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Second order relative entropy in holographic theories Schulz-Beach, Jonathon Matthew 2016

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Second Order Relative Entropy inHolographic TheoriesbyJonathon Matthew Schulz-BeachB.Sc., The University of Guelph, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Jonathon Matthew Schulz-Beach 2016AbstractRecently, there has been growing recognition that the tools from quantuminformation theory might be well-suited to studying quantum gravity in thecontext of the gauge/gravity correspondence. It is exploring this connectionthat is the main motivation for the work in this thesis.In particular, we focus on holographic field theories which possess clas-sical spacetime duals. The aim is that certain conditions on the classicalduals will narrow down the types of field theories that can be holographic.This will give a better understanding of the limitations and robustness ofthe gauge/gravity correspondence.We do so by computing the canonical energy for general perturbationsaround anti-de Sitter spacetime, which is dual to quantum Fisher informationin the field theory. We go on to prove the positivity of canonical energy anddiscuss the addition of matter fields. We further show that our result canbe interpreted as an interaction between scalar fields living in an auxiliaryde Sitter spacetime. We concluded with a summary of progress and futurechallenges for this program.iiPrefaceThe results of Chapters 3 and 4 have appeared in the peer reviewed Journal ofHigh Energy Physics, under the title “Entanglement entropy from one-pointfunctions in holographic states” [1] and are based on research performed bymyself in collaboration with Jaehoon H. Lee, Charles Riddeau and MarkVan Raamsdonk. All gravitational calculations were done by myself, andthis thesis only includes such parts.Figures 2.3 and 4.3 were created by myself and originally used in [1].Section 3.1.4 is an edited version of an appendix in [1] which was also writtenby myself.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . 81.3 Holographic entanglement entropy . . . . . . . . . . . . . . . 111.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 Holographic relative entropy . . . . . . . . . . . . . . . . . . 162.2 Canonical energy . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Relative entropy of general perturbations . . . . . . . . . . 313.1 Gravitational contribution to relative entropy . . . . . . . . . 313.2 Scalar field in AdS3 . . . . . . . . . . . . . . . . . . . . . . . 383.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Emergent de Sitter spacetime from entanglement . . . . . 424.1 Emergent de Sitter dynamics from entanglement . . . . . . . 424.2 Interactions in de Sitter . . . . . . . . . . . . . . . . . . . . . 444.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48ivTable of Contents5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51vList of Figures1.1 A traditional picture of two entanglement particles. If oneparticle is measured to have the red spin, then the other willalso have the red spin. . . . . . . . . . . . . . . . . . . . . . 51.2 Examples of a bi-partitioned system. The subregion A isshown for each of the two examples: a) a spin chain wherered lines represent entangled pairs. b) a continuous field the-ory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 The entanglement entropy for a generalized Bell pair describedby equation (1.7). . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Comparison between Escher’s artwork and proper geodesicsin the hyperbolic disk. To understand a spatial slice of AdS3,imagine that the bats/angels in Escher’s disk are have thesame area on. It is merely a property of the disk that morebats fit near the boundary. The geodesics are perpendicularto the boundary in b) and c). . . . . . . . . . . . . . . 101.5 The minimal Ryu-Takayanagi surface stretching into the bulkspacetime. The surface γA associated to the region A hasminimal area with respect to any other surfaces anchored toA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 (a) The Poincare patch of the AdS3 cylinder. The time t isrepresented by the height of the cylinder and z the radius.(b) AdS3 spacetime as a cylinder. The red line representsthe Ryu-Takayangi surface γA which has minimal area and isanchored to the boundary region A. . . . . . . . . . . . 132.1 The domain of dependence DB of the ball-shaped region B.The Killing vector ζB is the timelike flow through DB. . . . 172.2 The tangent plane to a density matrix σ. . . . . . . . . . . . 21viList of Figures2.3 The Rindler wedge RB associated with the ball-shaped bound-ary region B. The blue lines indicate the flow of ζ, and thered lines ξ. The surface Σ lies between B and the extremalsurface B˜. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1 Penrose diagram for dSd spacetime. The entangling regionB is in blue with the light cone reaching out to the uniquebulk point (R, xi). The wavy line represents the field δ(1)Spropagating from the bulk to the boundary. . . . . . . . . . . 444.2 The size of the entangling region determines the bulk depthin the dS geometry. There is a one-to-one mapping betweenbulk points and spherical regions on the asymptotic futureboundary I+. The light cone from a bulk point reaches theboundary on a spherical region. . . . . . . . . . . . . . . . . 454.3 A visual interpretation of (4.7) as a Feynman diagram. TheδS± field propagates from a point on asymptotic past infinity(I−), interacts with another δS± with a vertex given by g3 =12cL2and produces another δS± field which reaches the bulkpoint (R, x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.1 A summary of progress and future directions for the auxiliaryde Sitter approach. Green boxes indicate solved problems,and red boxes indicate present challenges. . . . . . . . . . . . 50viiAcknowledgementsFirstly, I would like to thank my supervisor Mark Van Raamsdonk for hisconstant guidance and inspirational passion for physics.I would also like to thank Philippe Sabella Garnier, Charles Rabideau,and Jaehoon Lee for helpful conversations throughout this project. Withouttheir help, much of this work would not of come to fruition.This work was supported in part by research funding from the NaturalSciences and Engineering Research Council of Canada.viiiDedicationTo my Mother,ixChapter 1IntroductionModern fundamental physics has largely been guided by the reductionistphilosophy; that is, to break matter apart and see ‘what it is made of’ on everdecreasing length scales. In quantum mechanics, the Heisenberg uncertaintyprinciple demands that we need correspondingly higher energies in order tolook at smaller distances. This has led to the creation of massive particleaccelerators like the Large Hadron Collider, which is currently reaching en-ergies of 13 TeV to probe lengths around 10−19 m, nine orders of magnitudepast the atomic scale.One of the remarkable discoveries of the reductionist program is that,sometimes, physics changes radically past a certain critical energy scale. Forexample, Fermi’s theory of beta decay describes the weak interaction well atlow energies, but it breaks down at energies near 290 GeV. In this case, thetheory was replaced by electroweak theory which accommodates this drasticchange by producing new particles, the W and Z bosons.In the case of gravity, a critical energy scale emerges because concentrat-ing very high energies in a small region results in the creation of microscopicblack holes. A crude estimate, due to Planck, puts the energies of this scaleat 1019 GeV, or equivalently, distances of 10−35 m. After this point increas-ing the energies only makes larger black holes and the reductionist programcompletely