The Harlow-Hayden conjecture and the

Papadodimas-Raju proposal

Christos Charalambous

Semester project in Theoretical Physics, Department of Physics,

ETH Zurich

November 30, 2014

Abstract

The following report begins with a brief introduction to the theory

of General relativity. Continuing, it presents the concept of a Black

hole and the strange eect related to these objects, termed the Hawk-

ing radiation, is explained. By accepting the existence of such an

eect is understood that a paradox arises in our current understand-

ing of the Black holes, the Black hole information paradox, where we

have an apparent loss of information. In the heart of this paradox

lies a contradiction between Quantum theory and General relativity

and it is believed that resolving this paradox may lead to a better

understanding of how the two theories can be brought together.

A few proposals about how to resolve the problem are presented.

More specically the Firewall solution is explained as well as the

Harlow-Hayden conjecture. In the last part of this report, a new

theory is presented which was developed by Papadodimas and Ragu,

and some parallels to the Harlow Hayden conjecture are stated.

1

Contents

I Introduction 4

II Preliminaries 8

1 What is a Black hole? 8

1.1 Introduction/ History . . . . . . . . . . . . . . . . . . . . . . 81.2 The metric of space and time . . . . . . . . . . . . . . . . . 81.3 General relativity . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Gravity and Einstein's eld equation . . . . . . . . . . . . . 111.5 The Schwarzschild Solution . . . . . . . . . . . . . . . . . . 13

2 Hawking radiation 15

2.1 The Unruh eect . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . 21

III Complementarity, Firewalls, the Harlow-HaydenConjecture and the Papadodimas-Raju proposal 30

3 Complementarity 30

4 Firewall 31

4.1 Scrambled states . . . . . . . . . . . . . . . . . . . . . . . . 314.2 The Firewall argument . . . . . . . . . . . . . . . . . . . . . 32

5 The Harlow-Hayden Conjecture 34

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1.1 Coarse-grained and Fine-grained quantities . . . . . . 345.1.2 Distillable entanglement . . . . . . . . . . . . . . . . 365.1.3 Page transition . . . . . . . . . . . . . . . . . . . . . 37

5.2 Reformulation of the Firewall argument . . . . . . . . . . . . 385.3 Strong Complementarity . . . . . . . . . . . . . . . . . . . . 405.4 The Harlow-Hayden conjecture . . . . . . . . . . . . . . . . 42

5.4.1 What is possible? . . . . . . . . . . . . . . . . . . . . 435.4.2 Why is Alice's computation slower than the Black hole

dynamics? . . . . . . . . . . . . . . . . . . . . . . . . 445.4.3 Scott Aaronson argument . . . . . . . . . . . . . . . 46

2

6 The Papadodimas-Raju proposal 47

6.1 Short description . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Bulk operators . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2.1 Choosing the bulk state . . . . . . . . . . . . . . . . 516.3 State dependence . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Conclusions 55

3

Part I

Introduction

Black hole information paradox has been puzzling physicists ever since Hawk-ing realized that black holes emit Hawking radiation. Essentially what makesphysicists uncomfortable is the fact that this phenomenon in conjugationwith the theory of general relativity, results in the violation of the gener-ally accepted fact that quantum theory is a theory governing matter in themicroscopic world and classical physics is the theory describing the worldmacroscopically, as it appears that an amplication of short distance quan-tum uctuations to macroscopic distances is taking place which results in aviolation of the Wilsonian limit. Therefore the problem really is that thereis no well established theory describing how classical physics emerges fromquantum physics.

The paradox itself can be stated in simple words as following:Matter in dierent initial pure states may collapse in black holes of the

same mass. Based on the no hair theorem, mass is the only attribute to de-scribe these black holes (assuming they are neutral and non-rotating just forsimplicity of argumentation) and hence they should look identical. So thereis an apparent loss of information about the initial pure state that one couldconsider to be hidden behind the event horizon of the black hole. Howeverwith the discovery of the Hawking radiation, according to which is possiblethat the entire Black hole radiates away, there is no more room to hide theseinformation and hence one stumbles upon the paradox. Mathematically theparadox implies the existence of a many to one map from the initial purestates to the nal state of radiation. This requires one to make the propos-terous statement that we have to abandon reversibility and hence unitarity,a cornerstone of physics since this assumption stems from the requirementthat when considering the system independently of its environment then itsevolution should be that of a closed system where we have conservation ofinformation.

The above apparent contradiction can be better understood in the con-text of a thought experiment involving an observer falling in a black hole.The paradox then translates to the observer being able to extract informa-tion from the Hawking radiation that has come out (by applying a unitarytransformation that unscrambles the nal state of the black hole in orderto obtain the initial state) before the observer actually falls in the hole. Toavoid this paradox the initial proposal was to make use of a feature of thequantum mechanics of noncommuting observables called complementarity.This essentially means that the two scenarios i.e. rst scenario the observerbeing outside the black hole making a measurement and second scenariothe observer falling in the black hole and making a measurement inside it,

4

can't be performed by the same observer. Susskind proposed that right atthe horizon there must exist a membrane which causes the following eect:An infalling observer will see the point of entry of the information as beinglocalized on the event horizon, while an external observer will notice theinformation being spread out uniformly over the entire stretched horizonbefore being re-radiated. This hence resolved the paradox.

The problem however was only solved temporarily as it was soon realizedthat the introduction of a second observer was to revive the paradox. Theproblem this time essentially translated into a violation of the monogamy ofentanglement. Hence to solve the problem it was suggested that a mysteriousmechanism should exist causing entanglement to break at some point in theevolution of the system and hence resolving the paradox. This mechanismwas called a rewall1 and it was placed at the event horizon of the black hole.This rewall basically caused the burning of the observer upon enteringthe Black hole and hence of any entanglement of his with the interior ofthe black hole. This was an idea that was very dicult to be digestedby many physicists due to the necessity to introduce something by hand.Hence there were many proposals to avoid the usage of a rewall in theyears that followed. The lesson learned from the introduction of the rewallwas the realization of the necessity to put some operational constraints onthe evolution of the system which will lead to a better understanding of theHilbert space describing the Black hole evaporation.

By investigating what the aforementioned constraints could be a newconstraint/criterion was understood that should hold. This is called thecomputationally accessible criterion which one could say that is an analogueof the causality criterion. The latter criterion, can basically be summarizedin the following sentence: 2 spacelike separated low energy observableswhich are not both causally accessible by some single observer do notneed to be realized even approximately as distinct and commuting opera-tors on the same Hilbert space. Causality criterion's meaning is basicallyunderstood as: whether quantum information is there or not depends on itspractical accessibility i.e. if there is indeed a causal connection between theobservable and the observer. This criterion is essentially what gave Bohr asan answer to the EPR paradox. In an analogous way the computationallyaccessible criterion can be dened as 2 spacelike separated low energy ob-servables which are not both computationally accessible by some singleobserver do not need to be realized even approximately as distinct and com-muting operators on the same Hilbert space, and is understood as sayingthat whether quantum information is there or not for a specic observerdepends on whether he can compute anything based on this information ornot. It is a stronger criterion than causality criterion as it also limits the

1essentially an analogue of the membrane proposed before

5

access to quantum information even if both of the two observables belong tothe past lightcone of the observer. This criterion can be seen as a criterionfor the validity of an eective eld theory2. Of course these two criteria arerestrictions to how the Hilbert space should look like but they do not givethe complete picture as they reveal what the Hilbert space is not allowed tobe but do not say what it should be.

Two proposals as to how one should actually understand the underlyingHilbert space are the strong complementarity and the standard comple-mentarity [7]. The rst one suggests that each observer has his own quan-tum mechanical theory which is precise for some observers and approximatein general. This allows for the introduction of a stationary observer thatstays outside the black hole for whom the Hawking radiation can be mea-sured and unitarity of the evaporation can be veried. At the same timeit allows for consistency conditions at the overlap of the causal patches ofthe two observers i.e. at the information that is physically available to bothof them. Based on this view, assuming a rewall is not necessary as theexperiment can not take place. However this idea is rather unsatisfying asthe existence of observers which have their own description of the universe,i.e. their own Hilbert spaces, approximate or not according to taste andwith no clear precise relationship between them, seems a rather inelegantfundamental framework.

On the other hand the second proposal, standard complementarity,which suggests the existence of a single Hilbert space on which the descrip-tions of the two observers are embedded (and hence unitarity is preserved),sounds more appealing. More specically, in the standard complementar-ity framework, each one of the observers has for each of the time slices hisown set of distinct operators that approximately commute. This frameworkacquires further momentum once its agreement with the AdS/CFT frame-work is observed. In this framework a semi-classical interpretation of anoperator according to one observer can be quite dierent from that of an-other observer, and most importantly some things that can be seen by oneobserver while they might not be visible to another.

This report assumes no knowledge of the reader about black holes oreven general relativity so the rst section of the next part of the report isdevoted to giving a brief introduction to these ideas. After establishing thisa relatively detailed description of the Unruh eect is given, which is theeect that is in the heart of Hawking radiation. Continuing a short descrip-tion of Hawking radiation is presented. Part III begins with a descriptionof the Black hole information paradox and hence an argumentation for theintroduction of the rewall idea. In the section that follows this, a pro-

2Eective eld theory should be described by operators that are related to observablesthat are measured in large scales, i.e. at the classical, limiting case of a quantum eldtheory.

6

posal on how to avoid the necessity of such an awkward structure as therewall is explained, and this proposal is called the Harlow-Hayden conjec-ture. In the part that follows the Harlow-Hayden conjecture, the proposalof Papadodimas and Raju is presented where the paradox is resolved fromanother perspective, one that makes use of the AdS/CFT correspondence.Some parallels are then drawn between the two proposals.

7

Part II

Preliminaries

1 What is a Black hole?

The following chapter mainly follows references [1] and [2]. Reference [3] isused for the description of the Schwarzschild solution.

1.1 Introduction/ History

According to Newton's theory of gravity, the escape velocity v from a dis-tance r from the center of gravity of a heavy object with mass m, is describedby

1

2u2 =

Gm

r(1)

What happens if a body with a large mass m is compressed to such anextend that the escape velocity from its surface would exceed that of light,or u > c? Are there bodies with a mass m and radius R such that

2Gm

Rc2≥ 1 (2)

This question was asked as early as 1783 by John Mitchell. The situationwas investigated further by Pierre Simon de Laplace in 1796. Do rays oflight fall back towards the surface of such an object? One would expect thateven light cannot escape to innity. Nowadays we know that to understandwhat happens with such extremely heavy objects, one has to consider Ein-stein's theory of General relativity, which describes the gravitational eldwhen velocities are generated comparable to that of light, in order to fullyunderstand what happens when such objects are considered.

1.2 The metric of space and time

General relativity is a geometric theory of gravitation published by Einsteinin 1916. It can be summarized as following: We consider the variation ofgravity in space and time and assume that information about it is encodedin a dynamical eld. General relativity essentially is the theory explaininggravity by ascribing this variation to the form of the metric tensor describingthe curvature of spacetime itself rather than some additional eld propagat-ing through spacetime. Therefore for Einstein the answer to the questionabove was equivalent to understanding how the curvature of spacetime was

8

related to the energy and momentum of the matter and radiation present inthe system under consideration.

In order to understand this profound insight of Einstein we need to un-derstand what a metric is, therefore we need to turn to a special case ofgeneral relativity where spacetime is assumed to be the simplest possiblei.e. at and more specically Minkowskian, the theory of Special Relativity.SR is based on the assumption that all laws of Nature are invariant under aspecial set of transformations of space and time3:

xµ′=

∑ν=0,...,3

aµνxν or x′ = Ax (3)

where xµ µ = 0, ..., 3 a lorentzian vector describing a spacetime point i.e.

xµ =

txyz

and A is a matrix representation of an element of the Poincare

group R1,3 ∼= SO(1, 3) the group of the most general linear transformationsof spacetime that leave

s2 =∑

µ,ν=0,...,3

ηµνxµxν (4)

invariant, where ηµν is a symmetric (0,2) tensor, called the Minkowski metric,that transforms under Poincare group transformations as

η′µν =(A−1

)αµ

(A−1

)βνηαβ (5)

and quantity s is called the invariant length of a lorentz vector xµ, the analogof distance in 3d.

