aa r X i v : . [ h e p - t h ] J a n arXiv:2101.nnnnJanuary 8, 2021 A holographic measure of quantum information exchange
Harvendra Singh
Theory Division, Saha Institute of Nuclear PhysicsHBNI, 1/AF Bidhannagar, Kolkata 700064, India
E-mail: h.singh [AT] saha.ac.in
Abstract
We estimate the net information exchange between adjacent quantum subsystemsholographically living on the boundary of
AdS spacetime. The information exchange is areal time phenomenon and only after long time interval it may get saturated. Normallywe prepare systems for small time intervals and measure the information exchange overfinite interval only. We find that the information flow between entangled subsystemsgets reduced if systems are in excited state whereas the ground state allows maximuminformation flow at any given time. Especially for
CF T we exactly show that a rise inthe entropy is detrimental to the information exchange by a quantum dot and vice-versa.We next observe that there is a reduction in circuit (CV) complexity too in the presenceof excitations for small times. 1 Introduction
Since the advent of AdS/CFT [1] the holography has produced simpler answers to manydifficult questions in strongly coupled quantum field theories. We consider the phe-nomenon of information exchange between two quantum subsystems having commoninterface. The information exchange between quantum subsystems is a real time phe-nomenon as they would be entangled. In quantum theory the information contained instate | ψ > , cannot be destroyed, cloned or even mutated. In bi-partite systems the infor-mation can be either found in one part of the Hilbert space or in the compliment. Thusthe quantum subsystems remain always entangled. Further it is generally understood thatthe exchange and sharing of quantum information is guided by unitarity and locality; seefor details [2, 3]. The time evolution in quantum theory is guided by the Hamiltonianflow. In this aspect all time dependent flows of isolated quantum system should essen-tially be unitary. Under these claims in the black hole evoparation process (includingthe Hawking radiation) it is expected that the curve for the entanglement entropy mustbend after the half Page time is crossed [4]. This certainly can hold true when a puresystem is divided into two small subsystems. But for the mixed states, say CFT dualsto AdS-black holes, it is not straight forward to answer this question. However, in recentmodels by coupling holographic CFTs to external matter (radiation) bath, and involvingnonperturbative techniques such as replicas and islands [5], people try to answer some ofthese questions. In this work we aim to measure in real time exchange of quantum information betweentwo adjacent subsystems (i.e. systems having common interface) for the field theoriesdual to AdS black hole spacetimes. The information exchange is driven by entanglementbetween states of the subsystems. As the quantum systems continuously exchange infor-mation almost infinite amount of information maybe exchanged over long time intervals.This leads to an overall information growth in time. The information exchange growswith time as ∝ ( δ d − − t d − ) for CF T d ground state. We show in this work that the sameis true even for excited state of CFT, but the information exchange growth is definitelylowered. Especially for small time intervals the loss in the information exchange goesquadratically with time. In later part of the work we also study related phenomenon ofcircuit complexity of quantum systems. The quantum complexity is understood to be ameasure of difficulty level in obtaining a target state | ψ f > from given initial state | ψ i > ,by using minimum number of possible gates (unitary operations). Here we only explorethe complexity as the volume (CV) conjecture [7]. There is an equivalent (CA) conjec-ture where complexity is equated with the SUGRA action evaluated on the Wheeler-DeWit patch [8]. The related first law like relation for complexity was proposed in [9] re-cently. We wish to show that CV complexity can put a bound on information exchangein quantum systems and two phenomena may be related very intrisically.We also aim to understand the question about reduction in overall information ex- See a detailed review on information paradox along different paradigms in [6] and list of referencestherein.
