Connections, holonomies, and holographic spacetime

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Theoretical Physics

Master Thesis

Connections, holonomies, and holographic

spacetime

by

Linda van Manen

11153989

September, 2020

60 credits

From 01-09-2019 to 24-09-2020

Supervisor:

Prof. Dr. J. de Boer

Examiner:

Dr. B. W. Freivogel

Institute of Theoretical Physics Amsterdam

University of Amsterdam

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We will review entanglement entropy and subregion duality in AdS/CFT and discuss a new perspective on the entanglement between reduced density matrices that descend from a common global state. Namely, entanglement as a connection that glues together entanglement wedges from the same global geometry. By utilizing parallel transport techniques, one can ‘build’ spacetime from entanglement and ultimately gain a better understanding of the curvature of AdS. After earlier found result for the geometric case, we present the first step into generalizing the formalism by exploring the charged sector of AdS. Hence, we have extended these ideas to an Einstein-Maxwell theory and show that the minimal surface gains an additional gauge symmetry. This is a U (1) gauge transformation along the minimal surface. Furthermore, we show that under a pure geometric transport, the internal U (1) coordinates are shifted along the minimal surface and discus options for probing U (1) holonomies.

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I would like to extend my gratitude to my supervisor Jan de Boer for his input and guidance throughout this year. My gratitude also goes out to Ricardo Esp´ındola for the hours we spend in front of a blackboard to understand parallel transport in AdS/CFT, and just as well for all the times Ricardo accompanied me during a coffee break. I would also like to thank my examiner and bachelor supervisor Ben Freivogel. His feedback during my bachelor project was of great value, and also thereafter I regularly reached out for his advice, which never disappointed me. I would like to thank him for his hospitality and his support and advice in general and particularly on finding a Ph.D. position. Last, but not least I want to thank Jasper Kager, Leonard Tokic, Niels de Visser, Amanda van Hemert, Jelle Conijn, Ferran Faura Iglesias, Jochem Kemp, and Zoran Milovi´c for all the hard work we did together and most of all for all the fun these people brought during my study for the last five years.

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a theory of gravity on a d + 1-dimensional AdS spacetime. Due to this duality, it is expected that local operators in the AdS spacetime are described by operators in the field theory. However, the question of how bulk1 locality is encoded in the boundary theory tended to be challenging. The search for the answer on how a given geometrical region in a dynamical AdS spacetime could emerge from a conformal field theory (CFT) has occupied many scientists over recent years [20–25,31–38]. Such as Hamilton et al. [24] who showed that in global coordinates it is possible to reconstruct a bulk operator if one has access to all data in the CFT. A formalism for bulk reconstruction that is known as the HKLL method2_{. They also showed that a CFT restricted to a} subset of the full AdS boundary is dual to a Rindler patch in AdS. The first attempt at AdS/CFT subregion dualities. Later it was shown that this formalism could be mapped to any causal wedge in AdS associated with an arbitrary boundary subregion [33]. It was also argued [33,34] that the reduced density matrix of a subset of the field theory can be utilized to describe the data in a causal wedge. In order to define the causal wedge, first consider the domain of dependence of a boundary subregion A. The domain of dependence is defined as the region containing all causal curves between two timelike separated points that run through A. Then, all causal curves through AdS that start and end on the causal domain define the causal wedge. A particular problem with this idea arose however with the introduction of a holographic relation for entanglement entropy. The by now famous relation proposed by Ryu and Takayanagi [14] is given by

S = Area of γ 4 G_Nd+2 .

A relation between the von Neumann entropy S in conformal field theory and the area of a mini-mal surface γ in the anti-de Sitter spacetime, which sparked the study of quantum information in the context of AdS/CFT. In recent years, a large amount of research has been conducted on the ideas of spacetime emerging from entanglement. As first elucidated in [19,20], entanglement is related to connecting AdS patches or more generally to the emergence of holographic spacetime. In fact, a boundary observable known as differential entropy is used to reconstruct AdS geometry, making it possible to probe the entanglement throughout the boundary state [22,55]. The AdS region probed by the entanglement inherent in the boundary density matrix generally extends fur-ther into the AdS spacetime than the causal wedge. This implied that the spacetime region that emerges from the reduced density matrix and by extension the entanglement entropy, is generally a larger region then the causal wedge. Motivated by this, a duality has been established between the reduced density matrix and a region known as the entanglement wedge [32–34,36,38]. Consider a boundary subregion A, then the entanglement wedge is defined as the domain of dependence of the AdS region bounded by the minimal surface γ and A. Strong arguments for this proposal were given by Jafferis et al. [38] who first showed that two boundary density matrices contain the same information as bulk density matrices through the holographic relation between relative entropies, a measure for distinguishability of two quantum states. Secondly, they proposed a gravitational dual for a particular non-local operator known as the modular Hamiltonian. Here, the bound-ary modular flow generated by the modular Hamiltonian is dual to the bulk modular flow in the entanglement wedge, with implications for the reconstruction of this particular subregion. This holographic relation between the bulk and CFT modular Hamiltonian is by now known as the JLMS relation3 [38] and is central for parallel transporting subregions in holographic theories.

Czech et al. [53] utilized the JLMS relation to propose a holographic relation between the bulk curvature and the modular Berry curvature of the space of modular Hamiltonian in a CFT. Here, the geometric connection that connects entanglement wedges is determined by the entanglement of its dual CFT state, making this a valuable tool in relating entanglement and the dynamics of spacetime. As an extension of this formalism, we studied the effects of a Maxwell field and its behavior under geometric parallel transport. Maxwell’s theory is in general an important factor

1_{AdS spacetime is also referred to as the bulk.} 2_{After Hamilton, Kabat, Lifschytz, and Lowe.} 3_{After Jafferis, Lewkowycz, Maldacena, and Suh.}

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mal surface. Not an uncommon result since the gauge symmetry, also called surface symmetry or edge mode, appeared in previous studies on systems with boundaries [56–59]. The full symmetry group now exists of two earlier found gravitational edge modes [53] and the U (1) edge mode. Here, the gravitational edge mode consists of the boost in the plane normal to the minimal surface and a diffeomorphism along the minimal surface. The U (1) edge modes commute with the boost, although it does not with the diffeomorphism. This correlation is again seen during the parallel transport process, as we find that the U (1) frame spanned by the internal U (1) coordinates is shifted along the minimal surface.

The setup of this thesis is as follows. In chapter 2 we will review the concepts related to subregion duality and the reconstruction of the entanglement wedge. This includes a discussion on entanglement entropy in QFT and holography, a general explanation of bulk reconstruction in global coordinates, the problem with subregion reconstruction, and how this is solved with a quantum error-correcting code [35,36]. The chapter is closed with a review on the holographic relation between modular Hamiltonians. In chapter 3 we focus on the parallel transport protocol. Here, we shortly address some familiar examples from physics, such as transporting vectors in general relativity, and fields in a gauge theory. In the latter, we give an intuitive explanation of how the gauge field can be perceived as a connection between local coordinate frames. The main part of the chapter is devoted to the explanation of fiber bundles. The fiber bundle is the mathematical tool used for parallel transport. The goal is to obtain a solid grasp on the mechanism of parallel transport in its most general form. The general treatment allows us to apply parallel transport to any physical problem at hand. After understanding how the entanglement wedge is reconstructed and having a good understanding of parallel transport we shall address the holographic parallel transport protocol for the entanglement wedge in chapters 4 and 5. In chapter 4 we concentrate on the AdS3/CFT2 scenario and show that the geometric connection for entanglement wedge is

determined by the boundary entanglement. In chapter 5, we discuss a slightly different approach to the parallel transport formalism, which holds in any dimension and show the obtained results of the study on holographic parallel transport in an Einstein-Maxwell theory.