fails. It appears that quantum gravity is more than just a theoryof “stuff” smaller than the Planck length. Indeed, such a length is not evenwell-defined since we cannot measure it. Although one might be temptedto argue that energies so incredibly large are irrelevant to physics, there aremany important questions which can only be answered by understandingphysics at the Planck scale; such questions include the nature of the BigBang, black hole singularities, and the ultimate fate of the universe.Despite the immense gap between current experiments and the Planckscale, there have been some remarkable hints as to what a theory of quantumgravity must look like. Perhaps the greatest clue was discovered by Beken-stein and Hawking around 1972, who were formulating an analogy betweenthe laws of thermodynamics and the laws of black hole dynamics [2, 3, 4].1Chapter 1. IntroductionTheir famous result says that the entropy of a black hole is proportional toits area throughSBH =A4GN(1.1)in natural units where c = ~ = 1.There are many remarkable things about the Bekenstein-Hawking for-mula. The most immediate observation is that black holes do indeed have awell-defined entropy (a measure of classical disorder) and hence obey certainthermodynamic relations. This suggests that gravity may emerge from ahitherto unknown microscopic theory, just as classical thermodynamic arisesfrom statistical mechanics. Secondly, the entropy is proportional to the sur-face area of the system and not the volume (as is the case with most thermo-dynamic systems). This property has been displayed in certain ground statesof condensed matter systems and suggests that the microscopic degrees offreedom are not defined in the full spacetime, but rather only a subspace ofit. In a sense, spacetime contains redundant features which do not contributeto the true physics of quantum gravity.The Bekenstein-Hawking formula also implies that the maximum amountof information that can be stored in any region of spacetime is proportionalto the (surface) area of that region. Due to the universality of black holes,whatever the true microstates of quantum gravity are, they appear to liveon the boundary of a spacetime region rather than in the region itself.A similar clue comes from classical general relativity (GR) itself. It haslong been known that, in GR, energy does not posses a proper local defini-tion but rather it can only be defined for an observer at asymptotic infinity[5]. If energy is to be identified with the Hamiltonian (a natural startingpoint for quantum theory), then it is suggestive that the true gravitationalHamiltonian is only defined at infinity.This idea is manifest in the holographic principle, which supposes that thecomplete description of the dynamics in some spacetime volume is completelyencoded in the boundary of the region. Much like a 3d movie in theatrestoday, the two-dimensional screen contains enough information so that thepolarized 3d glasses reconstruct the full 3d images for the viewer.The gauge/gravity duality is a precise realization of a holographic the-ory. It conjectures that a complete theory of quantum gravity in (d + 1)-dimensional spacetime is equivalent to a d-dimensional quantum conformalfield theory (CFT) on the asymptotic boundary of that spacetime. Thishas been explicitly demonstrated for certain theories; the most celebratedof which is the so-called AdS/CFT duality between type-IIB superstringtheory on AdS5×S5 and N=4 super-Yang-Mills theory in Minkowski space-2Chapter 1. Introductiontime (author?) [6]. Remarkably, this duality concretely relates a stronglycoupled quantum theory, where calculations may be intractable, to a low en-ergy classical gravitational theory where results may be possible. Exploitingthis strong/weak coupling duality has yielded tremendous applications tofar-reaching areas of physics; from superconductors to fluid dynamics [7, 8].Although the original conjecture was specific to string theory, there issubstantial evidence that the gauge/gravity duality holds for a much largerset of theories [9, 10, 11, 12, 13]. In fact, it may be that any consistentquantum gravity theory in asymptotically AdS space can be defined througha CFT on its boundary. Understanding precisely which CFTs can producesa dual spacetime is a very important question for exploring quantum gravity.In recent years, it has been recognized that the language of quantum in-formation theory might be well-suited for addressing questions about gauge/gravity.In particular, quantum entanglement in a quantum field theory (QFT), a keyquantity in information theory, has been shown to be fundamentally linkedto the geometry of the dual spacetime [14, 15, 16, 17]. In fact, without en-tanglement in the quantum theory, it has been argued there could be no dualspacetime [18, 19]. In a sense, entanglement is the glue that holds spacetimetogether.The deep connection between entanglement and geometry was strengthenby the realization that certain conditions of entanglement in a CFT couldbe translated to restrictions on gravitational solutions in the bulk spacetime[20]. Remarkably, even Einstein’s equations emerge (to first order) from thelaws of entanglement [21, 22]. Further known properties of entanglementcan be translated into constraints on gravitational physics, including theaveraged null energy condition [20]. The entanglement structure of a CFTgives provides insight regarding which field theories can produce consistentgravitational spacetimes. It is understanding this restriction that is the mainmotivation for the work in this thesis.The primary focus of this thesis is the holographic computation of thesecond order change in entanglement entropy of a CFT state when perturbedaway from the vacuum. This change is known as quantum Fisher informationwithin the literature and serves as a metric between states [23]. In calcu-lating the bulk dual to this quantity, we establish that (to second order)the entanglement entropy of such states can be represented as a smearingfunctional over products of one-point functions, i.e. expectation values. Wefurther explore connections between this result and a recent proposal foremergent dynamics in de Sitter spacetime [24].In the remainder of this Chapter we review various aspects of holographic31.1. Entanglemententanglement entropy. We start with a brief review of quantum entangle-ment and its useful properties. We then proceed to discuss the AdS/CFTdictionary and holographic entanglement entropy from Ryu-Takayangi sur-faces. We conclude with an overview of subsequent Chapters and a summaryof the main calculations of the thesis.1.1 Entanglement“I would not call that one, but rather the characteristic trait of quantummechanics; the one that enforces its entire departure from classical lines ofthought.”-Erwin Schrödinger on entanglementQuantum entanglement is perhaps the most curious aspect of quantummechanics. It allows for two particles to be perfectly correlated without anyphysical communication between them. This “spooky” action at a distancewas one of the most conceptually troubling aspects of early quantum the-ory. Einstein never truly accepted this idea and, along with Podolsky andRosen (EPR), he claimed that quantum mechanics could not be considered a“complete” description of reality (author?) [25]. This paradox loomed overtheoretical physics for nearly 30 years until John Bell showed that any “com-plete theory”, in the sense of EPR, must inherently be nonlocal (author?)