We consider planar
AdS d +1 geometry having a Schwarzschild black hole in the IR, de-scribed by the metric ds = L z − f dt + dx + · · · + dx d − + dz f ! (1)with the function f ( z ) f = 1 − z d z d , (2)where z = z is location of the horizon and the boundary is at z = 0. L is the radius ofcurvature of the AdS space, which is taken sufficiently large in string length units (i.e. L ≫ l s ) so as to suppress the stringy effects. The boundary field theory for AdS d +1 spacetime describes d -dimensional CFT d at finite temperature dπz [1]. We are interestedin measuring the entanglement information exchange in the CFT staying at the locationof interface between two subsystems. The interface divided the whole system into twosubsystems and dynamically is a ( d −
2) dimensional spatial plane, say between system A and the compliment B , as shown in figure (1). We intend to calculate the area ofa codimension-2 extremal hyper-surfaces embedded in bulk geometry (1) and anchoredprecisely at the interface, at x = x I , between two large semi-infinite subsystems. Sincethere is homogeneity and translational invariance, these results will apply to any constant3igure 1: The interface of two subsystems located at x = x I . shift in location of the interface ( x I → x I + a ). (The subsystems in question can begeneralized to a strip.)We now find codimension-2 surface ending at the interface x = x I , and having time-dependent embedding as shown in figure (2). As per holographic prescription [14, 15]generic codimension-2 extremal surfaces can be described by an action functional [16] I ≡ Z dzz d − s f + ( ∂ z x ) − f ( ∂ z x ) (3)We shall be working only for d >
2. From eq.(3) it follows that the extremal surfacessatisfy following equations of motion dx dz ≡ P ff / q E f − P + z d − dx dz ≡ − Ef / q E f − P + z d − (4)The parameters ( E, P ) are two integration constants. Since we are interested in knowing4igure 2:
The RT surface between time t i and t f . Since | ∂ z x | ≤ along the extremalcurve, we get a cusp-point at z = z i . the information exchange across the interface, which is a time dependent phenomenon,we will set x = constant = x I (along with P = 0). In this way we are selecting purelytime-dependent ( d −
2) dimensional extremal surfaces anchored at the interface x = x I at the boundary time x (0) = t f . The t f refers to some late time event at the boundary.(In case of the strip we could take x = l/ dx dz ≡ − Ef / q E f + z d − (5)Upon integration (and setting E ≡ /z d − i ) one obtains τ ≡ t f − t i ≡ Z z i dz ( z/z i ) d − f q f + ( z/z i ) d − (6)5here t i is an initial time event and given by the condition t i = x ( z ) | z = z i . Throughabove eq.(6) the constant z i gets related to time interval τ between two successive eventsoccurring at the interface. To emphasize, since | x ( z ) ′ | ≤ z = z i is at best a cusp -pointwhere the extremal surface starts and it ends at z = 0, see the figure (2) (It is quite unlikein the HEE nomenclature involving static RT surfaces, there z ∗ rather turns out to bethe turning point of the RT surface.). Since there is a time translation invariance in theCFT, one may also set t i = 0 and t f = τ . Thus every where by saying time we shall onlymean the time-interval.Hence the net (entanglement) information exchanged during time interval τ , across x = x I interface, can be obtained by evaluating the area of extremal surface in (6). From(3) I E = A d − L d − G d +1 Z z i δ dzz d − q f + ( zz i ) d − (7)We can note that the information exchanged is dimensionless quantity. The δ acts ascut-off to regularize UV divergence in the boundary CFT. At z = z i the time coordinateon extremal surface corresponds to the boundary event ( x ( z i ) = t i , x = x I ). Note I E isproportional to the cross-sectional area of the interface between the subsystems given by A d − = l l · · · l d − and it is large. As the integrand becomes singular as we go near to AdSboundary, we would regularize it by the contributions of x = constant , x = constant surface and single out the divergent piece. Thus we obtain I E | x = x I = A d − L d − G d +1 Z z i δ dzz d − ( 1 q f + ( zz i ) d − − − d − z d − i + I UV = A d − L d − G d +1 z d − i Z dξξ d − ( 1 q f + ξ d − − − d − + I UV (8)where ξ ≡ zz i , whereas the singular UV contribution is I UV = A d − L d − G d +1 d − δ d − whichis positive definite and diverges. It involves only the area of the interface/boundary.