Moving to General relativity will be easier if we rst study the eect ofletting the metric depend on the coordinates on the equations of motion ofa particle. Consider the most general coordinate frame, i.e. let the origi-nal coordinates (t,x,y,z) be completely arbitrary, mutually independent anddierentiable functions of four quantities u = uµ, µ = 0, ..., 3. By dier-entiable we mean that every point is surrounded by a small region wherethese functions are to a good approximation linear. This implies that x+dxexists (for dx arbitrarily small) and furthermore that (4) now becomes:

ds2 = ηµνdxµdxν = gµν(u)duµduν (6)

4where

gµν(u) =∂xα

∂uµ∂xβ

∂uνgαβ(x) (7)

3just like a coordinate transformation in 3d space leaves distances invariant4Einstein summation is implied here. Also ds2is essentially equal to the tensor product

dx⊗ dx

9

i.e. a relation for innitesimal transformations. If we were still talking aboutthe original coordinate system ∂xα

∂xµ= δαµand hence gµν(x) = gµν(x) and hence

the only transformation possible for the metric would be one by an elementof the Poincare group, i.e. coordinate transformations would have a trivialeect. However assuming that we are in this new coordinate system u (7) isvalid.

In the original coordinates xµ the motion of the particle would be:

dxµ(τ)

dτ= constant (8)

d2xµ(τ)

dτ 2= 0 (9)

assuming the metric was the original Minkowskian metric i.e. spacetime wasat. However in the new coordinate system the above equations take thefollowing form:

dxµ

dτ=∂xµ

∂uλ∂uλ

∂τ(10)

d2xµ(τ)

dτ 2=

∂2xµ

∂uλ∂uκduκ

dτ

duλ

dτ+∂xµ

∂uλd2uλ

dτ 2(11)

⇒ d2uλ

dτ 2+ Γµκλ(u)

duκ

dτ

duλ

dτ= 0

where

Γµκλ(u) =∂uµ

∂xα∂2xα

∂uκ∂uλ(12)

and is called the connection eld. We can see clearly that if we are talk-ing about a 4d at spacetime the above quantities u can't be other thana dierent coordinate parametrization of the original coordinates5. Henceequations (8) and (9) would be recovered. However, if the spacetime is notat there will not exist a coordinate system such as the original one wherethe coordinates were mutually independent and hence equation of motionwould be equation (11) with a slightly modied connection eld.

1.3 General relativity

From the above section it is clear that we would like our spacetime to be at,i.e. Minkowskian as the transformations related to this spacetime preservethe laws of nature. However we saw that it is enough if this transformationshold at the innitesimal level. Hence there is still the freedom of allowing anon at spacetime on a larger scale. It is then that we have to start talking

5there are no more degrees of freedom for u to be anything else

10

about the notion of manifolds and how spacetime geometry can be buildout of them. An n-dimensional manifold is a topological space that has aneighborhood that is homeomorphic to the Euclidean space of dimension n.

To obtain equations of motion for motion on the combined set of mani-folds, the spacetime geometry, we now need to know how the local equationsof motion have to be transformed from one manifold to the other. Thereforewe need to know how objects like this ∂µ = ∂

∂xµtransform. This partial

derivative in at spacetime is a map from a (k,l) tensor to a (k,l+1) tensor.It can be shown that an object that does the same thing, but in curvedspacetime, and is also coordinate independent is the following:

∇µVν = ∂µV

ν + ΓνµλVλ (13)

This equation is true for dierentiation of objects with indices upstairs (vec-tors). For objects with indices downstairs (one-forms) which represent mapsfrom (k,l+1) tensors to (k,l) tensors, the dierentiation takes the followingform:

∇µων = ∂µων − Γλµνωλ (14)

6

Connection Γµκλ in equations (13) and (14) is not the same as before. It isa quantity which was inserted in order to make ∇µV

ν and ∇µων transformas tensors. The connection essentially is the key object that tells us howthese objects change when transported from one tangent space (manifold)to the other. It can be shown that the connection can be given solely byconsidering the metric and the equation that arises is the following:

Γµκλ =1

2gρσ(∂µgνρ + ∂νgρµ − ∂ρgµν) (15)

Although Γµκλ itself does not transform as a tensor there exist an objectconstructed solely from the connection that transforms indeed like a tensor,which means that this object respects the symmetries of spacetime. This isthe Riemann curvature Rµ

καβ given by:

Rµκαβ = ∂αΓµκβ − ∂βΓµκα + ΓµασΓσκβ − ΓµβσΓσκα (16)

and if this is contracted we obtain the Ricci curvature Rκα = Rµκµα and from

a further contraction the Ricci scalar R.

1.4 Gravity and Einstein's eld equation

If we assume frame where gµν is time independent and a particle that is atrest in this coordinate frame then we can check that the particle will undergo

6reason is because one-forms

11

an acceleration based on equation (11)7:

d2xi

dτ 2= −Γi00 =

1

2gij∂jg00 (17)

where we see that this acceleration is independent of the mass of the par-ticle as expected by the Weak Equivalence Principle of Einstein which isthe physical principle that lead Einstein to have the inspiration of relatinggravity to curvature of spacetime. The gravitational potential in the caseabove is−1

2g00. It is possible to assume that the metric is a dierentiable

function of the coordinates x, i.e. metric is gµν(x) and in the above examplewe could let g00(x) take the shape of the gravitational potential of the earthand hence understand the motion described by (17) as the free-fall motion.

There is still the missing link of determining the gravitational potentialof the earth (which is related to its mass). We know that not all objects onor near the earth move in straight lines i.e. we don't have a at spacetimenear earth and hence Ricci curvature does not vanish. This means thatwe have to nd the connection eld surrounding earth and generally weneed an equation that will connect this connection eld to a heavy object.Einstein gave the answer to this problem. He came up with the followingeld equation:

Rµν −1

2Rgµν = −8πGTµν (18)

where Tµν is the energy-momentum tensor which acts as the source of thegravitational eld. In at spacetime the components of this tensor are de-ned as follows:

T00 = −ρ(x)

where ρ(x) the energy distribution,

Ti0 = T0i

the matter density which equals to momentum density, and Tij is the tensionwhich for a gas or liquid is

Tij = −pδijwhere p is the gas or liquid's pressure. The continuity equations is:

∂iTiµ − ∂0T0µ = 0

If now curved coordinates are considered then continuity equation for theenergy momentum tensor becomes:

gµνDµTνα = 0 (19)

7where latin indices run as i=1,2,3

12

where we see that an extra term containing the connection eld Γλαβ appearswhich plays the role of the gravitational eld which adds or removes energyand momentum to matter. 8With this equation in mind one can check thatby taking the covariant derivative of both sides of equation (18) the followingequation is obtained:

gµνDµRνα −1

2DαR = 0 (20)

which is called the Bianchi identity and is valid for the Ricci tensor in-dependently of any assumptions about the connection i.e. the spacetimecurvature.

1.5 The Schwarzschild Solution

In 1916 Schwarzschild managed to nd an exact non-trivial solution for equa-tion (18) (i.e. a metric that satised it) for the most obvious application ofa theory of gravity, i.e. the application to a spherically symmetric gravita-tional eld [3]. The metric he obtained is understood as giving the uniquestatic9 vacuum solution with an event horizon (it will be explain shortly).This is expressed in the Schwarzschild coordinates as following:

ds2 = gµν(x)dxµdxν = −(1−2GM

rc2)dt2+

dr2

1− 2GMrc2

+r2dθ2+r2sin2θdϕ2 (21)

This metric is describing an object called a spherically symmetric black hole.The puzzling thing about this metric is the fact that there is a singularityfor r = rs = 2GM

c2where r is the radius of a sphere. However nothing special

should happen at rs and this can be understood if equation (21) is rewrittenin the so called Eddington-Finkelstein coordinates as following:

ds2 = (1− rsr

)du2 − 2dudr − r2dθ2 − r2sin2θdϕ2 (22)

where we see that the singularity has been moved to r=010. u here is equalto:

u = t∗ + r

with t∗ = t+ 2GM ln | r2GM− 1 |

The above suggest that the singularity at rs we previously had is justa coordinate singularity. This is essentially a consequence of the dieomor-phism of the spacetime manifold. The fact that the singularity at r=0 can

8This is in agreement with the artifact of Special relativity that energy and matter areequivalent.

9by static spacetime we mean a stationary spacetime i.e. one that has no time depen-dence and in addition has a time reection symmetry. There also exist solutions that donot satisfy this property. These metric describe rotating black holes.

10c=1

13

not be removed by nding another coordinate system can be conrmed bythe singularity theorem of Penrose which is based on the concept of trappedsurfaces. Intuitively the fact that there exists a singularity at r=0 can beunderstood by looking at a Finkelstein diagram where we can see that thelight cones are distorted as they approach r=0. To conrm this we needto study the ingoing and outgoing null geodesics which dene a light cone.Past the r=2M point (letting constant G=1) all light cones face towards theline r=0 and at r=0 the light cones future is all spacelike. Therefore noevolution in time from that point onwards.

Figure 1: Finkelstein diagram. Light cones are dened using outgoing andingoing null geodesics. The ingoing radial null geodesics correspond to linesof constant u. The outgoing radial null geodesics correspond to lines ofconstant v = t∗ − r − 2 ∗ 2GM ln | r

2GM− 1 |describing the trajectories of

photons created in the hawking radiation phenomenon. One can easilyobserve that at r=2M the outgoing is just parallel to the time axis meaningthat the particle will not manage to move further away to larger radii. It istrapped for ever there. For smaller radii the particles future lies entirely on

the curvature singularity and hence it is bound to hit it at some point.

rs is called the event horizon of a black hole. Why this is called theevent horizon is understood when one considers the outgoing light rays andobserves that they should satisfy dr

du= 1

2(1− rs

r)11. We see that for r = rs,

drdu

vanishes and hence light rays can't escape. So whatever is inside the blackhole can't aect any events outside the black hole. For r < rsthe outgoinglight rays are dragged inward to decreasing r and eventually reach r=0. Onemore thing to note about this metric is that it is asymptotically at as itbecomes the familiar Minkowskian metric in the limit M → 0.

11for light rays dsdu = 0 i.e. no notion of time for photons

14

2 Hawking radiation

One important conclusion that can be drawn from the above observationsabout the Schwarzschild metric which describes a static black hole is thefact that since no light rays can escape from the interior of the black holethen no information can be obtained about it. More specically there is atheorem called the no hair theorem which postulates that all black holes12 can be described by three external observables, their mass, their electriccharge and their angular momentum. However Hawking managed to showin 1974 that Black holes do indeed emit radiation. This then implies that itmust be in principle possible to obtain information about the interior of ablack hole. Hence the famous Black hole information paradox emerges.

To understand this paradox we rst need to understand how this radia-tion predicted by Hawking emerges. In order to do so is better if we begin bya slightly simpler phenomenon, the Unruh eect, of which Hawking radiationcan be seen as a generalization. The proof of the Unruh eect that followsis based on reference [1]. The description of Hawking radiation follows [1],[3] and [4].

2.1 The Unruh eect

The reason we turn rst to Unruh eect is because this is an eect thatappears even in at spacetime where equations are simpler and we have abetter understanding of it. The Unruh eect can be summarized as following:Vacuum in Minkowski space appears as a thermal state with temperatureTU = ha

2πwhen viewed by an observer with acceleration a.. More explicitly,

if we are given a QFT vacuum in Minkowski spacetime on which all inertialobservers agree upon, and if we further consider applying the covariant for-malism of QFT in curved spacetimes on the Minkowski spacetime describedby non-cartesian coordinates (such as the ones describing the motion of anaccelerated observer) then we can see that the vacuum in this latter descrip-tion of the QFT will not be the same as the initial one even though thespacetime is always the same, i.e. Minkowski spacetime.

