BBehind the Horizon in AdS/CFT
Erik Verlinde and Herman Verlinde Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands Department of Physics, Princeton University, Princeton, NJ 08544, USA (Dated: November 19, 2013)We extend the recent proposal of [2] of a CFT construction of operators inside the black hole in-terior to arbitrary non-maximally mixed states. Our construction builds on the general prescriptiongiven in the earlier work [5]. We indicate how the CFT state dependence of the interior modes canbe removed by introducing an external system, such as an observer, that is entangled with the CFT.
Introduction
The question whether AdS/CFT admits the construc-tion of operators that describe the interior of a black holehas been a focus of attention over the past year [1–3]. Ina recent paper [2], Papadodimas and Raju proposed anelegant prescription, that deserves further scrutiny.The PR proposal employs the CFT realization of fieldoperators in the exterior geometry [4] φ CFT ( t, Ω , z ) = (cid:88) (cid:96), ω> (cid:0) b (cid:96),ω f (cid:96),ω ( z, Ω , t ) + h . c . (cid:1) (1)where f (cid:96),ω ( z, Ω , t ) are exterior mode functions and b (cid:96),ω are single trace operators in the CFT, that in the largeN limit satisfy the free field commutation relations (cid:2) b ω , b † ξ (cid:3) = δ ωξ . (2)Here, and in the following, we suppress all quantum num-bers of the mode operators b except for the frequency ω .Let (cid:12)(cid:12) Ψ (cid:11) denote some typical equilibrium CFT statewith total energy E . Expectation values of external op-erators b ω with respect to (cid:12)(cid:12) Ψ (cid:11) look like thermal expec-tation values with inverse temperature β . AdS/CFT du-ality identifies (cid:12)(cid:12) Ψ (cid:11) with the state of an AdS black holewith mass M = E . Following [2], we may then define theinternal operators (cid:101) b ω and (cid:101) b † ω via the mirror property (cid:101) b ω A α (cid:12)(cid:12) Ψ (cid:11) = e − β ω A α b † ω (cid:12)(cid:12) Ψ (cid:11) (3) (cid:101) b † ω A α (cid:12)(cid:12) Ψ (cid:11) = e β ω A α b ω (cid:12)(cid:12) Ψ (cid:11) (4)where A α denotes an arbitrary polynomical of the ex-ternal b ω operators. PR then proceed to give a statedependent construction of the mirror operators via [2]: (cid:101) b P Rω = g mn | u m (cid:105)(cid:104) v n | , g mn = (cid:104) v m | v n (cid:105) (5) | v n (cid:105) = A n | Ψ (cid:105) , | u n (cid:105) = e − β ω A n b † ω | Ψ (cid:105) where A n denotes a complete basis of operators made outof the b ω oscillators. The PR construction works for pure states, but introduces a non-linear state dependence. This non-linear state dependence is rather worrisome.Suppose we add an external system X that interacts withthe CFT, in such a way that asymptotic AdS modes canleak out. This is a model of black hole evaporation. Overtime, the CFT then evolves into a mixed state, that isentangled with the external system (cid:12)(cid:12) Ψ (cid:11) = (cid:88) i (cid:12)(cid:12) Ψ i (cid:11) CFT (cid:12)(cid:12) ψ i (cid:11) X (6)The standard rules of quantum mechanics require thatCFT observables can act on typical entangled states ofthis form. This means that they have to act linearly oneach state | Ψ i (cid:105) CFT that appears in the sum (6). Thedefinition (5) therefore does not work for mixed CFTstates like (6), and needs to be generalized.In this note, we give a definition of interior operatorsfor mixed CFT states with an arbitrary density matrix ρ = (cid:88) ij ρ ij (cid:12)(cid:12) Ψ i (cid:11)(cid:10) Ψ j (cid:12)(cid:12) . (7)Our construction is a direct application to AdS/CFTof the prescription given in [5], and works for anynon-maximally mixed state, with von Neumann entropy S vN = − tr( ρ log ρ ) sufficiently less than the maximal en-tropy S BH . In our set up, we are also forced to introducesome state dependence, but of a relatively mild form.The discussion below will be succinct. More detailed ex-planations can be found in [5]. Equilibrium State
