Entanglement entropy and black holes

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Entanglement entropy and

black holes

Leon Schoonderwoerd

6280080

leonschoonderwoerd@gmail.com

Report, Bachelor project Physics (15 EC)

conducted between 31–03–2014 and 30–06–2014

Under supervision of:

Jan Pieter van der Schaar

Second assessor:

Ben Freivogel

INSTITUTE FORTHEORETICALPHYSICS- INSTITUTE OFPHYSICS

FACULTY OFSCIENCE

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1 SUMMARY 1

1 Summary

This thesis considers entanglement entropy in quantum field theory. The concepts of en-tanglement, entropy and entanglement entropy are introduced. The entanglement entropy is computed for some ‘toy systems’ of coupled harmonic oscillators. An expression for the entanglement entropy for a massless scalar field is derived, with the surprising result that the entropy depends on the area of the considered region. A short introduction to the sub-ject of black holes and black hole entropy is given. The entanglement entropy from QFT is compared to black hole entropy, and some implications are discussed.

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2 SAMENVATTING 2

2 Samenvatting

Kwantumverstrengeling is een kwantummechanisch effect waarbij twee systemen op zo’n manier met elkaar gecorreleerd zijn, dat geen van deze systemen beschreven kan worden zonder het andere systeem in deze beschrijving mee te nemen. Het is één van de meest fun-damentele, maar tevens ‘rare’ en moeilijk te begrijpen aspecten van de kwantummechanica. Om op een kwantitatieve manier na te kunnen denken over verstrengeling hebben we een grootheid nodig die de mate van verstrengeling tussen twee systemen kan meten. Een grootheid die dit doet is de verstrengelingsentropie. entropie wordt vaak geïnterpreteerd als de hoeveelheid extra informatie die nodig is om een systeem volledig te kunnen beschrij-ven. Verstrengelingsentropie kan op een vergelijkbare manie geïnterpreteerd worden als de hoeveelheid informatie in het volledige systeem die niet bevat is in één van de deelsystemen. In deze scriptie zullen de concepten verstrengeling, entropie en verstrengelingsentro-pie geïntroduceerd worden. Vervolgens zullen er een aantal ‘proefberekeningen’ gedaan worden aan systemen van gekoppelde harmonische oscillatoren (een specifiek kwantum-mechanisch systeem dat erg geschikt blijkt om te redeneren over verstrengelingsentropie). Daarna zal bekeken worden hoe verstrengelingsentropie zich gedraagt binnen de kwantum-veldentheorie. Hierbij zal gekeken worden naar een regio in de driedimensionale ruimte. De verstrengeling treedt dan op tussen het (kwantum)veld bínnen en het veld buiten de regio.

Met de resultaten van deze berekeningen zullen we een specifiek fysisch systeem be-schouwen: het zwarte gat. Een zwart gat is een kosmologisch object dat gekenmerkt wordt door een zeer grote massadichtheid. Door deze dichtheid kan licht binnen een bepaalde straal rond het zwarte gat niet meer ontsnappen; vandaar dat het gat ‘zwart’ genoemd wordt. Het blijkt dat zwarte gaten ondanks deze eigenschap straling uitzenden, en daardoor een beschouwd kunnen worden alsof ze een temperatuur hebben. Dit leidt vervolgens tot de mogelijkheid om met behulp van de wetten van de thermodynamica de entropie van een zwart gat uit te rekenen.

De entropie van een zwart gat vertoont grote overeenkomsten met de verstrengelingsen-tropie uit de kwantumveldentheorie. De conclusie van deze scriptie is dan ook dat het goed mogelijk is dat de entropie van een zwart gat in essentie hetzelfde is als verstrengelingsen-tropie, waarbij het veld binnen het zwarte gat is verstrengeld met het veld daarbuiten.

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CONTENTS 3

3 Introduction

The quantum mechanical universe is one that seems intuitively strange. We are all familiar with examples of this weirdness like ‘Schrödinger’s cat’ or the uncertainty principle. This weirdness, of course, arises from the fundamental differences between quantum mechanics and the classical physics that we are intuitively familiar with.

There is nothing that illustrates this fundamental difference better than the quantum me-chanical notion of entanglement. In the words of Leonard Susskind[13]: “The phenomenon

of entanglement is the essential fact of quantum mechanics, the fact that makes it so different from classical physics.”

To be able to fully understand entanglement, we need a measure of the amount of en-tanglement in a given system. A quantity that does precisely this is called the enen-tanglement

entropy. This is a generalization of the notion of entropy from thermodynamics.

As it turns out, entanglement entropy can also be applied to explain the entropy con-tained in black holes. Before the historical paper by Hawking[6], it was thought that black

holes violated thermodynamics by only ‘sucking op’ matter and radiation while emitting nothing. Thus, the total entropy of a system containing a black hole would decrease. Since [6], however, black holes are thought to radiate according to a black body spectrum

deter-mined by the black hole mass_[10_].

Black hole radiation implies that black holes have a temperature. This in turn implies the existence of black hole entropy, saving the second law of thermodynamics. It can be shown that the entropy contained in a black hole is proportional to the area of the event horizon[10]. This in contrast to entropy in thermodynamics, which scales as the volume of

the system.

This writing will attempt the following things. Firstly, we will review basic knowledge about entanglement and entropy (section4). Secondly, we will calculate the entanglement entropy of some trial systems consisting of 2 respectively N harmonic oscillators (section5). Thirdly, we will move into the realm of quantum field theory (QFT) and derive the general area-dependence of entanglement entropy (section6). Lastly, we will use this knowledge to provide a possible explanation for black hole entropy and its dependence on area (section

7).

4 ‘Traditional’ entanglement entropy

4.1 Entangled states and the density matrix

Two quantum mechanical states are said to be entangled when neither of the states can be described by itself, without including the other state in this description. Some well-known

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4 ‘TRADITIONAL’ ENTANGLEMENT ENTROPY 5

examples of entangled states are the Bell spin-states, for instance |Φ+〉 = p1

2 |↓A↓B〉 + |↑A↑B〉 . (1)

In this notation,|↓A↓B〉 is taken to mean the tensor product |↓〉A⊗ |↓〉Bwhere both system A

and system B (which can be, for example, two electrons) are in the state|↓〉. Systems A and B are independent of each other, which means we can observe both systems independently. In this state, something special is happening. If observer A measures system A, he in-stantly knows what the outcome of a measurement on B will be: if A measured an|↑〉 state, B would get another|↑〉, if A measured a |↓〉, B would get another |↓〉. Therefore, by per-forming a measurement on system A, the wave function of system B is collapsed. System A and system B are entangled.