[26]. Since locality is a cornerstone of fundamental physics, it seems that“spooky” action at a distance is here to stay. 1However bizarre entanglement is, it is an immensely important part ofmodern physics. It is paramount to modern applications in quantum cryp-tography, quantum teleportation, and very generally in quantum computing[27]. Recently, it has also found use in condensed matter physics where itmay be used to characterize special quantum phase transitions where conven-tional order parameters fail. Such examples include superconducting phasesand topological order (author?) [28, 29, 30].Despite the widespread applications of entanglement entropy in modernphysics, the main motivation for this thesis is of a fundamental nature. It hasrecently been established that, in the context of holography, entanglemententropy in a CFT is dual to a surface in the bulk with minimal area [14, 15,31]. In this way, the geometric features of spacetime are intimately linked tothe entanglement structure of a CFT. Furthermore, it has been argued that1Of course this is unnecessary in certain interpretations of quantum mechanics, suchas if one thinks of an entangled pair as “one particle” in two difference places.41.1. Entanglement|Ψ〉 = 1√2(|↑〉|↓〉 − |↓〉|↑〉)Figure 1.1: A traditional picture of two entanglement particles. If one par-ticle is measured to have the red spin, then the other will also have the redspin.without entanglement in a CFT there would be no dual spacetime [19, 18].In this thesis we aim to better understand this connection.In the following sections we define the entanglement entropy and list someof its properties. We then discuss the area law and give a simple examplebefore moving on to the AdS/CFT correspondence.1.1.1 Basics of entanglement entropyA composite quantum state |Ψ〉 ∈ H, composed of substates |ψi〉 ∈ Hi, issaid to be entangled if it cannot be factorized as a product of the substates,|Ψentangled〉 6=∏i|ψi〉 . (1.2)By the superposition principle, an arbitrary state is a linear combination ofsuch product states |Ψentangled〉 =∑j cj∏i |ψi〉 .A more elegant way to characterize entanglement is through the densityoperator ρ, which is simply the outer product of the wavefunction,ρ = |Ψ〉〈Ψ|. (1.3)The density matrix is called pure if one of its eigenvalues is unity, otherwiseit is called a mixed state. A pure state has no entanglement and represents asingle quantum state, whereas a mixed state represents a statistical ensembleof states.As a statistical ensemble, one can define an associated entropyS = −tr(ρ log ρ) (1.4)51.1. EntanglementA(a) One-dimensional spin chain.AA¯(b) Quantum field theory in thecontinuum.Figure 1.2: Examples of a bi-partitioned system. The subregion A is shownfor each of the two examples: a) a spin chain where red lines represententangled pairs. b) a continuous field theory.called the von Neumann entropy. Since the density matrix has non-negativeeigenvalues (due to requiring positive probabilities), the entropy is also non-negative, and vanishes if and only if ρ is a pure state. The von Neumannentropy quantifies the amount of uncertainty about which state the systemis in. In the limit h → 0, the von Neumann entropy reduces to the usualthermal entropy.Now consider a quantum system in a pure state ρ. If we divide thesystem into two parts, A and B as in figure 1.2 then the total Hilbert spaceis simply the direct product between the two subspaces H = HA ⊗HB. Thestate of the system A without any reference of B, is obtained by tracing outall degrees of freedom in B. This defines the reduced density matrix for A,ρA =∑i〈iB|ρ|iB〉 = trB(ρ). (1.5)An observer confined to the region A will only be aware of the effectivedensity matrix ρA. This is a natural thing to consider in the context of blackhole physics where the interior of the black hole is inaccessible to an outsideobserver.The entanglement entropy of A is defined as the von Neumann entropyof the subsystem,SA = −trA(ρA log ρA). (1.6)This gives a direct way to characterize the entanglement between two regions.In the case of black holes, the entanglement entropy measures the amountof information hidden inside the black hole.As per figure 1.2, the entanglement entropy depends on both the systemand the imaginary entangling surface.61.1. EntanglementIf the system was originally in a mixed state, the entanglement entropywould no longer measure only entanglement. Instead it would count bothclassical and quantum correlations. This is to be expected because in thehigh-temperature limit it will reproduce the classical thermal entropy whichhas no entanglement [32, 33, 34]. For this reason we will only considerglobally pure states so that we isolate the quantum effects.We now provide a simple example before discussing entanglement entropyin a more general QFT.ExampleA typical example of an entangled state is the generalized Bell pair|Ψ〉 = √α| ↑〉A| ↓〉B +√1− α| ↓〉A| ↑〉B (1.7)where α ∈ [0, 1] enforces the normalization 〈Ψ|Ψ〉 = 1. The correspondingreduced density matrix isρA = trB(ρ) =(α 00 1− α)(1.8)and the entanglement entropy isSA = −α logα− (1− α) log(1− α) . (1.9)As shown in figure 1.3 the entropy vanishes at α = 0,1 and achieves a maxi-mum at α = 12 .Area lawIn a field theory the entanglement entropy is generally divergent. This isdue to the infinite number of degrees of freedom in a field. Fortunately,understanding the leading order divergences in the entanglement entropycan give profound insight into the behaviour of the underlying field theory.Consider a d-dimensional QFT on flat spacetime. For simplicity, wewill work on a constant time slice t = t0 which defines the spacelike (d −1)-dimensional submanifold N . The entanglement entropy of a subregionA ⊂ N can be computed through (1.6). To regularize the entropy, weintroduce an ultraviolet (UV) cutoff parameter , i.e. the lattice spacing.The entanglement entropy of a region A is thenSA = γArea (∂A)d−2+ (subleading terms) (1.10)71.2. AdS/CFT correspondence0 0.2 0.4 0.6 0.8αSAFigure 1.3: The entanglement entropy for a generalized Bell pair describedby equation (1.7).where γ is a constant that depends on the system in question [35, 36]. Thisis known as the area law. The fact that the entanglement is dominated bythe contribution from the boundary ∂A comes from the intuitive picture thatthe entanglement between A and A¯ is strongest near the boundary. Thismust be the case because the interactions are governed by local Hamilto-nian dynamics, and hence entanglement between A and A¯ is predominatelygenerated by local effects near ∂A.There are some systems which violate the area law. For example, anymetal or material with a Fermi surface (author?) [28]. Another commonexample is for a 2d CFT, where the entanglement entropy for region of lengthL in an infinitely long spin chain with lattice spacing  isSA =c3logL, (1.11)where c is the central charge of the CFT [37, 33]. Whereas this violates thenaive area law, it acquires a simple geometric meaning in holography.In the next section we cover the essentials of AdS/CFT which will beused later in the thesis.1.2 AdS/CFT correspondenceThe AdS/CFT (anti-de Sitter/conformal field theory) conjectures that aquantum theory of gravity in (d+1)-dimensional spacetime is equivalent to aquantum field theory with conformal symmetry in d dimensions. Maldacenaoriginally formulated the duality between superstring theory on AdS5 × S581.2. AdS/CFT correspondenceand strongly coupled N=4 SU(N) super-Yang-Mills theory in Minkowskispacetime (author?) [6]. Since then, a substantial amount of evidence hasdemonstrated that it is in fact more general [9, 10, 11, 12, 13]. In practice,exploiting this duality has led to novel insights about strongly coupled fieldtheories from gravity, as well as new hints towards a complete theory ofquantum gravity.It is all but impossible to describe all the ramifications of the AdS/CFTcorrespondence as, at present, the original paper by Maldacena has over11,000 citations. With applications in condensed matter [33, 38, 39, 8, 40],numerical relativity, and quantum information [41, 42, 17, 43, 44, 45, 16]; theAdS/CFT has proven to be one of the most significant theoretical discoveriesin the last 20 years.In the following section, we will discuss AdS spacetime and the holo-graphic dictionary between two dual theories.Anti-de Sitter spacetimeAnti-de Sitter (AdS) is a maximally symmetric spacetime with constant neg-ative curvature. As such, it is also a solution to Einstein’s equations witha negative cosmological constant. It is the Lorentzian version of hyperbolicspace, just as Minkowski space is to Euclidean space.The asymptotic symmetry group of AdSd+1 is SO(2, d), which happens tobe isomorphic to the conformal group in d dimensions [46]. This is one of themany reasons which suggest that AdS may be related to a lower-dimensionaltheory with conformal symmetry.AdS in three dimensions can be visualized as a cylinder as shown infigure 1.6. A constant time slice of three-dimensional AdS is referred to asthe hyperbolic disk, which was made famous by the artwork of Escher [47].Figure 1.4 compares Escher’s artwork with spatial geodesics on the disk.One coordinate patch for AdSd+1 isds2 =L2z2(dz2 + dxµdxµ) (1.12)with the index µ = 0, 1, ..., d. The dual CFT is said to live on the boundaryof AdS, that is the limiting surface as z → 0 (the boundary of the cylin-der). This is not a literal statement, as the two theories are distinct (andyet are supposedly different manifestations of the same theory). The radialcoordinate z can be interpreted as the length scale of the CFT in the senseof renormalization group (RG) flows (or energy scale). Small z correspondsto high energies in the field theory. Likewise, physics deep in the bulk (large91.2. AdS/CFT correspondence(a) Escher’s disk. (b) Geodesic tesselations. (c) Geodesicsstarting at onepoint.Figure 1.4: Comparison between Escher’s artwork and proper geodesics inthe hyperbolic disk. To understand a spatial slice of AdS3, imagine thatthe bats/angels in Escher’s disk are have the same area on. It is merely aproperty of the disk that more bats fit near the boundary. The geodesics areperpendicular to the boundary in b) and c).z) is related to the low energy infrared (IR) properties of the CFT. To dealwith the divergence in the metric at z = 0, one usually imposes a cutoff atz = . This amounts to the CFT having a lattice spacing (or UV cutoff) of.Holographic dictionaryFor two theories to be considered dual to each other, there must be a mapbetween the physical spectra of each theory. Bulk quantities in AdS shouldbe able to be written in terms of field theory observables and vice versa.In QFT, all properties of the theory are contained within the generatingfunctional (partition function) Z. In this way, the AdS/CFT correspondencecan be neatly phrased as the equivalence between the partition functions ofboth theories,ZAdS = ZCFT . (1.13)For strongly coupled field theories, the gravity action can be approxi-mated by low energy gravity (classical gravity), so thatZAdS ≈ eS0 , (1.14)where S0 is the classical action for the gravitational theory.101.3. Holographic entanglement entropyFor there to be an equivalence between observables, every classical fieldφα in the gravitational side must be associated to an operatorOα in the CFT.More precisely, the boundary value of the field φα(z = 0) = φα0 , couples toOα as a source term. That is, ZCFT = 〈e´φα0Oα〉. In this sense, the operatorsOα source the bulk fields φα at the boundary.We can then compute the connected n-point correlation functions of anoperator Oα from〈Oα1(x1)...Oαn(xn)〉 =δδφα10· · · δδφαn0logZCFT (1.15)=δδφα10· · · δδφαn0S0 . (1.16)It is convenient that, in order to compute quantum correlation functions,one only needs to take derivatives of a classical action. Although this formalequation gives a powerful computation tool, it does not address some of theconceptual aspects of a bulk/boundary correspondence. In the next sectionwe will explore a more recent entry in the holographic dictionary which hasa more direct link to geometry.1.3 Holographic entanglement entropyIn this section we introduce a generalization of the Bekenstein-Hawking for-mula due to Ryu and Takayangi (author?) [14]. Motivated by black holeentropy, they proposed that the area of a certain minimal area surface (ofcodimension 2) is exactly the entanglement entropy of region in the dualCFT. In taking the appropriate limits, the Bekenstein-Hawking entropy maybe interpreted as such a surface [11]. This provides a method to compute en-tanglement entropy in a CFT by calculating the area of an extremal surfacein AdS spacetime. We will only present a heuristic argument, and a moreformal derivation can be found in (author?) [15].We would like a way to holographically calculate the entanglement en-tropy on a time slice of a CFT between a subregion A and its complimentA¯, who share a boundary ∂A. Firstly, we imagine extending the divisionbetween A and A¯ to regions in the bulk spacetime with an imaginary surfaceγA. The boundary ∂γA exactly matches ∂A on the boundary as shown in fig-ure 1.5. Inspired by the Bekenstein-Hawking formula, we hope that the areaof such a surface will reproduce the entanglement entropy of A. Of coursethere are infinitely many choices for γA with very different areas, and so weneed a method to choose a single surface. Ryu and Takayanagi proposed111.3. Holographic entanglement entropyAA¯γAAdSzFigure 1.5: The minimal Ryu-Takayanagi surface stretching into the bulkspacetime. The surface γA associated to the region A has minimal area withrespect to any other surfaces anchored to A.that the proper surface, A˜ should be the surface possessing minimal area aswill be argued later. The surface with the minimal area which is anchoredto ∂A is called the Ryu-Takayangi surface and is denoted by A˜.The entanglement entropy SA of the CFT is then hypothesized to begiven bySA =Area(A˜)4G. (1.17)In AdS spacetime, the leading order contribution to the area in (1.17) comesfrom the boundary. In general, (1.17) diverges asSA =14GNArea(∂A)d−2+ · · · (1.18)which is consistent with the area law in (1.10). In this way, the Ryu-Takayangi claims to describe the entanglement entropy of a region to allorders.The Ryu-Takayangi formula is only applicable to static spacetimes orconstant time slices. In the dynamic case, there is a modified prescriptionby Hubeny-Rangamany-Takayanagi [31]. In this thesis, we will only be con-cerned with static spacetimes so the simpler Ryu-Takayangi method willsuffice. There have also been extensions to include quantum corrections [48],as well as a formal proof of (1.17) [[49]].The elegance of calculating entanglement entropies from minimal areasurfaces is best illustrated with an example. In the following section, we121.3. Holographic entanglement entropyAdS3FuturePoincareHorizonPastPoincareHorizont = 0(a)AdS3B B˜(b)Figure 1.6: (a) The Poincare patch of the AdS3 cylinder. The time t is rep-resented by the height of the cylinder and z the radius. (b) AdS3 spacetimeas a cylinder. The red line represents the Ryu-Takayangi surface γA whichhas minimal area and is anchored to the boundary region A.compute the entanglement entropy of a region in a 2d CFT which is dual topure AdS spacetime.1.3.1 Example in AdS3/CFT2As a simple example, let us consider calculating the entanglement entropy foran interval x ∈ [−R,R] of a two-dimensional CFT using the Ryu-Takayanagiformula. The dual theory is pure AdS3 spacetime, which makes it is easyto calculate the area of extremal surfaces. It is convenient to work in thePoincare patch of AdS3 as illustrated in figure 1.6. The line element in thesecoordinates isds2 =1z2(dz2 + dt2 − dx2). (1.19)As required for the Ryu-Takayangi prescription, we take a constant time131.4. Overviewslice t = 0. The geodesic equation for z = z(x) is thenz′′ z + z′2 + 1 = 0 (1.20)which has solutionsz2 = R2 − x2 . (1.21)From this we see that the geodesics in AdS3 are circles of radius R, as shownpreviously in figure 1.4. The area (or in this case length) of this minimalgeodesic surface is given byA =ˆdx√−g = 2ˆ Rdz√1 + z′z= 2 ln(2R)+O(1) (1.22)where  is a cutoff near the z = 0 boundary. The entropy then preciselymatches the result for a 2d CFT in (1.11) with L = 2R and a central chargeofc =32GN. (1.23)The elegance of this approach is remarkable. By merely minimizing thesurface area for a anchored surface in AdS spacetime, we have computed theentanglement entropy for a region in a CFT.1.4 OverviewAs demonstrated by the simplicity of the Ryu-Takayangi formula, the ge-ometric nature of entanglement is profoundly powerful. The bulk space-time encodes information about the nonlocal entanglement properties of aCFT state. We can then hope to ask which CFT states have entangle-ment structure that is consistent with having a holographic dual. It is clearthat in general, the space of all entanglement entropies for spherical re-gions, S = {S(R, xµ)∀xµ ∈ Rd, R > 0}, is much larger than the space of allasymptotically AdS metrics. It is then plausible that there exists a subsetof S which represents the entanglement entropies which can be possessed bya holographic CFT.One method to characterizing these subsets is to study arbitrary pertur-bations to asymptotically AdS which are dual to state perturbations of theCFT vacuum. We know from the standard AdS/CFT correspondence thatbulk fields will be sourced by expectation values of CFT operators 〈Oα〉.Propagating these into the bulk should determine the full fields and metric141.4. Overviewφµ(z, x) at least to some finite distance into the bulk. From this, the entan-glement entropy of any region can be constructed from the Ryu-Takayanagiformula. Following this logic, we see that the entanglement of regions ina CFT can be determined merely by knowledge of the one-point functions〈Oα〉. This provides a stringent test on whether a CFT possesses a classicalgravitational dual.In this thesis we present some explicit results for the entanglement en-tropies allowed for theories with a classical dual. For states near the vac-uum, the first order result is known to hold universally for all CFT. In thiscase holography does not place a constraint on the allowed entanglemententropies. Continuing to second order is it less clear if there are any ad-ditional constraints coming from holography. To find an explicit condition,we make use of the recently recognized equivalence between quantum Fisherinformation and bulk canonical energy [50].The remainder of this thesis is organized as follows. In Chapter 2, we in-troduce relative entropy, a natural extension of entanglement entropy, whichhas some useful properties for studying quantum gravity. We then discussthe gravitational dual to relative entropy, known as canonical energy. Fol-lowing that, Chapter 3 provides the main calculation of the thesis, that is,the second order relative entropy for a CFT state perturbed from the vac-uum. We go on in Chapter 4 to explore connections between this result andemergent dynamics in de Sitter space. We conclude in Chapter 5 with adiscussion of open questions and future directions for research.15Chapter 2BackgroundThis chapter introduces the technical aspects of calculating the quantumFisher information in a CFT. We first review the fundamental features ofentanglement for a spherical region and its physical interpretations. Wemotivate the first law of entanglement and introduce quantum Fisher in-formation. Furthermore, we discuss the gravitational dual to holographicstates, and define the canonical energy. Finally, we review the holographicdictionary between Fisher information and canonical energy.2.1 Holographic relative entropy2.1.1 Relative entropyThis section is concerned with formally defining the entanglement, and rel-ative, entropy for a holographic state.Consider a d-dimensional CFT which possesses a state |Ψ〉. For anyspherical region B, the reduced density matrix is obtained by tracing outthe degrees of freedom associated to the compliment of B,ρB = trB¯ (|Ψ〉〈Ψ|) .The density matrix ρ must be positive semi-definite and Hermitian sothat it has non-negative probabilities. Any such operator can be written asa exponential of another Hermitian operator throughρvacB =1Ze−HB (2.1)where HB is the modular Hamiltonian and Z = tr(e−HB ) is the usual parti-tion function.2 The normalization constant Z could have been absorbed intoHB through an additive constant and ensures that tr(ρ) = 1. Throughoutthe rest of this chapter we will set Z = 1 without loss of generality.2The term modular Hamiltonian originates from axiomatic quantum field theory, whilein condensed matter it is often called the entanglement Hamiltonian.162.1. Holographic relative entropyBDBζBFigure 2.1: The domain of dependence DB of the ball-shaped region B. TheKilling vector ζB is the timelike flow through DB.The explicit form of the modular Hamiltonian is only known in speciallocal examples. In the case that B is a spherical spatial region of radius R,the modular Hamiltonian is given by the well-known expressionHB = 2piˆBdd−1xR2 − x22RTtt(x) (2.2)where 〈Ttt(x)〉 is the CFT stress tensor.The modular Hamiltonian generates a conformal Killing vector ζ, whichacts within the causal diamond DB (the causal past and future of B) asshown in figure 2.1. For a circular region, the Killing vector is explicitlyζ =piR[(R2 − (t− t0)2 + |x− x0|2)∂t − 2(t− t0)(xi − xi0)∂i](2.3)where i runs over spatial indices. Using ζ, a covariant way to write themodular Hamiltonian isHB =ˆBζµ〈Tµν〉ν (2.4)where ν is a differential form related to the Levi-Civita tensor throughν =1(d− 1)!ν1...νd−1dxν1 ∧ ... ∧ dxνd−1 . (2.5)One can easily check that with the Killing vector in (2.3), the covariantexpression in (2.4) reproduces (2.2).172.1. Holographic relative entropyFor a state |Ψ〉 perturbed away from the vacuum state |0〉, we define thedifference in entanglement entropy as∆SB = S(ρB)− S(ρvacB ) .While the entanglement entropy is typically divergent, the entropy differencehas the benefit of remaining finite. Similarly we write the difference in theexpectation value of the modular Hamiltonian as∆〈HB〉 = tr(ρBH)− tr(ρvacB HB) .Another quantity of great importance is the relative entropy between twoquantum states. Considering a state ρ and a reference state σ, the relativeentropy is defined asS(ρ||σ) = tr(ρ log ρ)− tr(ρ log σ).Notice that this definition is not symmetric in ρ and σ; however, it hasmany useful properties. It can be shown that in general the relative entropyis non-negative;S(ρ||σ) ≥ 0, (2.6)where the equality holds if and only if the states are identical (ρ = σ). Fur-thermore, as the region on which ρ is defined increases in size, the relativeentropy increases; that is to say, it is monotonically increasing with vol-ume. The positivity and monotonicity properties make the relative entropya particular useful quantity to study.Relative entropy can be physically interpreted in the context of thermo-dynamics. For a thermal density matrix ρth = e−H/T /Z, the relative entropybetween any state ρ1 and the thermal one is the difference between the freeenergiesS(ρ1||ρth) = 1T(F (ρ1)− F (ρth)) , (2.7)where as usual F = 〈E〉−TS. The temperature in this expression is only thatof the initial thermal state so ρ1 can be arbitrary (it need not be thermal).In the case where we know the reduced density matrix ρ explicitly, it isuseful to write the modular Hamiltonian asHB = − log ρ (2.8)which coincides with the definition from (2.1) up to a normalization constant,Z = 1. The relative entropy then reduces to182.1. Holographic relative entropyS(ρ||σ) = tr(ρ log ρ)− tr(ρHB)+ tr(σ log σ)− tr(σ log