Complexity growth rates for AdS black holes with dyonic/ nonlinear charge/ stringy hair/ topological defects
CComplexity growth rates for AdS black holes with dyonic/ nonlinear charge/ stringyhair/ topological defects
Ali ¨Ovg¨un
1, 2, 3, 4, ∗ and Kimet Jusufi
5, 6, † Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4950, Valpara´ıso, Chile Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University,Famagusta, North Cyprus via Mersin 10, Turkey Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, Macedonia Institute of Physics, Faculty of Natural Sciences and Mathematics,Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia (Dated: January 30, 2018)In a seminal paper by Brown et al. [Phys. Rev. Lett. , no. 19, 191301 (2016)] a new conjecturewas proposed, namely it was argued that the quantum complexity of a holographic state is equalto action of a Wheeler-DeWitt patch in the late time limit suggesting that the fastest computer innature are the black holes. Motivated by this conjecture, in the present paper, we study the actiongrowth rate for different types of black holes such as dyonic, nonlinear charge, stringy hair, blackhole with a global monopole and a cosmic string. In general we find that action growth rates of theWheeler-DeWitt patch is finite for these black holes at the late time approach and satisfy the Lloydbound on the rate of quantum computation. Furthermore, in the case of a charged as well as theneutral black hole with a global monopole and a conical defect we show that the form of the Lloydbound relation remains unaltered but the energy is modified due to the nontrivial global topologyof the spacetime.
PACS numbers: 04.40.-b, 95.30.Sf, 98.62.SbKeywords: Black holes, AdS-CFT correspondence, Conformal field theory, Quantum complexity, Topologicaldefects
I. INTRODUCTION
Nature of humans is to always want to peel back thelayers of the unknown. We look up into the sky and tryto understand many unknowns from the beginning of theuniverse to smallest particles. Are there any limitationof our knowledge? We need more powerful helping toolsto gain more information about the unexplored nature.Today, one of the biggest challenges that humans haveis to build a quantum computer that is the topics of the-oretical computer science with the helps of mathematics,namely computational complexity theory which nowa-days motivates theoretical physicist [1–8, 10]. Quantumcomplexity theory is used to shed some lights on the prob-lems that quantum computers can solve.Since Maldacena enunciated the AdS/CFT correspon-dence which is also known as a gauge/gravity duality [11],holographic duality gains more interest [12–28]. Blackholes are formed after the gravitational collapse in Anti-de Sitter (AdS) space-time which is dual to thermallyconformal field theory. Maldecena and Susskind havemade first relation between the Einstein-Podolsky-Rosen(EPR) correlation with the Einstein-Rosen bridge (alsoknown as wormholes) [29]. In the other words, they havetried to connect quantum mechanics with gravity which ∗ † kimet.jusufi@unite.edu.mk is called ER=EPR relation. The importance of this re-lation is to allow the communications between Alice andBob from opposite sides of the wormhole [32, 33].Recently Brown et al. have argued that the number ofquantum gates, which is proportional to quantum com-plexity, is dual to the size of the Einstein-Rosen bridgefor AdS black holes in the dual boundary CFT [5]. Inparticular this suggest that quantum complexity help usto understand better the black hole physics especiallyholographic duality and information paradox [34]. Onthe other hand, it supports to build quantum computers.This conjecture is known as “complexity=action” (CA),namely