Pick a collection of modes b ω that span the free fieldHilbert space of the outside region, extended over some Other somewhat unsatisfactory aspects of the PR mirror map isthat it postulates the Unruh form of the CFT state as an ex-act rather than a statistical property and does not manifestlycommute with hermitian conjugation. The requirement that themirror of the adjoint equals the adjoint of the mirror is upheld,but only within exponential accuracy, thanks to the KMS prop-erty of the equilibrium state | Ψ (cid:105) . a r X i v : . [ h e p - t h ] N ov large but finite time interval and cut-off at the stretchedhorizon. Consider the class of special CFT states (cid:12)(cid:12) Ψ (cid:11) of energy E that (to very high accuracy) are annihilatedby all external lowering operators. b ω (cid:12)(cid:12) Ψ (cid:11) = 0 , ∀ ω. (8)This condition selects states that do not contain any sin-gle trace excitations in the chosen time interval. As longas E (cid:29) (cid:80) ω ω , there are many such states (cid:12)(cid:12) Ψ (cid:11) . Theyare all non-equilibrium states: from the bulk perspective,they describe black holes with a singular horizon, withboundary conditions prescribed by the Boulware vacuum.We can think of the (cid:12)(cid:12) Ψ (cid:11) as eigen states of the deformedCFT Hamiltonian H ( λ ) = H CFT + λ (cid:80) ω ωN ω , with N ω the number operator of b ω , with the property that theenergy E remains finite in the large λ limit.By repeatedly acting with single trace raising operators b † ω on the vacuum state | Ψ (cid:105) , we can now build a free fieldFock space with orthonormal basis states (cid:12)(cid:12) n (cid:11) = A n (cid:12)(cid:12) Ψ (cid:11) , (cid:10) n (cid:12)(cid:12) m (cid:11) = δ nm . (9)To leading order in 1 /N , the CFT Hilbert space decom-poses into a direct sum H CFT = ⊕ i H i of distinct copiesof the b ω Fock space, with each sector H i built on a dif-ferent vacuum state | i (cid:105) with b ω | i (cid:105) = 0. As emphasizedin [2], it is reasonable to assume that at finite N , thelabel n runs over a large but finite set of basis states.Any general outside operator O , given by some poly-nomial in the b † and b oscillators, can be characterizedby its action on the basis operators A n viaˆ O A n = (cid:88) m O nm A m (10)where the coefficients O nm = (cid:104) m | ˆ O| n (cid:105) are real c-numbermatrix elements. We will use this notation momentarily.Now consider the following ‘quantum quench’ scenario[6]: imagine that for t <
0, the Hamiltonian is given bydeformed CFT Hamiltonian H ( λ ) with λ (cid:29)
1. At t = 0,we suddenly turn off the deformation: we set λ = 0 andlet time evolve with H = H CFT . Each non-equilibriumstate (cid:12)(cid:12) Ψ (cid:11) then relaxes into a thermal equilibrium state (cid:12)(cid:12) Ψ (cid:11) = e − iτH (cid:12)(cid:12) Ψ (cid:11) (11)where τ denotes the relaxation time. τ is expected to beof order of a few times the scrambling time.The above discussion mirrors the set up in [5]. Next,following [5], we use the fact that | Ψ (cid:105) can always be de-composed as follows (cid:12)(cid:12) Ψ (cid:11) = (cid:88) n> A n C n (cid:12)(cid:12) Ψ (cid:11) (12) where, to the required accuracy, C n satisfy the properties (cid:2) C n , A m (cid:3) = 0 , (cid:88) n C † n C n = . (13)Equation (12) expresses | Ψ (cid:105) as a sum of terms, given bya factor in the b † Fock space, times a factor in the sectorof all b vacua. The C n act within this vacuum sector,and thus commute (to leading order in 1 /N ) with all op-erators that act within the b † Fock space. Physically,we can interpret C n as acting within the black hole inte-rior, defined as the complement to the outside Fock spacespanned by the A n ’s.The C n are the AdS/CFT analogues of the Kraus op-erators employed in [5, 7]. The second relation in (13)is the standard unitarity condition. Note that (11) isa linear map from the space of Boulware vacuum states | Ψ (cid:105) to the space of equilibrium states | Ψ (cid:105) . Therefore, the internal C n operators enjoy the same level of stateindependence as the external A n operators .The requirement that (cid:12)(cid:12) Ψ (cid:11) defines a thermal state givesuseful statistical information about the operators C n (cid:10) Ψ | C † m C n (cid:12)(cid:12) Ψ (cid:11) = w n δ mn . (14)with w n the Boltzman weight w n = e − βE n Z , (cid:88) n w n = 1 . (15)The relation (14) expresses the thermal equilibruim, ordetailed balance, between the external free field radiationmodes and the internal state of the black hole. Interior Operators