At first glance, this thought experiment may not seem to be very different from classical physics. However, there is a big difference. In classical physics, there is always a way for a third, external observer to know in advance what results both observers will get. In quantum mechanics, this is not so. There is no way to know in advance what observer A or B will measure, but as soon as one observer makes his measurement, the outcome of the second measurement is determined.

The collapse of the wave function of system B after system A is observed happens instan-taneously and over any distance. This phenomenon, called “spooky action at a distance” by Albert Einstein, forms an example of why entanglement is so inherently weird. Many de-bates have been sparked in this context, regarding for example locality and hidden variable theories. For the purpose of this paper, it is not necessary to go into these debates, so we will not risk doing so.

A more mathematical definition of entanglement goes as follows. We again look at the Bell state in equation1. We now propose that this 2-system state is merely the (ten-sor)product of the state of system A and the state of system B:

|Φ+〉 = |φ〉A⊗ |ψ〉B, (2)

where|φ〉Aand|φ〉Bare arbitrary states:

|φ〉A= a |↑〉 + b |↓〉 (3a)

|ψ〉B= c |↑〉 + d |↓〉 . (3b)

However, it turns out that there is no way to pick the coefficients a, b, c and d so that this product returns the Bell state in equation1. Therefore, neither of the component systems A and B can be described completely without describing the other system; A and B are, as we

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have seen before, entangled.

To better understand the nature of entanglement, we need a way to quantify the degree of entanglement in a given state. One quantity that does precisely this is entanglement entropy. Before we can dive into this subject, however, we first need to understand the concept of the density matrix.

4.1.1 The Density Matrix

So far, we have seen one example of an entangled state: the Bell state of equation 1. In general, entangled systems cannot be described by a single state|ψ〉. Rather, the system can be described in terms of its density matrixρ [13]. The density matrix gives a

proba-bilistic description of the system. Diagonalizing the (N-dimensional) density matrix gives an expression ρ =        ρ1 0 . . . 0 0 ρ₂ . . . 0 .. . ... ... ... 0 0 . . . ρ_N        , (4)

where theρi’s are the probabilities of the system being in state|i〉; they are the eigenvalues

ofρ.

When it is possible to express the system in terms of a single state vector|ψ〉, we can still compute the density matrix as

ρ = |ψ〉 〈ψ| , (5)

which can be generalized to a description of a mixed state:

ρ =X

n

p_n|ψn〉 〈ψn| (6)

where the pnare probabilities for the statesψn[4]. This notation and corresponding

inter-pretation give rise to the following properties of the density matrix:

• ρ†= ρ (7a)

• Tr(ρ) = 1 (7b)

• ρ ≥ 0, (7c)

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The last inequality (equation7c) means that every eigenvalue ofρ is nonnegative. The last two equations shown above emphasize again the probabilistic nature of the density matrix: all probabilities are greater than or equal to 0 and sum to 1

The density matrix can be used to calculate the expectation value〈O〉 of any operator O, also called the average density of O. As shown in[9],

〈O〉 = Tr(ρO). (8)

4.1.2 Reduced Density Matrix

The density matrix formalism is built upon the ability to describe composite, entangled sys-tems. However, it is also possible to consider parts of a composite system separately. This can be done by using the reduced density matrix.

For a system consisting of two subsystems (creatively named 1 and 2), the reduced den-sity matrices are defined as:

ρ1≡ Tr2(ρ) (9a)

ρ2≡ Tr1(ρ). (9b)

The reduced density matrix are interpreted as containing all information about only one of the subsystems_[9_{]. The reduced density matrices allow for operations on either one of}

the subsystems. For example, if operator O works on system 1 alone, 〈O〉ρ= Trρ(O ⊗ I)

= 〈O〉ρ1= Tr(Oρ1).

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Here, I is the identity operator. The notation O⊗ I means that operator O works on the first subsystem, while the operator I works on the second.

Reduced density matrices can give an insight into the correlation between the variables of subsystem 1 and 2. If we define

ρ12= ρ1⊗ ρ2, (11)

we can check ifρ = ρ₁₂. If this is not the case,ρ contains a certain amount of correlation between its subsystems; the subsystems are entangled

We would like to have some quantitative measure of the ‘amount’ of entanglement that is present in a system. One quantity that does this is, as stated in the introduction, entan-glement entropy. Before we can consider this, however, we have to review our knowledge of entropy in general.

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4.2 Classical entropy

The notion of entropy in classical physics pops up most notably in thermodynamics and statistical physics. The entropy (S) of a system is interpreted as the amount of information needed to know everything about the system. It is, therefore, a measure of information.

One way to compute the entropy of a given system is with the formula1

S= log Ω (12)

where k_Bis the Boltzmann constant andΩ is the number of possible microstates in the sys-tem. This formula holds only when every microstate is (in approximation) equally probable. If the assumption of equal probabilities does not hold (which in quantum mechanics is often the case), the entropy can be calculated as2

S= −X

i

p_ilog p_i (13)

where the sum is over all the microstates of the system and the pi denote the probabilities

of each microstate to occur.

It should be clear that in classical physics, the entropy of a system is closely related to the (number of) microstates of the system. If we imagine a system in three dimensions, it is easy to see that in general, the number of microstates (and therefore the entropy) scales with the volume of the system. Consider, for example, a three dimensional Ising model. If we increase the size of the box, the number of spins increases proportional to the volume, so the amount of microstates does as well.

Equation13is a general result for a classical system. It also closely resembles the result we will obtain when we try to calculate the entropy of a quantum mechanical entangled system. However, the volume-proportionality of entropy is, as we will see, something that does not hold for all quantum systems.

4.3 Entanglement Entropy

The quantum mechanical analogue of equation13(setting k_B= 1) is

S= −

N

X

i=1

ρilogρi (14)

[13]. Here, the ρiare again the probabilities for the system to be in state|i〉, as we have seen

before in equation4. This expression can be seen to be the sum of the diagonal elements of 1_{Note that throughout this thesis, constants are set to 1 for brevity and easy reading. Full versions (including}

all constants) of important equations can be found in appendixA. This equation corresponds to equation56.

