THE BLACK HOLE

INFORMATION PARADOX

Jan Schneider 1

29th January 2017

contents

1 Prerequisites 21.1 Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 von Neumann entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Gedankenexperiment - A first sign of trouble 42.1 Gedankenexperiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The qubit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Hawking Radiation in the Qubit Model 63.1 Quantum Vacuum & Qubit Model . . . . . . . . . . . . . . . . . . . . . . . 73.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Slicing spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Robustness of the Argument 13

1 jan_ thorben. schneider@ stud. uni-heidelberg. de

1

prerequisites 2

prologue

This report was carefully crafted for the compulsory master seminar “Quantum field the-ory in curved spacetime” at the University of Heidelberg. Albeit the diligent effort toexpress only clear and correct information, chances are an error might have sneaked itsway in. Feel free to contact me.

The black hole information paradox is puzzling physicists for four decades now. Untiltoday there was no solution put forward that the physics community agrees on.The story of the black hole information paradox, at least the story I will tell you, begins in1965, when the Kerr-Newman solution was found. It is the most general black hole solu-tion to Einstein’s field equations. However, it is only characterised by three (quantum)numbers: mass M , charge Q, and angular momentum J . In following years a conjectureby Wheeler was proposed that black holes are completely defined by these three numbers.He coined the phrase “black holes have no hair”. Although theorists still owe us a rigorousformulation of a theorem, today it is dubbed as the “no-hair theorem”. Still, we can sensethe long casting shadow of this no-hair “theorem”. It would imply that one were not ableto distinguish a black hole entirely made up of anti-matter from one which has sameM , Q,and J but is formed by matter. This “information destroying” property of a black hole isfundamentally in disagreement with our understanding of physics. We believe if we knewthe exact (micro)state of a system, we should be able to infer its initial state at least inprinciple.It was around that time when Stephen Hawking could proof in a rigorous manner thatin a classical context, there is no process which can lower the surface area of a blackhole

(dAdt ≥ 0

). While this is somewhat reminiscent of the second law of thermodynamics,

Hawking just thought of it as a purely mathematical analogy. Nonetheless, Jacob Beken-stein was inspired by this fact in 1972. He formally promoted the surface area of a blackhole to a real thermodynamic entropy [1]. Now the surface area of a black hole in Plancklengths would count the available microstates which form the same macrostate. And therewe rescued our information from being lost to the black hole! All black holes with sameM ,Q, and J might look the same, but in fact they might very well be made up of a differentmicrostate.However, in his landmark paper from 1975 Hawking argued that this association is notenough to conserve information [2]! Astonishingly, he derived the mechanism by whichthe temperature of a black hole arises only by means of quantum field theory in curvedspacetimes. Three years earlier Bekenstein argued the surface area truly is a classicalentropy, and now Hawking shows that the accompanied temperature to that entropy onlyarises in a quantum mechanical framework. By the same process the black hole also losesmass and thus decreasing in surface area.For the rest of this report I hope to explain Hawking’s argument.

1 prerequisites

First I want to make sure the reader is equipped with all the tools required to investigatethe paradox.

prerequisites 3

1.1. Density Operator

Definition 1 (Density Operator). A density operator ρ on a Hilbert space H is definedby three properties:

(i) ρ = ρ†

(ii) ρ ≥ 0, i.e. ∀ψ ∈H : 〈ψ|ρψ〉

(iii) ρ is a trace class operator and tr(ρ) = 1

Note: This definition implies that all eigenvalues are real, non-negative, and in the closedinterval from 0 to 1.