2 The background

Nowadays string theory is familiar under the majority of people. It is found all over YouTube and popular scientific blogs for string theory enthusiasts and possibly mentioned in high schools during physics classes. Overall it has one strong message to say: string theory is a theory of everything . . . , but what is everything?

By everything, string theorists mean all the fundamental forces in our universe and the in-teractions between them [3–5]. Today we have a description of the electromagnetic force, the weak force, and the strong force, however, one force remains a mystery; gravity. String theory is signed up to unify all four fundamental forces but has not quite realized this. The philosophical interpretations of string theories, however, just as the technical details of the theory shall not be addressed in this thesis as it is not strictly needed for the work presented here. Nevertheless, we do wish to spend a few general words on the topic, since this thesis nonetheless finds itself under the “category” string theory. Afterwards, we shall shortly discuss the relevant branch that emerged from superstring theory; the AdS / CFT conjecture.

The most elementary element in string theory is not a point particle, but a one-dimensional string, satisfying symmetries such as conformal invariance and general covariance. The strings occur in both open and closed forms, where the open strings are associated with the standard model forces. The closed strings, however, describe gravitational interactions. Besides bosonic excitations, which are described by the 26- dimensional bosonic string theory, one would like to describe the nature of fermions. For this, one would need to resort to the supersymmetric

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10-dimensional string theory4_.

In 1997, Maldacena [1] proposed the conjecture that type IIB superstring theory in AdS5 ×

S5_{is dual to N = 4 U (N ) super-Yang-Mills theory in 3 + 1 dimensions, where N = 4 denotes the}

number of supersymmetries the theory has. Roughly speaking it is a duality between a strongly coupled 4 - dimensional gauge theory and a gravitational theory in 5 - dimensional AdS spacetime. The AdS spacetime stands for the anti-de Sitter spacetime, a spacetime with constant negative curvature. The spacetime is named after the Dutch astronomer who in 1917 solved the Einstein equation with a constant positive curvature, known as the de Sitter spacetime. The AdS space-time, also referred to as the bulk, has a natural notion of a spatial boundary. The gauge theory is then said to “live” on the 4-dimensional boundary, and hence also referred to as the boundary theory. The boundary theory has a conformal symmetry, containing Poincar´e invariance and scale invariance. Such theory is in general known as a conformal field theory, i.e., CFT.

The AdS/CFT conjecture is a wide area integrated into (almost) every branch of physics [3]. Understanding every aspect of AdS/CFT in its full glory would take multiple projects on its own. Hence, in the remainder of this chapter, we shall restrict ourselves to an overall introduction to the topic, and focus on the duality between thermodynamics and black holes. For the interested reader, we suggest reading [3–5] and encourage the reader to dig into the overwhelming amount of literature available on the world wide web.

2.1 Anti- de Sitter / Conformal field theory (AdS/CFT) conjecture

The conjecture claims that a four-dimensional gauge theory is related to five-dimensional theories of gravity. In this sense, AdS/CFT is often referred to as a holographic theory, similar to an optical hologram that encodes a three-dimensional image on a two-dimensional object. In general the holographic theory encodes a d-dimensional theory into a (d-1)-dimensional theory. One of the first holographic studies, showing that a five-dimensional gravitational theory corresponds to a four-dimensional field theory, was the study of black holes [3,6,8,62]. A black hole is a finite-temperature system and thus has a notion of entropy. Here, the black hole entropy is proportional to the area of the black hole horizon. This relation is vastly different from the usual statistical entropy which is proportional to the volume of the system. However, a four-dimensional volume is an five-dimensional area. This implies that if a black hole can be described by a four-dimensional field theory, the black hole must “live” in five-dimensional spacetime.

2.1.1 Black holes and thermodynamics

Black holes have a tendency to “eat” everything that dared to come to close. As matter falls into a black hole, the black hole surface area A grows as [3]

A = 16πG

2_M2

c4 ,

with M its mass. Classically nothing escapes a black hole, hence its area can not decrease. This is similar behavior shown by thermodynamic entropy and thus one could ask if the black hole horizon area is an equivalent notion as thermodynamic entropy. It turns out that a black hole analogy can be found for every thermodynamic law [3]. However, there are several problems in constructing this analogy.

Starting with the zeroth law of thermodynamics, stating that a system will eventually reach thermal equilibrium and the temperature T becomes constant in equilibrium. We find a similar black hole law stating that a black hole, which initially may be asymmetric, will become spherical symmetric. Implying that the gravitational force will remain constant once symmetric. It is thus the gravitational force per unit mass or gravitational acceleration on the horizon, known as the surface gravity, that corresponds to the temperature. Continuing with the first law, we consider

4_{There are five consistent superstring theories, denoted as the type I, type IIA, type IIB superstring theories,}

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the fact that a black hole surface grows proportional to its mass, κdA ≈ GdM , and recall that mass equals energy. Compared to the first law of thermodynamics, dE = T dS, a black hole should indeed have a notion of temperature, here represented by the surface gravity κ = _4GMc4 . Be that as it may, a black hole with a notion of temperature, should have thermal radiation. Hence, a contradiction occurs, since classically nothing should emerge from a black hole. Secondly, horizon area and entropy have different dimensions. This is easily solved, however, by dividing the area with a squared length term.

In 1975, Hawking [6] showed that black holes indeed emit black body radiation due to quantum mechanical effects. This is now famously known as Hawking radiation and allows us to construct the first law of black holes as

SBH =

A kB

4 l2 pl

. (1)

Here, lpl is the Planck length to secure the correct dimensions, and kB is the Boltzman constant.

The first law of black holes is also referred to as the Beckenstein-Hawking formula. Hence it is also named after Jacob Bekenstein, who in 1972 first conjectured that black holes should have entropy [8]. Other problems have emerged however by introducing Hawking radiation, namely, these quantum effects violate the classical law that states that the area of the event horizon of a black hole cannot decrease. The problem with an evaporating black hole is that the emitted radiation does not contain the information that has fallen into the black hole. This means that once a black hole has disappeared all the information inside the horizon has disappeared with it, resulting in the black hole information paradox [6,7], which will be outside the scope of this thesis.

An important observation about the “area law” for black holes, analogue to the first law in thermodynamics, is that black hole entropy is proportional to the area, while statistical entropy is proportional to the volume of a system. Hence, it suggests that a black hole can be described by a statistical system with one spatial dimension lower than the gravitational theory. Based on this special example of holography, Ryu and Takayanagi proposed a different “area law” in AdS/CFT [4,14]. A relation between the von Neumann entropy in CFT and a minimal area in AdS, which sparked the study of quantum information in the context of AdS/CFT.

2.2 Holographic entanglement entropy

Consider two subregions A and its complement B which are in an entangled state |ψi. The amount of entanglement between a subsystem A and its complement is quantized by the von Neumann entropy given by [9,10]

SA= − Tr ρAlog ρA. (2)

Here, ρAis the reduced density matrix for a subregion A and is obtained by tracing out B from the

density matrix ρ = |ψi hψ|. An observer with only access to the region A will not have access to the information about the full entangled state. Instead, he or she will have access to the state of the degrees of freedom in A that is encoded in the reduced density matrix. This is analogous to the black hole scenario, where the region inside the horizon is inaccessible to observers outside. Given previous discussion, it is natural to ask about the relation between the von Neumann entropy and thermodynamic entropy. As mentioned before the thermodynamic entropy scales with the volume of a system. The von Neumann entropy, however, will not as will become clear in the next section.

