)

The closest neighbor spacing probability density is PCN(s) =2 exp(−2s) [14] for a Poissonian sequence. The rescaled off-diagonal V-matrix element w has a probability density of anexponential as discussed in Sect. II E. The probability densityof u, Pu(x), can be calculated from the relation

Pu(x) =

∫ ∞0

ds

∫ ∞0

dw 2e−2se−wδ(x−

[4 +

s2

Λw

]−1).

(65)As defined in [58], let

fΛ(x) =x

Λ(1− 4x), (66)

13

10−6 10−5 10−4 10−3 10−2 10−1

x

10−2

100

102

104

PS

2(x

)

(a)

0.0 0.1 0.2 0.3 0.4 0.5x

10−3

10−1

101

103

PS

2(x

)

(b)

FIG. 6. Two views of the S2 probability density of the E ⊗ E typeinitial state ensemble. In (a) log-log plot illustrating the heavy tailwhereas in (b) log-linear which is better for illustrating the endpointbehaviors at 0 and 1/3. The blue dashed line shows the theory pre-diction, Eq. (69). Λ = 10−6 and NA = NB = 50. The red solidline is the histogram obtained via numerical averaging of individualtime evolved curve of initial states after saturation (t > 2), and theblack solid line via Eq. (33). Both sets of data reveal a cutoff aroundO(Λ). The vertical magenta dotted line and green dot-dashed lineshows x = 1/3 and 1/2, respectively.

then upon carrying out the integral over w

Pu(x) =

∫ ∞0

ds exp(−2s− s2fΛ(x)

)[ 2s2

Λ(1− 4x)2

],

(67)which is similar to Eq. (84) in [58] with appropriate variabletransformations. From Eq. (63), u can be written in terms ofx = S2 as

u =1

6

(1−√

1− 3x)

(68)

and the solution inconsistent with the non-degenerate pertur-bation theory is discarded. This gives the S2 probability den-sity as

PS2(x) = Pu

(1

6

(1−√

1− 3x))∣∣∣du

dx

∣∣∣. (69)

Figure 6 illustrates PS2(x) for Λ = 10−6, where a heavy-

tail like distribution can be seen with S2 covering a range ofO(Λ) to O(1). Notice that around S2 = 1/3 the local maxi-mum predicted by Eq. (69) appears. However, some slight de-viations indicate that the assumption of just two unperturbedeigenstates participating shows some small corrections due totriple degeneracies that can occur with low probability.

For this ensemble of initial states, a broad distribution ofS2 suggests that equilibrium is not really achieved due to theheavy-tail-like behavior. In addition, due to a dearth of un-perturbed eigenstates participating in the time evolved state,a relaxation is not possible. As a result, a large ensemble ofinitial states are necessary for convergence to the average en-tanglement production curve.

2. E ⊗ C and E ⊗ R type initial states

Consider an initial product state whose one of the subsys-tem coherence measures is zero in the preferred basis and theother is non-zero

|α(0)〉 = |j〉 ⊗( ∑|k〉∈B′

B

zB,k |k〉), (70)

where without loss of generality c(2)A is taken to be vanishing.

For KB � NB an expression for the linear entropy can bederived as follows. The time evolution of the above state usingthe Schmidt decomposition in Eq. (60) is given by

|α(t; Λ)〉 =∑k

∑l

√λl,jk(t; Λ) |(jk)l〉 , (71)

where any phase factor that may appear is absorbed into|(jk)l〉. The reduced density matrix constructed out of thistime evolved state reads as

ρA(t; Λ) =∑k,k′

∑l,l′

zB,kz∗B,k′

√λl,jk(t; Λ)λl′,jk′(t; Λ)

× trB(|(jk)l〉 〈(jk′)l′ |). (72)

Note that the set of states {|(jk)l〉}must be energetically closeto the level corresponding to the index pair (j, k) as mentionedearlier, which amounts to having small energy differences be-tween them. Let (jl, kl) be the index pair of the ket |(jk)l〉,then for chaotic bipartite Floquet systems considered here theindex pair satisfies

jl + kl ≈ j + k (73)

for NA ≈ NB by the virtue of having approximately thesame uniform mean level spacing for both the subsystems.This translates to the index pair having a structure (jl, kl) =(j ± l, k ∓ l) for l = 1, 2, 3, . . . where l runs up to NA/2typically and energy differences θjlkl−θjk are approximately2π/NANB on average [58]. Furthermore, for very small per-turbations it is enough to keep first and second largest Schmidteigenvalues in Eq. (60) and neglect the rest [18]. This corre-sponds to only including the closest neighbor, |(jk)2〉, of |jk〉,

14

and the rest are insignificant. However, this argument breaksdown when triple degeneracies occur in the unperturbed spec-trum. Statistically speaking, this is of very low probabilityand contributes to higher order than O(

√Λ). Based on these

arguments, the partial trace that appears in Eq. (72) can becomputed as follows. For k = k′ and l = l′ the partial traceterm is |jl〉 〈jl|, and that for k = k′ and l 6= l′ it vanishessince kl = kl′ is forbidden according to Eq. (73). In thek 6= k′ case, the energy difference between levels (j, k) and(j, k′) is roughly 2π/NB , and so the probability that the set{|(jk)l〉} and {|(jk′)l〉} share a common unperturbed eigen-state or more whose product λl,jkλl′,jk′ is also significant isquite small for KB � NB . Hence, these contributions can beneglected. As KB increases, eventually this argument breaksdown.