The density operator, or synonymously density matrix, is the most abstract object whichcharacterises a quantum mechanical state. It incorporates both the notion of a pure state- a ray in Hilbert space - and the notion of an ensemble - a mixed state. These notionsare expressed in the following equations:

ρ = |ψ〉〈ψ| ⇔ ρ is pure. (1)

ρ =∑

i

pi |ψi〉〈ψi| ⇔ ρ is mixed. (2)

tr(ρ2)≤ 1 and tr

(ρ2)< 1 ⇔ ρ is mixed. (3)

ρ ∝ 1 ρ is maximally mixed. (4)

1.2. von Neumann entropy

Definition 2 (von Neumann entropy). The von Neumann entropy is a proper extensionof the Gibbs entropy to quantum mechanical systems. In natural units it is defined by

S(ρ) = − tr(ρ ln(ρ)) = −∑

i

pi ln(pi). (5)

The von Neumann has the following properties:

(i) S(ρ) ≥ 0 and S(ρ) = 0 ⇔ ρ is pure.

(ii) S(ρ) ≤ ln(d) with d = dim (H ) and S(ρ) = ln(d) ⇔ max. mixed.

(iii) S(Uρ U †) = S(ρ) for all unitary U .

In the language of the von Neumann entropy a pure state has zero entropy. A mixedstates has non-zero entropy. Property (iii) is nice, because it means in particular thatunitary time evolution never changes the value of the entropy. This is in agreement withour understanding that quantum mechanics is completely reversible.

1.3. Bipartite Systems

Definition 3. A system AB is called bipartite if

HAB = HA ⊗HB.

gedankenexperiment - a first sign of trouble 4

Any linear operator factorises on a bipartite system. For example the trace:

trHAB(MA ⊗MB) = trHA

(MA) · trHB(MB)

This gives us the very natural definition for the partial trace:

Definition 4 (Partial Trace). Given a bipartite system AB with subsystems A, B, then

trB : MA ⊗MB 7→ trHB(MB) ·MA. (6)

The tensor product can be made explicit:

MA ⊗MB =∑ijkl

cijkl |ai〉〈aj | ⊗ |bk〉〈bl| (7)

This yields the explicit description for calculating the partial trace:

trB MA ⊗MB =∑ijkl

cijkl · 〈bk|bl〉 · |ai〉〈aj | (8)

The partial trace is a tool of utmost importance. In a situation where one has onlyaccess to a subsystem, applying the partial trace to the full density matrix and ‘tracingover’ the degrees of freedom which are not accessible to the observer yields a density matrixwhich describes the correct state on and all accessible information in the correspondingsubsystem. We call this a reduced density matrix.A bipartite system can be entangled. While the state of the entire system might be pure,the state in the subsystem A (or B) - for an entangled system - is mixed. Expressing thiswith the von Neumann entropy means that the entropy of a system might be zero, yet theentropies of its subsystems can be positive. To quantify this more, we shall introduce theentanglement entropy.

Definition 5 (Entanglement Entropy). The entanglement entropy of a subsystem A isdefined by the von Neumann entropy of the reduced density matrix ρA:

Sentanglement(A) = S(ρA) =: S(trB(ρAB)) (9)

2 gedankenexperiment - a first sign of trouble

Now that we are fully equipped to tackle the paradox, we shall encounter it for the firsttime. Later, I will try to explain it with a different approach.

2.1. Gedankenexperiment

Suppose we have a black hole. Also, suppose we have a machine which produces entangledpairs of particles (arguably the much easier part to imagine). Of each pair, we shall throwone into the black hole while we keep the other partner. And we proceed at a rate withwhich the black hole loses mass via Hawking radiation. The mass and thus the surface areaof the black hole will stay constant, yet the black hole’s number of microstates evidentlyincreases with every particle we have thrown in. Thus, we have lost the connection betweenentropy and the surface area, because every proper entropy should actually count thenumber of microstates. Now the reader might say, (s)he was never a fan of Bekenstein’sformula in the first place. However, it gets worse than that!

gedankenexperiment - a first sign of trouble 5

2.2. The qubit model

To see why things get worse, we need to turn our attention to the entangled pairs. A stateof such an entangled pair might be given by

|ψ〉 = 1√2

|0〉A ⊗ |0〉B︸ ︷︷ ︸=:|00〉

+ |1〉A ⊗ |1〉B︸ ︷︷ ︸=:|11〉

(10)