2.2.1 Entanglement in QFT

We will begin our discussion on entanglement entropy in QFT by considering a discrete lattice system [9,11]. The lattice will have a lattice spacing and has degrees of freedom localized on the lattice sites. It is assumed to have a finite-dimensional Hilbert space Hαon each site, with α the

site index. A pure quantum state will then be an element of the Hilbert space which is written as a tensor product,

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The entropy between two points on the lattice is obviously not of interest. Tracing out every point in a field except for two will decohere the system extensively. Any entanglement that was present would consequently get destroyed. Instead, the field is split into two regions A and its complement region B. The boundary ∂Ais known as the entangling surface and the Hilbert space

factorizes as a tensor product,

H = HA⊗ HB.

Again, the reduced density matrices are given by ρA = TrBρ and ρB = TrAρ, and have equal

eigenvalues if the total system is in a pure state. Consequently, the entanglement entropy for both regions is equal, i.e., SA= SB. This implies that the entropy depends on a shared property

of the two regions. Since they do not have equal volume, V ol(A) 6= V ol(B), the entanglement entropy can not scale with volume. They do, however, share a lower dimensional surface ∂A= ∂B,

suggesting that the entanglement entropy of a pure state in a local QFT is equal to the area of the boundary surface,

SA=

Area of ∂A

d−1 . (3)

A continuous QFT is now obtained by letting the lattice spacing go to zero, → 0. From (3) in the lattice system discussion above, it is expected that entanglement entropy in field theories are ultraviolet divergent5_{. It might seem natural to assume a factorization of the Hilbert space} again. However, due to subtleties at the boundary ∂A this is not as straightforward as one might

think [9]. The notion of a subsystem as a spatial region can be defined by considering a globally hyperbolic spacetime manifold M in a relativistic quantum field theory [9]. Note that a Cauchy slice Σ is an acausal set i.e., no two distinct points on Σ are causally related. Furthermore, Σ has a causal domain covering the entire manifold, D(Σ) = M . By causal domain is meant the region that contains all causal curves between any two timelike separated points that cross Σ. The Heisenberg equations of motion evolve observables in time and, due to the causal structure, can be used to evolve any observable onto a given Cauchy slice Σ. Therefore, any observable can be constructed out of observables on Σ by using the equations of motion. This suggests that if one wants to split observables on M , it is sufficient to split the observables on Σ. This combined with the axiom of relativistic quantum field theory stating that spacelike-separated observables commute, i.e. any two subregions on Σ have independent degrees of freedom, makes Σ the nat-ural candidate to split the system into subsystems. Similar to the global case, any observable in the causal domain D[A] of A ⊂ Σ can be constructed (again, by using the equations of motion) from observables on A alone. Hence, HA and ρA are associated with the region D[A] and not

solely A. In other words, a region A0 of a different Cauchy slice Σ0, with the same causal domain D[A0] = D[A] has the same Hilbert space H0A= HA, ρ0A= ρA, and therefore S(A0) = S(A).

In what follows, the system will have a Cauchy slice at t = 0, which is split into two parts. Part A defined as the x > 0 line and part B given by x < 0. The origin is thus the entangling surface and the causal domain of the subregions are the right and left Rindler wedge, respectively. An observer in Rindler space will observe a thermal state. It is then to be expected that the reduced density matrix associated with, let us say, the right Rindler wedge, denoted as ρR, has a similar

expression as the Gibbs state from thermodynamics, albeit with an Unruh temperature [9,10]. The reduced density matrix of a Gibbs state can be represented by a Euclidean path integral on an interval with a length equal to the inverse temperature β [10]. As is shown below, one can also express the reduced density matrix associated with the half line as a Euclidean path integral [9,10].

2.2.2 Euclidean path integral formalism

For the construction of the reduced density matrix consider the free scalar field φ(x, t). In analogy with quantum mechanics, the scalar field theory is quantized by representing the states |Φi as wavefunctions Φ[φ] defined by

Φ[φ] = hφ|Φi .

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Here, |φi denotes an eigenstate of the operator field ˆφ(x, t = 0). Furthermore, the wavefunction is split into

φ(x) = φA(x) + φB(x),

where φA(x) has support on part A and φB(x) only on part B. The eigenstate |φi is rewritten

accordingly as

|φi → |φA, φBi ∈ HA⊗ HB,

where the states |φAi form a basis of HA and similarly for |φBi and HB. Hence, the matrix

elements of the density matrix ρA= T rB|Φi hΦ| are written as

hφA| ρA|φ0Ai =

Z

d[φB] hφA, φB|Φi hΦ|φ0A, φBi .

The integral is a similar procedure as taking the trace over B since φB is equal in both the bra

and ket.

Let us now consider the ground state |Φ0i. This state can be represented as a Euclidean path

integral over all fields ˜φ(x, itE), with tE < 0 subject to the boundary condition ˜φ(x, 0) = φ(x)

and ˜φ(x, −∞) = 0. The Euclidean path integral is obtained by analytically continuing the time to negative Euclidean time in the lorentzian path integral, i.e., t = itE, with −∞ < tE< 0 [9,10].

Hence, Φ0[φ] = 1 N Z φ|˜tE =0 ˜ φ|_{tE =−∞} D[ ˜φ] e−S[ ˜φ].

To obtain a path integral representation of the density matrix ρA, first note that the conjugate

state hΦ0| has a similar path integral definition with support on tE > 0. The reduced density

matrix has thus a path integral over fields defined for all values of tE. The trace over part B

implies that |Φ0i and hΦ0| are glued together at tE = 0 along x < 0. The boundary condition

along region A is kept free and the field values for the two limits are ˜φ(x > 0, tE= 0+) = φAand

˜

φ(x > 0, tE = 0−) = φ0A.

2.2.3 Minkowski ↔ Euclidean ↔ Rindler space

As demonstrated in the previous section, the vacuum state in Minkowski space can be represented in terms of a Euclidean path integral. Subsequently, a reduced density matrix can be constructed with the path integral representation and a trace over the region B. If one would map the Cartesian coordinates to polar coordinates (r, θ), it becomes clear that ρA is the state of a field

theory constricted to the Rindler wedge [9]. In doing so, the cut plane can be described by r ∈ [0, ∞), θ ∈ [0, 2π], and the point θ = 0 and θ = 2π with any given r are distinct, due to cut. Let θ be the Euclidean “time” coordinate such that the cut plane is a time interval of length 2π over the half line r ≥ 0. After a Wick rotation θ = iχ, one obtains the maps

x = r cosh χ, t = r sinh χ, with −∞ < χ < ∞, which are contained within the Rindler wedge.

To see this from another perspective and get a better understanding of the implications, note that the vacuum state is Lorentz invariant. Hence, it is not only annihilated by the Hamiltonian H, but also for instance, by the boost generated by K which leaves the origin invariant [9,10]. The Hamiltonian and boost generator are associated with the killing vector fields,

H ↔ ∂ ∂t, K ↔ x ∂ ∂t− t ∂ ∂x,

which shows that the Hamiltonian generally represents time translations. Likewise, in Rindler coordinates (σ, τ ), K generates translations in Rindler time τ via

x = 1 ae

aσ_{cosh aτ,} _{t =} 1

ae

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with a the proper acceleration, and which holds in the right Rindler wedge (0 < x < |t|). In other words, K plays the role of a Hamiltonian for the Rindler observers. A similar relation is obtained for the left Rindler wedge by changing the sign in the equation for x. An important observation is that one can also change from the right to the left Rindler wedge by shifting the Rindler time by an imaginary amount, τ → τ ± iπ/a. This maps x to −x and also reverses time. Note that the t = 0 plane in Minkowski plane which coincides with the tE = 0 plane after the

Euclidean rotation, is mapped to itself. In Rindler coordinates this plane is described by τ = 0 and τ = ±iπ/a. Hence, there exist two ways to analytically continue from the right to the left Rindler wedge, depending on the sign chosen.

To make it explicit, analytically continue the time coordinate to imaginary values τ = iη/a, and introduce a new coordinate ρ = eaσ_{/a. Then x remains real, although t becomes imaginary:}

x = ρ cos η, t = −iρ sin η.