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5 COUPLED HARMONIC OSCILLATORS 9

a matrix. Thus, we arrive at the most commonly used formula for entanglement entropy:

S= −Tr ρ log ρ (15)

where logρ is taken to mean the matrix with entries log ρ_{i j}.

As stated in section4.1, entanglement entropy is a measure of the degree of entangle-ment between two systems. As shown in_[9_{], S ≥ 0, since ρ}_i ≥ 0 for all i. Furthermore,

S= 0 only if our system is in a pure state, i.e. if there is only one nonzero ρ_i. As[9] puts it:

“The more non-zero entries there are,[...] the greater the entropy.” If we compare this with equation12(where the entropy increased with a larger amount of microstates, and was 0 if there was only 1 microstate), we see that this definition of entanglement entropy is a good quantum mechanical analogue to the classical definition.

We can extend the notion of entanglement entropy to reduced density matrices. If we define S(ρ₁) as the entanglement entropy in subsystem 1 (and S(ρ₂) and S(ρ₁₂) accordingly, we get from[9]:

S(ρ₁₂) = S(ρ₁⊗ ρ2) = S(ρ1) + S(ρ2)

≥ S(ρ).

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where the final step is an equality for uncorrelated subsystems, and an inequality for cor-related subsystems. In the latter case, the inequality arises because the full density matrix

ρ contains more information about the system: it ‘knows’ about the correlations between

subsystems 1 and 2. This information is not present inρ₁₂. Thus, entropy works in this way as well: the larger the entanglement between subsystems 1 and 2, the greater the inequality. Now that we have our formalism in place, it is time to apply it. This we will do starting from the following section.

5 Coupled Harmonic Oscillators

To arrive at a full QFT description of entanglement entropy, we need to first calculate the entanglement entropy of some more ‘traditional’ quantum mechanical systems. These are a system of 2 coupled harmonic oscillators, or a system of N coupled harmonic oscillators. More complete derivations of these systems can be found in appendixB.

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5.1 Two oscillators

The Hamiltonian for a system of two coupled harmonic oscillators is3

H=1₂p2₁+ p₂2+ k₀(x₁2+ x₂2) + k₁(x₁− x2) 2_.

(17) This hamiltonian can be diagonalized using a suitable change of variables to get

H=1₂(p₊2+ p₋2+ ω₊2x₊2+ ω2₋x2₋), (18)

which is evidently a Hamiltonian for two uncoupled harmonic oscillators. Therefore, the corresponding ground state wave function is

ψ0= π− 1 2(ω₊ω −) 1 4_exp₋1 2 ω+x 2 ++ ω−x2− . (19)

For reasons which will become clear later, we wish to compute the reduced density matrix for only one of the two oscillators, which we will now call the “outside” oscillator. We do this by constructing the ground state density matrixρ = |ψ₀〉 〈ψ0|. We then trace over one

oscillator (the first or “inside” oscillator) to obtain the reduced density matrix for the second, “outside” subsystem, whose diagonal elements are:

ρout(x2, x02) = Z +∞ −∞ dx₁ψ₀(x₁, x₂)ψ∗₀(x₁, x0₂) = π−1 2(γ − β) 1 2_exp−1 2γ(x 2 2+ x202) + β x2x02 . (20)

In this last expression, we haveβ =1₄(ω₊−ω−)2/(ω++ω−) and γ−β = 2ω+ω−/(ω++ω−).

To easily compute the entropy of this subsystem, it is useful to compute the eigenvalues ofρ_out. To do this, we have to solve the following equation:

Z +∞

−∞

dx0ρ_out(x, x0)f_n(x0) = p_nf_n(x). (21) whose solutions are guessed in[11]:

p_n= (1 + ξ)ξn (22a) f_n(x) = H_n(α12x) exp αx 2 2 (22b)

where the H_nare Hermite polynomials,α =Æγ2− β2andξ =_γ+αβ . 3

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By looking back at equation14, we can see why the eigenvalues ofρ_outare useful: we can directly use them to calculate the entanglement entropy of the one-oscillator subsystem. This entropy is

S= − log(1 − ξ) − ξ

1− ξlogξ. (23)

It should be obvious that there is no a priori distinction between the oscillator we picked as “outside” oscillator, and the “inside” oscillator. Therefore, the entire procedure is the same for both subsystems. So is its conclusion: both subsystems have an exactly equal amount of entanglement entropy.

5.2 N Oscillators

Having derived the entanglement entropy present in a 2-oscillator system, we will now gen-eralize this result to a system of N oscillators. We will start with the Hamiltonian, which for N oscillators is4 H= 1₂ N X i=1 p2_i +1₂ N X i, j=1 x_iK_{i j}x_j. (24) Here, the p_i are momenta for each oscillator and K is a real, positive, symmetric matrix denoting the coupling between all oscillators. K is diagonalizable: K= UTK_DU, where K_D is diagonal and U is orthogonal.

After diagonalizing this Hamiltonian (see appendixB.2), the ground state wave function is, just as in the 2-oscillator case, the product of N independent wave functions:

ψ0= π− N 4(det Ω) 1 4_exp−1 2~x · Ω · ~x (25) where~x is an N-vector and Ω is the ‘square root’ of K: Ω = UTpK_DU.

Just as before, we now divide the full system in an “inside” (oscillators 1 to n< N) and an “outside” (oscillators n+ 1 to N) part. We want an expression for the “outside” reduced density matrix, so we trace over the “inside” part to get:

ρout= Z +∞ =∞ n Y i=1 dxiψ0(x1, . . . , xn, xn+1, . . . , xN)ψ∗0(x1, . . . , xn, x0n+1, . . . , xN0). (26)

Note thatρ_outdepends on variables(x_n₊₁, . . . , x_N, x0_n₊₁, . . . , x_N0).

To compute this integral, we define the vectors ~x and ~x0(see equations85aand85bin appendixB.2) and write forΩ:

Ω =   A B BT C   (27) 4_Equation₅₉_{in appendix}_A_.