The learned reader immediately sees subsystem A is entangled with subsystem B in such away that if either one subsystem is measured in |0〉 (|1〉) the other must be in |0〉 (|1〉), too.Still, the state on the hole system is pure, i.e. the entropy of AB vanishes (S(ρAB) = 0).The density matrix of AB is given by

ρAB = |ψ〉〈ψ| = 12 (|00〉〈00|+ |11〉〈11|+ |00〉〈11|+ |11〉〈00|) . (11)

This yields the reduced density matrix for subsystem A

ρA = trB(ρAB) =2∑

i=0〈iB| ρAB |iB〉 (12)

= 〈0B| ρAB |0B〉+ 〈1B| ρAB |1B〉 (13)

=12 |0A〉〈0A|+

12 |1A〉〈1A| . (14)

This reduced density matrix is in diagonal form and proportional to the identity, thusmaximally mixed. We can read off the eigenvalues pi and compute the entanglemententropy:

S(ρA) = − tr(ρA ln ρA) = −∑

i

pi ln pi = −2 · 12 ln 12 = ln 2 (15)

This is a sensible result as the (exponential of the) entropy counts the number of entangledstates in subsystem A just as expected. So we learned that the particle in A has an entropyof S(A) = ln 2 > 0. We also know that entropy is an extensive quantity, i.e. N entangledparticles have an entropy of S = N ln 2. This entropy cannot decrease.Let us now return to our thought experiment, where we shall stop throwing things intothe black hole and let the evaporation proceed. We have collected N entangled particles.However, after the evaporation completed, there is nothing to be entangled with, leftaside a black hole! Yet, the quanta have non-zero entropy implying that they must be in amixed state, although we started with Equation 10, which is evidently a pure state. Thisevolution - from pure to mixed - is in direct contradiction with the Schrödinger equation.In density matrix language it is called von Neumann equation:

∂ρ

∂t= − i

h[H, ρ] ⇒ ρ(t) = U(t) ρ(0)U †(t), (16)

where U is the unitary time evolution operator. Inspecting the trace of ρ2 yields

tr(ρ(t)2

)= tr(ρ(t)ρ(t)) = tr

(Uρ(0)U †Uρ(0)U †

)= tr

(ρ(0)2

), (17)

which implies that pure states evolve into pure states, and mixed states into mixed states.This apparent non-unitary time evolution is often called the “black hole information para-dox”.

hawking radiation in the qubit model 6

comments One might object and propose that the infalling B quanta imprint theirentanglement onto the outgoing Hawking radiation. But this cannot happen, for theequivalence principle dictates all freely falling observers encounter nothing special at theevent horizon. Additionally, the (Kretschmann) curvature for a Schwarzschild black holeat the Schwarzschild radius is of order K ∝ 1

(GM)4 . Hence, the curvature is not too large toexpect QFT in curved backgrounds to break apart. We will later consider these thoughtswith more rigour and formulate a precise statement what we expect from a QFT in curvedbackground in the neighbourhood of the horizon to do.Moreover, we have had a glance at where to fix the information paradox: As we have

seen, the problem arises when we let the black hole evaporate away while we are still leftwith a set of quanta (the ones which escaped) with non-zero entropy due to entanglement.If we were able to somehow ‘get rid’ of the entanglement, e.g. imprint the entanglementof early escaping particles on later ones, such that the set of all escaping particles is ina pure state (S(ρ) = 0) but the earlier particles are entangled with the later ones, thenwe would have encountered an evolution from a pure state to a pure state. Here it iscrucial to recall that a state on a system might be pure (i.e. S = 0) but it can have anentangled subsystem (i.e. S > 0). Thus, it is a fallacy to believe entropy decreased; itis merely incorporated in the entanglement of early particles with later ones. The curvewhich describes this very course is named after D. Page.Here a brief overview of all involved entropies: The Bekenstein entropy corresponds tothe surface area of the black hole. The, let me call it Hawking entropy, corresponds tothe entropy of the radiation field (i.e. to the entanglement of the radiation with the BH).Over the lifetime of a black hole these three curves can be described as:

Bekenstein entropy: GM2 → 0Hawking entropy: 0 → GM2

Page curve: 0 → GM2

2 → 0

The page curve will for the first half follow the Hawking radiation, as the entanglementof the early photons increases. By the time of the crossing of Bekenstein and Hawkingentropy the page curve will turn and start following the Bekenstein entropy as the earlierphotons are no longer entangled with the black hole but rather with the later escapingparticles. The Page curve can only rise to roughly half of the Bekenstein entropy, becausethere is a limit to the rate with which the black hole can get rid of its entanglement withthe earlier escaped particles - namely only with the produced particles. Thus, we alsoexpect the Page curve to turn at around half of the lifetime.

3 hawking radiation in the qubit model

In an attempt to understand the black hole information paradox further, we shall turnourselves towards the Hawking radiation process. We leave our thought experiment behindto learn that the problem already arises when considering the Hawking radiation by itself.This section will closely follow [3].

hawking radiation in the qubit model 7

3.1. Quantum Vacuum & Qubit Model

Over the course of this seminar we have seen that vacua in curved backgrounds do mostlikely not agree with each other. They are connected via a Bogolyubov transformation.[4] and [5] as well as [2] have argued, the vacuum state for an infalling observer is givenby

|0〉infalling ∝ exp

∞∫0

dω2π e

− ω2T a†ω b

†ω

|0a〉 |0b〉 (18)

However, the exact treatment of this state is rather intricate and not particularly disclosing.For pedagogical reasons we will consider a simple model of the thermal Hawking radiation:a qubit model. Hence, we will think of Equation 18 as an infinite tower of excitations anddiscard all but the zero and first level of excitation:

|ψ〉 = 1√2(|0a〉 ⊗ |0a〉+ |1a〉 ⊗ |1a〉) (19)

This is of course not by pure chance reminiscent of Equation 10. [3] and [5] have arguedthat Equation 19 incorporates the key trouble making aspects of Hawking radiation withregards to the paradox. The central point we want to make is that through the Hawkingprocess, entangled pairs of particles are created. Equation 19 simply reduces the infinitelymany entangled states to only two.

3.2. Assumptions

Before we can continue our evaluation of the information paradox, we need to make clearwhat exactly we are assuming, because it is these assumptions which are being challengedby the paradox. In the next section we will follow along the lines of [3] and define ourassumptions.

solar system limit Physicists believe that in the limit of ‘solar system physics’the effects of quantum gravity become negligible. When spacetime curvatures are of theorder like in our solar system, we can perform experiments in our labs and safely ignorethe effects of quantum gravity. Applying QFT to a curved geometry is a priori not aproblem. In this section we want to clarify what the assumptions are. For that purposewe shall define the solar system limit as:

Definition 6 (Solar system limit). Under N ‘niceness’ conditions one can specify anyquantum state on an initial space-like slice. A Hamiltonian evolution operator then givesthe state on a later space-like slice. Furthermore, we assume locality: Any influence of astate in one region on the evolution of a state in another region goes to zero as the distanceapproaches infinity.

Definition 7 (Niceness Conditions). For our ‘niceness’ conditions we assume that aquantum state is defined on a space-like slice. This slice shall be smooth in the followingsense: The intrinsic curvature (3)R as well as the extrinsic curvature K of this slice ismuch smaller than the Planck length to the power of minus two:

(3)R l−2P l and K l−2

P l . (20)

hawking radiation in the qubit model 8

Moreover, the four-curvature (4)R in the neighbourhood of the slice shall be smaller than:

(4)R l−2P l . (21)

And lastly, the shift and lapse vectors which describe the transition of one slice to a latermay only vary slowly:

dN i

ds l−1P l and dN

ds l−1P l . (22)

In returning to the clarification what we expect from a QFT in the vicinity of thehorizon, I keep my promise. We will call this expected behaviour ‘information free’.