Hence the new coordinates correspond to the conventional polar coordinates on the Euclidean plane, which corresponds to the analytic continuation to imaginary times from Minkowski space. It follows that the analytic continuation to Euclidean time for Rindler and Minkowski observers is directly related and describes the same Euclidean space. In particular, what corresponds to boosts in Minkowski space is now just a rotation in Euclidean space and the Euclidean coordinate η is periodic in the sense that η and η + 2π correspond to the same point. This is precisely the characteristic of a thermal state [9,10], described by the density matrix

ρ =e

−βH

Z , with Z = Tr e

−βH_,

where the inverse temperature β is periodic. Compared to the Gibbs state, we observe that ρAof

the half line is a Gibbs state with β = 2π, and the Hamiltonian is replaced by the generator of τ translations (i.e., boosts in Minkowski space) [9,10]. Thus the reduced density matrix associated with the right Rindler wedge is given by,

ρA=

1 Ze

−2πK_, _{with Z = Tr e}−2πK_. ₍₄₎

Before going to the entropy, let us introduce one more notion. It is an operator known as the modular Hamiltonian or entanglement Hamiltonian defined by the equation [9,10]

ρ =e

−Hmod

Z . (5)

The name modular Hamiltonian comes from the fact that it plays an analogous role as the com-bination βH in the thermal density matrix. However, the modular Hamiltonian is generally a complicated, non-local operator which may have a different physical meaning than the usual Hamiltonian. In appendix (A) we discuss this operator further and show that the modular flow generated by Hmod can be mapped to the causal domain of a spherical subregion on a Cauchy

slice in Minkowski space [21,37]. This shows that it is also possible to construct a reduced density matrix for a spherical subregion in Minkowski space. The situation discussed in the appendix is of much greater interest though. It is one of the few situations where the modular Hamiltonian can be described as a local operator and has a relatively simple explicit expression. To get ahead of things, it is also an explicit example that allows us to construct a holographic relation for the modular Hamiltonian, and thus the reduced density matrix [19,20,37].

Compared to the definition of the modular Hamiltonian (5), it is clear that the modular Hamiltonian in Rindler space is the boost generator: Hmod= 2πK. Henceforth, K will be used

as the notation for the modular Hamiltonian.

2.2.4 The replica trick

To compute the von Neumann entropy one has to be able to take the logarithm of the reduced density matrix. This might be a complicated task, therefore it is convenient to introduce a general

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class of entropies known as the R´enyi entropies. The R´enyi entropies are defined as Sn(ρ) =

1

1 − nlog(Tr ρ

n_). ₍₆₎

The R´enyi entropies are often easier to compute and the von Neumann entropy can be recovered by taking the limit n → 1:

S(ρ) = lim n→1 1 1 − nlog(Tr ρ n_{) = lim} n→1− ∂ ∂nlog(Tr ρ n_),

where is the last step L’Hopital was used. By substituting (4) into the R´enyi entropy we obtain the expression in terms of the partition function Z:

Sn(ρ) = 1 1 − nlog Zn Zn 1 . (7)

For the computation of Sn(ρ), Tr ρn (or equivalently z_znn 1

) is required, which can be obtained with the Euclidean path integral discussed above [9,10]. ρn _{is then constructed as n copies of the cut}

plane representation of ρA. The upper edge of the cut on the first plane is glued to the lower edge

of the cut on the second plane, where the upper plane of the second plane is glued to the lower plane of the third plane again. This continues all the way to the lower edge of the cut of the nth plane which is then glued to the upper plane of the first plane, completing the circle. The result is a n-sheeted surface with a branch cut along A connecting two sheets, which is recognized as the Riemann surface for z1/n_{, where z = x + it is the complex coordinate. This method of writing}

Tr ρn _{is known as the replica trick. An important point is that the surface has a singularity at the}

origin. This singularity leads to a divergence in the path integral which is cut off by introducing a UV regulator. On this n-sheeted surface, the partition function Zn has a representation in terms

of the path integral over the fields φ

Zn=

Z

Dφ e−S[φ].

2.2.5 Gravitational dual of entanglement entropy

Now we are in a position to calculate the entanglement entropy in CFTd+1 from the gravity

theory on AdSd+2. As was laid out in the previous section, the Renyi entropies in CFT are

com-puted through the partition function. The gravitational dual is then straightforwardly constructed through the bulk-to-boundary relation. A fundamental principle from the AdS/CFT conjecture, which is expressed by the equivalence of the partition function in both theories [1,3–5],

ZCF T = ZAdS.

The focus in this section will be on the case without supersymmetry and omit the matter terms in the action. Hence, the strong coupled CFT is dual to classical AdS with a partition function of the Einstein Hilbert action

ZAdS = e−sEH, where SEH= −1 16πGd+2_N Z dd+1x√g (R + 2Λ),

with Gd+2_N is the Newton constant, R the Ricci scalar, and Λ the (in this case) negative cosmological constant. The boundary Riemann surface Rn is characterized by the presence of a deficit angle

δ = 2π(1 − n) on ∂A [4,10]. It is technically complicated, however one should find a (d + 1)-dimensional geometry that asymptotes to Rn and the deficit angle at r → ∞. The problem

simplifies in AdS3, since it has no propagating degrees of freedom, which implies that solutions

are given by patches of spacetime glued together [4]. The n - sheeted pure AdS3 geometry is

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assumed that this result generalize to a AdSd+1 geometry [14]. With this assumption, the Ricci

scalar becomes a delta function

R = 4π(1 − n)δ(γA) + R(0),

where R(0) is the Ricci scalar for pure AdSd+2 and δ(γA) is the delta function localized at the

codimension, with δ(γA) = ∞ for x ∈ γA and δ(γA) = 0 otherwise. The entanglement entropy is

obtained by substituting this action into the partition function SA= − ∂ ∂nlog Z_n Zn 1 _n=1= − ∂ ∂n h(1 − n)Area(γ_A) 4Gd+2_N i n=1= Area(γA) 4Gd+2_N . (8) The codimension surface γAis constructed such that ∂γ = ∂A. Notice that this is a surface in

a time slice, which is a Euclidean manifold. Although there are infinitely many different choices of how to construct γ, it was proposed by Ryu and Takayanagi [14], that one needs to choose the minimal area surface. Hence the variation of the area vanishes and the area takes a minimal value. This procedure singles out a unique minimal surface γA. Here we emphasize that γ is a

static minimal surface inside AdSd+2, whose endpoints are anchored at ∂A as is shown in figure

(1). The minimal surface, from here on also referred to as the RT surface has a similar look as the black hole area law of Bekenstein and Hawking (1). The RT relation can be regarded as a generalization of the black hole area law since, in the presence of an event horizon such as the AdS Schwarzschild black hole solutions, the minimal surface tends to wrap the horizon.

γ

Ab

Bb

A

B

Figure 1: A constant time slice. The boundary CFT is split into two subregions A and B. The entropy of the subregion A is equal to the minimal surface γ, which ends on ∂A. Additionally, en-tanglement between quantum fields in the region Ab and quantum fields in Bbwill give corrections

to the area law by Ryu and Takayanagi.

The RT surface (8) is limited in a couple of aspects [9]. It requires a classical Einstein gravity and the bulk has a time-reflection symmetry which leaves the boundary spatial region A invari-ant. For a general spacetime, without any symmetries assumed and a general boundary region, the entropy is given by the area of the extremal6spacelike surface γext(t). Here, γext(t) is thus the

extremal surface in the entire Lorentzian spacetime. This generalization is known as the Hubeny-Rangamani-Takayanagi (HRT) formula [15].