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where A is n× n, C is N × N and both B and BT are(N − n) × (N − n). Using these new definitions, we can compute the outside density matrix. We ignore prefactors, since the properties of the density matrix mean that the eigenvalues of_ρ_out(which we are ultimately interested in) must sum to 1. The outside density matrix is:

ρout(~z, ~z0) ∼ Z +∞ −∞ d_{~y exp}−12 ~x · Ω · ~x + ~x0· Ω · ~x0 = exp−12 ~z · γ · ~z + ~z 0 · γ · ~z0 + ~z · β · ~z0 (28)

whereβ = 1₂BTA−1B andγ = C − β. Note the similarities between this expression and equation20.

As[11] states, the eigenvalues of ρ_outremain unchanged under a change of coordinates. Therefore, we can use the following transformations without altering the eigenvalues:

γ = VT_γ DV (29a) x= VTγ 1 2 Dy (29b) β0_{= γ}−12 D VβV T_γ− 1 2 D (29c)

whereγDis diagonal and V is orthogonal. Under this transformation, equation28becomes:

ρout∼ exp −1 2 y· y + y 0 · y0 + y · β0· y0 . (30)

By now setting y= Wz, with W orthogonal and WTβ0Wdiagonal, we obtain

ρout∼

N

Y

i=n+1

exp−1₂(z2_i + z_i02) + β0z_iz0_i (31) whereβ_i0 is the i-th eigenvalue ofβ0. Looking back at equation20, we observe that each term in the above product is equal to equation20when we set_{γ = 1 and β = β}_i0. From this, we can infer that for every i there is an amount of entropy present given by equation

23[11]. Therefore, the total amount of entropy in this system can by calculated as

S= N X i=n+1 S(ξ_i) (32) where ξi= β0 i 1+q1− βi02 . (33)

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6 ENTANGLEMENT ENTROPY IN QFT 13

Now that we have derived an expression for the entanglement entropy in a system of N coupled harmonic oscillators, we are well prepared to start our treatment of entanglement entropy in the context of quantum field theory. This will be the subject of the following section.

6 Entanglement entropy in QFT

6.1 Introducing the field

We start our analysis of entanglement entropy in quantum field theory with the Lagrangian density for a massive scalar fieldϕ [7],

L =12ηµν∂µϕ∂νϕ − 1 2m

2_ϕ2_. ₍₃₄₎

Here,ϕ = ϕ(~x, t) can be considered the QFT equivalent to the position (operator) in quan-tum mechanics. This Lagrangian can be plugged into the Euler-Lagrange equation to obtain the equation of motion:

0=∂ L ∂ ϕ − ∂µ ∂ L ∂ (∂µ_ϕ) = −η_µν∂µ_∂ν_{ϕ − m}2_ϕ, (35) with which ϕ + m2ϕ = 0 (36)

where =_{∂ t}∂22− ∇2. Note that equation36is the Klein-Gordon equation.

To get a better feeling for working with fields instead of regular variables, we expand the fieldϕ in Fourier modes:

ϕ(~x, t) =

Z

d3~k ϕ_k(t) expi~k· ~x. (37) Using this, we can rewrite equation36to obtain

¨ ϕk+ (k 2_{+ m}2_)ϕ k= ¨ϕk+ ω 2 kϕk= 0 (38)

whereω2_k= k2+ m2. We herein recognize the equation of motion for a harmonic oscillator. Therefore, the fieldϕ at a given position in real space can be interpreted as the sum over an infinite amount of harmonic oscillators, one for every value of k.

At first glance, it appears that this result provides us with a bridge between QFT and quantum mechanics, and we can proceed with a further analogue of the derivations in sec-tion5.2. However, since the harmonic oscillators here exist only in momentum space, there

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6 ENTANGLEMENT ENTROPY IN QFT 14

is no clear way to ‘trace’ over a certain region of real space. Therefore the derivation above serves only as an illustration, but has no further uses in this writing.

To obtain an expression for the entanglement entropy in QFT, we need to go from the Lagrangian to th Hamiltonian formalism in quantized space. This will be the subject of the next subsection

6.2 Quantization and entanglement

To go from the Lagrangian to the Hamiltonian formalism for a massless scalar field, we write the Hamiltonian density:

H = π ˙ϕ − L =12π

2_{(~x, t) +}1

2|∇ϕ(~x, t)|

2 ₍₃₉₎

where we used equation34without the mass term. Hereπ(~x, t) = ∂ L_{∂ ˙}_ϕ = 1₂ϕ is the canon-˙ ical momentum corresponding to the fieldϕ. π and ϕ obey the canonical commutation relations:

ϕ(~x), ϕ(~y) = π(~x), π(~y) = 0 (40a)

ϕ(~x), π(~y) = iδ(~x − ~y). (40b)

If we integrate the Hamiltonian density over all real space, we obtain the full Hamilto-nian: H= Z d3~x H =1₂ Z d3~x π2(~x) + |∇ϕ(~x)|2 . (41) To derive the entanglement entropy in this case, we have to work in quantized space. We discretize space into a 3-dimensional lattice with spacing a, making the ultraviolet cutoff

M= a−1. We then put the entire system in a cubical box of sides L= (N +1)a (so the infrared cutoff isµ = L−1. This quantization results in the redefinition of the Hamiltonian (equation

41) as a sum: H= a 2 N X i, j,k=1 h πi, j,k+ 1 a2 n 3ϕ_i2_{, j,k}+ ϕ2_i_+1,j,k+ ϕ2_i_{, j+1,k}+ ϕ2_i_{, j,k+1} − 2ϕi, j,k(ϕi+1,j,k+ ϕi, j+1,k+ ϕi, j,k+1 oi (42)

where ϕi, j,k is taken to meanϕ(~xi, j,k, t) with ~xi, j,k = (ia, ja, ka). This equation can be

rewritten to resemble more closely equation24:

H= a 2 N X i, j,k=1 πi jk+ 1 2a N X i, j,k=1 N X l,m,n=1 ϕi jkKi jk,l mnϕl mn, (43)

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7 ENTANGLEMENT ENTROPY AND BLACK HOLES 15

which is very similar to the Hamiltonian of a series of harmonic oscillators (equation24). HereK is a matrix element containing all ‘coupling constants’ between the different ‘posi-tions’_{ϕ. Therefore, the resulting treatment of entanglement entropy reduces to the} treat-ment of entangletreat-ment entropy for N oscillators in section5.2.

A second way to quantize the Hamiltonian in equation41is given and calculated by[11],

using spherical harmonics and a spherical quantization region instead of a cubical one. Both these methods should result in the same final expression for the entanglement entropy.