Definition 8 (Information free horizon). A point on the horizon is called ‘informationfree’ if there exists a neighbourhood in which the evolution of field modes with wavelengthλ and lP l λ . K−1/2 (K is curvature at horizon) is given by a semiclassical (i.e. withoutquantum gravity) equation of motion of a QFT on empty curved space up to terms whichvanish as mP l

MBH→ 0.

3.3. Slicing spacetime

t

x

Figure 1: Exemplary space-like slice and its corresponding later slices (from black over red toyellow) in Minkowski spacetime.

In this subsection we want to elaborate more on the space-like slicing (or foliation) ofspacetime. As QFT in a general spacetime evolves quantum states on space-like slices toquantum states on space-like slices, we need to clarify how we slice through our geometry.In Figure 1 and Figure 2 a set of proposed space-like slices were drawn into Minkowski

hawking radiation in the qubit model 9

"t"

"t"

"t"General spacetime

Figure 2: Exemplary space-like slice and its corresponding later slices (from black over red toyellow) in an exemplary, generic spacetime. Note that such an evolution could nothappen in flat space. Slices with curvature would eventually after some time steps havelight-like regions and then time-like regions. Also note the direction “t” merely pointsinto the time-like direction; it is not to be confused with a co-ordinate labelled t.

spacetime and a generic spacetime, respectively. This shall illustrate an exemplary foli-ation.We want to consider only an astronomical black hole with mass MBH , no charge or an-gular momentum, long after the gravitational collapse formed the black hole. To goodapproximation we use the Schwarzschild geometry,

ds2 =

(1− rs

r

)dt2 −

(1− rs

r

)−1dr2 − r2 dΩ2 . (23)

We make the following choice of ‘nice’ slices:

1. For r > 2rs (outside): t = t1 = const. We shall call this region 1.

2. For r < rs (inside): r = r1 = const. and 14rs < r < 3

4rs. (region 2)

3. In between we choose a smooth connector that satisfies our ‘niceness’ conditions.(connector region)

Note that inside the black hole the time-like and space-like directions flip, thus ourchoice is in fact a space-like slice. On such a space-like slice we can follow the slice inregion 2 to early times where the black hole has not formed yet. Accordingly, there ismatter on the space-like slice.

For a later time this slice will undergo the following transitions:

1. For region 1, the ‘time-like direction’ is straightforward. The internal geometry willnot change in this region. We simply shift the co-ordinate t by an amount ∆t.

hawking radiation in the qubit model 10

2. For region 2, we find a similar, yet somewhat ‘orthogonal’, straightforward evolution.Again, the internal geometry will not change; we simply shift the co-ordinate r byan amount ∆r.

3. Given the previous considerations for the evolution of region 1 and 2, the connectorregion must ‘stretch’. This stretching is the essential feature of particle productionat the horizon.

Please note that while the stretching is the key feature for particle production, it is notinherent to gravity. Any geometry with a horizon will produce Hawking radiation.As mentioned earlier, we want to consider only real astronomical black holes which formedby gravitational collapse. Such a black hole geometry may be illustrated with the helpof the Penrose diagram in Figure 3. Another, hopefully illuminating, sketch providesFigure 4. In both figures you can see the same two slices whereas Σ2 is the correspondinglater slice to Σ1.

colla

psingmatter

horizon

Figure 3: Penrose diagram for an astronomical black hole. Two of our ‘nice’ slices are drawn - Σ1and Σ2. Original: [3]

hawking radiation in the qubit model 11

collapsing matter

b2

"t"

"t"

"t"

b1

a1

b1a1

a2

Figure 4: Illustration of the chosen slices in (approximate) Schwarzschild geometry. Again, thetime-like direction “t” is not to be confused with the co-ordinate t. The particle pairs(ai, bi) seem to be pulled out of the vacuum for an observer sitting infinitely far away.Original: [3]

3.4. Time evolution

We are now ready to inspect the time evolution of our quantum state on the initial ‘nice’slice (Σ1) to a later slice (Σ2). We will work in the limit ∆r → 0 and ∆t → 0.The state of the matter inside the black hole long after gravitational collapse is (|φM 〉).