Putting all this together, it can be shown that the reduceddensity matrix in Eq. (72) is approximately diagonal in theBA basis, and the largest Schmidt eigenvalue of ρA(t; Λ) inEq. (72) is

λ1(t; Λ) ≈∑k

|zB,k|2λ1,jk(t; Λ), (74)

and rest of the (relevant) Schmidt eigenvalues are

λl(t; Λ) = |zB,k|2λ2,jk(t; Λ), (75)

where for each l = 2, . . . ,KB + 1 there is a correspondingpair (j, k) in {λ2,jk(t; Λ)}k=1,...,KB

, however the ordering isa priori unknown. This gives the linear entropy as

S2(t; Λ) ≈ 2∑k

|zB,k|2λ2,jk(t; Λ)− 2∑k

|zB,k|4λ22,jk(t; Λ),

(76)

where λ1,jk(t; Λ) ≈ 1 − λ2,jk(t; Λ) is used, and the cross-terms λ2,jk(t; Λ)λ2,jk′(t; Λ) for k 6= k′ are neglected sincethey contribute to O(Λ) on average. Upon ensemble averag-ing and rearranging

〈S2(t; Λ)〉 ≈[C(2; t) + 2〈c(2)

B 〉C2(2; t)]√

Λ, (77)

where the result 〈λ22,jk〉 ≈ C2(2; t)

√Λ from [18] is used;

C2(2; t) is defined in App. D. Expanding the averaged linearentropy for short-time gives

〈S2(t; Λ)〉 = 4πt√

Λ +O(t3), (78)

which is same as that of E ⊗ E type initial states. The averagelinear entropy after long time saturates to

〈S2〉 ≈(5 + 3〈c(2)

B 〉8

)√Λ , (79)

which is obtained by taking t→∞ limit of Eq. (77).Relatively good agreement between the theory prediction

and numerical data is found as illustrated in Fig. 7 where C-type (|αKB

〉C, in (a)) and R-type (|αKB〉R, in (b)) states are

considered for subsystemB initial states. The deviations fromthe theory curve that are seen in the plot can be understood by

0 1 2 3t

0.000

0.002

0.004

0.006

〈S2(t

;Λ)〉

(a)

KB = 2

KB = 4

KB = 6

0 1 2 3t

0.000

0.002

0.004

0.006

〈S2(t

;Λ)〉

(b)

KB = 2

KB = 4

KB = 6

FIG. 7. Ensemble averaged linear entropy 〈S2(t; Λ)〉 for Eq. (70)-type initial states versus t for the RMT transition ensemble ofEq. (13) (a) E ⊗ C and (b) E ⊗ R-type. Both use NA = NB = 50and Λ = 10−6. The theory prediction is given by Eq. (77) in blacklines, K = 2 (dotted), 4 (dashed), 6 (dot-dashed), and short timebehavior Eq. (78) in a black solid line.

examining the probability density of S2 as shown in Fig. 8. Abroad heavy-tail-like behavior similar to that of E ⊗ E initialstates is found. Thus, for this type of initial states, equilibriumis a questionable notion. This shows that the convergence tothe averaged linear entropy curve of Fig. 7 is quite slow andrequires a large sample size. Furthermore, relaxation is notoccurring and no self-averaging is apparent. Hence no equi-libration occurs even though one of the subsystems has non-vanishing coherence.

3. C ⊗ C and R ⊗ R type initial states

Consider the initial states of the form Eq. (35) whose par-ticipant unperturbed eigenstates are from the subset B′AB =B′A ⊗ B′B . As discussed earlier, for the Λ → 0+ regime thetime evolved state after a long time remains largely within thesubspace described by B′AB of the full Hilbert space. Thisscenario gets modified as the interaction strength increaseswhere the rotation of eigenstates becomes increasingly rele-vant. However, for sufficiently small interaction strengths, a

15

10−5 10−4 10−3 10−2 10−1 100

x

10−2

100

102

104PS

2(x

)(a)

KB = 2

KB = 4

KB = 6

10−6 10−5 10−4 10−3 10−2 10−1

x

10−2

100

102

104

PS

2(x

)

(b)

KB = 2

KB = 4

KB = 6

FIG. 8. Probability density of S2 values for initial state type Eq. (70)(a) E ⊗ C and (b) E ⊗ R type. Various KB values are shown forNA = NB = 50 and Λ = 10−6.

perturbed eigenstate |jk(Λ)〉 consists of just |jk〉 and its en-ergetically closest neighbor |(jk)2〉, similar to the earlier as-sumption used for time evolved state |jk(t; Λ)〉. Thus for each|jk〉 in the subset B′AB , there is a corresponding set of suchclosest neighbors {|(jk)2〉}. The set {|(jk)2〉} can be dividedinto two subsets, where one of the subsets has eigenkets out-side of B′AB and the other contains the eigenkets that are alsoin B′AB . The elements in these subsets are a priori unknown.This situation brings some non-trivial effects on the entangle-ment produced and its fluctuation about the equilibrium value〈S2〉 compared to Λ→ 0+ regime and is discussed ahead.