To construct the interior operators for equilibriummixed states ρ , we need to introduce a mild form of statedependence. Consider the same quantum quench sce-nario, but now we prepare the CFT in some initial mixedstate ρ satisfying the Boulware vacuum conditions b ω ρ = ρ b † ω = 0 , (16)and with Schmidt decomposition ρ = (cid:88) ¯ i p i (cid:12)(cid:12) ¯ i (cid:11)(cid:10) ¯ i (cid:12)(cid:12) . (17)To this density matrix ρ , we associate a subspace of theCFT Hilbert space, defined via H code = (cid:76) ¯ i H ¯ i (18)where H ¯ i denotes the b ω Fock space built on top of thebasis state | ¯ i (cid:105) in the Schmidt decomposition of ρ . Wecall H code the code subspace. In the following, we con-sider typical initial mixed states ρ , whose code subspace H code that is much larger than the b ω Fock space, butmuch smaller than the total CFT Hilbert space.Let P denote the projection operator onto H code . Inother words P is the smallest projection operator thatleaves ρ invariant and commutes with the b ω oscillators P ρ = ρ P = ρ , (cid:2) P , A n (cid:3) = 0 . (19)After the quench, the CFT settles down in an equi-librium state ρ = e − iτH ρ e iτH = (cid:80) n,m A n C n ρ C † m A † m .Basic statistical reasoning shows that the matrix elementof C † m C n between any pair of typical basis states in H code takes the form (cid:10) ¯ i (cid:12)(cid:12) C † m C n (cid:12)(cid:12) ¯ j (cid:11) = w n δ mn δ ¯ i ¯ j . (20)The diagonal form (20) arises via a simple mechanism:the operators C † m and C n are large complex random ma-trices, whose product decomposes as a sum of many termswith different phase factors. This sum typically averagesout to zero, except when there is constructive interfer-ence. Thermodynamics fixes the overall normalization.The statistical property (20) plays a key role in the con-struction of the interior operators, and holds with expo-nential accuracy, provided that dim H code (cid:28) dim H CFT .Though easily verified, it hides a small puzzle. The C n alllower the energy of the CFT state, and thus map a biggerto a smaller Hilbert space. Indeed, C n is non-invertibleon H CFT . However, when restricted to any small sub-space H code , it becomes effectively invertible. This is thekey observation that will allow us to build internal b † ω oscillators, which, in spite of the fact that they lower theCFT energy, look like raising operators.In [5], motivated by the analogy with quantum errorcorrecting codes [7], we introduced recovery operators R n and their hermitian conjugates R † n via R n = P C † n √ w n ; R † n = C n √ w n P . (21)Using (20), we deduce that (up to exponentially smallcorrections) the recovery operator R n acts on the equi-librium state | Ψ (cid:105) via R m (cid:12)(cid:12) Ψ (cid:11) = √ w m A m (cid:12)(cid:12) Ψ (cid:11) (22)provided that | Ψ (cid:105) ∈ H code . So by acting with the R n ’s,one can recover the quantum information contained inthe original vacuum state | Ψ (cid:105) .Following [5], we define the mirror (cid:101) O of the generaloperator ˆ O specified in (10) as follows (cid:101) O = (cid:88) m,n O ∗ mn R † n R m (23)Note that these operators act linearly on all of H code . Mirror Property