[11] uses numerical methods to calculate an expression for the entanglement entropy in

this system:

S= 0.30M2R2, (44)

where M is the ultraviolet cutoff as defined earlier and R is the radius of the spherical region over which is traced. The constant 0.30 is just a numerical constant of order 1, so it can be ignored (since proportionalities are what interest us at this point). We can thus generalize this result to

S=R

2

a2. (45)

Here, the ultraviolet cutoff is specified as a−1, a being the lattice spacing. We observe that this expression scales with R2, or in other words, with the area of the sphere.

This result is perhaps surprising. If we recall section4.2, we see that for most systems, the entropy scales with the number of microstates. As such, the entropy commonly depends on the volume of the system instead of the area. This proportionality to the area is what makes entanglement entropy interesting in the context of black holes, which we shall see in the following section.

7 Entanglement entropy and black holes

7.1 Schwarzschild black holes

A black hole is a cosmological object (or, more precisely, a region of spacetime) in which mass is so condensed and thus gravity so strong that light cannot escape from it. For a spherical object of a certain mass, the radius below which the object becomes a black hole is called the Schwarzschild radius, named after[? ]. We will now follow the derivation (from [10]) that leads to this radius.

We consider first the Schwarzschild metric, which is a solution to Einstein’s field equa-tions for general relativity and gravity. This metric describes the curvature of spacetime

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around a spherical (nonrotating), uncharged mass in a vacuum:

ds2= g₀₀(t, r)c2dt2+ g₁₁(t, r) dr2− r2(dθ2+ sin2θ dφ), (46) with5 g₀₀= 1−2G M c2r (47a) g₁₁= − 1−2G M c2r −1 , (47b)

where G is the gravitational constant and M is the mass of the central object. In this metric (as should be the case), ds2> 0 is a timelike interval, while ds2< 0 is spacelike.

We can define the proper time for this system as

dτ = 1−2G M r c2 1₂ dt, (48)

where dτ is a proper time interval and dt is a coordinate time interval. We observe that for

r→ ∞, the two coordinates agree completely. Thus, we can interpret the time coordinate t as being the proper time at an infinite distance from the central mass. In this interpretation, the factor1−2G M_{r c}2

1₂

is the factor with which time nearer to the center is slowed down. At r= 2G M_c2 , the dilatation factor becomes 0. That means that for a clock at that radius, time runs infinitely slower than for a clock at r= ∞. This radius is called the Schwarzschild radius:

r_s= 2G M

c2 = 2m, (49)

where m = G M_c2 is called the geometric mass and has units of length. The Schwarzschild radius can be interpreted as the radius at which the escape speed from the area within it is the speed of light. Therefore, it is the radius from within which nothing, including light, can escape6. The area where r_{= r}_sis called the event horizon; the region within it is a black hole. This means that any object with mass M and radius R< r_s(M) will be a black hole.

7.2 Black hole thermodynamics

Black holes have always been interesting thermodynamical objects. As stated in the intro-duction, black holes have once been thought to violate the second law of thermodynamics, a problem that was only solved when Hawking radiation[1,6] was discovered.

5_{Note that here, constants like G and c are not omitted.}

6_{An interesting sidenote: this also means that it is impossible for an observer to see anything ‘fall into’ a black}

hole. As the object comes nearer to the horizon, the light (information) leaving it will be slowed down proportion-ally.

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Hawking radiation is the emission of black-body radiation by a black hole. A possible explanation for the existence of this radiation is the Unruh effect[7]. The Unruh effect is

the effect in which an observer who moves with constant acceleration observes black body radiation coming from a vacuum, while a stationary observer sees nothing.

The Unruh effect gives rise to a black-body spectrum

n(E) = exp E T_U − 1 −1 , (50)

where E is the energy of the emitted radiation and7

T_U= a

2π (51)

is the Unruh temperature. Here, a is the proper acceleration in physical units.

The Unruh effect takes place in Rindler metric, which is an accelerated ‘version’ of the Minkowski metric[7]. The important thing here is that both these metrics describe a

uni-formly flat spacetime. Since we are interested in black hole radiation, we need a corre-spondence between flat spacetime and the curved spacetime around a black hole. This correspondence is given by_[7_{]. When we replace the acceleration in equation}51with the surface gravityκ =_4m1, m being the mass of the black hole, we obtain8:

T_H= 1

8πm. (52)

This expression is the hawking temperature, or the effective temperature of a black hole. Since black holes now have both a mass (and thus an energy) and a temperature, we can calculate the black hole entropy from the first law of thermodynamics[7]:

dE≡ dm = THdSBH, (53)

where the subscript BH either stands for Black Hole or Bekenstein-Hawking. We can thus calculate dS= 8πm dm = d(4πm2), with which9

S= 4πm2= A 4, (54) 7_Equation₆₀_{in appendix}_A_. 8_Equation₆₁_{in appendix}_A_. 9 Equation62in appendixA.

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where A= 4πr_s2= 16πm2is the area of the Schwarzschild event horizon in terms of the Planck length10 l_P= ħhG c3 1₂ . (55)

We see that the entropy of a black hole depends on the area of the Schwarzschild hori-zon.11 This, and the very existence of black hole entropy, pose a set of problems. These will be addressed in the following section.

7.3 Problems with black hole entropy

As stated, there are several problems with the notion of black hole entropy, the most inter-esting12of which are13:

1. Why is black hole entropy dependent on area? As the derivation in section 7.2is semi-classical and entropy classically depends on volume, this is a very unexpected result.

2. Where does black hole entropy come from? If we compare equation54to equation

12, the questions arise what and where the black hole microstates are.

These points are interesting because they address the foundations of any theory regard-ing black hole entropy. In general, the ‘deeper’ an explanation can go, the more interestregard-ing it is. It turns out that the questions stated here are, in most explanations, closely related. That is, from a theory that answers question 2, there often follows an answer to question 1. An additional problem with these questions is that they are concerned mostly with what goes on inside the black hole horizon. Because the inside of a black hole is per definition inaccessible, there is no way to test theories regarding it. Therefore, any explanation of black holes considering the region inside the horizon, can be judged only by its logical consistency and application in other fields. Here, we again encounter the above made point about the ‘deepness’ of a theory.