Numerical data reveal that the effect of eigenstate rotationis quite evident during the initial entanglement productionphase. For any Λ > 0+, a linear-in-time growth is found,which happens to be the same as that of E ⊗ E initial statesgiven in Eq. (58) for t ≈ 0. The theory derived for Λ→ 0+ inEq. (38) fails for this initial entanglement growth phase andafter including the correction due to the rotation of eigen-states, the entanglement production for short times is givenby

〈S2(t; Λ)〉 ≈ 4πt√

Λ + 2〈c(2)A 〉〈c

(2)B 〉t2 , (80)

which quantifies a competition between quadratic entangle-ment growth and the generally expected linear behavior. De-

0.000 0.002 0.004 0.006 0.008 0.010t

0.0000

0.0001

0.0002

0.0003

〈S2(t

;Λ)〉

(a)

K = 2

K = 4

K = 50

0.000 0.002 0.004 0.006 0.008 0.010t

0.000

0.005

0.010

0.015

0.020

0.025

〈S2(t

;Λ)〉

4π√

Λ(b)

Λ = 10−6

Λ = 10−5

Λ = 10−4

FIG. 9. Short time behavior of ensemble averaged 〈S2(t; Λ)〉 for R⊗ R type initial states for NA = NB = 50. In (a) Λ = 10−6 withvarious K values. The black solid line represents Eq. (58), and otherblack lines indicate the combined theory in Eq. (80) corresponding tovarious K values. In (b) K = 50 with the Λ values indicated in thelegend. The black solid line represents Eq. (58) and the other blacklines correspond to various Λ values of the combined theory given inEq. (80). All the curves are scaled by 4π

√Λ.

pending on the circumstances, the quadratic or linear termmay dominate, and the ratio of the terms generates a crossovertime scale given by

t∗ =2π√

Λ

〈c(2)A 〉〈c

(2)B 〉

(81)

at which the linear and quadratic growth terms contributeequally. For t � t∗ a prominent linear growth is seen andfor t > t∗ it becomes increasingly quadratic. An interest-ing aspect of the expression for the crossover time scale isthat for small enough Λ and large enough quantum coher-ence in the initial state, t∗ → 0, the linear regime collapses,and short time entanglement production effectively displayspurely quadratic behavior. On the other hand, for values ofΛ and quantum coherence leading to a t∗ which is an ap-preciable fraction of unity (greater than the short scaled timeregime), this quadratic regime ceases to exist, and only thelinear regime behavior results. Note that quadratic growthhas been found to exist in strongly coupled holographic sys-tems [61, 62], where quadratic behavior transitions to a linear

16

0.0 0.2 0.4 0.6 0.8 1.0

〈c(2)A 〉 〈c

(2)B 〉

10−4

10−3

10−2

10−1

〈S2〉−〈c

(2)

A〉〈c(2

)B〉

FIG. 10. The difference of 〈S2〉 and initial state coherence〈c(2)A 〉〈c

(2)B 〉 as a function of the initial state coherence for various

interaction strengths Λ = 10−6 (#), 10−4 (�), 10−3 (5), 10−2 (♦)are shown where dimensionality NA = NB = 50. The horizontalblack lines show Eq. (59) for various Λ where the solid line corre-sponds Λ = 10−6 and dotted line is for Λ = 10−2.

growth, and in random local Gaussian circuits [63].The crossover between linear and quadratic entanglement

growth during the initial phase of the time evolution is shownin Fig. 9. In Fig. 9 (a) the average entanglement growth forfixed Λ = 10−6 and initial states of type R ⊗ R with vari-ous K values are shown (C ⊗ C type initial states show simi-lar behavior and are not displayed here), where the crossovertime scales t∗ = 0.056, 0.017, 0.0068, for K = 2, 4, 50, re-spectively. For small KA and KB values, the initial statesare more like E ⊗ E type and they show a prominent linear-in-time behavior for a long period during the initial phase asopposed to the initial states whose KA ∼ NA and KB ∼ NBwhere the linear-in-time behavior transitions to a quadratic be-havior rather quickly. As the interaction strength increases, adominant linear-in-time entanglement growth is found stretch-ing to longer and longer times predicted by the time scalet∗. This is displayed in Fig. 9 (b) where K = 50 andΛ = 10−6, 10−5, 10−4 whose crossover time scales are t∗ =0.0068, 0.021, 0.068, respectively. Here 〈S2(t; Λ)〉 is scaledby 4π

√Λ so that all the curves for various Λ have the same

slope for easier visual comparison.The influence of eigenstate rotation can be seen in the

difference of 〈S2〉 and 〈c(2)A 〉〈c

(2)B 〉 for various interaction

strengths as shown in Fig. 10. This difference is expectedto be of the order O(

√Λ) based on what is known for the

E ⊗ E case in Eq. (59), illustrated in the plot as horizontalblack lines. As the initial state coherence increases, there areincreasing deviations from this expectation due to the correc-tions that strongly depend on the initial state coherence. Fornear-maximal initial state coherence, the eigenstate rotationhas negligible affect on the saturation value 〈S2〉, and the dif-ference vanishes.

Furthermore, the probability density of S2 becomes in-creasingly localized as the interaction strength increases,which is not surprising. The change in shape is more pro-

10−5 10−4 10−3 10−2 10−1 100

x

10−2

10−1

100

101

102

103

PS

2(x

)

(a)

0.88 0.90 0.92 0.94 0.96x

10−1

100

101

102

103

PS

2(x

)

(b)

FIG. 11. Probability density of S2 values for R ⊗ R type initialstates for (a) K = 2, (b) K = 50. Λ = 10−6 (blue), 10−4 (red),10−3 (black), 10−2 (green), 1 (magenta), 10 (cyan).