To prove this its enough to consider the simplest case the 2-D Minkowskispacetime. The applicability to the 4-D case is guaranteed by sphericalsymmetry. The metric in the inertial coordinates reads:

ds2 = −dt2 + dx2 (23)

However as we are interested in observers with uniform acceleration thenit is better to describe the metric in a dierent set of coordinates, namely

12even black holes that are solutions of the Einstein-Maxwell equations of gravitationand electromagnetism in general relativity

15

Rindler (polar) coordinates (ξ, η) that are related to the inertial coordinatesas following:

t =1

aeaξsinh(aη) (24)

x =1

aeaξcosh(aη) (25)

with x >| t |13, where a is a constant, that represents the uniform acceler-ation of an observer in the Rindler coordinates. Then the metric in thesecoordinates is:

ds2 = e2aξ(−dη2 + dξ2) (26)14 which makes the causal structure of the spacetime more apparent. Youcan also see this by the introduction of the future and past event horizonsH+andH− respectively (a concept that will be explained soon). Graphicallythe space parametrized in the two dierent coordinates looks as following:

Figure 2: 2-dimensional Minkowski space in Cartesian coordinates andRindler coordinates. Notice that only the two wedges I and IV are

parametrized by the latter set of coordinates because of the requirement thatx >| t | and are also not parametrized simultaneously. Region I is the onethat is accessible by an observer accelerating in the positive x directionwhile region IV is the region that is accessible by an observer moving

backwards in time. Constant η lines can be understood as lines of constantradius away from the origin and constant ξ lines can be understood as lines

of constant angle from the axis.

13so that the observer has a real valued acceleration. This can be seen by the relationthat gives the trajectory of the accelerated observer: x2(τ) = t2(τ) + 1

α2 where α themagnitude of the acceleration and τ the parametrization of the trajectory

14−∞ < η, ξ < +∞

16

One more advantage of these coordinates is the fact that the Killingvector is also more obvious15. It is just

∂η =∂t

∂η∂t +

∂x

∂η∂x = a(x∂t + t∂x) (27)

which is the killing vector associated with a boost in x-direction (analogousto rotational symmetry in Euclidean space). The use of the killing vectorwill become apparent soon but in short the idea is that we will try to ndboost eigenstates with support in the Rindler wedges I & IV that we willcall right and left wedges respectively and then we will try to express thevaccuum states in terms of modes corresponding to these boost eigenstates.The end result of this process will be the appearance of the Unruh eect.

We also assume a massless scalar eld φ in this spacetime. This willsatisfy the massless Klein-Gordon equation which in Rindler coordinates is:

∂µ∂µφ = e−2aξ(−dη2 + dξ2)φ = 0 (28)

A solution to this equation is a normalized wave gk(ξ, η) = (4πω)−12 e−iωη+ikξ

where ω =| k |. Now it can be shown that along the surface t=0, ∂η is a hy-persurface orthogonal timelike Killing vector 16, except for the single pointat x=0 where it vanishes. This vector can therefore be used to dene a setof positive and negative-frequency modes, on which we can build a Fockbasis for the scalar eld Hilbert space. In region I of gure 1 the solutionis ne as the plane wave dened in this region has positive frequency withrespect to the future-directed17 killing vector18 which in region I is ∂η, i.e.∂ηgk = −iωgk . However in region IV the future-directed Killing vector is∂−η = −∂η therefore we get negative frequencies19. The only way to avoidthis problem (and hence be able to dene +ve frequency plane wave solu-tions which correspond to particles in the entire 2d plane) is to introducetwo separate sets of modes:

g(1)k (ξ, η) =

1√4πω

e−iωη+ikξ (29)

if in region I and g(1)k (ξ, η) = 0 if in region IV, and

g(2)k (ξ, η) =

1√4πω

e+iωη+ikξ (30)

15A killing vector is a vector eld that generates a symmetry of the spacetime16which means that η is a timelike killing vector, i.e. η · η < 0 and ∇µην +∇νηµ = 017note that only future directed Killing vectors can be described in this set of coordi-

nates18i.e. particles move forward in time antiparticles backwards19Here we could note that we can't describe regions II and III when using Kruskal

coordinates and this solves a potential doubled up spacetime related problem

17

if in region IV and g(2)k (ξ, η) = 0 if in region I. The two modes are actually

related due to the CPT symmetry of the free scalar eld. To relate them wehave to use the CPT antiuinitary operator Θ and the relation between thetwo is expressed as following:

Θ†g(1)k Θ = g

(2)k

The conjugate of these plane waves are also solutions and by analyticallyextending all 4 solutions to the rindler wedges II & III, we obtain a completeset of basis modes for any solutions to the wave equation throughout thewhole of spacetime, i.e. we are in a position to express the free scalar eldas:

φ(ξ, η) =∫dk [b

(1)k g

(1)k (ξ, η) + b

(1)†k g

(1)∗k (ξ, η) + b

(2)k g

(2)k (ξ, η) + b

(2)†k g

(2)∗k (ξ, η)]

(31)But eld φ can also be expanded in the original coordinates (t, x) i.e.

φ(t, x) =∫dk [akfk(t, x) + a†kf

∗k (t, x)] (32)

20Now we want these solutions to be normalized and hence we need an innerproduct. To dene the inner product such that it will be invariant undertransformations that leave the space of solutions invariant, we rst need tostudy the summetries of the space of solutions. The Klein Gordon lagrangianis the following:

L = (∂µφ(t, x))†∂µφ(t, x)−m2φ†(t, x)φ(t, x)

where µ ∈ t, x. One can then easily see that the lagrangian is invariantunder φ(t, x) −→ eiaφ(t, x) where a ∈ U(1). Using the Noether currentdenition we can nd the Klein Gordon current and then from that a con-served quantity called conserved charge which is equal to the integration ofthe noethern current over the whole space and nally we can assign thisconserved value to the inner product that we are looking for. Therefore theinner product takes the following form:

(φ1, φ2) = −i∫

Σ(φ1∂tφ

∗2 − φ∗2∂tφ1)dx (33)

where Σ is a spacelike hypersurface over which integral is taken. Then wesee that both the modes in the Rindler coordinates and the modes in theoriginal Minkowski coordinates satisfy some commutation relations:

(g(1)k1, g

(1)k2

) = δ(k1 − k2) (34)

20Both coecients b and a are operators as we know from 2nd quantization

18

(g(2)k1, g

(2)k2

) = δ(k1 − k2) (35)

(g(1)k1, g

(2)k2

) = 0 (36)

and similarly for fk and f∗k . This allows us to build the Fock space indepen-

dently using one time the states in the Rindler coordinates and the othertime using the original Minkowski coordinates, as the above suggest thatthe sets describing the space of solutions to the KG equation are composedof an orthonormal complete basis. However the ground state in one coordi-nate system will not look the same in another coordinate system, i.e. theMinkowski vacuum satisfying

ak | 0M >= 0 (37)

will be described as a multiparticle state in the Rindler representation andvice versa for Rindler vacuum | 0R > satisfying:

b(1)k | 0R >= b

(2)k | 0R >= 0 (38)

Essentially the explanation for this lies in the fact that an individual Rindlermode can never be written as a sum of positive frequency Minkowski modesbecause at t=0 the Rindler modes only have support on the half-line and toexpand such a function we must include necessarily negative frequency planewaves as well (i.e. plane waves that move backwards in time, antiparticles,something which makes it dicult to dene a well dened number operator).

To overcome this lets see how we can build a well dened plane wavealong the whole surface t=0 out of plane waves in Rindler coordinates. Todo this rst we need to express plane waves g

(1)k and g

(2)k in terms of the

Minkowski coordinates:

g(1)k =

aiωa (−t+ x)i

ωa

√4πω

(39)

g(2)k =

a−iωa (−t− x)−i

ωa

√4πω

(40)

Hence from this we can build the following normalized plane waves:

h(1)k (t, x) =

(eπω2a g

(1)k + e−

πω2a g

(2)∗−k )√

2sinh(πωa

)(41)

h(2)k (t, x) =

(eπω2a g

(2)k + e−

πω2a g

(1)∗−k )√

2sinh(πωa

)(42)

which can be checked to be well dened at t=0. These solutions are indeeda mixture of Rindler modes with support on both positive and negative

19

frequencies. Therefore now eld φ can be expanded in the new modes as:

φ =∫dk [c

(1)k h

(1)k + c

(1)†k h

(1)∗k + c

(2)k h

(2)k + c

(2)†k h

(2)∗k ] (43)

where because h(1)k and h

(1)k are well dened at t=0 the following holds:

c(1)k | 0M >= c

(2)k | 0M >= 0 (44)

Now c(1,2)k are related to b

(1,2)k by a Bogolyubov transformation as following:

b(1)k =

(eπω2a c

(1)k + e−

πω2a c

(2)†−k )√

2sinh(πωa

)(45)

b(2)k =

(eπω2a c

(2)k + e−

πω2a c

(1)†−k )√

2sinh(πωa

)(46)

and therefore now we are in a position to evaluate the following (well de-ned now) number operator in region I which corresponds to the number ofparticles a uniformly accelerated Rindler observer observes:

< 0M | n(1)R (k) | 0M >=< 0M | b(1)†

k b(1)k | 0M >=

1

e2π ωa − 1

δ(0) (47)

which is a Planck (thermal) spectrum of particles with temperature:

TU =a

2π(48)

In obtaining the last equality in equation (47) we really considered contin-uous wave packets rather than discrete single modes. Otherwise instead ofa delta function we would have obtained a Kronecker delta δ−k,−k which isequal to 1.

So we have seen how plane wave solutions expressed in Rindler spacecan be combined to form well dened plane wave solutions for which theannihilator operators annihilate Minkowski vacuum. This relation betweenoperators can also be seen as a relation between the states i.e. the Minkowskivacuum state and the Rindler states. To make it more explicit, the resultabove implies that locally (a requirement in order to be able to use theBogolyubov transformation) Minkowski vacuum can be expressed as a sumof Rindler Fock states that are the tensor product of states | nR > and

| nL > corresponding to the modes g(1)∗k and g

(1)k that appear on the left

and right of the Rindler horizon respectively (i.e. modes with -ve and +vefrequencies) as it can be seen on gure 3. The sum has the following form:

| 0M >=∑n

dn | nR > ⊗ | nL > (49)

20

Figure 3: Positive g(1)k and negative g

(1)∗k frequency modes near the Rindler

horizon. There is an innite number of modes there and hence an inniteamount of entanglement

A vacuum that can be expressed in this form i.e. in terms of corre-lated/entangled states is called the Boulware vacuum and the correlatedstates are called the Boulware quanta. Entanglement is however not enoughto have a at spacetime and hence a smooth infalling experience across theRindler or later across the black hole horizon [10]. The reason is that ifproduct states are used i.e. if dn = 1 for only one n, then it appears thatwe have an innite amount of energy on the horizon. The state has to bemixed.

This means that locally we have correlations between positive energyRindler quanta and negative energy Rindler quanta. Therefore states inMinkowski Hilbert space H can also be expressed as states in the tensorproduct of a Hilbert space dened in the left of the Rindler Horizon HLwhichcontains the -ve frequencies and a Hilbert space dened in the right of theRindler Horizon HR which contains the +ve frequencies 21, i.e.

∃ map I : H → HL ⊗HR (50)

2.2 Hawking radiation

The essence of the Hawking eect lies exactly in this correlated structureof the vacuum state at short distances. These correlations become trulyapparent in the Hawking eect where we have a quantum eld propagatingin the background of a stationary black hole. In this case instead ofhaving an evolution from a stationary region to another as we had before,we rather have the evolution of a stationary region as the one described

21More specically H = HL ⊗ HR modulo the degrees of freedom at exactly x=0 i.e.d.o.f. at the boundary which we are not dealing with here.

21

by Minkowski vacuum outside the black hole to a Cauchy surface that willbe composed from the event horizon of the black hole and the lightlikefuture. What really happens is that rather than the quanta stay in eithersite of the horizon forever, the outgoing quanta outside the horizon moveaway to innity and their correlated partners on the other side fall into thesingularity because of the presence of curvature of spacetime which resultsin acceleration of objects in this spacetime.

To understand this eect, rst we assume that an observer sees a Minkowskivacuum if he is near the event horizon of a black hole. For suciently smalldistance, more specically for observed length and timescales of the ordera−1 2GM where a is the observer's acceleration magnitude, the acceler-ation is still small compared to the scale set by Schwarzschild radius, whichessentially sets the radius of curvature of spacetime near the horizon, there-fore this is a valid assumption to make because spacetime eectively looksat (also remember that the event horizon is just a coordinate singularityand not a true singularity of spacetime therefore there is nothing specialabout it, so we expect to have vacuum there).