To give an answer to the questions mentioned above, several different interpretations of black hole entropy have been offered throughout the years. Most of these attempt to describe the way inner microstates of a black hole lead to the black hole entropy. This has been tried through microstates of internal matter (unlikely, since the space inside a black hole is thought to be mostly empty) or degrees of freedom in charge and rotation (invalid, since Schwarzschild black holes neiher carry charge nor rotate).

10_{Which, in our system of units, is conveniently equal to 1.} 11

A historical note: the derivation in section7.2was originally performed in reverse. The proportionality of black hole entropy on area was first suggested by[1]. Later, [6] derived the expressions for Hawking radiation and confirmed[1]’s ideas.

12

In my opinion, of course.

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More advanced theories try to explain black hole entropy through the multiplicity of horizon gravitational states (for example,[2]) or with the use of string theory (e.g. [12]).

None of those, however, lie within the scope of this writing, although they might be both interesting and valuable.

Equation 54 shows a dependence of the entropy on the area of a system, which we have seen before in equation45. This could mean that entanglement entropy is a possible explanation for black hole entropy. This is the explanatory candidate that we will go with in this thesis. The following section will investigate the idea that black hole entropy arises through entanglement in QFT.

7.4 Black hole entropy as entanglement entropy

It is very clear that equations45and54are very similar. In fact, when we set the lattice spacing in equation45to the Planck length, they are exactly equal (up to a numerical con-stant of order 1, which we ignored previously). The reduction of the lattice spacing to l_P makes physical sense when we view the Planck length as the smallest possible length scale at which conventional physics and QFT can be applied.

The question now is what the similarities between entanglement entropy and black hole entropy could mean. The simple answer is that black hole entropy is entanglement entropy. This would mean that there is a degree of entanglement between the scalar field inside and outside the black hole.

The theory that black hole entropy is entanglement entropy has several merits. Firstly, because we trace out the inside region of the black hole, we avoid the problem of the inac-cessibility of the inside of the black hole. secondly, entanglement entropy quite readily offers an answer to the questions in section7.3. As stated in_[3_{], the things that are entangled are}

field degrees of freedom (DOF) just in- and outside the black hole horizon. This immediately answers question 1. Also, the number of DOF-pairs at the horizon obviously scales with the area of the horizon. Therefore, question 2 is answered indirectly.

A third merit of he QFT explanation for black hole entropy is its simplicity and generality. Computing the entropy of a region is, although sometimes technically difficult, conceptually easy. The theory works for any region and any massless (Klein-Gordon) field (as proven by [8]) and is consistent with all of QFT.

There are, unfortunately, a few limitations to our theory. The first, and most important of these, is that the proportionality of entanglement to area only works for ground- or low-energy states of the quantum field. As computed in_[3_{], when higher-energy states of the}

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cor-8 CONCLUSION 20

rections to the ‘area law’. It is uncertain if this is reflected in the way black hole entropy behaves.

Furthermore, the result of section6is, as stated, only equal to the entropy of a black hole up to a numerical constant. This is great for showing that the theories are similar, but the factor of 0.30 in equation44cannot be completely ignored. As[3] suggest, the numerical

factor may be due to a specific choice in prefactor while discretizing 3-space. In this case, another discretization scheme might solve this problem. However, it might also be the case that this cannot be done. In that situation, the QFT explanation is in serious trouble. Further research is needed to settle this.

With this, we have come at the end of this thesis. The next section will summarize all that we have talked about and repeat the general conclusion.

8 Conclusion

Let us start this final section with a summary of everything that has come before. In section

4 we started out introducing the concepts of entanglement and the density matrix. We then introduced entropy and entanglement entropy. In section5we derived results for the entanglement entropy for ‘toy systems’ of coupled harmonic oscillators. In section6 we moved on to quantum field theory. In section 7we turned our attention to the entropy contained in black holes and tried to explain this with our earlier results.

The final conclusion of this thesis can be formulated as follows: “QFT entanglement entropy is a possible candidate for the explanation of black hole entropy”. This idea has been investigated in section7.4. Both the merits (avoiding untestable statements on black hole interiors, explaining key questions and being simple and generally appliccable) and limitations (different behavior for higher-energy states, numerical differences between en-tanglement entropy and black hole entropy) have been discussed.

It is advised that more research is conducted in the fields of QFT entanglement entropy and black hole entropy. Such research should be focused on eliminating the limitations of the current theory, or finding a different, more powerful theory (some examples of which have been briefly named in the second part of section7.3). Research like this has the po-tential to significantly increase our knowledge about quantum ‘weirdness’ (as we called it in the introduction), black holes or other fundamental but hard to understand components of nature.

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A NOTATIONS AND CONVENTIONS 21

Acknowledgements

A project like a bachelor’s thesis is almost never completed without help. This writing forms no exception to that rule. This section gives some attention to those who helped me complete my thesis.

First and foremost, I would like to thank my advisor, Jan Pieter van der Schaar, for being my guiding light throughout the process of writing. Without him, this thesis would not be. Secondly, I woul like to name Ben Freivogel for being my second reviewer and for pointing out holes in my knowledge. Thirdly, I thank Tim Bakker, for many useful conversations and discussions on technical details. Lastly, I express my gratitude to Emiel Woutersen, for proofreading an giving some useful feedback.

Appendix A

Notations and conventions

This appendix will contain a list of all notations and conventions used in this writing in table

1. It will also list full versions (including all constants) of important equations found in the main text.

Table 1: A list of (notational) conventions used throughout this paper. Constants Constants (e.g. ħh, kB, c, G) are set to 1; we will work

in natural units throughout. The only exception to this is section7.1.

Matrices Matrices are printed bold in section4.2. Anywhere else they are printed as regular capital letters.

Operator ‘hats’ Operator hats such as in ˆH, ˆxare omitted everywhere.

There will now follow a list of important equations found in the main text, where all constants are included.