the complexity of holograpic state is equal to ac-tion of a Wheeler-DeWitt patch (on-shell action evalu-ated on a bulk region) in the late time limit. When find-ing the action growth rate, the main difficulties is to findthe contribution of boundary terms [6, 8].Very recently, many papers in the literature have beendevoted to this conjecture to check the action growth rateand the relation between Lloyd bound [77] in differenttypes of black holes whether it is valid or not [10, 26, 35–76]. Furthermore, Lehner et al. provide new method tofind the growth rate of the action using null boundaries[35]. In this paper we shall follow the original methodproposed by Brown et al. in Ref.[6].The relation of complexity C with the spatial volume V which is for the Einstein-Rosen bridge is given firstlyby Stanford and Susskind as follows [9]: a r X i v : . [ g r- q c ] J a n FIG. 1: For AdS black hole when the time increases, the WDWpatch gains a red slice and it loses a green slice for a pair oftimes ( t L , t R ). The figure is taken in Ref. [6]. C ∼
VGl . (1)Note that the l is the anti de-Sitter (AdS) radius andthe G is Newton’s constant. Susskind et al. led the waythrough several works in which the duality of CA hasbeen shown to be related with the action of the black holein the Wheeler-DeWitt patch in a following way [5, 6, 8]: C = Aπ (cid:126) . (2)An important effect of CA duality is to show thebounded quantum complexity growth rate which isknown as Lloyd bound [77]: d C dt ≤ Eπ (cid:126) , (3)with the average energy density E .Moreover, the surface of the wormhole where there islinearly growing patch provides the relation d C /dt ≈ T S for t >> /T , in which T is the temperature and S standsfor the entropy of the black hole. This formula also givesus a hint of a growth rate of qubits so that the contactbetween the quantum complexity and quantum informa-tion theory can be shown [30, 31].There is now a growing consensus that CA dualityis in the same chain of quantum complexity and ac-tion growth. In a short period of time, the seminal pa-per of CA duality inspired many other authors resultingwith many research papers dedicated to this problematic[41, 46, 48, 50, 56, 61, 62, 65, 67–69, 72, 73, 76]. Someexact results of the action growth rate are given as fol-lows:neutral BH : d A d t = 2 M ; (4)rotating BH : d A d t = [( M − Ω J ) + − ( M − Ω J ) − ] ; (5)charged BH : d A d t = [( M − µQ ) + − ( M − µQ ) − ] . (6) Here ± stands for the outer and inner horizons of theblack hole.In this paper, our goal is to check the validity of thecomplexity by calculating the action growth rate for thedyonic AdS black hole, AdS black hole with nonlinearsource, AdS black hole with stringly hair, AdS black holewith global monopoles and cosmic strings. To do so, wecalculate first the boundary term of the action and thenthe bulk action. Afterwards, we obtain the result of totalaction growth which is related to expected results and wecarry out the main calculations of this paper to obtainthe Lloyd bound. Then we compare these results with aoriginal paper of Brown et al. [6]. It will be interestingto see the differences between the calculations of com-plexity of dyonic AdS black hole, AdS black hole withnonlinear source, AdS black hole with stringly hair, AdSblack hole with global monopoles and cosmic strings fromthe original RN-AdS black hole case. The CA conjecturesgive us good results to understand the quantum compu-tation. Note that in late time approximation, complex-ity grows linearly in time. Moreover, main contributionscome from the patch from behind the horizon, known asWdW patch shown in Fig. 1. It is supposed that, thislinear action growth generates the interaction betweenquantum