nounced for KA � NA, KB � NB , where it transi-tions from the broad heavy-tail like behavior to a localizedGaussian-like probability density around Λ = 10−2 as shownin Fig. 11 (a). On the other hand, for KA ∼ NA, KB ∼NB the density is a narrow, distorted Gaussian-like shape,which becomes even narrower as the interaction strength isincreased; see Fig. 11 (b). This also is not surprising becausefor small KA, KB and Λ → 0+ the probability density isbroad whereas for large KA, KB it is quite localized as seenin Figs. 3 (a), 4 (a). So as the interaction strength increases, forinitial states with small KA, KB more and more unperturbedeigenstates are perturbatively added leading to a more local-ized Gaussian-like distribution by the virtue of central limittheorem along with self-canceling of terms involving oscil-lating eigencomponents with random frequencies that appearin the S2 expression. This also means that as the interactionstrength increases, more and more initial states relax to theequilibrium as observed in Figs. 12 (a) and (b) where bothC ⊗ C and R ⊗ R type initial states are shown, respectively.As the interaction strength increases, a significant change inσ2(S2) occurs only for K = 2. Although, for most K, a dropoccurs across the intermediate strength perturbation regime.Interestingly, for maximally coherent C⊗ C type initial states(K = 50), the relaxation measure is nearly constant and muchless than unity, even extending to the intermediate and strong

17

10−5 10−3 10−1 101

Λ

10−6

10−5

10−4

10−3

10−2

10−1

100σ

2(S

2)

(a)

K = 2

K = 6

K = 10

K = 14

K = 22

K = 50

10−5 10−3 10−1 101

Λ

10−5

10−3

10−1

σ2(S

2)

(b)K = 2

K = 6

K = 10

K = 14

K = 22

K = 50

FIG. 12. The relaxation measure σ2(S2) as a function of Λ for(a) C ⊗ C and (b) R ⊗ R type initial states. Various K values withNA = NB = 50.

regimes, implying that the system attains thermal equilibriumeventually. Note that the time taken to reach saturation getsshorter and shorter as Λ increases. In other words, the maxi-mal coherence in the initial product states is an alternate pathto thermalization of the system under time evolution with Λdetermining how rapidly it reaches thermal equilibrium re-gardless of whether the eigenstates of the system are thermalor not.

B. Intermediate perturbation regime

This regime can be viewed as an interpolation or transitionfrom the weak to strong interaction regime, where a weak de-pendence on the initial state coherence is observed which van-ishes as the interaction strength increases. From this observa-tion, the expression for 〈S2(t; Λ)〉 derived in the study [18]for E ⊗ E initial states in the non-perturbative regime

〈S2(t; Λ)〉 ≈ 〈S∞2 〉[

1− exp

(−C(2; t)

√Λ

〈S∞2 〉

)], (82)

can be used as a decent approximation to the entanglementproduced for any initial product states. This is illustrated in

0.0 0.5 1.0 1.5 2.0t

0.0

0.2

0.4

0.6

0.8

1.0

〈S2(t

;Λ)〉

(a)

K = 2

K = 50

0.5 1.0 1.5log10(K)

−7

−6

−5

log

10(v

aria

nce

)

(b)

FIG. 13. (a) Ensemble averaged linear entropy 〈S2(t; Λ)〉 as a func-tion of rescaled time t is shown for K = 2, 50 R ⊗ R type initialstates. The black solid curve shows the E ⊗ E type result in Eq. (82)for the non-perturbative regime. (b) The two fluctuation measures,σ2(S2) and σ2(S2) for C ⊗ C (+ and ×, respectively) and R ⊗ Rtype (� and #, respectively). For both (a) and (b) NA = NB = 50and Λ = 1.

Fig. 13 (a) for Λ = 1. The saturation values for K = 2, 50can be seen to be slightly higher than that the theory curve inEq. (82) and K = 50 curve saturates above that of K = 2.Note that in the strong interaction regime discussed previouslyin Sect. IV B the above equation was used to show that theaverage entanglement generated is largely independent of theinitial (product) state.

The probability density of S2 for Λ = 1 is highly localizedin contrast to the ones in the weak perturbative regime; seeFig. 11. The initial state coherence appears to play a role in thewidth of the density. The K = 50 case is almost identical tothat of Λ = 10 in the strong interaction regime. Furthermore,the fluctuation measures obtained numerically for C ⊗ C andR ⊗ R type initial state ensembles show that a majority of theinitial states eventually attain the equilibrium value 〈S2〉 andremain there as shown in Fig. 13 (b).

18

C. Discussion

It is worthwhile to collect all the results discussed so far toget an overall picture of how relaxation and equilibration de-pend on the quantum coherence in an initial product state andthe interaction strength between the subsystems. Together, thecoherence and transition parameter govern the entanglementgrowth and its saturation to an equilibrium (if it exists) andthe nature of the equilibration.

In the ultra-weak interaction regime, where the rotation ofeigenstates are neglected, an existence of equilibrium can beestablished based on the equilibrium measure where the equi-librium becomes sharper as the average coherence in the ini-tial state ensemble increases. The coherence of the initialstates also affects its ability to equilibrate. A weak to strongequilibration is found to occur as the average coherence ofthe initial state ensemble increases. In particular, for maxi-mally coherent initial product states the system thermalizes af-ter sufficiently long time and remains thermalized for almostall the initial states as indicated by negligibly small fluctua-tions about the equilibrium. Furthermore, a quadratic initialentanglement growth is seen in this regime characterized bythe average coherence of the ensemble under consideration.

A similar story can be said for weak interaction regime, ex-

cept for the new effects that arise due to the rotation of eigen-states. Firstly, for initial states whose coherence is vanishingan equilibrium is questionable and hence no relaxation can beseen. The initial entanglement growth is linear-in-time and ispurely due to the rotation of eigenstates and the rate is dic-tated by the transition parameter. The average entanglementsaturation value is perturbative and is also dictated strongly bythe transition parameter. Secondly, for initial state ensembleswith non-vanishing average coherence, a competition betweenlinear and quadratic growths occurs in the initial phase. Whilethe linear-in-time behavior is same for any kind of initial stateensemble, the quadratic growth depends strongly on the av-erage coherence. Furthermore, the entanglement saturationshows strong dependence on the transition parameter for ini-tial state ensembles with low average coherence while a weakdependence is seen for ensembles with high coherence.