Then we have to make the following observation. In the discussion forthe Black hole spacetime coordinates we saw that the original coordinatesdid not cover the entire space, as they predicted a singularity at the eventhorizon, and we managed to avoid this by nding the Finkelstein set ofcoordinates that do indeed describe the entire spacetime, apart from thepoint r=0, and are understood to be a more general set of coordinates.Now there is a set of coordinates that is even more general than this asit has the additional property that allows us not only to describe future-directed paths passing the event horizon but also past-directed. In additionin this set of coordinates one can study what happens at the event horizonmuch easier as they allow the denition of Schwarzschild modes which areconned in a region close to the event horizon and vanish at innity. Thisset of coordinates is called the Kruskal coordinates and is given by:

ds2 =32G3M3

re−

r2GM (−dT 2 + dR2) (51)

for xed θ and ϕ coordinates22. Therefore the observation we referred to atthe beginning is the similarity of these coordinates to the Rindler coordinatesin Minkowski vacuum as given in (26) with the main dierence being thatthe metric now is depended on r, a necessary condition to have a curvedspacetime. Below you can see a diagram of the Schwarzschild spacetimegiven in Kruskal coordinates23:

22again spherical symmetry allows us to do so without losing generality23notice that the whole space is described by these coordinates now

22

Figure 4: Schwarzschild spacetime in Kruskal coordinates. Notice itssimilarity with the Minkowski spacetime in Rindler coordinates. The basicdierence is that now regions II and III are also described and that is

where the singularity lies. Note also that there is still no communicationbetween the regions I and IV, no signal can be sent from one region to theother. The black and white holes in regions II and III respectively work as

a non-traversable wormhole between the two at regions.

The spacetime is divided in 4 regions:1) Region I is the original spacetime which is observable by physical

instruments. At T=X is the event horizon2) Infalling matter enters region II and will fall into the singularity at

r=03) Region III is the time reversal of region II. It is called the white hole

region4) Region IV has the same properties as region I which means that is

describing an asymptotically at region inside radius r = rs.It is still true that the metric takes a constant value at the event horizon

i.e. there is no singularity there. If one evaluates the components of thecurvature tensor, will nd that these diverge as r−3, i.e. at r = 0 there willindeed be a singularity, and what physically happens as we approach it isthat strong tidal forces appear and will shread apart anyone approachingthe singularity.

The observation of the similarities with the minkowski metric space leadsus to the conclusion that we have the same quanta again appearing in thewhole space, with the case that they appear right at the event horizon (inboth future and past event horizons) being particularly interesting. In thisscenario, the particles split following the outgoing geodesics shown in gure[1] in the Finkelstein diagram.

23

In what follows, we want to have a good understanding of the space thatwe are going to study therefore we want to make the asymptotic behavioraccessible to study. In addition we are only interested in the causal structureof the system (i.e. we want to describe null geodesics that dier only by amultiplication by a scalar to be assigned the same description) therefore weare after a way to simplify the description of the space so that reduntantinformation will not appear. To do so we need to apply a conformal com-pactication which is a transformation that takes advantage of symmetriesto do exactly what we want, to avoid redundant information. This is atransformation of the metric of the form:

gab = Ω2gab

with Ω a positive scalar function of the space time, which preserves thecausal structure of the spacetime and at the same time allows us to in-vestigate behavior at innity. Mathematically speaking the eect of thecompactication is to establish that the boundary of the spacetime is well-dened, which was not before as energy diverged at innity due to inniteredshifting. Pictorially the transformation has the following form:

Figure 5: Conformal transformation of space S where gab lives to spaceΩ(S) where gab lives. Ω(S) is a subspace of a bigger space S ′ and its maindierence from S is the fact that now the boundaries of the space are well

dened.

To be more rigorous, this compactication allows us to dene a properHamiltonian formulation of general relativity in an asympotically at space-time by casting the energy as a boundary integral and hence allowing thedenition of the procedure of evaporation which otherwise would not havebeen possible.

This conformal transformation can be read from spacetime diagrams aswell. The resulting spacetime diagrams (after applying the transformation)are called Penrose diagrams. Here you can see Penrose diagrams for theMinkowski spacetime and the Schwartzschild spacetime respectively:

24

Figure 6: Penrose diagram for Minkowski spacetime. i0 represents spacelikeinnity where spacelike geodesics begin and end. i−represents timelike pastwhere timelike geodesics begin. i+ represents timelike future where timelikegeodesics end. I−represents lightlike past where lightlike geodesics begin andI+represents lightlike future where lightlike geodesics end. Regions I and IIare assumed (because they are not really, it is just the way we represent the

space in Rindler coordinates that induces these event horizons) to beseperated by an event horizon where the positive and negative modes

discussed in the previous section appear on the right and left of the horizonrespectively.

Figure 7: Penrose diagram for Schwarzschild spacetime. This diagram isobtained by a conformal transformation of the initial innite space in gure4 preserving the causal structure of the space so that everything has thesame interpretation as before. Compare to gure 6 the dierence is the

appearance of singularities in regions II and IV representing the area inside

25

the event horizon of a black hole and a white hole (basically an object thatreects everything) respectively.

Considering the part of the penrose diagram that corresponds to theobservable by physical instruments spacetime is enough for our purpose:

Figure 8: Black hole during the process of Hawking radiation. Note thatonly outgoing waves are present at the lightlike future I+and only incomingwaves reach the lightlike past I−. Also on these surfaces, we have after thecompactication a Hilbert space describing a free quantum eld theory. The

above is part of a Penrose diagram.

Hawking explicitly proved the phenomenon that appears at the eventhorizon by following an observer with a purely positive free-fall frequencymode traveling on a geodesic starting from future null-innity I+ and movingtowards the future event horizon of a collapsing Black hole H+ and thenreecting on the singularity (using geometrical optics), traversing the pastevent horizon H− and nally reaching the past-null innity I−24. In doingso Hawking was in a position to check if the so called free-fall vacuum at thehorizon results indeed from a generic state that exists prior to collapse ofthe matter that forms the black hole. This sounds like a solid expectationas the initial state is assumed to be the vacuum of ultra high frequencymodes, the time and length scales of which are much shorter than those ofthe collapse. He conrmed this by showing that at the past null innitythe mode associated with this observer was still purely positive free-fallfrequency and hence concluding that the observer was still in vacuum when

24In this process one has to match the metric in the at spacetime outside the blackhole to the metric inside the black hole

26

it reached the past null innity and hence he must have been in vacuum inbetween as well25. He showed at the end that the Minkowski vacuum looksto an outside observer as a thermal state again with temperature Tr = ar

2π

just like in the Unruh eect.However we have a small dierence here because now we can't assume

that at innity spacetime still looks at and hence we can't assume thatacceleration is uniform. The radiation near the horizon will propagate (dueto this non-at spacetime geometry) with an appropriate redshifting of itstemperature, as the acceleration on which the temperature depends on willbe redshifted. It will be given by:

T∞ =Vr≈event horizon

V∞Tr≈event horizon = lim

r→2GM

Vrar2π

=κ

2π(52)

where Vr≈event horizon and V∞ the redshifting factor at event horizon andinnity respectively which is given as26:

Vr =√−KµKµ =

√1− 2GM

r(53)

where K is the timelike killing vector ∂t. Also ar = GMr√r−2GM

is the magni-

tude of the acceleration of the free falling observer and κ = limr→2GM(Vrar)is the surface gravity which essentially gives the acceleration that an ob-server at innity has to cause to a particle in order to keep it at a positionnear the event horizon. This redshifting of the radiation converges to a nitevalue as we move towards innity and not to 0 as it would have happened ifwe were in at spacetime. The eect described above is called the Hawkingeect. One important property of these photons worth mentioning here isthe fact that the outgoing Hawking radiation state is independent of theinitial state of the photons. This was also true in the Unruh eect. Thereason we mention this is to note that these photons will not provide to hidethe information after the black hole has evaporated.

25if we assume that an eective eld theory description of the system in the in betweentime is valid

26V∞ ≈ 1

27

Figure 9: Particles and antiparticles near the Black hole event horizon.The particle escapes at innity while the antiparticle is being dragged insidethe black hole until it hits the singularity. The radiation corresponding to

particles that escaped in this way is called Hawking radiation.

A rst consequence of this eect is the fact that since the Black holeis radiating it must be losing energy and consequently mass. Since thisprocess does not seem to have an end then this means that a Black holewill be evaporating until it disappears. This however can not be conrmedpractically because even a small black hole of size of order 10−13cm takesabout the present age of the universe to evaporate. Hence any black hole ofa signicant size will take way much more time than the age of the universeto evaporate.

A second consequence is the violation of the no hair theorem for Blackholes which says that a stationary Black hole can be completely describedby its mass, its charge and its angular momentum27. Obviously when wemeasure a property of the outgoing particle then we will obtain informationabout its entangled counterpart which is inside the Black hole. Hence weobtain information about the Black hole other than the 3 mentioned abovewhich should not be possible since light can't escape from a Black hole. Thisis the famous Black Hole information paradox.

Considering the above problems one is forced to choose one of 3 possiblecases of what could be the end result of the process of evaporation viaHawking radiation:

1. We can accept that when the black hole that is being evaporatedreaches the planck size, for no obvious reason at the moment, theevaporation process is forced to stop leaving behind this small blackhole which we will call a remnant. However if we want to maintainthe purity of the initial state of the black hole then we should expectthat an extraordinary amount of entanglement entropy should existin the black hole. If we are to accept that black hole entropy reallymeasures the number of microstates a black hole can be into then sinceentropy is proportional to the area of the black hole then we see thatthe size of the black hole bounds the entropy it can have. Howevera black hole of planckian size would be expected to have a very highenergy (temperature) and hence also very high entropy, much higherthan the one allowed. Hence we would face a contradiction.

2. Another possibility would be for the black hole to evaporate com-pletely. In this case however from energy conservation arguments, the

27which they can only be measured indirectly, i.e. measurement does not require infor-mation to come to the observer in the form of a signal

28

nal burst of photons can not have sucient entropy in order to purifythe earlier radiation and hence the radiation will be in a mixed state.This violates quantum mechanics and it implies that information islost.

3. And the last but the one that is going to be supported in what followsis that the state of the Hawking radiation coming out of the blackhole is not really in a thermal state. It only looks like that for smallsubsystems, i.e. when a small number of photons is observed. But ifthe whole of the Hawking radiation is seen together after the processof evaporation has taken place then one can conrm that the state isindeed still a pure state. The information is essentially carried out insubtle correlations.

There are reasons to believe that perhaps the 3rd option is the more prob-able to be true. A serious problem that one realizes to arise if he studiescarefully the process under which Hawking radiation comes out, is that ofmodes appearing very near the event horizon and more specically within aplanckian distance a region where we expect to have an innite number ofmodes as it was explained before. This problem is related with a UV diver-gency or else called a short distance divergency. These modes are expectedto have energies larger than the Planck limit and hence lie in a regime ofphysics where the laws we know are not expected to hold as the eect ofgravity on quantum physics should become signicant. Currently we do nothave a theory to describe this. It is possible to nd a limit on the time themodes come out of the black hole after the initial shell falls in, after whichthese high energy modes are expected to appear. There are various sugges-tions to avoid this problem but there are still a lot of physicists that arenot satised with the proposed solutions and consider the actual solution ofthis problem to hide also the answer to the black hole information paradox.This problem is termed the transplanckian problem.

29

Part III

Complementarity, Firewalls, the

Harlow-Hayden Conjecture and

the Papadodimas-Raju proposal

It can be shown that the Black hole information paradox essentially arisesfrom the conict of quantum theory with general relativity therefore under-standing this paradox may give us an important cue as to how the two canbe put together. In what follows, some proposed ideas to resolve the paradoxare examined. References [5], [6] and [7] are the ones that are related to thematerial presented here.

3 Complementarity

Black hole complementarity is essentially a set of axioms/postulates regard-ing Black holes [5]. These are:

1. The process of formation and evaporation of a black hole, as viewedby a distant observer, can be described entirely within the contextof standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoingHawking-like radiation.

2. Outside the stretched horizon28 of a massive black hole, physics canbe described to good approximation by a set of semi-classical eldequations and more specically those of a low energy eective eldtheory with local Lorentz invariance.

3. To a distant observer, a black hole appears to be a quantum systemwith discrete energy levels.

4. A freely falling observer experiences nothing out of the ordinary whencrossing the horizon. 29

28A stretched horizon is a membrane of length the Planck length where measurementsof position can't be performed accurately due to uncertainty principle

29This means both that any low-energy dynamics this observer can probe near hisworldline is well-described by familiar Lorentz-invariant eective eld theory and alsothat the probability for an infalling observer to encounter a quantum with energy E 1

rsis suppressed by an exponentially decreasing adiabatic factor as predicted by quantumeld theory in curved spacetime.