ρ)= − S(ρ)− 〈HB〉ρB(λ) + 〈HB〉ρB(0) + S(σ)= ∆〈HB〉 −∆S.The positivity of relative entropy then immediately implies the constraint∆〈HB〉 −∆S ≥ 0. (2.9)For a sufficiently small perturbation of the vacuum state, this implies to firstorderδ(1)S(ρ||σ) = 0, (2.10)which gives the so-called first law of entanglementδ(1)S = δ(1)〈HB〉. (2.11)Adding the temperature back into this expression yields the usual thermo-dynamic equation TδS = δE, which applies to states near equilibrium.The second order, δ(2)S(ρ||σ) is not symmetric under exchanging ρ andσ.If we consider a state ρ = σ+δρ1+δρ2, where δρ1 and δρ2 are independentperturbations, may define an inner product2〈δρ1, δρ2〉σ = δ(2)S(σ+δρ1 +δρ2||σ)−δ(2)S(σ+δρ1||σ)−δ(2)S(σ+δρ2||σ) ,(2.12)which is symmetric under (δρ1,δρ2). Notice that in the case that δρ1 = δρ2 =12δρ, we recover〈δρ, δρ〉σ = δ(2)S(σ + δρ||σ) . (2.13)The inner product in (2.12) is clearly non-negative, symmetric, and van-ishes only if δρ1 = δρ2 = 0. We can think of it as a metric on the space ofstates perturbed away from the reference state σ. The metric lives on thetangent space to σ and is the known as quantum Fisher information. It isalso sometimes called the Bures, or Helstrom metric; However, we will stickto the general terminology of quantum Fisher information.Our principle motivation to study Fisher information comes from theproperties of relative entropy, although there are many other reasons it isan significant quantity. For example, the quantum Fisher information playsan important role in quantum state estimation, that is, how to approximatethe state ρ given a set of measurements on n copies of the quantum state[23, 51].192.1. Holographic relative entropy2.1.2 Quantum Fisher informationConsider a one-parameter family of states |Ψ(λ)〉 where λ = 0 denotes thevacuum state. Such an expansion is always possible for well-behaved states[50]. We consider a perturbative expansion around the vacuum state in λ:ρ(λ) = σ + λ δρ (2.14)where ρ(0) = σ. The relative entropy is thenS(ρ||σ) = tr((σ + δρ) log(σ + δρ))− tr((σ + δρ) log σ)We expand the logarithm using a Taylor series;log(σ + δρ) = log(σ0) + λddλlog ρ(λ)∣∣λ=0+λ22d2dλ2log ρ(λ)∣∣λ=0+O(λ3)so that the relative entropy becomesS(ρ||σ) = λtr[σddλlog ρ∣∣λ=0]+λ2tr[δρddλlog ρ∣∣λ=0+σ2d2dλ2log ρ∣∣λ=0]+O(λ3)(2.15)The first term vanishes since tr[σ ddλ log ρ∣∣λ=0] ∼ tr [σσ−1δρ] = tr [δρ] = 0.Now consider the related quantity ddλtr[ρ ddλ log ρ]which is also identicallyzero for any λ since ddλ log ρ acts as the formal inverse operator to ρ.We thenevaluateddλtr[ρddλlog ρ] ∣∣λ=0= tr[(ddλρ)ddλlog ρ] ∣∣λ=0+ tr[ρd2dλ2log ρ] ∣∣λ=0= tr[δρddλlog ρ∣∣λ=0]+ tr[σd2dλ2log ρ∣∣λ=0].This is nearly the second term in (2.15).Using this expression to simplify (2.15), we haveS(ρ||σ) = −λ22tr[σd2dλ2log ρ∣∣λ=0]+O(λ3) (2.16)orS(ρ||σ) = λ22tr[δρd2dλ2log ρ∣∣λ=0]+O(λ3) . (2.17)We define the right-hand-side of this expression at order λ2 as the quantumFisher informationFσ(δρ, δρ) = 〈δρ, δρ〉σλ2=12tr[δρd2dλ2log ρ∣∣λ=0], (2.18)202.1. Holographic relative entropyδρ2δρ1σFigure 2.2: The tangent plane to a density matrix σ.which we recognize as the inner product from (2.12). To get a better senseof what (2.18) means, we naively disregard the ordering so thatFσ(δρ, δρ) = 12tr(δρσ−1δρ) . (2.19)This is a much more illuminating expression. We see that Fσ is quadraticin the perturbation δρ, and as such it is clearly positive as required thepositivity of relative entropy. Note that because our constraint in (??) islimited to second order, the Fisher information is precisely the O(λ2) termin the relative entropy.δ(2)S(σ + δρ||σ) = λ2Fσ(δρ, δρ) (2.20)Using the above results, the total entanglement entropy of a region B ina field theory with a state expanded as (2.14) isSB = SvacB + λˆBζµ〈Tµν〉ν − λ2Fσ(δρ, δρ) +O(λ3) . (2.21)This expression is exact for any CFT, however, it is expected that the struc-ture of the O(λ2) term contains information about whether the CFT is holo-graphic [1]. Whereas the linear term is given by the vacuum expectationvalue of the stress tensor, the Fisher term generally may not have a localVEV.2.1.3 Gravitational relative entropyIn this section we discuss the gravitation dual of relative entropy.212.1. Holographic relative entropyConsider a one-parameter family of geometries M(λ) which are dualto states |Ψ(λ)〉 in a holographic CFT. We take the origin λ = 0, to bepure AdS spacetime with the dual vacuum state |Ψ(0)〉. Now consider aspherical region B, which lives on a spatial slice of the boundary of theunperturbed spacetime. The Rindler wedge RB associated with B is definedas the intersection of the causal past and causal future of DB; the causaldiamond. The boundary of the Rindler wedge is a bulk surface, B˜, whichpossesses a minimal area in the sense of Ryu and Takayanagi, that is ∂B˜ =∂B. The boundary Killing vector ζ can be extended into the bulk wedgesuch that it is null on the (bulk) boundary of the wedge as illustrated infigure 2.3; we denote this new vector by ξ.On the boundary, ζ generates a notion of time within the causal dia-mond. Likewise, the bulk ξ gives a notion of time to the full Rindler wedge.By a diffeomorphism, the wedge RB can be mapped to the exterior of aSchwarzschild-AdS black hole where B˜ acts as the horizon. This fact willbe useful (since there exist many powerful tools) to understand black holespacetimes.zxtBDBΣ B˜Figure 2.3: The Rindler wedge RB associated with the ball-shaped boundaryregion B. The blue lines indicate the flow of ζ, and the red lines ξ. Thesurface Σ lies between B and the extremal surface B˜.The extension to a perturbed spacetime with λ 6= 0 spacetime is clear; foreach perturbed region B(λ) we associate the Ryu-Takayanagi surface B˜(λ)222.1. Holographic relative entropyin M(λ) as the extremal surface which satisfies ∂B˜(λ) = ∂B(λ). We maynow relate each perturbed CFT state to a geometry which is perturbed awayfrom a hyperbolic black hole in AdS.In analogy with the CFT case, we wish to compute the relative entropy,Sgrav(g(λ)||g(0)), between the perturbed metric and pure AdS. Motivatedby the CFT result we defineSgrav(g1||g0) = ∆Egrav −∆Sgrav , (2.22)where ∆Egrav is the difference in gravitational energy between M(0) andM(λ), and ∆Sgrav is the entropy difference.The notion of gravitational entropy can be identified with the area of anextremal area surface via the Ryu-Takayanagi formula, so thatSgrav(λ) ≡ 14GNArea[B˜(λ)], (2.23)and hence ∆Sgrav = Sgrav(λ)− Sgrav(0).We may also define the gravitational energy Egrav, the analogous quan-tity to 〈HB〉, by utilizing some key results from holography. For a (d +1)−dimensional spacetime, an asymptotic expansion near the boundary canbe parameterized in Fefferman-Graham coordinates asds2 =`2z2(dz2 + dxµdxµ + zd Γµν(z, x)dxµdxν)(2.24)where Γµν(z, x) remains finite in the z → 0 limit. In the remainder of thissection we set the AdS scale to unity, ` = 1. The expectation value ofthe CFT stress tensor is related to the asymptotic behaviour of the metricthrough〈Tµν〉 = d16piGNΓµν(z = 0, x) . (2.25)Thus, from the first law of entanglement for a spherical region, we havethat the change in the modular Hamiltonian in B is∆〈HB〉 = d8GNˆ|x|<Rdd−1xR2 − |x|22RΓtt(z = 0, x) , (2.26)since 〈HB〉 = 0 for pure AdS. The right hand side of (2.26) only depends onbulk quantities, so we define it as ∆Egrav. Then for holographic states, wemay write the relative entropy between two metrics g(λ), g(0) completely interms of bulk quantities through (2.23) and (2.26).232.2. Canonical energyThe gravitational relative entropy is also important from the prescriptiveof black hole thermodynamics. To first order, it gives the first law of blackhole mechanics δEgrav = δSgrav. The second order contribution was recentlydiscussed in [52] to understand the stability of black branes and black