Regardless, we assume the effect this matter has on the region around the horizon forreasons of locality to be negligible: Measured on the space-like slice (both Σ1 and Σ2) thismatter is far, far away from the horizon. Hence, the initial state within our qubit modelis given by:

|ψ〉 = |φM 〉 ⊗1√2( |0a1〉 |0b1〉+ |1a1〉 |1b1〉) (24)

Please note the tensor product of the matter state with the qubit state ensures locality. Itimplies that the matter and the vacuum at the horizon are not entangled and there is no“spooky action at a distance”.

As we assume a black hole long after gravitational collapse we expect the matter state|φM 〉 not to change in the next time step.

|φM 〉+∆t−−→ |φM 〉 (25)

Now the crucial part; how will the state in the region of the horizon change?We established in item 3 that the region of the slice near the horizon must undergo somekind of stretching. This will cause the recently created pair (in Figure 4 called a1 and b1)to move away from the region in addition to their own propagation in the time step. Forreason of locality we do not want the newly created pair (in Figure 4 called a2 and b2) toinfluence the previous pair.

hawking radiation in the qubit model 12

One might think of our qubit model like pearls on a chain: In every time step thehorizon will produce a new pair of pearls and place them in the centre of our chain1. Eachof the particles will be entangled with its partner and it cannot influence any earlier one.Hence, the state on the next slice is given by

|ψ〉 +∆t−−→|ψ〉 ⊗ 1√2(|0a2〉 |0b2〉+ |1a2〉 |1b2〉)

= |φM 〉 ⊗1√2(|0a1〉 |0b1〉+ |1a1〉 |1b1〉)⊗

1√2(|0a2〉 |0b2〉+ |1a2〉 |1b2〉) .

(26)

Now that we know the state of the next slice, we can compute the entropy of all quantawhich escaped the black hole - the set of a quanta:

ρa1,a2 = trM ,b1,b2(ρ) = trM (ρM )︸ ︷︷ ︸=1

· trb1,b2(ρa1,a2,b1,b2)

=12

1∑i=0〈ib1 | 〈ib2 |

[|0a10b1〉〈0b10a1 |+ |1a11b1〉〈1b11a1 |

+ |0a10b1〉〈1b11a1 |+ |1a11b1〉〈0b10a1 |]⊗ [1↔ 2]

|ib1〉 |ib2〉

=12(|0a1〉〈0a1 |+ |1a1〉〈1a1 |

)⊗ 1

2(|0a2〉〈0a2 |+ |1a2〉〈1a2 |

)

=14

(1 00 1

)⊗(

1 00 1

)=

14

1(

1 00 1

)0(

1 00 1

)

0(

1 00 1

)1(

1 00 1

) =

14

1

11

1

(27)

Since the density matrix for the a quanta is diagonal, the entanglement entropy (Equa-tion 9) is readily computed:

Sa1,a2 = −4∑

i=1pi ln pi = −41

4 ln(1

4

)= 2 ln 2 (28)

Expectedly, we found that the entropy of two pairs is double the entropy of one.In fact, after N time steps we have the state,

|ψ〉 +N ·∆t−−−−→ |ψ〉 = |φM 〉 ⊗1√2(|0a1〉 |0b1〉+ |1a1〉 |1b1〉)

⊗ 1√2(|0a2〉 |0b2〉+ |1a2〉 |1b2〉)

. . .