states. Hence, one can consider the derivationof the time only depends on the mass/energy of the blackhole/quantum states which saturates the Lloyd bound onthe growth of the complexity [5, 6, 8, 61].In the present article we wish to compute the actiongrowth of the dyonic/nonlinear charge, stringly hair andglobal monopoles black holes in four-dimensions. Ourwork is organized as follows: in section II the actiongrowth rate of the black holes with dyonic charge arediscussed. In section III, we compute the complexityand action growth rate of the black holes with nonlin-ear charge. Then we repeat calculations for the blackhole stringy hair in section IV. Then in section V, wecompute the action growth rate of the charged RNAdSblack hole with a global monopole. Finally in SectionVI, we discuss RNAdS black hole with a conical defect.Finally, we conclude our work in Section VII. II. ACTION GROWTH RATE OF THE DYONICADS BLACK HOLES
Here, we consider the following action for the dyoniccharged AdS black hole: [78, 79] A = 116 π ˆ d d x √− g [ R − − F µν F µν ] , (7)where R is the Ricci scalar and Λ stands for the cosmolog-ical constant. It is noted that F µν is the electromagneticfield tensor. The electromagnetic tensor is modified togive a dyonic property, where the magnetic charge ap-2ears. The electromagnetic 4-potential is A = (cid:18) − q E r − + q E r + (cid:19) dt + ( q M cosθ ) dφ (8)in which q E and q M are respectively, electric and mag-netic charges. The metric function yields f ( r ) = 1 − Λ r − mr + q E + q M r , (9)The spacetime is given by ds = − f ( r ) dt + dr f ( r ) + r d Ω , (10)in which d Ω is the line element of a (2)-dimensionalhypersurface.To calculate the action growth rate [6], we use the ac-tion of bulk and the boundary terms as follows: A = A bk + A bd = 116 π ˆ d x √− g [ R − − F µν F µν ]+ 18 π ˆ ∂M d x √− hK, (11)where h is the induced metric for hypersurface and K isthe trace of the extrinsic curvature. For the dyonic AdSblack hole, the action growth of the bulk is calculated asfollows: d A bk d t = Ω π ˆ r + r − r (cid:20) − l − F (cid:21) dr = − Ω πl ( r − r − ) − Q r − − r − − ) , (12)where Q = q E + q M . The extrinsic curvature with themetric is K = 1 r ∂∂r (cid:16) r (cid:112) f ( r ) (cid:17) = 2 r (cid:112) f + f (cid:48) √ f . (13)Second we find the contribution from the YGH surfaceterm within WDW patch at late time approximation asfollows: d A bd d t = 18 π ˆ r + r − d x √− h K. (14)Resulting withd A bd d t = Ω π (cid:20) r (cid:112) f (cid:18) r (cid:112) f + f (cid:48) ( r )2 √ f (cid:19)(cid:21) r + r − (15)= 3Ω πl ( r − r − ) + Q r − − r − − ) + 2Ω π ( r + − r − )Hence we obtain the total growth rate of action fordyonic AdS black hole isd A d t = 2Ω π (cid:18) r + − r − + r − r − l (cid:19) . (16) We rewrite above result in more compact wayd A d t = Q (cid:18) r − − r + (cid:19) . (17)by using the mass M : M = 2Ω π (cid:18) r + + r − + 1 l r − r − r + − r − (cid:19) , (18)and charge Q : Q = 2Ω π r + r − (cid:18) l r − r − r + − r − (cid:19) . (19)The growth rate of action within WDW patch for dyonic-AdS black hole at late time approximation is obtainedsimilarly the seminal paper of Brown et al. [5, 6]d A d t = ( M − µ + Q ) − ( M − µ − Q ) , (20)in which, µ − = Q/r − and µ + = Q/r + stand for thechemical potentials at inner/ outer horizons. Moreoverit satisfies the Lloyd bound [77].In particular the quantum complexity growth rate isbounded by d C d t ≤ Eπ (cid:126) , (21)where E is the average energy of the quantum state re-lating to the ground state. Hence, the dyonic charge hasregular impact on the complexity similarly charged AdSblack hole. III. ACTION GROWTH RATE OF THE BLACKHOLES WITH A NONLINEAR SOURCE