For intermediate and strong interaction regimes, little tonegligibly small dependence on the initial states is observed,including the ones with vanishing coherence. In theseregimes, a strong equilibration is seen. A noteworthy pointhere is that the initial state ensemble with maximal coherencethermalizes regardless of the strength of the interaction buthow rapidly it thermalizes is controlled by the transition pa-rameter. Larger the transition parameter, shorter it takes tothermalize. Table I summarizes the key results.

TABLE I: Various initial state ensembles and nature of the equilibrium and initialentanglement growth in various interaction regimes.

Initial state ensemble Interaction regimeUltra-weak Weak Intermediate Strong

E ⊗ E, E ⊗ C, E ⊗ R no equilibration, no equilibration, strong equilibration, strong thermalization,(vanishing coherence) initial linear growth initial linear growth initial linear growth initial linear growth

C ⊗ C, R ⊗ R weak equilibration, weak equilibration, strong equilibration, strong thermalization,(low coherence) initial quadratic growth late linear to quadratic crossover initial linear growth initial linear growthC ⊗ C, R ⊗ R strong equilibration, strong equilibration, strong equilibration, strong thermalization,

(high coherence) initial quadratic growth early linear to quadratic crossover initial linear growth initial linear growthC ⊗ C strong thermalization strong thermalization strong thermalization strong thermalization

(maximal coherence) initial quadratic growth early linear to quadratic crossover initial linear growth initial linear growth

VI. SUMMARY AND OUTLOOK

In this paper, some of the direct consequences of quantumcoherence towards the entanglement production, equilibrationand thermalization are presented. In an earlier study [18], theentanglement production of special E ⊗ E type initial stateswas found to depend only on a universal transition parameterΛ, and the rescaled time t, which as shown in this study, gener-alizes the scope of these universal quantities to generic initialstates. As guided by perturbation theory, various perturbationregimes were identified.

In the ultra-weak regime Λ → 0+ where the diagonal ele-ments of the perturbation matrix are relevant and the effect ofrotation of eigenstates is neglected, it was analytically shownthat the entanglement production is a Gaussian as a functionof time and saturates to the product of coherence measures(a coherence measure based on the l2-norm) of the subsys-tems in their respective preferred eigenstate basis. This im-plies an unusual initial quadratic time dependent increase in

the entanglement. Whether quadratic or linear behavior is tobe expected is contained in Eq. (80). It was found that for ini-tial (product) states whose subsystem coherence measures arenear maximal after long time the system saturates to the maxi-mal entanglement. Furthermore, such states show thermaliza-tion as evident from the distribution of infinite time average oflinear entropy and the relaxation measure, which is a remark-able result noting that the interaction between the subsystemsis ultra-weak, and the full system eigenstates are barely differ-ent from the non-interacting case. This shows that quantumcoherence acts as a resource for equilibration and thermaliza-tion.

In the weak regime, where the eigenstate rotation is rele-vant in the perturbation theory, it was found that the probabil-ity density of the infinite time linear entropy average of ini-tial states whose product of subsystem coherence measures isvanishing has a broad heavy-tail like behavior and shows norelaxation. Entanglement production is of the order O(

√Λ)

on average and shows a slow convergence to the average due

19

to the heavy-tailed nature of the distribution. In the initial en-tanglement growth phase, it can be shown that the entangle-ment production is linear in time and the rate is proportionalto√

Λ. This initial linear-in-time entanglement growth is seenregardless of the initial state coherence and is universal, evenfor large Λ. The effect of eigenstate rotation is also evident inthe entanglement saturation values, especially for initial stateswhose coherence is close to minimal. On the other hand, ini-tial states with near-maximal coherence are not dependent onthis. Lastly, in the intermediate regime and beyond, the entan-glement production appears to have little to no dependence onthe initial product states and shows an exponential behavior.

In the present study, an RMT transition ensemble was usedto mimic a bipartite system whose subsystems are fully quan-tum chaotic. For a real dynamical system either single-particleor many-body, features like eigenstate scarring [64], k-bodyinteractions in case of many-body systems (see the review[43]) and other dynamical features may give rise to some sys-tem specific deviations to the universal features presented inthis paper. It would be interesting to know how the univer-sal features presented here would change if the subsystemsare not fully chaotic, where the notion of the transition pa-rameter may not exist due to selection rules and existence oflocal integrals of motion (see the review [9]). As seen in thisstudy, the initial product state coherence plays a crucial rolein determining the fate of the interacting bipartite system atlong times. This study may shed light on understanding thephase transition towards thermalization not just with the inter-action strength, but also due to coherence in the initial state.Moreover, it would be interesting to understand the entangle-ment production in a semiclassical sense similar to the earlierfidelity studies that related the transition parameter to the clas-sical action diffusion coefficient and the phase space volumein the ultra-weak perturbation regime [56, 57].

ACKNOWLEDGMENTS

We are grateful for Washington State University’s KamiakHigh Performance Computer at the Center for InstitutionalResearch Computing, which was used extensively for the nu-merical calculations of the present study. This research waspartially funded by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) – 497038782.