Equation12: S= k_BlogΩ. (56) Equation13: S= −k_BX i p_ilog p_i. (57) Equation17: H= 1 2m p₁2+ p2₂ + ħh 2 2m k₀(x2₁+ x₂2) + k₁(x₁_{− x}₂)2 . (58)

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B FULL DERIVATIONS 22 Equation24: H= 1 2m N X i=1 p2_i + ħh 2 2m N X i, j=1 x_iK_{i j}x_j. (59) Equation51: T_U= ħha 2πck_B. (60) Equation52: T_H= ħhc 3 8πk_BG M. (61) Equation54 S=1 4 k_Bc3 ħ hG A. (62)

Appendix B

Full derivations

B.1 Two Harmonic Oscillators

As stated in section 5.1, we start our computation for two harmonic oscillators with the corresponding Hamiltonian:

H=1₂p2₁+ p₂2+ k₀(x₁2+ x₂2) + k₁(x₁_{− x}₂)2 . (63)

To diagonalize this operator, we introduce a variable substitution:

x₊= x1_p+ x2 2 (64a) x₋= x1p− x2 2 (64b) ω+= Æ k₀ (64c) ω−= Æ k₀+ 2k₁. (64d)

When we use these to recompute the momenta, we get:

p₊= i d dx₊ = d x₁ dx₊ p1+ d x₂ dx₊p2 (65a) p₋= i d dx₋ = d x₁ dx₋ p1+ d x₂ dx₋p2 (65b)

which can be rewritten as

p₁= i d dx₁ = d x₊ dx₁ p++ d x₋ dx₁ p− =p1 2(p++ p−) (66a)

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B FULL DERIVATIONS 23 p₂= i d dx₂ = d x₊ dx₂ p++ d x₋ dx₂ p− =p1 2(p++ p−) (66b) to finally obtain: p2₁+ p₂2= p₊2+ p₋2. (67) Plugging all these new variables into the Hamiltonian, we can now write down the di-agonalized hamiltonian:

H=1₂(p₊2+ p₋2+ ω₊2x₊2+ ω2₋x2₋). (68)

This is the Hamiltonian for two uncoupled harmonic oscillators. Therefore, the associ-ated ground state wave function is the product of two independent wave functions, which are derived in any quantum mechanics textbook (e.g.[5]):

ψ+ 0= ω + π 1₄ exp−1₂ω+x2₊ (69a) ψ− 0= ω − π 1₄ exp−1₂ω₋x2₋ (69b) ψ0= π− 1 2(ω₊ω −) 1 4_exp₋1 2 ω+x 2 ++ ω−x2− . (70)

Writingψ₀as a state vector in the position basis, we now have:

|ψ0〉 =

Z +∞

−∞

dx₁dx₂ψ₀(x₁, x₂) |x₁〉 |x2〉 (71)

and the ground state density matrix/operator can be constructed as:

ρ = |ψ0〉 〈ψ0| = Z+∞ −∞ dx₁dx₂dx0₁dx0₂ψ₀(x₁, x₂)ψ₀∗(x₁0, x₂0) |x₁〉 |x2〉 〈x01| 〈x10| (72)

Now, the reduced density matrix for one oscillator (which we will call the ‘outside’ oscil-lator and label as osciloscil-lator 2) is:

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B FULL DERIVATIONS 24

where the trace is over the states of the second, ‘inside’ oscillator (labeled as oscillator 1). Because in general,

|ψ〉 = Z +∞

−∞

dx₁dx₂ψ(x₁, x₂) |x₁〉 |x2〉 (73)

we can calculate the outside density matrix as:

ρout= Z +∞ −∞ dx〈x|ρ|x〉 = Z +∞ −∞ dx dx₁dx₂dx0₁dx0₂ψ₀(x₁, x₂)ψ₀∗(x₁0, x₂0) 〈x|x₁〉 |x2〉 〈x01| 〈x20|x〉 = Z +∞ −∞ dx dx₁dx₂dx0₁dx0₂ψ₀(x₁, x₂)ψ₀∗(x₁0, x₂0) 〈x|x₁〉 |x2〉 〈x02| 〈x10|x〉 = Z +∞ −∞ dx dx₁dx₂dx0₁dx0₂ψ₀(x₁, x₂)ψ₀∗(x₁0, x₂0)δ(x − x₁)δ(x − x₁0) |x₂〉 〈x20| = Z +∞ −∞ dx₂dx₂0ψ₀(x₁, x₂)ψ∗₀(x₂x0₂) |x₂_{〉 〈x}0₂_| (74)

Note that in the last step, integrating over the two delta functions results in x₁= x₁0 in the final integral.

Having arrived at this, we can now compute the (diagonal) matrix elements ofρ_out in the position basis:

ρout(x2, x20) = 〈x2| ρout|x02〉 = Z +∞ −∞ dx₁ψ₀(x₁, x₂)ψ∗₀(x₁, x0₂) (75)

Our next job is to solve this integral. Using Wolfram Mathematica, we find pω+ω− π Z +∞ −∞ dx₁exp −1 2ω+x 2 1− 1 4ω+(x 2 2+ x202) − 1 2ω+x1(x2+ x 0 2) − 1 2ω−x 2 1− 1 4ω−(x 2 2+ x022) + 1 2ω−x1(x2+ x 0 2) =pω+ω− π p 2π p_ω ++ ω− exp − 1 8(ω+₊ω₋) ω 2 +(x2− x02) 2_{+ ω}2 −(x2− x20) 2 + 2ω+ω−(3x 2 2+ 2x2x02+ 3x022) = π−1 2(γ − β) 1 2_exp −ω 2 ++ 6ω+ω−+ ω 2 − 8(ω₊+ ω₋) (x 2 2+ x022) + (ω+− ω−) 2 4_(ω₊_{+ ω}₋_{) + x}₂x₂0 = π−1 2(γ − β) 1 2_exp −γ 2(x 2 2+ x202) + β x2x02 (76)

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The eigenvalues of this density matrix are the solutions to Z +∞

−∞

dx0ρ_out(x, x0)f_n(x0) = p_nf_n(x). (77) They are at the present only found by guessing[11]:

p_n= (1 + ξ)ξn (78a) f_n(x) = H_n(α12x) exp αx 2 2 . (78b)

Lastly, we can compute the entropy of this subsystem from equation14:

S= − ∞ X n=1 ρnlogρn = − ∞ X n=1 (1 − ξ)ξn log(1 − ξ)ξn = −(1 − ξ) ∞ X n=1 ξn_{log(1 − ξ) + n log ξ} = −(1 − ξ)  log(1 − ξ) ∞ X n=1 ξn + log ξ ∞ X n=1 nξn   = −(1 − ξ) log(1 − ξ) 1 1− ξ+ log ξ ξ (1 − ξ)2 = − log(1 − ξ) − ξ 1− ξlogξ (79)