30

All of these postulates seem to be reasonable if we consider each one on itsown. However contradictions arise when we require that they all hold at thesame time. In the discussions that follow, 1 and 2 are assumed to be trueand the eort of all the proposed resolutions of the Black hole informationparadox is to somehow bring into the picture postulate 4.

4 Firewall

The problems for the complementarity solution to the black hole informationparadox arise as soon as we introduce a second observer in the picture.

4.1 Scrambled states

To start studying the inconsistencies that arise in complementarity we rstneed to dene the concept of scrambled states. Lets consider a Black holethat is formed from the collapse of some pure state and then it continuouson decaying. From postulate number 1 we understand that the state of theHawking radiation that is being emitted has to also be a pure state so thatinformation is not lost and hence we have a unitary evolution indeed. Soassume that the original pure state of N qubits is | Ψ0 >=| 000...000 > andthe unitary evolution is a randomly chosen 2N × 2N matrix U chosen fromthe group invariant Haar measure, which has the property that it scramblesthe original state or in other words, it thermalizes it [6]30. A scrambledstate is dened as the state for which any small subsystem has essentiallyno information, where by small subsystem is implied a subsystem that isdescribed by half the total number of qubits. But how do we determinewhether a state has suciently small amount of information to be consideredscrambled? To do this we need to dene a way to measure distance of statesand then quantify the above statement by measuring how far the state isfrom the maximally state. The distance measure that we can use to do sois the operator trace norm dened as:

‖M ‖1= tr√M †M

The actual state chosen using the matrix U mentioned above is:

| Ψ >= U | Ψ0 >=∑i

| ψi >E ⊗ | i >L (54)

where in the 2nd equality E and L refer to early and late radiation respec-tively i.e. to not yet scrambled and scrambled part of the radiation. The

30the idea behind using this random matrix is that the process of evaporation is socomplex that we can just assume that the state of the end result is randomly chosen fromthe set of possible states

31

states | i >L represent a complete basis of the small subspace L as after thescrambling we know that the scrambled subset must contain no informa-tion31 i.e. the reduced density matrix for this part should be ρL = 2−

M2 IL

where IL =∑i | i >L the identity matrix and M the number of qubits in

the scrambled part of the state i.e. the qubits in the late radiation. Alsothe structure of | Ψ > in terms of a sum of tensor products of early andlate radiation suggests that early and late radiation are entangled. The factthat we have the appearance of the small subsystem in the late radiation isguaranteed by the page theorem which says that for any bipartite Hilbertspace HA ⊗HB we have:

∫dU ‖ ρA(U)− IA

| A |‖1≤

√√√√ | A |2 −1

| A || B | +1

where | . |denotes the dimension of the corresponding hilbert space we seethat if indeed | B || A | as is the case we have now then the state is boundto be close to the maximally mixed state for any U chosen.

4.2 The Firewall argument

From the consideration of scrambled states above we see that the structureof the state of the Hawking radiation (which is split into early and late ra-diation) shows us that a qubit in the late radiation is manifestly maximallyentangled with a qubit in the early radiation therefore we can construct op-erators acting on early radiation the action of which on | Ψ > is equal toa projection onto any given subspace of the late radiation. This essentiallymeans that we can obtain information about late radiation from measure-ments on the early radiation. Another feature of the structure of the Hilbertspace describing the states of the Hawking radiation is the fact that the totalradiated energy is nite therefore the dimension of the Hilbert space mustbe nite and hence this is what makes it possible to compute a time thatallows us to distinguish between late and early radiation. This time as wewill see shortly is called the page time.

Now consider an outgoing Hawking mode in the later part of the radi-ation. This can be a localized wavepacket with width of order rs

32. Frompostulate 2 we can assign to this mode a unique lowering operator b

(1,2)k and

hence measure the number of photons that will be present in a given modeof the late radiation by projecting onto eigenspaces of b

(1,2)k

†b(1,2)k . Also from

31the physical meaning of this is that the available states to be occupied in the earlysubspace are much less in number than those in the late subspace therefore we assumethat the late subspace has almost no information

32Harlow is [9] considered a wavepacket extended in some time dierence dt insteadbut it is essentially the same idea

32

postulate 2 we can relate this Hawking mode to one at earlier times as longas we stay outside the stretched horizon which guarantees local lorentz in-variance as the space is almost at. This mode now would have a muchhigher energy as the mode would be blue shifted while propagated to a fardistance outside the event horizon physically meaning that energy should begiven in order to move the mode to an outer region.

In addition consider an infalling observer with the associated set of in-falling modes ω. We know from Hawking radiation eect above, i.e. fromequations (45) and (46) that33:

b(1) ∝∫ ω

0[Bc(1)

ω + Cc(2)†−ω ]dω (55)

b(2) ∝∫ ω

0[Bc

(2)k + Cc

(1)†−k ]dω (56)

where c(1,2)k and c

(1,2)†−k lowering and raising operators respectively for these

modes of the infalling observer and b(1,2) are annihilation operators corre-sponding to wavepackets rather than single frequencies as we had before.And it is exactly here that the contradiction arises. If state | Ψ > is an

eigenstate of b(1,2)†b(1,2) then it can't be also an eigenstate of c(1,2)k

†c(1,2)k and

hence state of the infalling observer can't be the Minkowski vacuum andhence postulate 4 would be violated, and as a consequence = observer at theevent horizon would observe high energy quanta. If now | Ψ > is an eigen-

state of c(1,2)k

†c(1,2)k then state is not eigenstate of b(1,2)†b(1,2) which means

that the evolution from the outgoing Hawking radiation at a later time tothe infalling observer is not unitary and hence postulate 1 is violated. In asentence the contradiction is the following: Purity of the Hawking radiationimplies that the late radiation is fully entangled with the early radiation, andthe requirement that the infalling observer sees nothing special at the eventhorizon implies that it is fully entangled with the modes behind the horizon.This constitutes a violation of the monogamy of entanglement. What thisactually means is that we are not allowed to use local eld operators on acurved spacetime because we should take into account the coupling of theeld with gravity.

This can also be seen as a violation of the strong subadditivity of theentropy:

SBH + SHA ≥ SH + SABH (57)

where A corresponds to a mode in the interior of the black hole, B to anearly Hawking mode i.e. to a mode in the near-the-stretched-horizon region(between the stretched horizon and the photon sphere) and H to an outgoing

33the dierence being that now we are averaging over a range of frequencies rather thanconsidering a single frequency

33

mode i.e. a mode in the black hole stretched horizon. H and A must bein a pure entangled state in order to give rise to a vacuum observed byan infalling observer which means that SHA = 0. Therefore SBHA = SB.Then (57) implies that SBH ≥ SB + SH . At the same time we know thatSB > SBH . Therefore we get that SH < 0 which is obviously a contradiction.

One could consider a rotating Black hole to check whether this contra-diction is still there. Initially he might think that this result holds only forlow angular momentum modes as the high angular momentum modes aretrapped behind a large barrier in the eective radial potential. However itis known that it is possible to mine energy from a Black hole by lowering anobject below the potential barrier and letting it absorb the trapped modes.Therefore one can show that the problem persists for rotating black holes aswell. It is proposed that the only way out of this problem is to accept theexistence of a Planck density of Planck scale radiation at the event horizon,which is called the rewall, that causes the observer to burn up and henceresolves the paradox.

5 The Harlow-Hayden Conjecture

The following chapter follows closely references [6] and [7].

5.1 Introduction

5.1.1 Coarse-grained and Fine-grained quantities

The system that describes Hawking radiation is a scrambled qubit as waspresented above. These kind of systems are characterized by quantities thatare of either of two avors, they either have coarse grained properties orne-grained properties [6]. The dierence between the two is basically thatthe degrees of freedom at time 0 and a later time t in a system characterizedby ne grained quantities will dier tremendously when even the smallestperturbation is applied on them, either in the initial conditions or on theHamiltonian. On the other hand coarse grained quantities are insensitive tosuch tiny perturbations.

A ne grained quantity in our system would be one that is measured byan operator that involves more than half the qubits, i.e. an operator on theearly radiation. The reason is the following: First one can observe that theoriginal qubit operators acting on each qubit alone are the Pauli operators34,hence any operator in the system should be expressible in the form of a sumof products of these operators. Furthermore we assume that we just wantto make a measurement on one of the qubits in the small system, which

34if we are to consider the simplest system possible

34

we know that it will be entangled to a hidden qubit in the big (remaining)system. It is possible to show then that the operator for this other hiddenqubit will not just be one of the original operators but it will rather bean extremely complicated combination of the original operators, involvingmore than half the qubits. Say that the operator acting on the single qubitin the small system is σ(1) and the complicated operator that is acting onthe qubit in the big system is τ(1)35. Then if we had measure the quantityσ(1) · τ(1) on the maximally mixed state and on the pure state respectively,assuming that it is in a singlet state, we would have obtained the followingresults:

< Ψmixed | σ(1) · τ(1) | Ψmixed >= 0 (58)

and< Ψpure | σ(1) · τ(1) | Ψpure >= −3 (59)

which means that it would allow us to distinguish between the two36. An im-portant thing to note here is that the operator τ(1) depends on the scrambledstate | Ψ > and hence to obtain it we need to know exactly what scramblingdynamics have taken place so that we will know | Ψ > accurately. Had weused an operator τ ′(1) corresponding to a scrambled state | Ψ′ > but withthe original state | Ψ > the result (58) would also be 0 and hence we wouldnot be able to distinguish the two states. This means that σ(1) · τ(1) isactually a ne-grained quantity.

However by postulates 1 and 2 we are supposed to assume that a typicalblack hole microstate, determined by using the Pauli matrices again, shouldbe a coarse grained quantity as coarse grained quantities correspond to timesymmetric states which is what we expect to have if 1 and 2 are true asall laws in semiclassical physics are CPT invariant. There is also anothermotivation why we would like to consider coarse grained quantities. Firstlets remind ourselves that the energy and entropy of a photon gas in 3+1dimensions are given by:

E ∼ V T 4

and

35This τ(1) is basically the corresponding operator that should be applied on the wholeof the scrambled state | Ψ > but that would have a non-trivia eect only on the bigsystem that would collapse the state in the same state that σ(1) would bring the wholestate if it had been applied on the whole of the state but with eect only on the smallsystem. So basically one would consider operators acting non-trivially only on | ψi >Eand operators acting nontrivially only on | i >L but that would both bring the initialstate at the same new state.

36we wouldn't be able to say which one is the pure and which one is the maximallymixed but we could say that they are dierent

35

S ∼ V T 3

Now if we assume that we are forming a black hole by compressing a gasof energy E which is equal to the mass of the black hole M (assumingnonrotating neutral black hole) into a volume V ∼ r3

s ∼ M3 with rs theschwarzshild radius, then we see that the entropy will be proportional to

S ∼ M32 ∼ r

32s ∼ A

34 . Now it was rst proposed by Bekenstein and then

conrmed by Hawking that the entropy of a black hole is proportional to itsarea, a principle also known as the holographic principle:

S ≤ A

4G

The entropy of a black hole can be seen as the number of dierent microstatesthe black hole can be into. So we see that simply by compressing a gas, wedon't get enough degrees of freedom to match what is expected in a blackhole. Therefore to account for this the only way to increase the entropyis if we built the remaining of the black hole up by sending photons insuch a way so that radiation will not start coming out before we nish. Todo this we are bounded by the laws of semiclassical physics so we have todo it in a CPT invariant way. This process will essentially be the inverse ofevaporation. So we see that another puzzle appears. In particular, the aboveimply that detailed microscopic physics occurring within the ne-graineddegrees of freedom, including the ow of microscopic information in theHawking emission process, cannot be seen in semiclassical eld theory. Thisis the origin of the apparent violation of unitarity in the Hawking evaporationprocess. The solution to this problem that we will see at a later point liesin the usage of two sided-schwarzschild on which ne-grained quantities willbecome coarse grained. This splitting of the degrees of freedom of the Hilbertspace into coarse-grained and ne-grained quantities is similar to the caseof dissipation. So if we manage to avoid having to implement this splittingit means that we manage to avoid dissipation by preserving the informationby transferring it to another space.