(The UV contribution is thus similar to the entanglement entropy in this aspect, but itis independent of local time information and forms the universal part.) The first termcontains all time information and it is finite. Let us pick small time interval (say the time period of a stop watch). We wish to measurenet information exchanged between two systems staying at the x = x I interface, andholding a stop watch. For small enough time the extremal surfaces (6) would lie in thevicinity of the asymptotic AdS region only, we can easily expect z i ≪ z . In these caseswe can evaluate the extremal area (8) perturbatively by expanding it around pure AdSvalue (i.e. treating pure AdS as a ground state of the CFT). For small τ values, we firstevaluate eq.(6) by making perturbative expansion of the integrand τ = z i Z dξξ d − √ R z di z d (1 + 12 R ) ξ d − + O (( z i z ) d ) ! z i b + ǫ b + i · · · ! (9)where ξ ≡ zz i and R ( ξ ) ≡ ξ d − . The dots indicate terms of higher order in expansionparameter defined as ǫ = ( z i z ) d ≪
1. The coefficients b , b , and i are precise integralquantities that are positive definite, but mostly smaller than unity. These numbers areprovided in the appendix. The series (9) can also be inverted to obtain the z i -expansion z i = ¯ z i (1 − ¯ z di z d b + i b + · · · ) (10)where we define ¯ z i = τb as the value specific to pure AdS for given interval τ . Theequation (10) summarizes the net effect of the metric deformations (with black hole) onthe z i -value perturbatively. One can easily see that new z i (with excitations) is smallerthan ¯ z i for pure AdS. Having obtained z i -expansion, an expansion can now be obtainedfor the information area functional. From the area integral (8), we get the finite part as I E = A d − L d − G d +1 z d − i Z dξξ d − (( 1 √ R −
1) + 12 z di z d ξ d − R + · · · ) − d − ! (11)The respective finite integrals can be separately evaluated at each order to give I E = A d − L d − G d +1 z d − i ( − c + z dc z d c + · · · ) (12)where coefficients c , c , ... are precise real values (see the appendix). We emphasize herethat c a ’s are all finite and positive for d >
2. Substituting z i expression from (10) andkeeping terms up to first order we determine I E = I UV − A d − L d − G d +1 c ¯ z d − i z di z d d −
22 2 b + i b − ¯ z di z d c c ! + · · ·≡ I AdS + I (1) (13)while I AdS = A d − L d − G d +1 d − δ d − − c b d − τ d − ! . (14)is the information exchanged in time τ for pure AdS d +1 . It is positive definite and isthe leading most term in the expansion. In large time limit τ → ∞ the I AdS divergesas expected. That is to say an infinite information is exchanged by the subsystems inthe CFT ground state. In other words the ground state constitutes maximum to theinformation mobility.
The subsequent term in equation (13) I (1) = A d − L d − G d +1 c ¯ z i z d − ( d −
2) 2 b + i b + c c ! (15)consists first order contribution of excitations to the information flow. The informationflow remains maximum for pure AdS, as we show next, the first order contribution (due7o excitations) is negative and tends to reduce the net information flow. The net changein the information exchange due to excitations is △ I E = − A d − L d − G d +1 τ Db z d (16)where D = (cid:16) − c + ( d − c b + i b (cid:17) >
0. Note coefficient D is positive definite for all d >
2. Thus the expression on the right hand side of eq.(16) is negative definite, suggestingthat the net information exchange for the excited CFT has decreased as compared to thezero temperature CFT. Also it can be noted that △ I E is quadratic in time at first order,suggesting that having black hole excitation in the bulk (or excitations in the CFT)decreases net information flow across the subsystem’s interface.From eq.(16) we may determine the rate of reduction in information flow relative tothe vacuum, as △ I E τ = − L d − A d − G d +1 τ Db z d < • The relative loss in information exchange due to excitations grows quadraticallywith time. • The information flow is proportional to cross-section area A d − of the boundary/interfacebetween the systems. • It is negative definite for all dimensions indicating that reduction in informationflow is an universal feature for excited CFT states.We will show in the next section that the higher order corrections, as being perturba-tively suppressed, cannot change this first order leading behaviour of an excited state.Nevertheless, the absolute sign of second order term will still be important to know here.