In this section, we use the black hole with a nonlinearsource to calculate the complexity. The total action withbulk term using the Einstein-power Maxwell invariant(PMI) gravity and boundary term is given by [80] A = A bk + A bd = 116 π ˆ M d x √− g (cid:18) R + 6 l + L P MI (cid:19) + 18 πG ˆ ∂M d x √− hK, (22)where L P MI = ( −F ) s and F = F µν F µν . It is notedthat, we use the special case of s = 3 / ds = − f ( r ) dt + dr f ( r ) + r d Ω , (23)3here d Ω is for the standard element on S , and themetric function is f ( r ) = 1 + r l − mr + 2 / q ln( r ) r , (24)Now we calculate straightforwardly the action growth ofthe bulk:d A bk d t = Ω π ˆ r + r − r (cid:20) − r + q l l r (cid:21) dr = Ω r − π l − Ω r +3 π l − q r +3 π + q π r − . (25) Then we obtain the extrinsic curvature: K = 1 r ∂∂r (cid:16) r (cid:112) f ( r ) (cid:17) = 2 r (cid:112) f + f (cid:48) √ f , (26)to calculate the YGH surface term within WDW patchat late time approximation as follows:d A bd d t = Ω π (cid:20) r (cid:112) f (cid:18) r (cid:112) f + f (cid:48) ( r )2 √ f (cid:19)(cid:21) r + r − = Ω (cid:0)(cid:0) r − l + 3 r − (cid:1) ln ( r − ) + (cid:0) − r − l − r − (cid:1) ln ( r + ) − r − l + r + l − r − + r +3 (cid:1) π (ln ( r − ) − ln ( r + )) l (27)Note that we use f ( r ± ) = 0. Using the late time ap-proximation, it can be read off the total growth rate of action for dyonic AdS black hole within WDW patch asfollows:d A d t = (cid:16) − √ (cid:0) r − l − r + l + r − − r +3 (cid:1) l (ln ( r − ) − ln ( r + )) (cid:17) / ( − r + + r − )2 l ( − ln ( r − ) + ln ( r + )) r − r + . (28)To write it in more compact way, we obtain the mass mm = (cid:0) r + l + r +3 (cid:1) ln ( r − ) − r − ln ( r + ) (cid:0) l + r − (cid:1) r − ) − ln ( r + )) l , (29)and total charge qq = (cid:113) − √ r − l − r + l + r − − r +3 ) l (ln ( r − ) − ln ( r + )) l (ln ( r − ) − ln ( r + )) (30)Hence, the total action growth rate for dyonic AdSblack hole becomesd A d t = ( m − µ + q ) − ( m − µ − q ) . (31)Here also it reduces to original charged AdS black holecase and satisfy the Lloyd bound. In particular the quan-tum complexity growth rate is bounded byd C d t ≤ Eπ (cid:126) , (32)where E is the average energy of the quantum state re-lating to the ground state. Therefore, the black hole witha nonlinear source reduces to normal charged AdS blackhole, when nonlinear term is gone. IV. ACTION GROWTH RATE OF THE ADSBLACK HOLES WITH STRINGLY HAIR
We consider an action in which gravity is coupled toelectrodynamic field as [85, 86] A = 116 π ˆ d x √− g [ R −
2Λ + L ( F ) + || J || ] , (33)where the string field is J = H (1 − Ω ) with H = − e σ ∆ σ , Ricci scalar curvature is R and Λ stands forthe cosmological constant. L ( F ) is the Lagrangian of lin-ear electrodynamics field given by L ( F ) = − F where F = F µν F µν , with F µν = ∂ µ A ν − ∂ ν A µ is the electro-magnetic field tensor. F = Qr .The spacetime of the black hole with stringy hair(BHSH) is recently found by Boos and Frolov [87]:d s = − f ( r )d t + f − ( r )d r + d ω , (34) dω = r e σ (cid:0) dθ + sin θ d ϕ (cid:1) , (35)where σ is constant which depends on θ and φ , but in thispaper we chose it as a σ . The interesting feature of thespacetime is that the metric is warped and distorted with4 ω . The radius of the black hole is located at f ( r + ) = 0,where the metric function is f ( r ) = 1 − Mr + Q r − r Λ . (36)It is noted that M is a mass of the black hole and thecharge of the black hole is Q . To study the complexityon the black hole with stringly hair, we write the totalaction with bulk term and boundary term as follows: A = A bk + A bd = 116 π ˆ d x √− g [ R − − F µν F µν ]+ 18 π ˆ ∂M d x √− hK, (37)Then we find the action growth of the bulkd A bk d t = Ω π ˆ r + r − e σ r (cid:20) − l − F (cid:21) dr = e σ (cid:20) − Q r + + Q r − + Ω r − πl − Ω r +3 πl (cid:21) (38)and we calculate the YGH surface term within WDWpatch at late time approximation resulting withd A bd d t = Ω π (cid:20) r e σ (cid:112) f (cid:18) r (cid:112) f + f (cid:48) ( r )2 √ f (cid:19)(cid:21) r + r − = 2Ω e σ πl ( r − r − ) + Q e σ r − − r − − )+ 2Ω e σ π ( r + − r − ) . (39)Finally we can write the total action growth rate as:d A d t = 2Ω e σ π (cid:18) r + − r − + r − r − l (cid:19) . (40)and after considering σ = 0, the growth rate of actionof the black hole with stringly hair reduces tod A d t = ( M − µ + Q ) − ( M − µ − Q ) . (41)In particular the quantum complexity growth rate isbounded by d C d t ≤ Eπ (cid:126) , (42)where E is the average energy of the quantum state re-lating to the ground state. It is noted that the effect ofthe stringly hair can be V. ACTION GROWTH RATE OF CHARGEDADS BLACK HOLE WITH A GLOBALMONOPOLE
A global monopole is an interesting object with a widerange of physical implications in the context of gravitytheory as well as quantum theory. It is speculated thatthese objects can arise during the phase transition of asystem composed by a self-coupling scalar triplet φ a inthe early universe. In particular, the simplest model ofsuch scenario can be studied by the following Lagrangiandensity [88, 89] L GB = − (cid:88) a g µν ∂ µ φ a ∂ ν φ a − λ (cid:0) φ − η (cid:1) , (43)with a = 1 , ,
3, with λ being the self-interaction term, η is known as the scale of a gauge-symmetry breaking.The field of such a system is given by φ a = ηh ( r ) x a r , (44)where x a = { r sin θ cos ϕ, r sin θ sin ϕ, r cos θ } , (45)such that (cid:80) a x a x a = r . Using the field equations andthe relation for φ a one can show that the problem reducesto a single equation for h ( r ) given as [88] f h (cid:48)(cid:48) + (cid:20) fr + 12 f ( f ) (cid:48) (cid:21) h (cid:48) − hr − λη h (cid:0) h − (cid:1) = 0 . (46)Interestingly, outside the core in the large limit ap-proximation one can take h ( r ) →
1, with the energy-momentum tensor given by the following relations T tt = T rr (cid:39) η /r and T θθ = T ϕϕ = 0. The global monopolemetric (also known as Barriola-Vilenkin metric) with cos-mological constant is given as follows [88, 89] ds = − f ( r ) dt + dr f ( r ) + r (cid:0) dθ + sin θdϕ (cid:1) , (47)where f ( r ) = 1 − πη − Mr + Q r + r l . (48)In the last expression M ≈ M core denotes the globalmonopole core mass, with M core ≈ λ − / η , note that fora typical grand unification scale η = 10 GeV. The totalaction if our system reads A = 116 π ˆ d x √− g ( R − − F µν F µν )+ ˆ d x √− g L GB + 18 π ˆ ∂M d x √− hK. (49)5ntroducing the following coordinate transformationinto the metric (47) given as [90] t → (1 − πη ) − / t,r → (1 − πη ) / r,M → (1 − πη ) − / M,Q → (1 − πη ) − Q, we find the following result ds = − f ( r ) dt + dr f ( r ) + (1 − πη ) r (cid:0) dθ + sin θdϕ (cid:1) (50)in which f ( r ) = 1 − Mr + Q r + r l . (51)We shall calculate now the contribution from the bulkaction asd A bk d t = ˆ d x √− g (cid:20) π ( R − − F µν F µν ) + L GB (cid:21) = Ω (1 − πη )16 π ˆ r + r − r Ξ( r, l, Q, η ) d r, (52)withΞ = − π η l + 96 π η r − r (8 π η − r l + 6 l + 2 Q r (53)with the Ricci scalar given by R = − π η l + 96 π η r − r (8 π η − r l . (54)Hence the action growth rate givesd A bk d t = Q (1 − πη ) (cid:18) r − − r + (cid:19) − r − r − l + 4 πη ( r + − r − )( l + r + r + r − + r − ) l (55)On the other hand, first we find that the extrinsic cur-vature remains unchanged due to the presence of a globalmonopole, namely we find K = 2 (cid:112) f ( r ) r + f (cid:48) ( r )2 (cid:112) f ( r ) . (56)Thus, the contribution from the YGH surface termyieldsd