5.1.2 Distillable entanglement

To make our arguments more solid it would be good to have a way to quantifythe amount of entanglement between two systems. Therefore we need todene the notion of distillable entanglement, which essentially means thenumber of Bell pair shared by two subsystems. In our case we are interestedin the distillable entanglement from the small subsystems that appear afterthe scrambling of the initial pure state, or the subsystems one nds in theMinkowski vacuum at either side of the horizon [6]. Note it does not have to

36

be exactly a Bell pair, i.e. we do not require to have maximal entanglementbecause then we would have almost none in our system as the scrambling ofthe initial pure system leads to almost maximally mixed states and hencethese subsystems are also not maximally entangled. Pairs that are arbitrarilyclose to Bell pairs are called regulated Bell pairs.

5.1.3 Page transition

Page transition is an important concept of the argumentation that will follow[6]. It is based on a theorem which says that in a scrambled system of Nqubits, given any small ε there is a number of qubits N(ε), such that ifN > N(ε) then the number of regulated Bell pairs is equal to the smaller ofM and N −M where M is the number of qubits in a scrambled subsystemof the system of N qubits. As M is varied between 0 and N the followinggraph for the number of regulated Bell pairs i.e. the distillable entanglementD is obtained:

Figure 10: Distillable entanglement from a scrambled state with respect toM the number of qubits of the small subsystem. The page graph aboveessentially encodes the evolution of the initial and nal hilbert spaces

describing a free scalar quantum eld theory.

The physical interpretation of this is that until the Page time, Hawkingradiation comes out in a maximally mixed state carrying no information. Allthe information about the initial pure state leaks from the black hole afterthe page time. The above triangular graph is rather a simplication of howthe evolution of the distillable entanglement should look like. Determininghow exactly this graph should look like requires knowing the exact process ofevaporation and knowing this would actually solve the black hole informationparadox as it would mean that we have a quantum theory of gravity.

37

5.2 Reformulation of the Firewall argument

Let H, B, R and A represent the near the stretched horizon region (betweenthe stretched horizon and the photon sphere), the black hole stretched hori-zon, the beyond the photon sphere region till innity and the the interiorof the Black hole respectively. Then a resume of the arguments included inthe subsection about the Firewall is the following:

1) In the infalling frame Bi and Ai are maximally entangled, where Bi

and Ai modes i in the respective regions.2) In the exterior frame Bi and HBi are maximally entangled, where HBi

is the subsystem of H where the modes that are partnered with B live.3) Maximal entanglement is monogamous.This implies that Ai and HBi should refer to the same degrees of freedom,

i.e.:A = HB (60)

Perhaps another way to say this is that H is the hologram at the horizon,that represents the interior A and this relation is expressed by a mappingH ⇔ A.

A rst problem that arises is the fact that this map must be non-linear,i.e. the relation between Ai and operators in H must depend on the initialstate | Ψ0 > since the particular form of HBi is state-dependent as explainedin section 5.1.1 (identifying τ(1) with HBi). However it is not really a verybig problem as linearity can be restored by embedding the system in a largersystem, e.g. one that contains the operators that create the shell.

Now based on the above relation, to have an uncorrupted black holeinterior it is necessary that the distillable entanglement D between B andH should be equal to the number of qubits in A. However as evaporationtakes place, distillable entanglement reduces until it reaches 0 (at the Pagetime), while the Black hole is still to disappear.

Evaporation is depicted schematically in the following gure:

Figure 11: The evolution of: the number of qubits at the stretched eventhorizon H, denoted by NH , the number of qubits in the near the event

horizon region B, denoted by NB and the number of qubits in the after thephoton sphere region R, denoted by NR.

38

It can be shown that as evaporation takes place, D remains maximal untilthe cusp point. Also until that point entanglement is only shared betweenH and B meaning that the radiation eld is entangled with the remaining ofthe black hole as evaporation takes place. The point of change is the pointwhere NH is half the total number of qubits. After this point D reduceslinearly to 0 at the page time:

Figure 12: Distillable entanglement between B & H.

while the Black hole has not necessarily evaporated by that time i.e. thenumber of qubits in region A is not zero yet. Therefore the necessity for aFirewall arises again.

Notice, however that implicitly in the above argumentation we assumedthat the interior modes, i.e. the interior degrees of freedom are build fromexterior degrees of freedom that are found near the event horizon, i.e. in B.Therefore need to examine what happens if this assumption is relaxed.

From conservation of amount of entanglement is understood that entan-glement between B and H moves to entanglement between B and R, i.e. toentanglement between the late and early radiation necessary to have a purestate of radiation at the end of the process. The following gure describesthe process:

Figure 13: Distillable entanglement between B & H contrasted to distillableentanglement between B & R with respect to time.

39

This means that the mapping we had before H ⇔ A has to be replacedby a map R⇔ A, i.e. the identication of the modes now should be:

Ai = RBi (61)

which implies a radically greater form of delocalization of information thanbefore, such as the one present in any holographic theory.

Now the problem that requires the existence of a rewall can be reformu-lated as following: Assume that an observer at R, Alice has in her disposala quantum computer, the input of which is the early half of Hawking radia-tion. She then uses it to output a qubit which we assume to be the distilledRBi . We further assume that she knows the exact initial state of the blackhole and the precise laws of evolution of all N degrees of freedom compris-ing the system. So if she has the degree of freedom RBi she can evolve itbackwards to take it with her in a journey towards the interior of the blackhole. If we further assume that this degree of freedom is one that would onlyappear well after the page time (as we know the degrees of freedom in thelate radiation are much more than in the early radiation) then the followingcontradiction arises: Alice would be in a position to jump into the blackhole, carrying RBi , in time to meet the original Ai and it partner Bi. Alicecould then check whether her version of Ai is entangled with Bi somethingthat would violate entanglement monogamy and hence require the existenceof a Firewall. 37

But there is a possible way out of the necessity of a Firewall. It mightbe physically impossible for Alice to distill RBi in time to bring it back tomeet Ai.

5.3 Strong Complementarity

Given that nobody has ever managed to really see an implementation ofblack hole complementarity or nd a rigorous mathematical description ofa rewall, it was natural for physicists to search for a better theory. Toexamine how it is possible to avoid having a Firewall, based on the argu-ment regarding the time it is required to distill RBi , we have to look atAlice's (infalling observer) and Bob's (exterior stationary observer) causalpast patches.

In looking in Alice's past patch we consider a space-like hypersurfacepassing through two degrees of freedom A and B. From complementarity weknow that they are entangled in Alice's frame. If we assume that these twomodes are well after page time, then this means that the slice dened bythe aforementioned space-like surface must intersect more than half of theoutgoing Hawking radiation.

37Note here however that apart from this contradiction no violation of the principlesof quantum mechanics is observed hence the experiment is in principle possible.

40

Figure 14: Alice's causal patch

Now we can consider a similar thing in Bob's frame. Bob will also seemode B, as well as the outgoing radiation that was seen by Alice but itwill not see mode A which is inside the black hole. Now consider again aspacelike hypersurface passing from B and RB. It is easy to conclude thatthe contradiction due to the violation of the monogamy of entanglementappears again.

Figure 15:Alice's causal patch superimposed with Bob's causal patch.

Consider also the stretched horizons from the perspective of each ob-server. They are in principle dierent. This is also depicted in gure 11.

41

But stretched horizons are dynamically involved in the production of Hawk-ing radiation. Therefore the two description of the production of radiationcannot be exactly the same.

Figure 16: Alice's causal patch superimposed with Bob's causal patch, nowincluding the stretched horizon for both of them as well.

At the level of coarse grained properties of the radiation, the descriptionsof Alice and Bob should agree on the overlap region that includes B but theywill not agree on RB i.e. on late radiation modes. That is because RB is anextremely ne-grained quantity and hence since the stretched horizons areeven a little bit dierent in the two causal patches the two descriptions ofRB should be dierent. Therefore for Alice the large-scale entanglements ofa pure but scrambled state will not be apparent. Similarly Bob's descriptionwill not include A.

Therefore the proposed solution is that both stories are complementaryin the quantum sense, i.e. each causal patch should have its own quantumdescription and therefore no observer can conrm both stories simultane-ously.

5.4 The Harlow-Hayden conjecture

Heisenberg's realization that it is operationally impossible to measure boththe position and momentum of a particle could have been dismissed on thebasis that not knowing how to measure both of the quantities at the sametime with innite precission should not be a reason why one should not beable to do it. However this would have been a profoundly wrong retort.The correct interpretation is that Heisenberg's operational limitation is an

42

essential part of the consistency of a new type of theory where particle nolonger means what it used to. Similarly complementarity can be viewed asa consistency condition, but it is however not the full story. It is not thenew theory, like the description of particles as waves was in the Heisenberg'stheory case. It is understood that physicists should still look for such theorywhich could then be checked based on the complementarity consistency test.

Elaborating on the above idea (about the proposed strong complementar-ity principle) Harlow and Hayden realized that even though the Gedanken-experiment described above (with Alice entering the Black hole and com-paring the entanglement of her distilled qubit with a qubit inside the backhole that we already know that is entangled with a mode in the photo-sphere) does satisfy quantum mechanics, it may not satisfy the principles ofrelativity both special and general [7] and hence they tried to take advan-tage of this to give a more sound description of the strong complementaritypresented above. In fact these two theories, special and general relativity,impose limitations on information such as locality and holographic limita-tions. Based on these limitations Harlow and Hayden conjectured that thequantum computer that Alice has at her disposal will require exponentialtime to distill RBi which is longer than the time it takes for any observer toreach the singularity of the black hole and hence it is in principle impossiblefor an observer to observe the violation of the monogamy of entanglement.

5.4.1 What is possible?

Let HR⊗HC be the Hilbert space of the Hawking radiation HR adjoined bythe Hilbert space of Alice's quantum computer HC . As it was explained insection 4.1 referring to the scrambled states concept, the state of the systemafter scrambling can be described as following:

| Ψ >=1√

| B || H |

∑b,h

| b >B| h >H UR | bh0 >R (62)

and hence what Alice has to do when we say that she is going to distill RBi

is to apply a unitary transformation Ucomp on HR⊗HC in order to undo URand put the bits which are entangled with B into an easily accessible formon system C. In order to do this Alice has to carefully choose the initialstate of her computer | Ψ >C so that she can achieve the following:

Ucomp : UR | bh0 >R ⊗ | Ψ >C 7−→| something > ⊗ | b >memory (63)

where | something >is any pure state of the computer and radiation minusthe rst k qubits assuming that state | b >is described by k qubits. It canbe shown that the probability of Alice picking the right initial state is

P = (2

ε)−2|C|(|R|2m(2k−1)−1) (64)

43

where ε 1 and m and k are the numbers of qubits in region H and Brespectively. It can also beshown that the time to nd such an initial stateis:

t ∼ e2 log( 2ε)|R||C||H||B| (65)

which essentially is the time to get close to any arbitrary state. Thereforeit is understood that this time is a double exponential in the entropy of thewhole system. By taking advantage of the structure on the variation of thestate, i.e. of the symmetries of the state space, it is possible to reduce thistime to a single exponential, but it is not possible to do better than this.This is the basis of the argumentation of Harlow and Hayden as they argue

that since for an averaged black hole in our universe it would take ∼ 10101040

to nd the state then there is no reasonable way to extract it.

5.4.2 Why is Alice's computation slower than the Black hole dy-

namics?

We could reverse the task, so that instead of looking for a particular initialstate we are now looking for the desired unitary evolution given the initialstate. The idea is to see whether Black hole dynamics put any constrain onUR so that Alice will be able to implement it faster.