The physical quantities such as energy or pressure can be obtained by expanding thebulk AdS geometry (1) in Fefferman-Graham asymptotic coordinates suitable near theboundary [17], also in [18]. The energy density of the CFT is given by △E = L d − πG d +1 d − z d (18)The pressure along all directions is △P = L d − πG d +1 z d (19)8he eq. (17) in terms of physical observables may also be written as − b △ I E πτ = τ A d − △ P x = τ △ F x (20)The negative sign indicates that there is an overall reduction in information flow or mo-bility and it is directly proportional to the force (pressure) generated by the excitations .However no steady state is reached over small times, as the rate of information loss △ ˙ I E only increases with time initially. However a steady state seems to have been reachedafter long times, where our perturbative approach would rather fail. The numerical plotsin figures (3), (5), (7) suggest that late time behaviour generically Lim τ →∞ △ I E → d . We may recall that the entanglement entropy usually rises whenever CFThas excitations [18]. This growth in HEE may appear to be related to loss in informationexchange between the subsystems. Nevertheless it is clear that CFT pressure plays vitalrole in (entanglement) information exchange (flow). In order to make this more robust we need to calculate higher order terms. Taking stepsas in the previous section, we calculate the second order terms in the expansion of theinformation integral, which we schematically denote as I ≡ I AdS + I (1) + I (2) + · · · (22)where I AdS and first order term I (1) have already been obtained. We focus on finding I (2) and its absolute sign in the next. In the first step, we obtain expansion for z i in terms of τ , as done in (9) and (10), up to second order z i = ¯ z i [1 − (2 b + i )2 b ¯ ǫ + 8 b ( d + 1) + 8 b ( d + 1) i + 2( d + 1) i b ¯ ǫ − b (8 b + 4 i + 3 i )8 b ¯ ǫ + O (¯ ǫ )] (23)where ¯ ǫ ≡ ¯ z di z d ≪ The rate of flow of charge carriers in presence of emf E behaves as j = σE in the steady state, whereconstant σ is the conductivity. (2) = L d − A d − G d +1 z d − i [ 3 c d − c (2 b + i )4 b − c d (2 b + i )4 b − c b ( d − − b + 8 b b − b d − b i − b di − i − di + 4 b i + 3 b i )]¯ ǫ (24)The parameters b a , i ab and c a have definite numerical values provided in the appendix. Itis not important to know them all. A significant result follows from here is that, the I (2) is positive definite for all d >
2. This has been thoroughly checked by us. The absolutesign of second order term is important as this will provide us with a bound. It leads to animmediate conclusion that the loss in information exchange will have a bound, namely b | △ I E | πτ ≤ △F x (25)where the net force F x is given by cross-section area of the interface times the entangle-ment pressure P x along the x direction. The force is in the transverse direction of theinterface between subsystems.To proceed further we now study individual case of CF T d . Let us discuss phenomenafor d = 3 , , d = 4 ( AdS )The relative information flow per unit area per unit time up to second order is obtainedas − b △ I E A τ = 4 πα τ (cid:18) − . α ( τb z ) + 2 . α ( τb z ) + O (¯ ǫ ) (cid:19) △ P x (26)where numerical values of some coefficients for the purpose of estimate are as α ≃ . , b ≃ . x component of CF T pressure exerted on the interface locatedat x = x I , is △ P x = L πG z while area of boundary is A . It is remarkable that theequation (26) is in the factorized form and may be written as − b △ I E A τ = 4 πα τ Q ( t, z ) △ P x (27)where the factor Q is Q = (cid:18) − . .