B.2 N Harmonic Oscillators

The hamiltonian for N oscillators is easily generalized from equation63:

H=1 2 N X i=1 p2_i +1 2 N X i, j=1 x_iK_{i j}x_j (80)

where K is a real, symmetric matrix with positive eigenvalues denoting all ‘spring constants’ the oscillators are ‘attached’ to. We know that K is a diagonizable matrix, since it can be written as K= UTK_DU, where K_Dis a diagonal matrix, and U an orthogonal matrix. Thus the Hamiltonian can be diagonalized with the following transformation:

x→ ˜x = U x (81a)

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with which the Hamiltonian becomes:

H= 1₂ N X i=1 p_i2+1₂ N X i, j=1 ˜ x_iK_{i j}˜x_j = 1 2 N X i=1 p_i2+1₂ N X i, j=1 UTx_iK_{i j}U x_j = 1 2 N X i=1 p_i2+1₂ N X i, j=1 x_iK_Di jx_j (82)

With this diagonalized Hamiltonian, the full ground state wavefunction becomes:

ψ0= Y i ψi 0 =Y i   q Kii_D π   1 4 exp −1 2xi q K_Diix_i =Y i  Ωii D π 14 exp −1 2xiΩ ii Dxi = π−N 4(det Ω) 1 4exp −1 2~x · ΩD· ~x (83)

where K_iimeans the i-th diagonal element of K,Ω is the square root matrix of K, given by

Ω = UT_pK

DU, and the ~x in the last expression is an N-vector.

Note that the transformation in equation81also diagonalizesΩ. The diagonal variant ofΩ is written here as Ω_D.

Just as before, we would like to calculate the ‘outside’ density matrix. To do this, we trace over the ‘inside’ system, which now consists of n≤ N oscillators. We get:

ρout= Z +∞ −∞ n Y j=1 d ˜x_j〈xj|ρ|xj〉 = Z +∞ =∞ n Y i=1 dx_iψ₀(x₁, . . . , x_n, x_n₊₁, . . . , x_N)ψ∗₀(x₁, . . . , x_n, x0_n₊₁, . . . , x_N0) (84)

where some steps are omitted; they are exactly the same as in the derivation of equation75, but for n oscillators instead of 1.

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To compute the above integral, we first introduce a series of vectors and specify the matrixΩ: ~x =              x₁ .. . x_n x_n₊₁ .. . x_N              =   ~y ~z   (85a) ~x0₌              x₁ .. . x_n x0_n₊₁ .. . x0_N              =   ~y ~z0   (85b) Ω =   A B B_T C   (85c)

With this notation, we can rewrite the integral, ignoring the prefactors:

ρout(~z, ~z0) = Z +∞ −∞ d~y exp −1 2 ~x · Ω · ~x + ~x 0_{· Ω · ~x}0 (86) We compute: ~x · Ω · ~x = ~y · A · ~y + ~y · B · ~z + ~z · BT · ~y + ~z · C · ~z ~x0_{· Ω · ~x}0_{= ~y · A · ~y + ~y · B · ~z}0_{+ ~z}0_{· B}T · ~y + ~z0· C · ~z0 (87)

and (with some help from Wolfram Mathematica) obtain the following result, where again prefactors are ignored:

ρout∼ Z +∞ −∞ d~y exp − ~y · A · ~y − ~y · B · ~z − ~y · B · ~z0−1 2 ~z · C · ~z + ~z 0_{· C · ~z}0 (88a) = exp1 4 z· B T_A−1_B · z + z0· BTA−1B_{· z}0+ z · BTA−1B_{· z}0 + z0_{· B}T A−1B· z − 2z · C · z − 2z0· C · z0i (88b) = exp−1 2z· γ · z − 1 2z 0_{· γ · z}0_{+ z · β · z}0 _(88c) = exp−1 2 x· γ · x + x 0 · γ · x0 + x · β · x0 ∼ ρout(x, x0) (88d)

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where after the first line vector arrows have been omitted. In the last step z has been re-named to x, which is now an N− n-vector. Also, γ = C − β and β =12B

T

A−1B. Generalizing equation77to the N-oscillators case yields an expression

Z +∞

−∞

d~x ρ_out(~x, ~x0)f_n(~x0) = p_nf_n(~x) (89) where~x here has dimension N − n + 1. This implies that under a transformation x → Gx, the eigenvalues of _ρ_out_{(x, x}0_{) and (det G)ρ}_out_{(Gx, Gx}0_{) are identical. Therefore, we can} transform our coordinates in any way without changing the eigenvalues.

We now use the following transformations:

γ = VT_γ DV (90a) x= VTγ− 1 2 D y (90b) β0_{= γ}−12 D VβV T γ− 1 2 D (90c)

where_γ_Dis diagonal and V is orthogonal. With these, we can rewrite equation88as:

ρout∼ exp −1 2 x· γ · x + x 0_{· γ · x}0_{+ x · β · x}0 = exp  − 1 2 Vγ− 1 2 D y V T_γ DV V T_γ− 1 2 D y +V γD− 1 2y0VTγ_DV VTγ_D− 1 2y0 + V γD− 1 2yβVTγ_D− 1 2y0 = exp −1 2 y· y + y 0_{· y}0_{+ y ·}_γ D− 1 2VβVTγ_D− 1 2 · y0 = exp−1 2 y· y + y 0 · y0 + y · β0· y0 (91)

where we used the diagonality of γD, the orthogonality of V and careful transposing of

matrix-vector products to arrive at the final answer.

By now setting y= Wz, where W is orthogonal and WTβ0W is diagonal, we obtain:

ρout∼ exp −1 2 W T z· W z + Wtz0· W z0 + WTz· β0· W z0 = exp−1 2 z· z + z 0_{· z}0_{+ z · W}T_β0_W_{· z}0 = N Y i=n+1 exp −1 2 z 2 i + zi02 + β0ziz0i (92)

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REFERENCES 29

whereβ_i0 means an eigenvalue ofβ0. If we now look back at equation76, we observe that each term in the above expression is equal to equation76when we setγ = 1 and β = β_i0. This means that for every i there is an amount of entropy given by equation79, and the total amount of entropy in the system can be calculated as S= P_iS(ξ_i).

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Entanglement entropy and black holes