The mirror operator (cid:101) O represents the Hawking partnerof the operator O , the degree of freedom behind the blackhole horizon that is entangled with O . In the notation(10), we can write the mirror property (3)-(4) as (cid:101) O A α (cid:12)(cid:12) Ψ (cid:11) = A α e − β H ˆ O † e β H (cid:12)(cid:12) Ψ (cid:11) (24)We would like to show that the operators (23) satisfy thismirror property for every state (cid:12)(cid:12) Ψ (cid:11) ∈ H code .The calculation is straightforward. Using the defini-tions (21) of the recovery property (22), we deduce that R † n R m (cid:12)(cid:12) Ψ (cid:11) = e β ( E n − E m ) A m C n (cid:12)(cid:12) Ψ (cid:11) (25)With the help of this intermediate result, we compute (cid:101) O A α (cid:12)(cid:12) Ψ (cid:11) = (cid:88) m,n O ∗ nm A α R † n R m (cid:12)(cid:12) Ψ (cid:11) = (cid:88) m,n e β ( E n − E m ) O ∗ mn A α A m C n (cid:12)(cid:12) Ψ (cid:11) = (cid:88) n A α (cid:0) e − β H ˆ O † e β H A n (cid:1) C n (cid:12)(cid:12) Ψ (cid:11) = A α e − β H ˆ O † e β H (cid:12)(cid:12) Ψ (cid:11) In the first line we used the definition (23) and that the R n ’s commute with A α . Next we use (25). The third stepis a direct application of the notation introduced in (10).To arrive at the last line, we use equation (12).The mirror property (24) guarantees that the opera-tors (cid:101) O satisfy the same operator product algebra as theexternal efective QFT operators O . For a more directverification, and for a quantification of the correctionsto this result, we refer to [5]. Note further that, unlikethe PR construcion [2], eqn (24) is a derived property,rather than a postulate or definition. Moreover, insteadof just for a single state, (24) holds for a large collec-tion of states, namely all states | Ψ (cid:105) = e − iτH | Ψ (cid:105) with | Ψ (cid:105) ∈ H code . This is why our definition of the mirroroperators extends to mixed states of the form (7). A Firewall Test
A typical argument in favor of a firewall scenario goesas follows. Suppose one has shown that a given equilib-rium state | Ψ (cid:105) has a smooth horizon. Now consider anew state obtained by acting with a unitary rotation (cid:12)(cid:12) Ψ new (cid:11) = U (cid:12)(cid:12) Ψ (cid:11) (26)where U commutes (to very high accuracy) with the ex-ternal operators b ω and with the Hamiltonian H . Thenew state then decomposes as (cid:12)(cid:12) Ψ new (cid:11) = (cid:88) n A n C n, new (cid:12)(cid:12) Ψ , new (cid:11) (27)with | Ψ , new (cid:105) = U | Ψ (cid:105) and C n, new = U C n U † . It isthen argued that, since C n, new looks different from theoriginal interior operators C n , the new state can not alsobe regular at the horizon.Our construction (23) of the interior operators pointsto a clear loophole in this argument. Consider the de-composition (12) of | Ψ (cid:105) . Since U is assumed to commutewith the Hamiltonian and the A n operators, it must alsocommute with the operators C n . In particular, U com-mutes with the internal operators C n, new = U C n U † = C n (28)So in fact, by adopting the definition (23) of the mirroroperators, while including both | Ψ (cid:105) and | Ψ new (cid:105) in thecode subspace, we see that both states are identified asstates with a smooth horizon.Conversely, if one insists on creating a firewall state byacting on | Ψ (cid:11) with an operator U that does not com-mute with C n operators, then U also does not com-mute with the Hamiltonian H . So firewall states arenon-equilibrium states [2, 8]. Concluding Remarks
We have shown how to construct CFT operators that,within any given small subsector of the full CFT Hilbertspace, have the correct semi-classical properties to beidentified with free field theory modes in the black holeinterior. The formulas (23)-(23) define linear operatorsthat satisfy the mirror property (3)-(4) for any state ordensity matrix within this subsector, and act as semi-classical free field oscillators up to exponential accuracy (cid:15) = e S code − S BH (cid:28) , (29)where S code = log dim H code . The mirror modes are theHawking partners of the outgoing modes: they initiallylive on the left wedge of an eternal black hole. However,they can, without much trouble, be analytically contin-ued to the region just behind the future horizon.The construction presented here follows the generalprescription in [5]. As explained there, the success ofthe construction relies on