Deviating from Einstein gravity, means we need to include higher derivative corrections to the bulk gravitational action [17]. Deviating from the classical limit, means quantum corrections of order G0

N (or N0) need to be included. Here, one can consider two regions AB and Bb as static

background geometry on which an effective field theory lives. The quantum fields in the divided bulk subregions might conceivably be in an entangled state as well, which gives rise to quantum corrections to the leading (classical) term in equation (8) [16]. Hence the RT surface is then given by

SA=

Area of γ

4Gd+2_N + Sqm+ O(GN), (9)

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with

Sqm= SAb+

δA 4GN

+ h∆Swaldi .

The change in the area is due to quantum corrections that cause a shift in the classical background. The last term is the expectation value of the Wald-like entropy [16,62].

Consider this holographic relation for entanglement entropy in the boundary theory. If the boundary data arises from the area of a surface in the gravitational dual, one could ask what the corresponding geometry is. In particular, what is the bulk metric, which leads to the given entanglement data? Let us now consider an alternative question: Suppose we are given a global state of the boundary CFT and in addition a spatial subregion A. Is there a specific region of the bulk geometry that can be described by the reduced density matrix ρA on the boundary? In

other words, can bulk fields be reconstructed with information from the reduced density matrix of a subregion in CFT? These questions led to what is now known as subregion duality and were first investigated in [32–34].

The current understanding is that the bulk region known as the entanglement wedge is dual to the reduced density matrix. The entanglement wedge is depicted in (4) and shall be further elaborated on in the next section. Before we explain this concept, we shall first address how bulk and boundary data are related and therefrom explain the reasoning for the entanglement wedge.

2.3 Bulk reconstruction

The AdS/CFT correspondence is usually defined as the equality of the partition functions of the bulk and boundary theories. A later version of the AdS/CFT dictionary gives us an overview of the duality between states and operators in the anti-de Sitter spacetime and the CFT on the boundary. On the most basic level, the conjecture can be thought of as an isomorphism between bulk and boundary Hilbert spaces [2,4]. Here the knowledge of the Hilbert space of a QFT on a spatial Sd−1_{is equivalent to the knowledge of the set of local operators. Hence, one could expect}

that any local bulk operator is isomorphic to a set of local CFT operators. The basic entry we need from the dictionary is the “extrapolate map”, which makes it possible to recover CFT cor-relation functions by extrapolating the insertion points of bulk corcor-relation functions [2,4,23,24,27].

In pure AdS, a bulk scalar field φ(r, t, Ω) with mass m corresponds to a scalar primary operator O(t, Ω) on the boundary with dimension ∆ by [27]

lim

r→∞r

n∆_{hφ(r, t}

1, Ω1) φ(r, t2, Ω2) . . . φ(r, tn, Ωn)ipure

AdS= h0| O(t1, Ω1) O(t2, Ω2) . . . O(tn, Ωn) |0i ,

(10) with |0i the vacuum state. This statement has been proven in [28] for interacting scalar fields. More generally, it is expected that for any semi-classical asymptotic AdS geometry g there is a dual state |φgi [27]. Although the bulk dual of a given boundary state is generally unknown, the

extrapolate dictionary is useful for the so-called “scattering experiments” [29]. Here, wavepackets close to the boundary are sent into the bulk, scattered around, and subsequently collected close to the boundary. The result will accordingly be given by the CFT correlator. This does, however, not cover all bulk information [27]. For example, the extrapolate dictionary does not directly pro-vide the correlator between bulk fields for finite values of r. This may be useful, however, for the description of a local bulk experiment [27]. Hence, one would need to develop the bulk-boundary dictionary further.

For a computation [27] consider the free scalar field in a semiclassical bulk, i.e., a bulk with a gravitational constant G << `d−1, where ` the AdS radius is. To have a semiclassical dual, it is essential for a CFT to have a parameter7 N >> 1 [2,27]. This controls the factorization of CFT

7_{So far, N is the central charge of CFTs with a bulk dual. See [}₂_{] for the role of N in the stringy origins of the}

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correlators, which can be dual to the perturbative parameter in the gravitational theory. The parameter N is an expansion parameter in CFT and related to the gravitational constant by

N2=`

d−1

G .

Considering the single expansion parameter, the most general bulk theory with Einstein gravity and scalar fields will have an AdS metric in global coordinates

ds2= −1 + r 2 `2 dt2+ dr 2 1 + r2 `2 + r2dΩ2_d−1,

where henceforth ` = 1. We shall work in the large N limit, i,e, G → 0. In this limit gravity is “switched off”, hence we can ignore gravity and consider the scalar field in a fixed background. In this limit, the action is

S = Z

dd+1x√−g (∂µφ∂µφ + m2φ2), (11)

and the free field equation is

( − m2)φ = 0,

with the D’Alembartian in AdS. This can be solved in terms of a mode expansion [2]. Note that the metric has a rotational and time translation symmetry. Hence, the solution may be written as the stationary wave solutions φωl(r, t, Ω) = eiωtψlω(r)Yl(Ω), where Yl(Ω) are the usual

spherical harmonics, which is an eigenstate of the D’Alembartian with eigenvalue l(l + d − 1) [2]. Substituting this in the field equation gives

(1 + r2) ψ00+d − 1 r (1 + r 2_{) + 2r}_ψ0₊ ω2 1 + r2 − l(l + d − 2) r2 − m 2_{ψ = 0,}

which at large r becomes

r2ψ00+ (d + 1) r ψ0− m2ψ = 0. (12) This has clearly polynomial solutions of the form r−a and after substituting r−a in (12) two solutions emerge with a = ∆, d − ∆. Here ∆ = d₂ +1₂√d2_{+ 4m}2 _{is the scaling dimension of the}

scalar field in AdS. Hence, the asymptotic solution to the field equation is given by

φ(r, t, Ω) = r∆−dK(t, Ω) + r−∆L(t, Ω). (13) The first part are the non-normalizable modes, whereas the latter part with r−∆ fall off are the normalizable modes. These are the ones of interest in defining a unitary field theory in AdS [2].

The full solution for ψlω(r) is given by the hypergeometric function [2]

ψlω(r) = r √ 1 + r2 l 1 √ 1 + r2 ∆ 2F1(a, b; c; r2 1 + r2), with a = 1 2(l + ∆ − ω), b = 1 2(l + ∆ + ω), (14) c = l +d 2.

The hypergeometric function converges if c is a positive integer for all _1+rr22 << 1 and if Re[c −

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−√d2_{+ 4m}2 _{< 0, with −}d2

4 ≤ m

2_{. Hence it diverges. The function will converge if and only if}

either a or b is a negative integer. In this case the function reduces to a polynomial. Consequently, ω is quantized as ωnl= ∆ + l + 2n, with n = 0, 1, 2, . . . , as can be seen from (14). The full solution

for φωl(r, t, Ω) is then given by

φnl(r, t, Ω) = 1 Nnl eiωnlt_Y l(Ω) r √ 1 + r2 l 1 √ 1 + r2 ∆ 2F1 − n, ∆ + l + n; l + d/2; r 2 1 + r2 , where Nnlis a normalization constant. The final result in terms of a mode expansion is then given

by [2] φ(r, t, Ω) =X nl anlφnl+ a†nlφ ? nl, (15)

where the coefficients anl and a†nl are the annihilation and creation operators.

This is, however, just half of the dictionary. One would also like a formulation of the bulk field φ(r, t, Ω) from the boundary theory, which is known as bulk reconstruction. The first approach to this was by Hamilton, Kabat, Lifschytz, and Lowe [23–25] and goes by the name HKLL method. To obtain this result, note that the extrapolate dictionary implies a boundary condition for the bulk field:

O(t, Ω) = lim

r→∞r

∆_{φ(r, t, Ω).} ₍₁₆₎

This relates the boundary value to a primary operator in CFT. Of course (16) is not an actual boundary condition since it relates two different spaces: an operator that acts on the CFT Hilbert space and the boundary value of a bulk field. The HKLL method actually constructs a CFT operator φCF T(r, t, Ω) which obeys the bulk equation of motion, with boundary condition (16).