558 ( τb z ) + O (¯ ǫ ) (cid:19) (28)and it is always smaller than unity. It may be an indication that effectively the timeperiod (or time per collision) gets shortened after inclusion of higher order terms in theperturbative expansion. An explicit negative sign on the lhs indicates that there is overallreduction in the information flow for the excited states. As the case here the CFT exci-tations are thermal, but actually that need not have been the case. The results would beidentical for all finite energy (IR) gravity perturbations in the bulk. − b △ I E A τ = 4 πα Q ( τ ) τ △ P x ≃ πα d − Q ( τ ) τ △ E≤ πα d − < Q <
1, and for a quantum state with energy △E and a half-life τ it is understood that τ △ E ≃ due to the uncertainty.Case-2: for d = 3 This case involves a 3-dimensional boundary theory which is dualof AdS . The spacetime AdS arises as near horizon geometry of multiple M2-branes in11-dimensional supergravity. The subsystem interface here is 1-dimensional. It is foundthat − b △ I E A τ = 4 πα τ (cid:18) − . α ( τb z ) + 1 . α ( τb z ) + O (¯ ǫ ) (cid:19) △ P x (30)where we have used numerical values, α ≈ . , b ≃ . CF T pressure △ P x = L πG z exerted on the subsystem boundary located at x = x I , wheras A is length of the interface.Case-3: for d = 6 The information flow per unit area per unit time up to second orderfor AdS bulk spacetime is given by − b △ I E A τ = 4 πα τ (cid:18) − . α ( τb z ) + 6 . α ( τb z ) + O (¯ ǫ ) (cid:19) △ P x (31)where we have α ≃ . , b ≃ . CF T pressure △ P x = L πG z exerted on the system boundary located at x = x I , while 4-dimensional subsysteminterface area is A . AdS
AdS case is special where boundary CFT is 2-dimensional and bulk geometry is theBTZ like black hole. The subsystem interface is point-like or a dot. Here we can get exactresult for the cusp value z i z i = z (1 − Sech ( τz )) (32)and finite part of information flow becomes △ I E = L G ln( s − Sech ( τz )) − ln(1 + s Sech ( τz )) − ln( τz ) ! (33)For pure AdS (taken as CFT ground state) the information flow is given by: I = L G ln( 2 τδ )11ollowing from (33) we get a variational principle for small changes δI E = Lτ G Coth τz Sech τz (2 + q Sech τz )2(1 + Sech τz + q Sech τz ) − z τ δ z (34)It is an exact expression. In the perturbative regime of small time interval ( τ ≪ z ) weobtain δI E ≃ − Lα τ G δ z (35)where α is a known constant and terms of order O ( τ z ) are dropped. Most importantpoint to note here is that the loss in information exchange remains quadratic for smalltime interval at the leading order. When large enough time ( τ → ∞ ) has elapsed weinstead find δI E ≃ − L G z δ z ! = − πT th δ M (36)where M is the energy density of the black hole and T th ∼ πz is horizon temperature. We can see all local time dependence disappears or wiped out at large time interval . Thelarge time relation (36) may be recasted purely in terms of thermal entropy density as δI E + 8 πT th δs th = 0 (37)It is an exact relation for the loss in information processed by the quantum dot (theinterface) and corresponding rise in the entropy for excited state of the system. Note herethe CFT system is 1-dimensional and the system interface is point-like. It suggests thatthe rise in entropy is always detrimental to the information flow, read conductivity, andvice versa. In the previous section we could make perturbative study of the information flow for finitetimes only. However the results at large times were elusive because the perturbation breaksdown. To understand the late time behaviour of information exchange we pick three casesof
AdS , AdS and AdS .For d = 3 the two plots are obtained in figures (3) and (4).In the fig.(3) the net information exchange for excited CF T rises sharply in tandemwith ground state growth and actually grows very close up to the ground state plot. Butat late time the two growths differ considerably. The difference of growth △ I is plottedin the figure (4). It depicts a net reduction in the entanglement growth for the excitedstate. The information loss initially grows quadratically in time and then slows down atlate times. The similar results for CF T are given in fig(5) and (6).12 t ⅈ Information Vs Time
Figure 3:
Net information exchange per unit area of the interface as a function of timefor d = 3 case. We have taken I UV = 25 , z = 5 , L = 1 , G N = 1 / . The upper curverepresents I AdS . It shows maximum information is exchanged in the ground state. t - - - - - - - dI Vs t Figure 4:
The loss in information flow with excitations for d = 3 . We have taken I UV =25 , z = 5 . These graphs show universal feature that information loss (with excitations) growsquadratically near t = 0, but it appears to get saturated at late times and curve getsflattened. These results for excited states of CF T can be found in figs.(7) and (8).There too the information loss curve behaves quadratically in time initially but unlikeprevious two cases it rebounds and information loss starts reducing at late times. Thisis a direct effect of dimensionality of the theory. The early time behaviour is consistentwith perturbative analysis where first order term is indeed quadratic in time. We now evaluate another time dependent quantity when the CFT is in excited state. Thecomplexity of quantum circuits is a measure of time evolution of the state of the systemas a whole. It can be defined holographically by the volume action functional describing13 .5 1.0 1.5 2.0 2.5 t ⅈ Information Vs Time
Figure 5:
Net information exchange per unit area of the interface as a function of timefor d = 4 case. We have taken I UV = 25 , z = 5 , L = 1 , G N = 1 / . The upper curverepresents I AdS . It shows maximum information is exchanged in the ground state only. t - - - - - - - dI Vs t Figure 6:
The loss in information flow with excitations for d = 4 . We have set I UV =25 , z = 5 . a time-dependent (codim-1) extremal surface [7] C E ≡ V d − L d G d +1 Z dzz d q f − − f ( ∂ z x ) (38)where V d − ≡ l l l · · · l d − is the net spatial volume of the CFT. In this sense quantumcomplexity is a bulk property of the CFT and extensive in nature. Note the difference thatthe volume complexity is dimensionful whereas information exchange I E is dimensionless.From (38) it follows that extremal surfaces have to satisfy following equation of motion dx dz ≡ z d /z dc f q f + z d /z dc (39)The z c is the integration constant and it will get related to the time interval on theboundary. To first solve x equation, let us choose an extremal surface having boundary14 .5 1.0 1.5 2.0 t ⅈ Information Vs Time
Figure 7:
Net information exchange per unit area of the interface as a function of timefor d = 5 case. We have taken I UV = 25 , z = 5 , L = 1 , G N = 1 / . The upper curverepresents I AdS . It shows maximum information is exchanged in the ground state. t - - - - dI Vs t Figure 8:
The loss in information flow with excitations for d = 5 . We have taken I UV =25 , z = 5 . value x (0) = t f , where t f is a time event on the boundary. The eq.(39) gives uponintegrating τ = t f − t i ≡ Z z c dz ( z/z c ) d f q f + ( z/z c ) d (40)where t i is some initial time given by t i = x | z = z c . Thus the constant z c gets relatedto time interval between two events on the boundary. There is overall time translationinvariance in the system. It is also clear that for small intervals we will have a situationwhere z c ≪ z .The expression for complexity during time interval τ , can be obtained by calculating15he area of the extremal surface from action (38) C E = V d − L d G d +1 Z z c δ dzz d q f + ( zz c ) d (41)The integrand in (41) becomes singular near the AdS boundary which is the UV divergenceof the CFT. We can regularize it by the contribution of x = constant surface and singleout the diverging UV part. Thus we obtain C E = V d − L d G d +1 z d − c Z dξξ d ( 1 q f + ξ d − − d − + C UV (42)The divergent part is given by C UV = V d − L d G d +1 d − δ d − , it is positive definite and propor-tional to spatial volume of the CFT. We will now evaluate C E perturbatively for small τ .First, by evaluating rhs of eq.(40) up to first order, we find z c = ¯ z c (1 − f + j f ¯ ǫ + O ( ǫ )) (43)where ¯ z c = τf and expansion parameter ¯ ǫ = ¯ z dc z d ≪
1. The coefficients f , f and j areall finite and given in the appendix. It is not immediately important to know them here.Using z c equation (43) and substituting in (42), we find the net change in complexity( C E − C AdS ) to be given as △ C E = V d − L d G d +1 d ¯ z c z d − ( d −
1) 2 f + j f + d d ! + higher orders = − V d − L d G d +1 τf D c z d + higher orders (44)Thus up to first order the change in complexity per unit volume per unit time is f △ C E V d − τ ≃ − πD c △ E (45)where D c ≡ ( d − d f + j f − d D c > d values. For example, for d = 4 one gets D c ≈ .