the effectiveness of a ‘recoveryoperation’, analogous to that used in quantum error cor-recting codes [7]. This recovery process works with highfidelity, provided the code subspace is small enough. Though this result gives evidence against the firewallscenario, the main elements of the paradox remain inplace. In the CFT model for black hole evaporation, inwhich the asymptotic AdS modes can leak into an ex-terior system X , the CFT state will eventually evolveinto a highly mixed state that saturates the Bekenstein-Hawking bound. At this point the recovery operationthat underlies the semi-classical reconstruction of the in-terior geometry breaks down.This is not surprising. The microscopic mechanismthat brings out black hole information is unlikely to beexplained by semi-classical physics. Information carryingradiation modes are not semi-classical: they correspondto errors in the recovery operation. However, this is nota breakdown of the correspondence principle, but the re-sult of a misguided attempt to capture the complete CFTHilbert space in terms of a single semi-classical reality.Indeed, the lesson that seems to be emerging, is that themechanism by which information can escape from blackholes is related to the fact that quantum black holes canbe in an incoherent superposition of many semi-classicalstates. The construction of the interior operators mustthen involve a measurement, that selects one of the pos-sible semi-classical realities.If one insists on writing the semi-classical interiormodes as linear operators on the whole CFT Hilbertspace, then this can be done as follows. Split the wholeCFT Hilbert space into a direct sum of code subspaces H CFT = (cid:76) k H ( k )code (30)Introduce an auxiliary external system (like the one men-tioned a moment ago or by adding an observer) that be-comes highly entangled with the CFT. In this set up, onecan write linear operators of the type [8] (cid:101) O = (cid:88) k (cid:101) O ( k ) P ( k ) X (31)where P ( k ) X projects on the particular subspace of H X that is correlated with the semi-classical CFT sector H ( k )code . The operators (31) satisfy the QFT operatorproduct algebra and act as linear operators on the fullCFT Hilbert space. In this sense, they are state inde-pendent. However, the operators (30) do depend on theentangled state of the combined system CFT plus X. Thistype of state dependence can be physically acceptable. This use of the exterior Hilbert space indeed looks reasonablegiven the physical fact that local Rindler modes at the horizonare virtual [8]. Local Rindler particles can only be made realby introducing an accelarated detector, and should therefore bethought of as modes created by the detector [11].
Acknowledgement
We thank Daniel Harlow, Juan Maldacena, Suvrat Rajuand Edward Witten for helpful discussions. The researchof E.V. is supported by the Foundation of Fundamen-tal Research of Matter (FOM), the European ResearchCouncil (ERC), and a Spinoza grant of the Dutch ScienceOrganization (NWO). The work of H.V. is supported byNSF grant PHY-1314198. [1] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, JHEP , 062 (2013) [arXiv:1207.3123 [hep-th]]. JHEP , 018 (2013) [arXiv:1304.6483 [hep-th]].[2] K. Papadodimas and S. Raju, arXiv:1310.6335 [hep-th];arXiv:1310.6334 [hep-th]; arXiv:1211.6767 [hep-th]. [3] D. Marolf and J. Polchinski, Phys. Rev. Lett. ,171301 (2013) [arXiv:1307.4706 [hep-th]][4] T. Banks, M. Douglas, G. Horowitz and E. Martinec,hep-th/9808016; D. Kabat, G. Lifschytz and D. A. Lowe,Phys. Rev. D , 106009 (2011) [arXiv:1102.2910 [hep-th]].[5] E. Verlinde and H. Verlinde, JHEP , 107 (2013)[arXiv:1211.6913 [hep-th]].[6] P. Calabrese and J. L. Cardy, Phys. Rev. Lett. ∼ preskill/ph229/[8] E. Verlinde and H. Verlinde, arXiv:1306.0515 [hep-th];arXiv:1306.0516 [hep-th].[9] S. L. Braunstein, S. Pirandola and K. Zyczkowski,Phys. Rev. Letters 110, (2013) [arXiv:0907.1190[quant-ph]].[10] Y. Nomura and S. J. Weinberg, arXiv:1310.7564 [hep-th].[11] W. G. Unruh and R. M. Wald, Phys. Rev. D29