Henceforth, the subscript shall be used if it clarifies the discussion, otherwise, the subscript is neglected as is commonly done in literature. For simplicity, we will focus on the case where ∆ is an integer. In this case, the solution becomes periodic in time [25]. Hence we can limit to the range −π > t > π. For the general case, we refer to [23,25].

Start with substituting the expansion (15) into (16): lim r→∞r ∆_{φ(r, t, Ω) = lim} r→∞r ∆X nl anlφnl+ a†_nlφ?nl= O(t, Ω). (17) Then lim r→∞r ∆_φ nl = lim r→∞r ∆ 1 Nnl eiωnlt_Y l(Ω) r √ 1 + r2 l 1 √ 1 + r2 ∆ 2F1 − n, ∆ + l + n; l + d/2; r 2 1 + r2 (18) = 1 Nnl eiωnlt_Y l(Ω)2F1 − n, ∆ + l + n; l + d/2; 1:= gnl(t, Ω) (19) such that (17) becomes

O(t, Ω) =X

nl

anlgnl+ a†nlg ?

nl. (20)

When ∆ is an integer, the modes gnl are orthogonal to all gnl? on the interval −π < t < π [25].

Hence, we can solve for anl (and similar for a†nl) by

anl= Z π −π dt Z dΩ g_nl? O(t, Ω).

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φ A

Figure 2: On the left: in global coordinates, the bulk field φ is reconstructed from the boundary theory by integrating the product of the smearing function and spacelike separated CFT operators. The smearing function has support on the purple shaded region. On the right: the Rindler wedge associated with the boundary region A. The Rindler wedge can be reconstructed solely from boundary operators in A.

Substituting the coefficient back into (17) and we find φ(r, t, Ω) = Z π −π dt0 Z dΩ0 X nl φnl(r, t, Ω) gnl?(t 0_{, Ω}0_{) + c.c.}_O(t0_{, Ω}0_), where φnl(r, t, Ω) gnl?(t

0_{, Ω}0_{) is real and thus equal to its complex conjugate. The final result is}

then given by φ(r, t, Ω) = Z π −π dt0 Z dΩ0K(r, t, Ω; t0, Ω0) O(t0, Ω0), with K(r, t, Ω; t0, Ω0) = 2(φnl(r, t, Ω) gnl?(t 0_{, Ω}0_)).

Here the boundary-to-bulk kernel K(r, t, Ω; t0, Ω0) is usually referred to as the smearing function8_. Note from (15) with ωnl = ∆ + l + 2n, where n = 0, 1, 2, . . . , that the solution for O(t, Ω) does

not contain any modes in range −∆ + 1 . . . ∆ − 1. Therefore if a term eimt _{is added, with m an}

integer between −∆ + 1 and ∆ − 1, the integral of its product with O would vanish [23]. This allows us to construct the smearing function such that it has support on the boundary region that is spacelike separated from φ(r, t, Ω). Thus the integral is over the set containing operators inside a complete spatial circle on the boundary as is shown in figure (2).

Unfortunately, there are a few obstacles with bulk reconstruction through the HKLL method [26]. First, it seems that the HKLL method gives a definition of bulk operators intrinsic to the boundary CFT, however, to know the smearing function, one would first need to solve the bulk equations of motion. This presumes an existing bulk geometry in which the scalar field acts as a probe. Secondly, the constructed operators are coordinate dependent and therefore not diffeomor-phism invariant. Well defined operators in a gravity theory, however, should be diffeomordiffeomor-phism invariant, since this is a gauge symmetry of the bulk. Hence, every element of the operator al-gebra should be diffeomorphism invariant. To construct a diffeomorphism invariant operator one would start with a local operator and need to add gravitational dressing [26]. This operation is non-unique and non-local. Lastly, a local bulk scalar field in global coordinates depends on all CFT operators at a fixed time. This is true even when the bulk field is pushed to the boundary. This contradicts with the idea of the smearing becoming more and more local as the bulk operator

8_{For a view on the construction of the smearing function in poincar´}_{e coordinates in AdS}

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approaches the boundary. Given the reconstruction discussed above, we could apply a coordinate transformation from global to Rindler coordinates. Then, the boundary operators are smeared over the Rindler wedge such that the smearing function becomes local as the bulk operator reaches the boundary. Note that the Rindler wedge only covers a part of the full AdS. Once more, one can solve the field equations in terms of a mode expansion. For AdS3this is done with the ansatz

φ(t, r, φ) = e−iωteikφ_f

k,ω(r) and shall be left as a fun exercise for the reader9.

Given the AdS3 metric in Rindler coordinates

ds2= −(r2− r2 +)dt 2₊ dr 2 r2_{− r}2 + + r2dχ2, one should find the solution [30]:

fk,ω(r) = r−∆ r2− r₊2 r2 −i˜ω/2 F∆ − i˜ω − i˜k 2 , ∆ − i˜ω + i˜k 2 , ∆, r+2 r2 .

Here −∞ < t, χ < ∞ and r+< r < ∞, with r+ the radial position of the Rindler horizon. The

AdS boundary is positioned at r = ∞ and we have defined ˜ω = ω/r+ and ˜k = k/r+. The mode

functions are real and satisfy [23]

fk,ω= f−k,ω= fk,−ω = f−k,−ω.

Finally, the field is expanded in Rindler modes as φ(t, r, φ) = Z ∞ −∞ dω Z ∞ −∞ dk aωke−iωteikφfωk,

and the Rindler boundary field is given by φ0(t, φ) = lim r→∞r ∆_{φ(t, r, φ) =} Z ∞ −∞ dω Z ∞ −∞ dk aωke−iωteikφ, such that aωk= 1 4π2 Z dt dφ e−iωteikφφ0(t, φ).

Formally, the smearing function can be written as a Fourier transform of the AdS-Rindler mode functions fk,ω(r) [23]:

K = Z

dk dω e−iωteikφfk,ω(r).

Although the mode functions grow exponentially with k, hence the smearing function diverges. This is not necessarily a problem, if we consider the smearing function as a distribution instead of a function [31]. Furthermore, the field modes are not quantized, since fnωis regular at r = 0 [23,30].

Consequently, one can no longer change the support of the integration region by adding modes. All modes contribute. Therefore it is expected to find well-localized boundary operators [23].

The interesting point of the AdS-Rindler reconstruction is that any field in the Rindler wedge, the smearing distribution has only support on a boundary subregion associated with the Rindler wedge. No information regarding the rest of the boundary is required. The problem with construc-tion in the Rindler wedge, however, is that it is coordinate dependent. To resolve this problem we can define a causal wedge, depicted in figure (4). To define the causal wedge consider the earlier discussed domain of dependence D[A] of a boundary region A. The causal wedge C[A] is then defined as the set of bulk points through which there exists a causal curve that starts and ends on D[A]. When the CFT Cauchy surface Σ is the t = 0 slice and the boundary subregion A is a semi-circle, then the causal wedge is the Rindler wedge. As discussed before, an HKLL smearing function exists on the Rindler wedge. Then, since the isometry of AdS allows us to map the Rindler wedge to any causal wedge, the HKLL method exists for reconstruction in any causal

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wedge. Any field inside the wedge can thus be constructed in terms of O in D[A]. No coordinates were introduced this way, so the reconstruction is coordinate independent. Hence, there exists a coordinate independent HKLL smearing function in Rindler coordinates. Unfortunately, we are not free of all problems yet.