A bd d t = (1 − πη )Ω π (cid:104) r (cid:112) f ( r ) K (cid:105) r + r − = 3(1 − πη )2 l ( r − r − ) + (1 − πη )( r + − r − )+ (1 − πη ) Q (cid:18) r + − r − (cid:19) . (57) For the total action growth rate we find, d A dt = d A bk dt + d A bd dt = (1 − πη )( r + − r − ) + [3(1 − πη ) − r − r − )2 l + 4 πη ( r + − r − )( l + r + r + r − + r − ) l . (58)From f ( r + ) = 0, we find M given by M = r + r l + Q l r + l . (59)If we use this equation, from f ( r − ) = 0 on the otherhand we find Q , as follows Q = r + r − l + r + r − + r r − + r r − l . (60)Finally, putting all these results together we obtaind A d t = 4 πη ( r + − r − ) + (1 − πη ) (cid:18) Q r − − Q r + (cid:19) , (61)or d A d t = (cid:2) M + 4 πη r + − (1 − πη ) µ + Q (cid:3) − (cid:2) M + 4 πη r − − (1 − πη ) µ − Q (cid:3) . (62)It is worth noting that in the last equation the chemicalpotentials on the horizons are given by µ ± = Q/r ± . Fur-thermore if we define the ADM charge which correspondsto the charge measured at infinity, given by Q = 14 π ˆ F µν d Σ µν = (1 − πη ) Q, (63)the above simplifies to d A dt = (cid:2) M + 4 πη r + − µ + Q (cid:3) − (cid:2) M + 4 πη r − − µ − Q (cid:3) . (64)In the limit of Schwarzschild-AdS black hole one has r − → r + → M , consequently µ + Q →
0, and µ − Q → M . It follows d A d t → M (cid:0) − πη (cid:1) . (65)Thus, we have shown that the presence of globalmonopole modifies the growth rate action for the neu-tral as well as the charged black hole. In particular thequantum complexity growth rate is bounded byd C d t ≤ Eπ (cid:126) (cid:0) − πη (cid:1) . (66)The last equation shows that the average energy of thequantum state relating to the ground state E , is modi-fied in the presence of a global monopole with the Lloydbound given relation d C d t ≤ E π (cid:126) . (67)where we have introduced the modified average energy ofthe quantum state given by E = (1 − πη ) E .6 I. ACTION GROWTH RATE OF RNADSBLACK HOLE WITH A COSMIC STRING
The spacetime metric of a RNAdS spacetime with acosmic string is given as [91] ds = − f ( r ) dt + dr f ( r ) + r (cid:2) dθ + (1 − µ ) sin θdϕ (cid:3) , (68)where f ( r ) = 1 − Mr + Q r + r l . (69)The total action of the system is given by A = 116 π ˆ d x √− g ( R − − F µν F µν ) + A string + 18 π ˆ ∂M d y √− hK. (70)In which the action associated to a cosmic string with-out internal structure aligned in the z -axes can be givenas follows A string = − ˆ d ζ √− γ T µµ (71)where T µµ = 2 µδ ( x ) δ ( y ), in which µ is the tension of thecosmic string. Furthermore, ζ a are coordinates on thestring world-sheet. First we calculate the contributionfrom the bulk action which givesd A bk d t = Ω (1 − µ )16 π ˆ r + r − r (cid:18) − l + 6 l + 2 Q r (cid:19) d r, (72)where the Ricci scalar is found to be R = − l . (73)Note that there is zero contribution from the cosmicstring action. Hence the action growth rate givesd A bk d t = Q (1 − µ ) (cid:18) r − − r + (cid:19) − (cid:0) r − r − (cid:1) (1 − µ )2 l (74)And the contribution from the YGH surface term isd A bd d t = (1 − µ )Ω π (cid:104) r (cid:112) f ( r ) K (cid:105) r + r − = 3(1 − µ )2 l ( r − r − ) + (1 − µ )( r + − r − )+ (1 − µ ) Q (cid:18) r + − r − (cid:19) . (75)Then we can get the total action growth rate, d A dt = d A bk dt + d A bd dt = (1 − µ )( r + − r − ) + (1 − µ )( r − r − ) l (76) It follows that d A dt = (1 − µ ) (cid:18) Q r − − Q r + (cid:19) , (77)in other words d A dt = [ M − (1 − µ ) µ + Q ] − [ M − (1 − µ ) µ − Q ] . (78)One can introduce the ADM charge in the spacetimebackground of a cosmic string given by Q = (1 − µ ) Q ,in that case the above equation simplifies to d A dt = [ M − µ + Q ] − [ M − µ − Q ] . (79)In the limit of Schwarzschild-AdS black hole we have µ + Q →