Before this a quick look on the time that the Black hole takes to evap-orate has to be taken so that one can be in a position to compare the twoand make reasonable conclusions. This time is needed as a function of thenumber of qubits required for the process. Generally this is not a very easytask, therefore lets consider a specic example, the Schwarzschild Black holein 3+1 dimensions. The entropy of this is proportional to the area of theblack hole (as was conjectured by Bekenstein and later proven by Hawking)which is proportional to M2 in Planck units, and it evaporates in a timeproportional to M3, the volume of the black hole. So since entropy is pro-portional to the number of qubits, this means that n ∼ M2 where n is thenumber of qubits and hence the time to evaporate is t ∼ n

32 . In the above we

assumed that the evaporation process is described by an adiabatic model.This means rstly that we need a Hilbert space in which we can have blackholes of dierent sizes:

H = ⊕nfn=0(HBH,nf−n ⊗HR,n) (66)

the dimensionality of which is nf2nf . A unitary transformation increases n

by one each time that is applied and a black hole is considered old whenn >

nf2. In addition one can observe that, if it is assumed that there is no

width in the energy of a black hole formed in collapse, then it is possible torecast the whole dynamics as unitary evolution on a smaller Hilbert space of

44

dimension 2nf , but with the subfactors changing in time, and hence expectto have the following type of evolution (if we assume a specic initial state):

Udyn | 00...00 >init=1√

| B || H |

∑b,h

| b >B| h >H UR | bh0 >R (67)

So to actually draw some conclusions now, we assume a particular initialstate, say | 00...00 >, and we want to see how we can obtain this matrixUdyn that can lead us to state (61). We can split this matrix in a productof a mixing matrix Umix :

Umix | 00...00 >init=1√

| B || H |

∑b,h

| b >B| h >H | bh0 >R (68)

and a matrix UR :

UR1√

| B || H |

∑b,h

| b >B| h >H | bh0 >R=

1√| B || H |

∑b,h

| b >B| h >H UR | bh0 >R (69)

As you can see this UR is acting on all of the qubits and since Alicedoes not have access to qubits in B and H while she is doing the distillationthis means that she has to brute force her construction of U †R and this willtake a lot of time. This is analogous to the problem of time reversing acomplex chaotic system when even a small amount of information is lostbefore reversing it.

By recasting the problem of distillation into a problem of error correction,which it makes sense to do so as we can view the problem of Alice's ignoranceabout the state of the qubits in B and H as her trying to correct for an errorin the data she received, we can then see how the exponential time couldpossibly arise. If further we allow for generic initial state, which means nowthat the evolution takes the form:

Udyn | i >=1√

| B || H |

∑b,h

| b >B| h >H UR(i) | bh0 >R (70)

then decoding takes necessarily exponential time in n which means that timedependence on n for distillation is at a higher scale than that of the time forevaporation, or dierently put the two processes belong to dierent classesof complexity. By using arguments from quantum computation theory wecan show that it is in principle not possible to achieve what the paradox issuggesting and hence problem is resolved without the need of a rewall.

45

5.4.3 Scott Aaronson argument

Scott Aaronson came up with yet another reason that the aorementioneddistillation should not be possible to be eciently implemented. He arguedbased on the impossibility of inverting one-way function. These are func-tions that are easy to evaluate but hard to invert, meaning that if we have amap f, which we think as a an example of one-way function, that maps m-bitstrings to n-bit strings then there should exist a polynomial sized circuit Ufsuch that:

Uf | x, 0 >=| x, f(x) >

but at the same time there should not exist a polynomial sized circuit Uf−1 6=U−1f such that:

Uf−1 | 0, f(x) >=| x, 0 >

To see how this translates to our problem imagine that we are given asthe initial state of the outside of the black hole the following:

| ψ >=1√

2 | H |

∑h1,h2

| h1, h2 >H (| h1·h2 >B| f(h1), h2, 0 >R + | h1·h2+1 >| f(h1), h2, 1 >)

which is a state with not even classical correlations between the qubits inH (near the event horizon) and the qubits in R (far away in the radiationeld). h1 · h2 denotes inner product of strings of qubits and then answermodulo 2. From previous discussions we can see that it is possible to nda circuit that implementing it will bring the state of the black hole to thisstate in a polynomial time. However if we assume that there exists also a U †Rthat will allow us to distill a qubit from R then we are to run into troubles.More specically we must then have that:

U †R | f(h1), h2, 0 >R=| h1 · h2, g(h1, h2) >R

U †R | f(h1), h2, 1 >R=| h1 · h2, g′(h1, h2) >R

which means that if we are given (f(h1), h2) then we will be able to determineh1 · h2 . But then using this knowledge we will be in a position to invertfunction f . For example by using the fact that we know f(h1) and also bydetermining h2 to be 1 for the rst qubit and 0 for the rest then we canknow what h1 will be.

The existence of such functions is crucial in quantum cryptography andis widely believed that they do exist. Hence the conclusion of the aboveparagraph is that the assumption that we can eciently distill a qubit fromR must be wrong.

46

6 The Papadodimas-Raju proposal

In quantum eld theory one naturally studies the correlation functions oflocal operators. The idea is that one of the local operators correspond tothe system to be measured and the other one to the state of the measuringapparatus which we can imagine moving on the spacetime. However incurved spacetime this is only possible if we have no coupling of the localoperator representing the apparatus with gravity, a scenario where we canmake sense of local ideas such as eld operator at a particular point inspacetime. As the apparatus has to be composed of matter and as there isno matter that will not couple to gravity, another quantity is a much moresuited candidate to study in order to determine the evolution of states onthe spacetime. This is the S-matrix and is dened as:

P (χ | ψ) =|< χ | S | ψ >|2

where | χ > is the in state i.e. state in innite past and | ψ > is the outstate i.e. state in innite future, and P (χ | ψ) is the probability of obtainingthe out state given the in state. This S matrix is basically the unitarytransformation of the state U that we were after when we were explainingthe harlow hayden conjecture.

So all the information we obtain are from this S-matrix. However incompactifying the space and hence reducing the information available wealter the system in such a way that causes problem in dening the S-matrix.The proposal that follows by Papadodimas and Raju tries to do exactlythis. To nd a well dened S-matrix that will describe the experience of theinfalling observer.

6.1 Short description

Another attempt to solve the black hole information paradox was broughtto light only recently by Kyriakos Papadodimas and Suvrat Raju [8]. Theireort is based on the preexisting principle mentioned before, called the holo-graphic principle which states that the degrees of freedom in a region ofspacetime scale with the area of its boundary. More specically the twotried to take advantage of a specic realization of this principle, namelythe AdS/CFT correspondence which conjectures that any actions appliedon the bulk of the theory which is assumed to be described by an anti-deSitter spacetime can be related to actions on the boundary of the spacetimedescribed by a conformal eld theory. So Papadodimas and Ragu assumedthat the black hole's interior is indeed described by an asymptotically anti-de Sitter spacetime 38 and they tried to use the exterior CFT operators, i.e.

38The denition of a space being asymptotically AdS is that its boundary is timelikeand that its internal part is desribed by: ds2 = 1

cos2ρ (−dt2 + dρ2) with ρ ∈ [0, π2 )

47

the ones that are found at the boundary of the spacetime, the event horizon,in order to nd how the interior operators should look like. One of the greatadvantages of using the AdS spacetime is that the cases of black holes wherethe problem appears reduce as it is possible to seperate the black holes asold and young black holes after noticing the following phenomenon: The ra-diation reaching all the way at the boundary of the spacetime i.e. at I+andI− surfaces reects back and hence if this happens faster than the time theblack hole takes to evaporate then this means that the black hole will neverdisappear as the radiation will contribute in building the black hole againonce it reaches the event horizon of the black hole. As was explained abovethe black hole information paradox appears only after the black hole hasevaporated away because otherwise we can just assume that the evapora-tion remaining outside the black hole is in a thermal state and hence itcontains no information. So the problem remains only in those black holesthat evaporation is faster than the event of the radiation reaching back tothe event horizon after reecting at the boundary which we are going to callyoung or small black holes. Note however that there is also a problem withassuming the spacetime to be Anti-de Sitter, and that is that there exists notrivial way of dening an S-matrix on it. Papadodimas and Raju proposalmay seem to be doing so but they only managed to do this for low energieswhere the spectrum of the Hilbert spaces is pretty clean.

Essentially the assumption of Papadodimas and Raju was that the de-grees of freedom inside the black hole and the degrees of freedom in earlyradiation are essentially the same, since they are related by AdS/CFT, andthis is how they proposed that the puzzle of the black hole informationparadox might be resolved. Hence this is another A = RB type approach toresolve the paradox (meaning that is also based on the fact that the degreesof freedom in the regions are essentially the same). Following this assump-tion, they showed that if gravity, which is the governing force of the evolutionof the state | ψ > describing the black hole, is to be understood as a processin a unitary quantum mechanical framework then the bulk local CFT opera-tors ΦCFT (x) which can be interpreted as spacetime points via the AdS/CFTshould appear in the cases that we want to study (or can currently study) inlow point correlation functions (i.e. few spacetime points in the bulk) in theblack hole state | ψ > of the form < ψ | ΦCFT (x1)...ΦCFT (xn) | ψ > wheren should be small39. This is a consequence of their assumption i.e. that weare allowed to use AdS/CFT correspondence. 40 Essentially the existence ofthe low point correlation functions mentioned above can be understood as

39in agreement with the structure of an anti de Sitter spacetime which is assumed todescribe a very sparse spacetime including very few spacetime points

40Much of the usefulness of the duality results from the fact that it is a strong-weakduality: when the elds of the quantum eld theory are strongly interacting, the ones inthe gravitational theory are weakly interacting and thus more mathematically tractable

48

coming from working in the framework of a low eective eld theory. A keyfeature of their end result was that the mapping between CFT operators andthe bulk-local operators ΦCFT (x) depends actually on the state of the CFTand this result links this work with what was concluded in the last section,as what it means is that the operator ΦCFT (x) has a physical interpretationonly at a given state, and hence indicates that the initial assumption mustbe correct.

6.2 Bulk operators

In the Harlow-Hayden conjecture it was shown that the interior operatorswere explicitly split into two categories the coarse and the ne graded oper-ators. From the perspective of this approach now there is no need for suchsplitting although the splitting does appear in simple cases. To nd the formof the interior operators we begin by studying the exterior operators whichwe are going to construct out of free eld operators.

We denote the free eld exterior operators, which are approximatelyequivalent to single-trace operators on the boundary, by Oi

ωn,m where i is aconformal primary index, ωn is a mode in frequency space and m is the an-gular momentum. Now for the interior operators denoted Oi

ωn,m , which arealso referred in the literature as mirror operators, we have two requirements:rst that they will commute with the exterior operators and second that theywill be entangled with them. Since we are in the eective eld theory regime,it is enough to consider a small algebra Λ41 subset of the algebra of thepossible operators, the set of operators of interest sort of. Intuitively, we arerestricting the possible measurements an infalling observer can perform tothe set of the ones that are easily computable by him. With the assumptionmade above, that the bulk operators that we are talking about describe aneective eld theory, Papadodimas and Ragu showed that the two conditionstranslate into:

[Oi1ω1,m1

, Oi2ω2,m2

]Oi3ω3,m3

...Oikωk,mk

| ψ >= 0 (71)

Oi1ω1,m1

| ψ >= e−βω12 Oi1

−ω1,−m1| ψ > (72)

with all the operators belonging to Λ. These equations only hold under theassumption that k N where N is the central charge of the theory. Theassumption is dropped when the state | ψ > is close to being the thermalstate.

Then we go a step further and we construct local CFT operators inside

41doesn't really have to be an algebra

49

the black hole based on the operators from eld theory:

ΦiCFT (t,Ω, z) =

∑m

∫ω>0

dω

2π[Oi

ω,mg(1)ω,m(t,Ω, z)+Oi

ω,mg(2)ω,m(t,Ω, z)+h.c.] (73)

where g(1)ω,m are the analytic continuations of the left-moving modes from

outside to inside the black hole42 and g(2)ω,mare right moving modes inside

the black hole. The right modes can be understood as the very energeticmodes that propagate through the infalling matter using geometric optics,as was explained in section 2.2, and reect at the singularity, hence thename mirror operators to the operators that accompany these modes. Animportant remark to make is the fact that the modes inside and outside theblack hole refer to the same degrees of freedom now. The above constructionphysically means that if we consider the CFT correlators:

< ψ | Φi1CFT (t1,Ω1, z1)...Φin

CFT (tn,Ωn, zn) | ψ > (74)

will see that they behave like those of a perturbative eld propagating inthe AdS-Schwarzschild geometry, where the AdS-Schwarzschild geometry isguaranteed by requiring that apart from the state being in equilibrium asit was mentioned above, also the energy at this state < ψ | HCFT | ψ >is much larger than the central charge N . In this limit gravity is classicaland operators map single states to single states and multiparticle states tomultiparticle states of the same number. In section 6.3 we will see that whenthis fails is exactly when the problem appears.