C B A Bb Ab φ A B C Ab Bb Cb φ

Figure 3: A Cauchy slice of AdS with CFT on the boundary. On the left, the boundary contains two subregions A and B, each with a causal wedge associated with it. The bulk operator φ is contained in both causal wedges and therefore should be represented in terms of CFT operators from region A and B. Hence, the only possibility for φ to have support on the boundary is if φ is related to operators inside the intersection of A and B. However, as can be seen in the figure, the field is not contained in the bulk region associated with C ≡ A ∩ B. On the right, the bulk field φ lies outside the casual wedge associated with any boundary subregion, although it can be reconstructed with A ∪ B, B ∪ C, or A ∪ C. The field should now be reconstructed with any of the three operators that lie in all these regions. There is no possible way that these operators are equal. This contradiction is solved by perceiving the AdS/CFT as a quantum error-correcting code.

There are multiple ways to portray the problem [35]. For example, consider a timeslice of AdS with two overlapping causal wedges C[A] and C[B] as is shown in figure (3). An operator φ is contained in both causal wedges, but not in their intersection C[C] ≡ C[A ∩ B]. The operator should then be reconstructed by operators with support on only A, but also with support on only B. For a CFT operator with only support on A to be equal to an operator on only B, however, it undoubtedly needs to have support on their intersection C. This is where the paradox arises since φ was chosen such that it lies outside the casual wedge of C. Consequently, the CFT operators should not have any support on the region C.

Another example is shown in the right figure of (3). Here, a bulk field φ lies outside the casual wedge associated with any boundary subregion, although it can be reconstructed in A ∪ B, B ∪ C, or A ∪ C. The mutual intersection of these regions are three points and can be extended to more regions, with more mutual intersection points. There is no possible way that they are all equal operators. To solve the paradox, Almheiri, Dong, and Harlow [35] conjectured that the AdS/CFT dictionary should be viewed as a quantum error-correcting code (QECC).

2.3.1 Quantum error correction

For an overview of quantum error correction [35], consider a spin system consisting of three qutrits. The qutrits can be thought of as the (three) degree of freedom on the boundary. Now let us say Alice wants to send a state

|Ψi −

2

X

i=0

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to Bob. The qutrit, however, can get lost or erased along the way, and thus the information will never reach Bob. So instead she constructs the following three qutrits:

|0i → |000i |1i → |111i |2i → |222i and send the state

˜ |Ψi = 2 X i=0 ai ˜i , with ˜0 = 1 √

3(|000i + |111i + |222i)

˜1 = 1 √

3(|012i + |120i + |201i)

˜2 = 1 √

3(|021i + |102i + |210i).

The states˜i form the logical Hilbert space or code subspace, whereas the basis states |ii span the physical Hilbert space. The QEC protocol has two remarkable properties. First, the reduced density matrix of any state ˜|Ψi is maximally mixed. Thus the information about the state can not be acquired through a single qutrit. Furthermore, Bob can always apply a unitary transformation on two of the three qutrits and retrieve the physical state. This is because the physical information is stored non-locally in the entanglement between the logical qutrits. Here, it is assumed that the probability of losing a qutrit is p, with p << 1, such that the probability of losing two qubits is negligible.

As an example, assume that Bob has access to the first two qutrits. Bob is then able to use an unitary transformation U12 that acts only on the first two qutrits as a permutation:

|00i → |00i |11i → |01i |22i → |02i

|01i → |12i |12i → |10i |20i → |11i ,

|02i → |21i |10i → |22i |21i → |20i

implementing (U12⊗ I3) ˜i = |ii ⊗ 1 √

3(|00i + |11i + |22i).

If Bob would act with this on an encoded message, he can recover the state |ψi: (U12⊗ I3) ˜|ψi = |ψi ⊗

1 √

3(|00i + |11i + |22i).

By symmetry, Bob can clearly construct the state if he has only access to the second and third qutrits, or to the first and third qutrits. In quantum information terminology, this algorithm is called a quantum error-correcting code and protects against single qutrit erasures.

We were however interested in the action of operators instead of retrieving states. Luckily, there exists an isomorphism between the operators on the physical space and the code subspace. An operator can be constructed that has the same action on the code subspace operator as the physical operator, but with non-trivial support on two of the three qutrits. Indeed, consider the operator that acts on the physical space as

O |ii =X

j

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Then there exists a three-qutrit operator ˜O which implements the same transformation on the code subspace: ˜ O ˜i = X j (O)ji ˜j .

It is then straightforward to check that the operator O12≡ U12† O U12, act as O12 ˜i = U † 12O U12 ˜i = U₁₂† O |ii ⊗√1

3(|00i + |11i + |22i) =X

j

(O)jiU12† |ji ⊗

1 √

3(|00i + |11i + |22i) =X

j

(O)ji

˜j ,

with O acting on the first qutrit. Thus, O12acts only on the first two qutrits, but with an action

equivalent to the action of O on the physical space. By symmetry, we are also able to construct O23 and O13, and have discovered three operators with nontrivial support on different qutrits, yet

have nevertheless the same action in the code subspace.

The paradox of reconstructing a bulk operator as discussed before can now be resolved by perceiving the bulk Hilbert space as a coded subspace and the boundary Hilbert space as the physical space. Consider again the right picture in figure (3). The operator φ can be represented as φAB, φBC, or φAC, with support on A ∪ B, B ∪ C or A ∪ C, respectively. This is, of course,

directly analogous to the example with three qutrits and the operators O12, O23, and O13. The

authors of [35] argues that this is not just an analog, but the protocol on how AdS/CFT repro-duces the bulk. Furthermore, as the bulk operator moves further away inwards along radial paths, the reconstructed wedge has to be taken larger and larger. This can be rephrased as saying that local bulk operators represent logical operations on an encoded subspace, that becomes better protected against localized boundary errors as the operator is moving inwards.

To conclude, HKLL provides a useful method of reconstructing bulk operators from CFT operators but has a couple of conceptual problems associated with it. Now that we have a description of how bulk and boundary operators are related, one can address the question of the gravitational dual of the reduced density matrix. We would like to know whether there is a natural bulk region that is determined by ρA independently of the choice of the global density

matrix. A first observation that was made earlier on, was that a density matrix ρA associated to

the degree of freedom in a boundary subregion A does not solely entail all information about the state of degrees of freedom in A. In fact, the density matrix allows us to compute observables localized in the domain of dependence of A [33]. Any operator in the causal diamond can be expressed in terms of the fields in A alone and therefore computed with the density matrix. Hence, in constructing the bulk dual of ρA, it is more natural to consider the causal diamond,

than the subregion A.

2.3.2 subregion duality

Considering earlier discussed ideas of bulk reconstruction, the first natural bulk region that might be dual to the reduced density matrix is the causal wedge [33,34]. An observer could easily send information into C and subsequently receive information back. Such observation corresponds to computing response functions, in which fields are locally perturbed in D[A] and observed later on. The reduced density matrix can be used for such computation [33]. However, as was argued by [36], the requisite bulk region should contain more information than the causal wedge. At

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the very least the region should be aware of the entanglement inherent in the reduced density matrix. As was shown before the boundary entanglement is dual to an extremal surface that is not necessarily contained within the causal wedge.

Motivated by the discussion above, a second proposal for the bulk dual of ρA that came to

be is a bulk region known as the entanglement wedge [19,20,35–38]. The entanglement wedge denoted as W [A], is defined as the domain of dependence of a region on the Cauchy slice bounded by the extremal surface and the boundary region A. See figure (4). W [A] thus corresponds to a bulk region whose geometry is probed by the entanglement at the boundary. This might im-ply that the boundary has access to all degrees of freedom inside W [A] [33]. Interestingly, the extremal surface, and thus the entanglement wedge, generally extends further into the bulk than the causal wedge. The HRT formula is then able to access information beyond the causal wedge. This has been called entanglement wedge reconstruction and the study on this topic has signifi-cant importance. One could consider the definition of a causal wedge as an event horizon [35]. It could never contain any points inside the horizon since all points in the causal wedge are in the future or past of the boundary subregion. All degrees of freedom in a CFT are then dual to the degrees of freedom outside the horizon. If entanglement wedge reconstruction is true, however, it is possible to reconstruct a bulk degree of freedom inside the horizon. However, all situations up till today where one can compute entanglement entropy in CFT directly, are situations where the entanglement wedge and causal wedge are equal [34].