Appendix A: Derivation of variances of V matrix elements

Starting from Eq. (18), the expression for the variance is

〈|Vjk,j′k′ |2〉 =⟨ ∑ab,a′b′

uA∗ja uB∗kb u

Aj′a u

Bk′b (2πξab)

× uAja′ uBkb uA∗j′a uB∗k′b (2πξa′b′)⟩. (A1)

Given the probability density of ξab, Eq. (14), and indepen-dence

〈ξab ξa′b′〉 = δab,a′b′1

12, (A2)

the variance reduces to the expression

〈|Vjk,j′k′ |2〉 =π2

3

⟨∑ab

|uAja|2 |uAj′a|2 |uBkb|2 |uBk′b|2⟩, (A3)

where Haar averaging over unitary groups remains to be per-formed. Using the known results [53] gives

〈|uAja|2 |uAj′a|2〉 =1 + δjj′

NA(NA + 1), (A4)

and similarly for subsystemB in Eq. (A3). Summing over thevariables a, b leads to the result in Eq. (19).

The covariance, 〈xjk xj′k′〉, in Eq. (23) is given by

〈xjkxj′k′〉 =〈Vjk,jk Vj′k′,j′k′〉√

〈|Vjk,jk|2〉√〈|Vj′k′,j′k′ |2〉

, (A5)

where making use of√〈|Vjk,jk|2〉 =

√〈|Vj′k′,j′k′ |2〉 and

Eq. (19) gives√〈|Vjk,jk|2〉

√〈|Vj′k′,j′k′ |2〉 =

4π2

3(NA + 1)(NB + 1).

(A6)That leaves the 〈Vjk,jk Vj′k′,j′k′〉 computation to be done.Starting from Eq. (18) and using Eq. (A2) gives

〈Vjk,jk Vj′k′,j′k′〉 =π2

3

⟨∑ab

|uAja|2 |uAj′a|2 |uBkb|2 |uBk′b|2⟩

(A7)and performing Haar averaging on the subsystem unitarygroups, as in Eq. (A3), establishes Eq. (23).

Appendix B: Details of numerical calculations

All the calculations presented in this article are based onrealizations of the random matrix transition ensemble definedin Eq. (13) using subsystem dimensionality NA = NB = 50.The sample size details of time evolution raw data for variousC⊗ C and R⊗ R type initial state ensembles and all Λ valuesare shown in Table. II as initial state sampled per realization× total number of realizations.

TABLE II. Various Λ values and corresponding initial state samplesper realization × number of realizations.

Λ C ⊗ C R ⊗ R E ⊗ E E ⊗ C E ⊗ R10−6 1250 × 5 2500 × 5 2500 × 20 2500 × 20 2500 × 510−4 1250 × 5 750 × 5 N/A N/A N/A10−3 750 × 5 750 × 5 N/A N/A N/A10−2 2500 × 5 2500 × 5 N/A N/A N/A1 2500 × 2 2500 × 2 N/A N/A N/A10 50 × 5 50 × 5 N/A N/A N/A

For the short time behavior depicted in Fig. 9 and for var-ious Λ values 1000 × 5 initial states were used instead. Inaddition, for Fig. 6 data generated for the S2 probability den-sity based on the infinite time average expression in Eq. (33)used 2500 × 10 initial states.

20

Appendix C: Eigenvector statistics of unitary ensemble

Consider an eigenvector |α〉 of anN -dimensional CUE ma-trix of unitary ensemble. Represented in some fixed basis |i〉,it is given by |α〉 =

∑Ni=1 zi |i〉, where zi are complex coeffi-

cients. The only constraint is the normalization,∑i |zi|2 = 1.

The probability density of the eigenvector components aregiven by [43, 65, 66]

P (z1, z2, . . . , zN ) =(N − 1)!

πNδ( N∑i=1

|zi|2 − 1), (C1)

for which reduced probability density can be found by inte-grating out N − l variables resulting in

Pl(z1, z2, . . . , zl) =Γ(N)

πlΓ(N − l)(

1−l∑i=1

|zi|2)N−l−1

.

(C2)Using the above reduced probability density various momentsof zi can be computed analytically. Below a list of usefulmoments relevant to the main text are given

〈|zi|4〉 =2

N(N + 1), (C3)

〈|zi|2|zj |2〉 =1

N(N + 1)for i 6= j, (C4)

〈|zi|8〉 =24

N(N + 1)(N + 2)(N + 3), (C5)

and

〈|zi|4|zj |4〉 =4

N(N + 1)(N + 2)(N + 3)for i 6= j.

(C6)

Appendix D: C(2; t) and C2(2; t) functions

The C(2; t) and C2(2; t) functions are defined in a previ-ous work [18] for the context of the ensemble averaged un-perturbed eigenstate time evolution, Eq. (60). The functionC2(2; t) is given by⟨∑

l>1

λ2l,jk(t; Λ)

⟩= C2(2; t)

√Λ, (D1)

where

C2(2; t) =

∫ ∞0

dw

∫ ∞−∞

dz

(4z

z2 + 4w

)2

exp(−w)

× sin4( t

2

√z2 + 4w

). (D2)

The function C(2; t) defined in Eq. (57) is given by

C(2; t) = πt(

3e−t2 − 1

2e−4t2

)+ π3/2erf(t)

(1

2+ 3t2

)+ π3/2erf(2t)

(1

8− 3t2

), (D3)

where erf(x) = (2/√π)∫ x

0e−t

2

dt is the error function.

[1] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and itsmechanism for generic isolated quantum systems, Nature 452,854 (2008).

[2] J. M. Deutsch, Quantum statistical mechanics in a closed sys-tem, Phys. Rev. A 43, 2046 (1991).