0, and µ − Q → M . It followsd A d t → M (1 − µ ) . (80)Thus, we have shown that the presence of a conicaldefect modifies the growth rate action for the neutralblack hole as well as the charged black hole. In particular,the quantum complexity growth rate is bounded byd C d t ≤ Eπ (cid:126) (1 − µ ) , (81)When a topological defect is introduced the globalspacetime topology becomes nontrivial, hence it is con-venient to introduce the ADM mass in the presence ofcosmic string which gives M = (1 − µ ) M , yieldingd A d t → M . (82)In that case, the Lloyd bound can be written asd C d t ≤ E π (cid:126) . (83)where the average energy of the quantum state in pres-ence of conical defects is modified as E = (1 − µ ) E . VII. CONCLUSION
In this paper, using the “complexity=action” (CA)conjecture the complexity growth rate is studied in theAdS black holes with dyonic/ nonlinear charge/ stringyhair/ topological defects.We have investigate the boundary term of the ac-tion and the bulk action to calculate the action growthrate. Afterwards, we obtain the result of the total actiongrowth which is related to results of the seminal paperof the Brown et al. [6] and we carry out the main calcu-lations of this paper to obtain the Lloyd bound. Furtherwe have explored the differences between the calculationsof complexity of dyonic AdS black hole, AdS black holewith nonlinear source, AdS black hole with stringly hair,7dS black hole with global monopoles and cosmic stringsfrom the original RN-AdS black hole case.For the black hole with dyonic charge, we found thatthe action growth rate of the black hole depend on thetotal charge where the dyonic charge is emerged.On the other hand, for the black hole with nonlinearsource as well as the black hole with stringly hair wefind that the action growth rate reduces to the familiarcharged black hole case reported in the literature. Inother words, the Lloyd bound is fulfilled in all three cases.On the other hand, in the case of RNAdS black holewith a global monopole we find that the quantum com-plexity growth rate is bounded byd C d t ≤ Eπ (cid:126) (cid:0) − πη (cid:1) . Thus, due to the presence of a global monopole theLloyd bound is slightly modified. Lastly, we have usedthe black hole with a conical defects (cosmic string) whichgive us a fruitful resultd C d t ≤ Eπ (cid:126) (1 − µ ) . Hence, the form of Lloyd bound relation remain unal-tered but the energy changes. This modification of theenergy, however, is to be expected due to the nontrivialglobal topology of the spacetime when topological de-fects are introduced. For this reason, we have used theADM charge, as well as the ADM mass in the total action growth rate. Another interesting way to find similar re-sult for the complexity growth rate of the WDW patch atlate time point is using the approach proposed by Lehneret al.’s method [35].More importantly, the CA conjectures provide inter-esting results to shed light on the quantum computa-tion. Furthermore complexity grows linearly in time ifone propose the late time approximation where the maincontributions come from the WdW patch. Hence the lin-ear rates of the action growth support the link betweenquantum states and it saturates the Lloyd bound on thegrowth of the complexity.This is another important evidence for the idea thatblack holes are the fastest computers and scramblers innature. It would also be very interesting to investigatethe complexity growth rate, which is link between space-time geometry and quantum entanglements [92], in dif-ferent gravity theories and different geometries to under-stand deeply it’s nature. We will leave it to our futureprojects.
ACKNOWLEDGMENTS
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