⊗ 1√2(|0aN 〉 |0bN

〉+ |1aN 〉 |1bN〉) ,

(29)

and an entanglement entropy of

S(ρai) = N ln 2. (30)

We have now established that within our bit model every pair adds to the entropy ofescaping particles an amount of ln 2. Furthermore, the black hole has gained the same

1 As with any other comparison, this one, too, does not go without warning.

robustness of the argument 13

amount of entropy via the ‘infalling’ particles.Suppose after N such time steps the black hole lost mass until it reaches a small size andmass such that our ‘niceness’ conditions are no longer satisfied: MBH ≈ mP l ⇒ K > l−2

P l .Consequently, we will stop evolving in time.Within our framework we cannot tell what is going to happen next. There are these

two options:

1. Remnant: Hawking radiation stops. There exists an object with limited volumeand energy (m ≈ mP l, l ≈ lP l) but an virtually unbounded amount of degrees offreedom. Surely, a remnant is nothing one encounters in ordinary physics. It hasbeen argued that such an object imposes several other complications to physics [6],[5], [3].

2. Pure to mixed: Hawking radiation continues. The black hole radiates away andwe have an evolution from a pure state to a mixed state. This time evolution is, asexplained above, in conflict with unitary time evolution in quantum mechanics.

The problem one faces now is usually called the “black hole information paradox” al-though in one scenario one actually does not encounter an information problem.

comment Please note that all the previous calculations would have gone through withas many levels of excitation as pleasing. Instead of the qubit state in Equation 19 on mightas well use

|ψ〉 = 1√n

(|0a〉 |0b〉+ . . .+ |na〉 |nb〉

). (31)

The amount of entropy one pair added would then be lnn. Consequently, we encounterthe problem as posed above.

4 robustness of the argument

Mathur shows in [3], section 5, how robust the qubit model under deformations is. Insteadof repeating, not to say copying, his argument, I want to pick out a clever piece of analysis.The von Neumann entropy (Equation 5) satisfies in a tripartite system ABC an inequalitycalled ‘strong subadditivity’:

S(ρABC) + S(ρB) ≤ S(ρAB) + S(ρBC) (32)

We shall now consider the newly created escaping particle as one subsystem (a), itsinterior partner as an other subsystem (b) and ‘everything else’ as the third subsystem(E), whereas everything else means all previously created quanta and the interior matter.With strong subadditivity, we can set up the inequality,

SabE + Sa ≤ SaE + Sab. (33)

However, the system ab is a pure state, namely the vacuum of Equation 18. Thus

Sab = 0 and SabE = SE . (34)

This, in turn, implies

SE + Sa ≤ SaE ⇔ Sa ≤ SaE − SE . (35)

references 14

Equation 35 says nothing less than that the difference in entropy before emitting aHawking quantum and after having emitted it is at least the entropy of the quantumemitted. Thus, we can infer that we follow the Hawking curve in its ascending, evergrowing path. However, to follow the Page curve we need to lose entropy. Equation 35implies to follow the Page curve one needs a negative entropy for Sa, if one assumes thesystem ab to be pure and thus assumes an information free (definition 8) horizon. Hence,we can never ‘get rid’ of our excessive entropy in the radiation field and are forced to runinto the paradox, again.As a matter of fact, for this whole analysis we never relied on the qubit model. We mightjust consider the subsystem a of all modes defined outside the horizon, the subsystem b ofall interior modes, and ‘everything else’. We can then argue by the purity of Equation 18that we still encounter an increase in entropy where we should observe a decrease in ordernot to encounter the paradox.

references

[1] Jacob D. Bekenstein. Black holes and entropy. Phys. Rev. D, 7:2333–2346, Apr 1973.http://link.aps.org/doi/10.1103/PhysRevD.7.2333.

[2] S. W. Hawking. Particle creation by black holes. Communications in MathematicalPhysics, 43(3):199–220, 1975. http://dx.doi.org/10.1007/BF02345020.

[3] Samir D. Mathur. The Information paradox: A Pedagogical introduction. Class. Quant.Grav., 26:224001, 2009. https://arxiv.org/abs/0909.1038.

[4] Robert M. Wald. On particle creation by black holes. Communications in MathematicalPhysics, 45(1):9–34, 1975. http://dx.doi.org/10.1007/BF01609863.

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