432 and f ≃ . .1 A bound on loss of information exchange The bound in the last section on relative information flow can also be stated differentlyin terms of complexity as (in d > b △ ˙ I E A d − ≤ α d LD c f △ C E V d − (46)where A d − is the area of the system’s interface and V d − is total volume of the system. △ C E measures the net loss in quantum complexity. The constants α d and D c depend onspacetime dimensions. (Especially for d = 4, we have α ≃ .
558 and D c ≃ . We explored net quantum information exchange or sharing in real time at the interfacebetween subsystems for
CF T d living at the boundary of AdS d +1 spacetimes. The in-formation exchange is a continuous phenomenon and only after long time intervals fullinformation exchange gets saturated. Generally we prepare systems for small time in-tervals only. We find perturbatively that the rate of information exchange between twosystems is always reduced when the systems have excitations for all d , while CFT groundstate has a maximum information exchange rate. The net information loss is directlyproportional to the force or pressure generated by the CFT excitations. We get a bound b | △ I E | πτ ≤ △F x where the force F x is in transverse direction to the interface. The inequality is saturatedonly at the first order of perturbation. Especially for 2-dimensional CFT, the large timerelation can be recasted in terms of thermal entropy density as δI E + 8 πT th δs th = 0It is an exact relation for the loss in information processed by the quantum dot (theinterface) and corresponding rise in thermal entropy for the excited state of the system.Note the CF T system represents 1-dimensional wire and the subsystem interface is onlypoint-like. It proves that the rise in entropy is always detrimental to the information flow(or conductivity) and vice versa.We further observed that there is an overall reduction in quantum complexity too inthe presence of excitations. While it is well understood that excited states add more17o the entanglement entropy due to rise in disorder. From this perspective we are ledto argue that the reduction in the complexity may be due to relative higher symmetryexcited states may have as compared to CFT ground state. The ground state is alwaysmore ordered but it maybe less symmetric as we observe in ferromagnetic systems withan spontaneously broken symmetry.We have got a new bound which states that rate of loss in the information flow dueto excitations will remain bounded by the corresponding loss in quantum complexity fora given state of the system. A Integral Quantities
Some useful integral values we have used in the calculation of information flow are beingnoted down here (where R ( ξ ) = 1 + ξ d − ) b = Z dξξ d − √ R ≃ .
284 (for d = 3) b = Z dξξ d − √ R ≃ .
133 (for d = 3) b = Z dξξ d − √ R ≃ .
086 (for d = 3) i = Z dξξ d − R √ R ≃ .
087 (for d = 3) i = Z dξξ d − R √ R ≃ .
052 (for d = 3) i = Z dξξ d − R √ R ≃ .
033 (for d = 3) c = Z dξξ − d (1 − √ R ) + 1 d − ≃ .
131 (for d = 3) c = Z dξ ξR √ R ≃ .
353 (for d = 3) c = Z dξ ξ d +1 R √ R ≃ .
077 (for d = 3) , (47)We have provided numeric values for d = 3 case only. For other d cases one can obtainthese values easily.Another set of integral coefficients which appear in complexity are (with R ( ξ ) = 1+ ξ d ) f = Z dξξ d √ R ≃ .
214 (for d = 3) f = Z dξξ d √ R ≃ .
117 (for d = 3) j = Z dξξ d R √ R ≃ .
080 (for d = 3) d = Z dξξ − d (1 − √ R ) + 1 d − ≃ .
599 (for d = 3)18 = Z dξ R √ R ≃ .
867 (for d = 3) (48)while f = Z dξξ d √ R ≃ .
173 (for d = 4) f = Z dξξ d √ R ≃ .
091 (for d = 4) j = Z dξξ d R √ R ≃ .
063 (for d = 4) d = Z dξξ − d (1 − √ R ) + 1 d − ≃ .
413 (for d = 4) d = Z dξ R √ R ≃ .
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