C[A] D[A] A CFT AdS γ W[A] CFT AdS A D[A]

Figure 4: On the boundary, we have the causal domain D[A] of the subregion A. On the left picture, the causal wedge C[A] is defined as the set of bulk points through which there exists a causal curve that starts and ends on D[A]. On the right, the entanglement wedge W[A] is depicted. It is defined as the causal domain of the region with boundary γ ∪ A.

That being said, the authors of [36] were able to prove that local operators, acting within a code subspace, can be reconstructed inside the entanglement wedge. In doing so they utilized the quantum error correction explained above, and a particular bulk formulation for the boundary modular Hamiltonian. This formulation was established by Jafferis, Lewkowycz, Maldacena, and Suh [38], who showed that besides the modular Hamiltonian, there exists a holographic relation for the relative entropy, making a strong case for the entanglement wedge reconstruction.

2.3.3 Holographic relation of the modular Hamiltonian

The relative entropy is a statistical measure of distinguishability between two density matrices ρ and σ, and it is defined as

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The relative entropy has multiple properties making it interesting to talk about. Properties such as positivity and monotonicity [12], and while entanglement entropy is not well defined for QFT’s, relative entropy’s have precise mathematical definitions [13]. Positivity of the relative entropy means it is non-negative and vanishes if the two density matrices are equal, i.e,

S(ρ|σ) ≥ 0, S(ρ|σ) = 0 → ρ = σ.

Monotonicity of relative entropy states that relative entropy decreases under inclusion, thus S(ρA|σA) ≤ S(ρ|σ),

where ρAand σAare reduced density matrices obtained by tracing out the same amount of degree

of freedom from ρ and σ, respectively.

Now, consider σ as a reference state and define the modular Hamiltonian Kσ = − log σ. The

relative entropy can then be rewritten as

S(ρ|σ) = ∆ hKσi − ∆ hSi ≥ 0, (21)

where the last inequality is guaranteed by the positivity of relative entropy. A first observation is that relative entropy is at least of quadratic order. Thus the contribution at linear order vanished for every σ. To understand this consider moving the reference state σ through a family of states ρ(λ), parametrized by λ such that ρ(0) = σ. Hence, the two density matrices coincide at λ = 0, which means S(ρ(0)|σ) = 0. However, due to the positivity of the relative entropy, S(ρ(λ)|σ) > 0 for both positive and negative λ. Therefor, if the relative entropy is a smooth function, its first derivative must vanish at λ = 0. This means that

δS = δ hKσi ,

which is known as the first law of entanglement [18].

An second important note to make is that a gravitational theory, with its diffeomorphism invariance, is a gauge theory. The definition of entanglement entropy is then ambiguous [11], since one can always add a local gauge transformation. It often occurs that two different descriptions of entanglement entropy give results that differ by the expectation value of a local operator [16], i.e. S(ρ) = Tr(ρ O) + ˜S(ρ). If this is the case then, through the first law of entanglement entropy, the two modular Hamiltonians are related by

S(ρ) = Tr(ρ O) + ˜S(ρ) → K = O + ˜K.

A holographic relation for the modular Hamiltonian can now be extracted from the HRT relation (9) with corrections of order GN. Without loss of generality, we consider the bulk to be

a code subspace Hcode= HAb⊗ HBb, where Ab is the bulk region with boundary γ ∪ A and Bbits

complement as was shown in figure (1). The states ρ are small perturbations around a reference state σ inside the bulk, and can be constructed as path integrals in an effective field theory that is perturbatively coupled to gravity [36]. To proceed, the authors of [16] observated that both the area term and wald-like entropy are expectation values of operators in the bulk effective theory. Hence, the holographic relation for entanglement entropy can be written as

S(ρA) = S(ρAb) + T r(ρAbAloc), (22)

where Aloc is an operator in the semi-classical quantized bulk theory given by

Aloc=

Aext

4G + . . . .

Here, the dots contain the corrections at higher order in 1/N [16], or the presence of general gravitational interactions such as the Wald entropy [62]. Then with equation (22) linearized about σ, and the first law of entanglement entropy, we can conclude that

Tr(δσAKσA) = Tr δσAb(A σ loc+ Kσ_Ab) . (23)

(25)

Here, δσ is an arbitrary perturbation, and the superscript on Aσ

loc is to emphasize that Aloc is

located at the surface defined by extremizing S(σAb) + Aloc. Equation (23) is linear in δσ, thus

after integrating we obtain

Tr(δρAKσA) = Tr ρσAb(A σ loc+ Kσ_Ab) , (24)

with ρ and σ arbitrary states within Hcode, and the superscript on ρσAb is to clarify that even

though there are other states ρ, the bulk Hilbert space is factorized at the quantum extremal surface for σ. Since, (24) holds for any density matrix in the code subspace this implies [36]

ΠcKσAΠc= Aloc+ KσAb, (25)

with Πc the projection operator onto the code subspace. This is the JLMS relation applied to a

code subspace.

Moreover, with the relative entropy (21), equation (22), and the JLMS relation (25), we find up to N0 _{order that [}₃₈_]

S(ρ|σ)CF T = S(ρ|σ)bulk.

Recall that relative entropy is a measure of the distinguishability of two quantum states. So the holographic relation for relative entropy can only exist if the boundary density matrices ρ and σ contain the same amount of information as the bulk ρ and σ [36,38]. As a consequence of the holographic relation for relative entropy, an observer with only access to A could for example measure if particles were added to the vacuum in the entanglement wedge in such way that the bulk relative entropy changes. From this, it should be clear that the entanglement wedge is the rational candidate for the dual of A.

2.3.4 Entanglement wedge reconstruction

To recapitulate, a Cauchy slice of the bulk is divided by the RT surface into a region Ab and Bb.

The entanglement wedge is defined as the causal domain of Ab and a smaller causal wedge can

be associated with the boundary region A. A particular case where the entanglement wedge is larger than the causal wedge is shown in figure (5). It is a case with two intervals in a CFT2,

where the total size of the intervals is larger than half the size of the entire system. An operator near the boundary of the causal wedge that is modular evolved will eventually develop a non-zero commutator with nearby operators outside the causal wedge. From a different viewpoint, an operator near the boundary of W[A] will have an approximate local modular flow [38]. It will propagate along the light lines emerging from the extremal surface and might be in causal contact with operators in C[A]. Hence, to reconstruct operators in the entanglement wedge, one needs to understand the modular flow.

One particular modular flow was already discussed in section (2.2.3). Systems in a thermal state or with a Rindler subregion have a time translation symmetry and a local modular Hamiltonian that generates Rindler time τ translations. In appendix (A) we also show that this is true for CFTs with a spherical subregion. The local bulk operators inside the entanglement wedge, which coincide with the causal wedge can be expressed in term of the operators in the boundary subregion by the earlier described HKLL method:

φ(X) = Z A dyd−1 Z dτ K(X; y, τ ) O(y, τ ), X ∈ Ab.

A natural proposal [38] to generalize this bulk reconstruction is to replace Rindler time τ by the modular parameter s. In other words, consider the modular flow of local boundary operators, defined as OA(x, s) ≡ U (s) OA(x, 0) U−1(s). The generalized AdS-Rindler reconstruction which

accounts for the non-locality of the modular Hamiltonian will then be φ(X) = Z A dx Z ds K(X; x, s) O(x, s),

Connections, holonomies, and holographic spacetime

Theoretical Physics

Master Thesis