[3] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E50, 888 (1994).

[4] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Fromquantum chaos and eigenstate thermalization to statistical me-chanics and thermodynamics, Adv. Phys. 65, 239 (2016).

[5] K. He and M. Rigol, Initial-state dependence of the quenchdynamics in integrable quantum systems. III. Chaotic states,Phys. Rev. A 87, 043615 (2013).

[6] E. J. Torres-Herrera and L. F. Santos, Effects of the interplaybetween initial state and Hamiltonian on the thermalization ofisolated quantum many-body systems, Phys. Rev. E 88, 042121(2013).

[7] M. Collura, M. Kormos, and P. Calabrese, Stationary entangle-ment entropies following an interaction quench in 1D Bose gas,J. Stat. Mech. , P01009 (2014).

[8] M. Rigol and M. Srednicki, Alternatives to eigenstate thermal-ization, Phys. Rev. Lett. 108, 110601 (2012).

[9] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Collo-quium: Many-body localization, thermalization, and entangle-

ment, Rev. Mod. Phys. 91, 021001 (2019).[10] R. Khare and S. Choudhury, Localized dynamics following a

quantum quench in a non-integrable system: an example on thesawtooth ladder, J. Phys. B 54, 015301 (2020).

[11] J. Rau and B. Muller, From reversible quantum microdynamicsto irreversible quantum transport, Phys. Rep. 272, 1 (1996).

[12] J. B. French, V. K. B. Kota, A. Pandey, and S. Tomsovic, Sta-tistical properties of many particle spectra v: fluctuations andsymmetries, Ann. Phys. (N.Y.) 181, 198 (1988).

[13] S. C. L. Srivastava, S. Tomsovic, A. Lakshminarayan, R. Ket-zmerick, and A. Backer, Universal scaling of spectral fluctua-tion transitions for interacting chaotic systems, Phys. Rev. Lett.116, 054101 (2016).

[14] A. Lakshminarayan, S. C. L. Srivastava, R. Ketzmerick,A. Backer, and S. Tomsovic, Entanglement and localiza-tion transitions in eigenstates of interacting chaotic systems,Phys. Rev. E 94, 010205(R) (2016).

[15] Z. Bai and S. Du, Maximally coherent states, Quantum Inf.Comput. 15, 1355 (2015).

[16] A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quan-tum coherence as a resource, Rev. Mod. Phys. 89, 041003(2017).

[17] T. Tkocz, M. Smaczynski, M. Kus, O. Zeitouni, andK. Zyczkowski, Tensor products of random unitary matrices,

21

Random Matrices: Theor. Appl. 1, 1250009 (2012).[18] J. J. Pulikkottil, A. Lakshminarayan, S. C. L. Srivastava,

A. Backer, and S. Tomsovic, Entanglement production by in-teraction quenches of quantum chaotic subsystems, Phys. Rev.E 101, 032212 (2020).

[19] W. H. Zurek and J. P. Paz, Decoherence, chaos, and the secondlaw, Phys. Rev. Lett. 72, 2508 (1994).

[20] P. A. Miller and S. Sarkar, Signatures of chaos in the entan-glement of two coupled quantum kicked tops, Phys. Rev. E 60,1542 (1999).

[21] D. Monteoliva and J. P. Paz, Decoherence and the rate of en-tropy production in chaotic quantum systems, Phys. Rev. Lett.85, 3373 (2000).

[22] A. Tanaka, H. Fujisaki, and T. Miyadera, Saturation of theproduction of quantum entanglement between weakly coupledmapping systems in a strongly chaotic region, Phys. Rev. E 66,045201 (2002).

[23] H. Fujisaki, T. Miyadera, and A. Tanaka, Dynamical aspects ofquantum entanglement for weakly coupled kicked tops, Phys.Rev. E 67, 066201 (2003).

[24] J. N. Bandyopadhyay and A. Lakshminarayan, Entanglementproduction in coupled chaotic systems: Case of the kicked tops,Phys. Rev. E 69, 016201 (2004).

[25] P. Calabrese and J. Cardy, Evolution of entanglement entropyin one-dimensional systems, J. Stat. Mech.: Theory Exp. 2005,P04010 (2005).

[26] H. Kim and D. A. Huse, Ballistic spreading of entanglement ina diffusive nonintegrable system, Phys. Rev. Lett. 111, 127205(2013).

[27] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli,R. Schittko, P. M. Preiss, and M. Greiner, Quan-tum thermalization through entanglement in an iso-lated many-body system, Science 353, 794 (2016),https://www.science.org/doi/pdf/10.1126/science.aaf6725.

[28] M. C. Banuls, J. I. Cirac, and M. B. Hastings, Strong and weakthermalization of infinite nonintegrable quantum systems, Phys.Rev. Lett. 106, 050405 (2011).

[29] C.-J. Lin and O. I. Motrunich, Quasiparticle explanation of theweak-thermalization regime under quench in a nonintegrablequantum spin chain, Phys. Rev. A 95, 023621 (2017).

[30] N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantummechanical evolution towards thermal equilibrium, Phys. Rev.E 79, 061103 (2009).

[31] J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog.Phys 81, 082001 (2018).

[32] M. Srednicki, Thermal fluctuations in quantized chaotic sys-tems, J. Phys. A 29, L75 (1996).

[33] L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, Theory of optical scintillation, J. Opt. Soc. Am. A 16,1417 (1999).

[34] E. Lubkin, Entropy of an n-system from its correlation with ak-reservoir, J. Math. Phys. 19, 1028 (1978).