Now something that we would also like to have is for the horizon of theblack hole to remain smooth. This translates into the following requirement:

< ψ | Oi1ω1,m1

...Oj1ω′1,m

′1...Ojl

ω′l,m′

l...Oin

ωn,mn | ψ >=

e−β2

(ω′1+...+ω′l)Z−1β Tr[e−βHOi1

ω1,m1...Oin

ωn,mnOjl−ω′

l,−m′

l...Oj1

−ω′1,−m′1] (75)

where Zβ is the partition function of the CFT at temperature β−1 and notingthat the mirror operators have opposite frequency and angular momentathan the operators outside the black hole. This can be seen as a KMScondition for the equilibrium states | ψ >. Equation (75) is the one to usein order to dene the states | ψ >.

The resolution of the strong subadditivity puzzle The resolution ofthe paradox lies exactly in equation (71). If we represent the set of earlyradiation modes which is a part of the exterior modes Oi

ωn,m as R , the modesright outside the event horizon as B and the set of modes right inside the

42obtained by solving the equations of motion from the boundary in the CFT

50

event horizon Oiωn,m as H, just as we did in section 5.2, then the problem

is that for an old black hole, B must commute with both R and H ordierently put, that [Oi1

ω1,m1, Oi2

ω2,m2] 6= 0. The solution to the problem lies

exactly in the fact that when [R,H] is computed within low point correlationfunction is equal to 0 which means that essentially the two are describingthe same degrees of freedom so there is no problem with the fact that B hasto commute with both sets.

6.2.1 Choosing the bulk state

Modes g(2)ω,m in equation (73) can't be obtained by the same equations of

motion that we used for the left moving modes. To obtain them we have towork in a 2-sided AdS-Schwarzschild wormhole space and hence need twocopies of the CFT theory [9]. What the construction of Papadodimas andRagu actually does is to simulate the propagation of these modes in the2-sided bulk to a single copy of CFT by taking advantage of the mirroringthat takes place at the singularity. The assumption of Papadodimas andRagu of the space being asymptotically AdS has the following eect on thestate of the black hole (which is analogous to the usage of at spacetime inMinkowski space and equation (49)): The equilibrium state | ψ > in theirtheory can be split into 2 parts the left and right part corresponding to eachone of the two copies of the CFT we should have. So a general state | ψ >which is called a Hartle-Hawking state, should have the following form:

| ψ >=∑i,j

Cij | i∗ >L| j >R (76)

where| i∗ >L:= Θ† | i >R (77)

with Θ a CPT transformation, indicating that the two CFT are just a copyof each other. Notice that state in equation (76) is indeed an entangledstate. However Papadodimas and Ragu in their eort to satisfy equation(75) they went a step further and they assumed that

Cij ∝ δije−βEi

2 (78)

i.e. that C is equivalent to the thermal state. However Harlow in his latestpaper spotted that if corrections are going to be introduced to the eec-tive eld theory then black hole CFT states are not compatible with beingequivalent to the thermal state, i.e. equation (75) is not satised any more.He could't though give an answer to the problem as to what C should beequal to.

51

6.3 State dependence

Assume that the state of the black hole| ψ > as described by an infallingobserver is of the form of (76) and that indeed satises equation (75) i.e.it is an equilibrium state. Let the set of equilibrium states as seen by aninfalling observer be E ⊂ HCFT

43. The operators that this observer woulduse to characterize the state | ψ > in E will belong to the small sabalgebraA of the set of all operators, which corresponds to a subset containing thecommuting (when inserted in low point correlation functions) operators andmirror operators mentioned before. Now for observers jumping in the blackhole too early or too late dierent subset of the operators in A would haveto be used so also the set of equilibrium states for them will in general alsobe dierent.

For each one of the states | ψ > we can build a linear subspace wherethe excited states will live which we call Hψ and is equal to A | ψ >. It iseasy to observe that for these excited states to make any physical sense itis important that the spaces Hψ should not intersect because if it happensto have Aα | ψ >= Aβ | ψ′ >=| χ > for | χ > an excited state then we cannot really tell whether this excited state corresponds to the physical realitydescribed by state | ψ > or | ψ′ >.

The same problem can be translated into the impossibility to implement aunitary measurement problem. All unitary state-dependent measurements,such as the one we expect to have here, correspond to the following process:

| i >S1| j >S2−→| i >S1 Ui | j >S2 (79)

where S1, S2 the systems under consideration and Ui a unitary matrix thatdepends on the state of the rst system. To be more precise in our exam-ple we are interested in a unitary measurement performed using a state-dependent operator. This process would be described as following:

| i >S | 0 >PA| 0 >Pf

−→| i >S | ai >PA| 0 >Pf

−→∑j′

Cij′ | j′ >S | ai >PA| f(ai, bj′) >Pf

(80)

where in the rst case a measurement of the observable corresponding tooperator A was performed (say position), entangling system S and the clas-sical pointer PA, and in the second case a measurement of the observablecorresponding to the quantum operator f(ai, B) was performed. PA and Pfare the classical pointers, the set of values our apparatus can take. In thelatter case of measuring operator f(ai, B) one can see that the operator tobe applied depended on the value of the rst measurement. Also this secondmeasurement corresponds to entangling system S and the classical pointerPA to pointer Pf . This process corresponds to the following quantum circuit:

43Remember that everything is simulated on a single copy of the CFT

52

Figure 17: Unitary process for measuring A and then conditionallymeasuring f. Time goes up.

It is good to note at this point that parallels could be drawn of this situa-tion to the case of spontaneous symmetry breaking. Nevertheless the pictureregarding the measurement process described above is only the classical pic-ture of the evolution, i.e. the process that one would normally perform in alab. This is denetily not the case in a measurement performed inside theevent horizon of a black hole.

Now lets turn to the actual problem in hand. Making a measurementon the degrees of freedom outside the event horizon is relatively easy be-cause we are still in the classical picture. We have in this case the scenariodescribed in equation (80) with the state dependent operator being Oi

ωn,m

which essentially depends on where the state describing the black hole isand what the mass of the black hole is. Intuitively this could be seen as theobserver measuring his position relative to the black hole is and how big theblack hole is before jumping.

However measuring the interior operators is much more tricky. We don'treally know how these interior operators should look like44 because we can'tjust evolve the operators outside using the equations of motion for the eldsand hence obtain the operators. We would need a theory of quantum gravityfor this as we explained above. So for the sake of understanding the dicul-ties regarding the description of the interior operators, we just assume theexistence of an operator A that can be used to distinguish these operators,and hence we label them by an index corresponding to the eigenvalues of Aas Oa, which are the state-dependent operators we were looking for. Thebig disappointment lies in the fact that in order for the infalling observerto make such a labeling and hence decide in which state the interior of theblack hole is, he would have to make a very sensitive measurement of theblack hole, due to the large number of degrees of freedom that exist in theinterior of a black hole45, and this is something that we would not expect

44Note that we don't assume an eective eld theory describing the degrees of freedomin the interior of the black hole this time

45note that we are not in the AdS/CFT regime any more so since we are moving tohigh energies we cant assume that the number of degrees of freedom in the interior issuciently small

53

him to be able to do because the number of degrees of freedom his apparatuscan determine for him would probably be much smaller than the number ofdegrees needed to characterize a state in the black hole. The problem justas previously could also be translated as the commutator of A with the op-erators inside and outside the event horizon being too large and hence notallowing us to use eective eld theory any more. So we conclude that aunitary measurement of the type described by equation (80) can not be per-formed, if we are to require eective eld theory to still hold. In fact one canshow that in general any form of unitary evolution of the system togetherwith an apparatus can not take place. The argument for this is essentiallythe fact that the linear subspaces Hψ determined above do intersect andhence is not possible to distinguish between excited states or equivalentlybetween interior operators and hence can't have unitary evolution. The keydierence between our case and the unitary evolution we described above inthe classical picture is the fact that this time the sizes of the systems areinterchanged, i.e. now the size of the system to be measured, the black hole,is much bigger than the size of the apparatus that the observer will carrywith him in order to make the measurement.

To make this last point more apparent, we use a more neat way to see thestate dependence problem arising when we assume the nonlinearity necessaryfor the Papadodimas Raju proposal, i.e. that the theory that we are lookingfor should be one of the A = RB form where the interior of the black holeis dened to be just whatever is entangled with the exterior. To see theproblem assume that the following state is the initial state of the black hole:

| ψ+ >=1√2

(| 0 >B| 0 >HR + | 1 >B| 1 >HR) (81)

and that | 00 >AB + | 11 >AB is the state at the smooth horizon at A andB. Now the following 3 states:

| ψ− >=1√2

(| 0 >B| 0 >HR − | 1 >B| 1 >HR) (82)

| χ± >=1√2

(| 0 >B| 1 >HR ± | 1 >B| 0 >HR) (83)

which are obtained just by a unitary transformation of the state in (81)should also be allowed to be used to dene the interior modes based onthe exterior modes and should also be expected to have a smooth horizon.However by simply taking superpositions of these 4 states we can createstates where B is pure and not entangled with anything. As it was mentionedin a previous section such product states have divergent energy at the eventhorizon.

54

7 Conclusions

In this project we began by giving a short introduction to general relativityand continued with describing what black holes are. We proceeded by de-scribing the eect of Hawking radiation. We then described a problem thatarises exactly due to this phenomenon, the black hole information paradox.We gave a number of ways one can see this problem. First we presented itas a violation of the monogamy of entanglement. This we showed that wasequivalent to violation of the strong subadditivity principle. In the last sec-tion of the report we also gave a description of the paradox as the apparentviolation of unitarity of quantum mechanics.

Several attempts to resolve the paradox were also presented in chrono-logical order. First the complementarity principle resolution was presented.In this approach only one observer was considered and it was soon realizedthat introducing a second observer revived the problem. Therefore anothersolution was proposed to the problem: the rewall approach. This was aresolution that did not really solve the problem but it rather put it underthe carpet. The necessity to add a rewall at the event horizon by handin order to solve the issues mentioned above was not really a satisfactorysolution. Following this idea of solving the problem was the idea based oncomputational arguments. It was argued that if the problem was presentedas an observer being able to extract information about the black hole beforeeven jumping in, then there was no need to nd a resolution as practicallythe computation that the observer would have to make to extract any use-ful information about the interior of the black hole would take him moretime than it would take this observer to fall into the black hole and hit thesingularity. Of course it is obvious that although this suggested that theproblem would practically almost never appear, this did not consist a su-cient explanation with respect to the theory of the paradox. Hence naturallyHarlow and Hayden were lead to the conclusion that what actually preventsthe problem of appearing is the fact that dierent observers have access todierent slices of the spacetime. Not all observations are available to them.

Although the last idea of Harlow and Hayden appeared to be on theright track for giving an answer to the problem, it was not supported bya rigorous mathematical theory. Papadodimas and Raju however came upwith an interesting idea of taking advantage of the AdS/CFT correspondenceto give Harlow's and Hayden's idea a more solid ground in our opinion. Theybasically showed that their idea could be correct as long as we assumed thatthe interior of the black hole was described by an anti-de Sitter space andwe restrict ourselves to the energies relevant for an eective eld theory forthe description of the degrees of freedom in the black hole. In Harlow's lastpaper [9]we see that he shares some of Papadodimas and Raju ideas andpartially agrees with the way they treated the problem. Of course as we

55

saw even this solution requires some assumptions to give an answer to theproblem. For a theory that will be capable of solving the problem completelyit is necessary to understand how gravity would behave in large energies andfor this we need a quantized theory of gravity.

56

References

1. Spacetime and Geometry: An introduction to General relativity, Sean

Carroll

2. Introduction to the theory of Black holes, Gerard t'hooft, Utrecht Uni-

versity 2009

3. Introductory lectures on Black hole thermodynamics, Ted Jacobson, In-

stitute of theoretical Physics, University of Utrecht

4. Particle creation by Black holes, S. W. Hawking, Commun. math. Phys.

43, 199-220 (1975)

5. Black holes: Complementarity or rewalls?, Ahmed Almheiri, Donald

Marolf, Joseph Polchinski and James Sully arxiv:1207.3123

6. Black hole complementarity and the Harlow-Hayden conjecture, Leonard

Susskind, arxiv: 1301.4505

7. Quantum computation vs rewalls, Daniel Harlow, Patrick Hayden, arxiv:

1301.4504

8. State-dependent Bulk-boundary maps and black hole complementarity,

Kyriakos Papadodimas and Suvrat Raju, arxiv: 1310.6335v2

9. Aspects of the Papadodimas-Raju proposal for the black hole interior,

Daniel Harlow, arxiv: 1405.1995v2

10. Jerusalem lectures on black holes and quantum information, Daniel

Harlow, arxiv: 1409.1231v2

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