[35] J. Aberg, Quantifying Superposition, arXiv e-prints quant-ph/0612146 (2006), quant-ph/0612146.

[36] T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying co-herence, Phys. Rev. Lett. 113, 140401 (2014).

[37] E. C. G. Sudarshan, Equivalence of semiclassical and quantummechanical descriptions of statistical light beams, Phys. Rev.Lett. 10, 277 (1963).

[38] R. J. Glauber, Coherent and incoherent states of the radiationfield, Phys. Rev. 131, 2766 (1963).

[39] M. A. Nielsen and I. L. Chuang, Quantum computation andquantum information, Cambridge University Press, Cambridge(2010).

[40] G. Styliaris, N. Anand, L. Campos Venuti, and P. Zanardi,Quantum coherence and the localization transition, Phys. Rev.B 100, 224204 (2019).

[41] N. Anand, G. Styliaris, M. Kumari, and P. Zanardi, Quan-tum coherence as a signature of chaos, Phys. Rev. Research3, 023214 (2021).

[42] C. E. Porter, Statistical Theories of Spectra: Fluctuations, Aca-demic Press, New York (1965).

[43] T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, andS. S. M. Wong, Random-matrix physics: spectrum and strengthfluctuations, Rev. Mod. Phys. 53, 385 (1981).

[44] A. Altland and M. R. Zirnbauer, Novel symmetry classesin mesoscopic normal-superconducting hybrid structures,Phys. Rev. B 55, 1142 (1997).

[45] A. Pandey and M. L. Mehta, Gaussian ensembles of randomhermitian matrices intermediate between orthogonal and uni-tary ones, Commun. Math. Phys. 87, 449 (1983).

[46] S. Tomsovic, Bounds on the Time-Reversal NoninvariantNucleon-Nucleon Interaction Derived from Transition StrengthFluctuations, Ph.D. thesis, University of Rochester (1986), re-port number 974, 1987.

[47] J. B. French, V. K. B. Kota, A. Pandey, and S. Tomsovic, Statis-tical properties of many particle spectra vi: fluctuation boundson n-n t-noninvariance, Ann. Phys. (N.Y.) 181, 235 (1988).

[48] O. Bohigas, M.-J. Giannoni, A. M. Ozorio de Almeida, andC. Schmit, Chaotic dynamics and the GOE-GUE transition,Nonlinearity 8, 203 (1995).

[49] S. Tomsovic, M. B. Johnson, A. Hayes, and J. D. Bowman, Sta-tistical theory of parity nonconservation in compound nuclei,Phys. Rev. C 62, 054607 (2000).

[50] O. Bohigas, S. Tomsovic, and D. Ullmo, Manifestations of clas-sical phase space structures in quantum mechanics, Phys. Rep.223, 43 (1993).

[51] S. Tomsovic and D. Ullmo, Chaos-assisted tunneling,Phys. Rev. E 50, 145 (1994).

[52] T. Herrmann, M. F. I. Kieler, F. Fritzsch, and A. Backer, En-tanglement in coupled kicked tops with chaotic dynamics, Phys.Rev. E 101, 022221 (2020).

[53] Z. Puchała and J. Miszczak, Symbolic integration with respectto the haar measure on the unitary groups, B Pol Acad Sci-Tech65, 21 (2017).

[54] M. L. Mehta, Random Matrices (Third Edition), Elsevier, Am-sterdam (2004).

[55] A. Peres, Stability of quantum motion in chaotic and regularsystems, Phys. Rev. A 30, 1610 (1984).

[56] N. R. Cerruti and S. Tomsovic, Sensitivity of wave field evolu-tion and manifold stability in chaotic systems, Phys. Rev. Lett.88, 054103 (2002).

[57] N. R. Cerruti and S. Tomsovic, A uniform approximation forthe fidelity in chaotic systems, J. Phys. A: Math. Gen. 36, 3451(2003).

[58] S. Tomsovic, A. Lakshminarayan, S. C. L. Srivastava, andA. Backer, Eigenstate entanglement between quantum chaoticsubsystems: Universal transitions and power laws in the entan-glement spectrum, Phys. Rev. E 98, 032209 (2018).

[59] A. Lakshminarayan, Z. Puchała, and K. Zyczkowski, Diagonalunitary entangling gates and contradiagonal quantum states,Phys. Rev. A 90, 032303 (2014).

[60] B. Jonnadula, P. Mandayam, K. Zyczkowski, and A. Laksh-minarayan, Entanglement measures of bipartite quantum gatesand their thermalization under arbitrary interaction strength,Phys. Rev. Research 2 (2020).

[61] H. Liu and S. J. Suh, Entanglement tsunami: Universal scal-ing in holographic thermalization, Phys. Rev. Lett. 112, 011601

22

(2014).[62] H. Liu and S. J. Suh, Entanglement growth during thermaliza-

tion in holographic systems, Phys. Rev. D 89, 066012 (2014).[63] Q. Zhuang, T. Schuster, B. Yoshida, and N. Y. Yao, Scrambling

and complexity in phase space, Phys. Rev. A 99, 062334 (2019).

[64] E. J. Heller, Bound-state eigenfunctions of classically chaotichamiltonian systems: Scars of periodic orbits, Phys. Rev. Lett.53, 1515 (1984).

[65] F. Haake, Quantum signatures of chaos, volume 54, SpringerScience & Business Media (2013).

[66] A. Lakshminarayan, S. Tomsovic, O. Bohigas, and S. N. Ma-jumdar, Extreme statistics of complex random and quantumchaotic states, Phys. Rev. Lett. 100, 044103 (2008).