holes.We now have the tools to compute the relative entropy of a holographicstate |Ψ(λ)〉 near the vacuum from the gravitational side. In the next sectionwe will focus specifically on the second order contribution to the relativeentropy known as canonical energy.2.2 Canonical energyIn this section we review the very basics of Wald’s canonical energy as pre-sented in [52]. We do not attempt to go into full mathematical detail; ratherwe only to introduce the general picture.Consider a smooth one-parameter family of geometries with a metric g(λ)perturbed away from pure AdS. It was shown in [50] that to first order inthe perturbation γ = dg/dλ|λ=0, the gravitational relative entropy obeysd2dλ2Sgrav (g(λ)||g(0))∣∣∣∣λ=0= WΣ(g, γ,£ξγ) + 2ˆΣξabd2dλ2Eˆµν (2.27)where b is defined in (2.5) and Eˆµν vanish for a solution of the full non-linear Einstein equations of motion coupled to matter. Following [52] werename WΣ(g, γ,£ξγ) to the canonical energy E because it plays the role ofa Rindler energy associated with the Killing vector ξ.There are a number of important properties of E . Firstly, it is indepen-dent of the Cauchy slice Σ, which makes it a conserved quantity analogousto an energy. For non-trivial perturbations, the canonical energy E is non-degenerate and positive definite.3 If E is negative, it implies that the space-time is unstable with respect to the perturbations. These properties wereused in [52] to probe the stability of black hole and black branes.The canonical energy is quadratic in the perturbative bulk fields and isgiven explicitly by as integral over a Cauchy slice Σ. It can be decomposedinto a geometric term and a matter term via3A notable exception is for a mass perturbation of a Schwarzchild black hole. It isknown to be stable, and yet the canonical energy is negative.242.2. Canonical energyE(δg1, δg1) = WΣ(δg1,£ξδg1)=ˆΣωfull(δg1,£ξδg1)=ˆΣωgrav(δg1,£ξδg1) +ˆΣωmatter(δφ1,£ξδφ1)=ˆΣωgrav(δg1,£ξδg1)−ˆΣξaT(2)ab , (2.28)where δφ1 represents the perturbation to all matter fields.In order to generalize these arguments to non-homogeneous spacetime,it will be necessary to promote the canonical energy to a bilinear form asE(δg1, δg2) = WΣ(g, δg1£ξδg2) , (2.29)for independent metric variations δg1, δg2. It can be shown that E(δg1, δg2)is always non-negative and only vanishes if and only if δg1 = δg2 = 0 [52].In this sense, the canonical energy serves as a natural metric on the space ofperturbations to a metric. It is then not surprising that its dual is quantumFisher information, a metric on the space of density matrices. This corre-spondence further strengthens the idea that the geometry of spacetime isfundamentally related to the entanglement structure of a quantum theory.2.2.1 DefinitionsThe full derivation of the canonical energy is fairly involved and is reviewedin [50, 52]. Here we only present the basics needed to understand the com-putation of E in practice.Implicitly, the above analysis depended on the extremal surface remain-ing in the same coordinate location as we vary λ. Fortunately, it was shownin [52] that it is always possible to choose coordinates such that the extremalsurface B˜ remains in the same coordinate location and that the Killing vectorξ satisfies the gauge conditionsξ|B(λ) = ζ, (2.30)ξ|B˜(λ) = 0, (2.31)£ξg(λ)|B˜(λ) = 0. (2.32)The third equation merely states the ξ continues to be a Killing vectoron the extremal surface B˜. The explicit formula for the bulk Killing vector252.2. Canonical energyisξ =piR(R2 − (t− t0)2 + |x− x0|2)∂t− 2piR(t− t0)z ∂z− 2piR(t− t0)(xi−xi0)∂i(2.33)where i runs over spatial indices.Consider the pure AdS metric g(0) and a perturbation h which satisfiesthe linearized equations of motion about g. We require that a new pertur-bation γ satisfies the gauge condition (2.32) and is related to h through thegauge transformationγ = h+£V g. (2.34)Our main task now is to determine the gauge field V which leaves thesurface B˜ extremal. We choose coordinates along the geodesic such thatthe index i runs over dynamic coordinates and the index A runs over static,constant coordinates,Xi = σi, XA = XA0 . (2.35)The metric perturbation has to satisfy two equations: the condition that ξis a Killing vector, and that the first order variation of A(X + δX, g + h)vanishes. These give£ξh|B˜(λ) = 0 , (2.36)∇iγiA −12∇Aγii|B˜(λ) = 0 , (2.37)evaluated on the extremal surface B˜(λ). These can be further broken downinto three equationsγiA = 0 , (2.38)γAB − δABγCC = 0 , (2.39)∇iγiA −12∇Aγii = 0 . (2.40)These three equations represent the differential equations for V , given aspecific metric perturbation h. Explicitly, they can also be written as(hiA +∇iVA +∇AVi) |B˜(λ) = 0 , (2.41)(hAD −12δADhCC +∇AVD +∇DV A − δDD∇CV CC) ∣∣∣∣B˜(λ)= 0 . (2.42)262.2. Canonical energyThe bilinear canonical energy is an integral of a form ω(g, γ1, γ2) over theregion Σ between B and B˜. Under the gauge transformation γ → h whichcontinues to satisfy the equations of motion, the integrand of the canonicalenergy only changes by a derivative of a form ρ(h, V ) defined in (2.46),ω(g, γ,£ξγ) = ω(g, h,£ξh) + dρ(h, V ) . (2.43)The integral over dρ is equivalent to a boundary integral through Stokes’stheorem. Moreover, the only contribution is from B˜ since the field V vanisheson B. The end result is that the canonical energy reduces to an integral overa two-form ω(g, h,£ξh) and a one-form ρ(h, V ),E(h, h) =ˆΣω(g, h,£ξh) +ˆB˜ρ(h, V ) , (2.44)the details of which can be found in [50].In summary, for some metric perturbation h in an arbitrary gauge, wefirst must find a field V which solves the conditions (2.41). We can thencalculate E in a the Rindler wedge RB. Because the integral over ρ is eval-uated at B˜, we only need to compute V on the surface and not throughoutthe entire region Σ.As per (2.44), the ω and ρ tensors are defined throughω(g, h,£ξh) =116piGNaPabcdef (£ξhbc∇dhef − hbc∇d£ξhef ) , (2.45)ρ(h, V ) = χ(γh, [ξ, V ])− χ(£ξh, V ), (2.46)withχ(A,B) =116piGNab(Aac∇cBb − 12A cc∇aBb +∇bAacBc −∇cAacBb +∇aAccBb),Pabcdef = gaegfbgcd − 12gadgbegfc − 12gabgcdgef − 12gbcgaegfd +12gbcgadgef ,[ξ, V ]a = ξb ∂bVa − V b ∂bξa ,and the  tensors are defined as in (2.5).2.2.2 Example for BTZ black holeIn this section we provide a sample calculation for the canonical energy ofa planar black hole geometry. We show this to become familiar with the272.2. Canonical energysteps involved before generalizing to a more complicated spacetime in laterchapters.Consider a 2-dimensional CFT which is dual to an asymptotically AdS3spacetime given by the metricds2 =1z2(dz2 + (1 + λz2/2) dx2 − (1− λz2/2) dt2) . (2.47)This represents the first order approximation to a planar black hole geometry.In these planar coordinates the perturbation is simply, Htt = λ, Hxx = λ;However, these coordinates are ill-suited to working with the geodesic surfaceB˜. Instead, we opt for polar coordinates x = r sin θ, z = r cos θ since theextremal surface is easily parameterized by constant R and θ ∈ [0, pi/2). Theperturbation Hµν then takes the formhtt =λ ,hrr =λ sin2 θ ,htr =λ r sin θ cos θ ,hθθ =λ r2 cos2 θ ,with all other components equal to zero. Since we are considering the t =0 surface, we may choose Vt = 0 to simplify the calculation. The six Vequations from (2.41) then reduce to three independent equations∂2θVr − 3 tan ∂θVr − 2Vr − 2 r sin2 θ = 0 , (2.48)∂rVθ + ∂θVr − 2 tan Vr + r sin θ cos θ = 0 , (2.49)2 ∂rVr +4rVr + 2− cos2 θ = 0 . (2.50)The final equation is an ordinary differential equation for Vr, so it can besolved immediately asVr =r(cos2 θ − 2)3+C1r2 cos2 θ+C2 sin θr2 cos2 θ. (2.51)The undetermined coefficients C1, C2, must vanish to keep Vr finite on thenboundary θ = 0, pi/2. The remaining components can now easily be solvedfrom (2.49). The result isVθ = −r2 sin θ (2 + cos2 θ)3 cos θ+C2r cos θ+ F1(θ) (2.52)282.3. Summarywhere F1(θ) is an undetermined function which will thankfully not enter thecanonical energy calculation.Making use of the definitions (2.45), the two-form contribution isω(g,H,£ξH) = −r4 cos3 θ2RGNdr ∧ dθ, (2.53)which integrates toˆΣω(g,H,£ξH) = − 12Rˆ R0ˆ pi2−pi2r4 cos3 θ dθ dr = −2R415. (2.54)The evaluation of ρ = ρrdr + ρθdθ only requires knowledge of ρθ, sincewe are concerned with the constant R surface. Explicitly evaluating ρθ atr = R givesρθ|B˜ = −R412GNcos3 θ(2 cos2 θ − 3) (2.55)which upon integration yieldsˆB˜ρ =745GNR4 . (2.56)Summing the contributions from´ω and´ρ gives the total canonical energyE = R445GN. (2.57)For pure gravity, (2.57) represents the sole contribution to the canonicalenergy associated with the Rindler wedge RB. This surprisingly simply resultagrees with the requirement that the relative entropy is positive. It was alsoconfirmed in [50] that this matches the CFT result for Fisher information intwo dimensions2.3 SummaryIn this chapter we introduced the concept of relative entropy in both thefield theory and gravitational theory. By considering only the second orderchange relative entropy for a state perturbed from the vacuum, we definedthe quantum Fisher information which acts as a metric on the space of per-turbations to a density matrix. On the gravity side, we defined the canonicalenergy, a natural sympletic form, which represents a metric on the space of292.3. Summaryperturbations to a black hole. We provided an example calculation for aBTZ black hole and the result matches that expected in [50]In the next chapter, we will use the canonical energy formalism to com-pute the quantum Fisher information for a general state perturbed from thevacuum.30Chapter 3Relative entropy of generalperturbationsIn this Chapter we describe how to find the second order relative entropyfor a state which is perturbed from the vacuum by solving the dual grav-itational problem. As discussed in [1], such states can be attained from alocal conformal transformation of the vacuum. On the gravitational side,this corresponds to a general linear perturbation of AdS spacetime.This Chapter is broken down into four main parts. First, we derivethe most general perturbations of AdS3 which are consistent with a 2d CFTdual. We then apply the procedure from Chapter 2 to compute the canonicalenergy for such a spacetime. Following that, we prove the positivity ofthis expression and perform some consistency checks. Lastly, we go on tocalculate the matter contribution to the canonical energy from a scalar fieldperturbation.3.1 Gravitational contribution to relative entropy3.1.1 General perturbation of AdS3Let us consider the constraints on the boundary data of AdS3 arising fromthe consistancy of the stress tensor in a two-dimensional CFT. For any statein a CFT, the stress tensor must be traceless and conserved,〈T µµ 〉 = 0 , (3.1)〈∂µTµν〉 = 0 . (3.2)In our two-dimensional case, we can express the constraints most simply interms of light-cone coordinates, x± = x± t, so that〈T+−〉 = 0 , (3.3)∂+〈T−−〉 = ∂−〈T++〉 = 0 . (3.4)313.1. Gravitational contribution to relative entropyThus, a general 2d CFT stress tensor can be described solely by the twofunctions, 〈T++(x+)〉 and 〈T−−(x−)〉. These can to be identified with theholomorphic and antiholomorphic components.Assuming the CFT is holographic, there exists a gravitational dual withthe metric given byds2 =1z2(dz2 + dxµdxµ + z2Γµνdxµdxν). (3.5)The CFT stress tensor is related to the asymptotic behaviour of the AdSmetric through (2.25), so thatΓ++(x, 0) = 8piGN`AdS〈T++(x+)〉 , Γ−−(x, 0) = 8pi GN`AdS〈T−−(x−)〉 .(3.6)Now consider a a small perturbation to the CFT vacuum governed by λ. Tofirst order we haveΓ++(x, 0) ≡ λh+(x+) , Γ−−(x, 0) ≡ λh−(x−) . (3.7)The z behaviour of Γ±±(x, z) is then determined by the Einstein equationsand the initial data from (3.7). To first order in λ, the radial component ofthe linearized Einstein equations becomes1z3∂z(z3∂zΓµν) + ∂ρ∂ρΓµν = 0 . (3.8)The first order solution of (3.8) is simply given by (3.7). Together theseresults give the most general form for an asymptotically AdS metric whichhas a consistent CFT on the boundary. In terms of the functions h±, themetric isds2 =1z2(dz2 + (1 + λz2h+)(dx+)2 + (1 + λz2h−)(dx−)2)(3.9)=1z2(dz2 − (1− λz2h+)(1− λz2h−) dt2 + (1 + λz2h+)(1 + λz2h−) dx2+2λz2(h+ − h−) dt dx)Notice that if h+ = h− = 12 we recover the BTZ black hole metric in(2.47).3.1.2 Gauge-fixed coordinatesNow that we know the most general form of the metric, we can compute thecanonical energy of a Rindler wedge. We will useE(h, h) =ˆΣω(g, h,£ξh) +ˆB˜ρ(h, V ) (3.10)323.1. Gravitational contribution to relative entropyas per (2.44). This formula assumes a gauge condition in which the coordi-nate location of the minimal surface B˜ remains fixed. These differ from theFefferman-Graham gauge conditions we have been using so far. Thus, wemust first find a gauge transformation,V , to bring the metric perturbationto the appropriate form. Finding such a gauge for an arbitrary h is hope-less, instead we opt to decompose h into its Fourier components and findthe corresponding Vˆk for each mode of h. To do so, we first note that anysolution to the linearized Einstein’s equations around AdS can be written asa Fourier integralhµν(t, x, z) = λˆ [δ+µ δ+ν hˆ+(k)eikx+ + δ−µ δ−ν hˆ−(k)eikx−]dk , (3.11)with the gauge choice haz(t, x, z) = 0. Note that Greek indices run overthe CFT dimensions while Latin indices run over the full AdS spacetimedimensions.Due to the linearity of the problem, V can also be decomposed into leftand right k-modes asVa(t, x, z) = λˆ [Vˆ +a (k)eikx+ + Vˆ −a (k)eikx−]dk . (3.12)With this parameterization, the gauge functions Vˆ ±a can be directly solvedthrough the differential equations in (2.41). In terms of polar coordinates(x = r sin θ, z = r cos θ), these become−2 tan θ Vˆr + ∂θVˆr + ∂rVˆθ = −hˆθr , (3.13)−2 tan θ Vˆt + ∂θVˆt + ∂tVˆθ = −hˆθt , (3.14)r∂rVˆt + r∂tVˆr + 2Vˆt = −rhˆrt , (3.15)2r∂rVˆr + 2r∂tVˆt + 4Vˆr = −r(hˆrr + hˆtt), (3.16)∂2θ Vˆr − 3 tan θ ∂θVˆr − 2 Vˆr = tan θ hˆrθ , (3.17)∂2θ Vˆt − 3 tan θ ∂θVˆt − 2 Vˆt = tan θ hˆtθ , (3.18)where Vˆa = Vˆ +a (k)eikx++ Vˆ −a (k)eikx− . Due to symmetry, the mode so-lutions are related through V +r (θ, r, t) = V −r (θ, r,−t) and V −t (θ, r, t) =−V +t (θ, r,−t), and so we need only focus on the right-moving componentx−.Equations (3.17) and (3.18) are ordinary differential equations for Vˆt(θ)and Vˆr(θ) which can be solved directly. The solutions for the right-moving333.1. Gravitational contribution to relative entropypart areV −t (t, r, θ) =F2(t, r)cos2 θ+F1(t, r) sin θcos2 θ− i eikx− (k2r2 cos2 θ − 2)2k3r2 cos2 θ,V −r (t, r, θ) =F4(t, r)cos2 θ+F3(t, r) sin θcos2 θ− e−ikx− (ik2r2 sin θ cos2 θ − 2kr cos2 θ + 2 i sin θ)2 k3r2 cos2 θ,where Fi(t, r) are undetermined functions.To find the canonical energy, we also need the derivatives ∂rVθ, ∂tVθ.These can then be obtained from (3.13), (3.14) respectively,∂tV−θ =eikx−2 k2 r cos θ(2 + k2r2 cos2 θ − 2 ikr sin θ − 2F1(t, r)e−ikx−)∂rV−θ =eikx−k3r2 cos θ(2kr sin θ + r3k3 sin θ cos2 θ + i(r2k2 cos2 θ − kr2 + 2)−2k3r2e−ikx−F3(t, r)).Now the solutions for Vt and Vr diverge at the boundary θ = ±pi2 unlesswe choose Fi(t, r) appropriately. Requiring these to be finite at the boundary,we obtainF1(t, r) = F4(t, r) =sin(kr)r2 k3e−i k t , (3.19)F2(t, r) = F3(t, r) = − i cos(kr)r2 k3e−i k t . (3.20)Now that we have V , we can compute the canonical energy by using(2.45), (2.46) and (3.10). Such calculations are best handled with symbolicalgebra packages.3.1.3 ResultsThe canonical energy as a momentum space integral in terms of the boundarystress tensor neatly splits into the sum of two termsE =ˆdk1dk2 Kˆ(2)(k1, k2) 〈Tˆ++(k1)〉〈Tˆ++(k2)〉 + {+↔ −} (3.21)with the kernelKˆ(2)(k1, k2) =256pi2R4GNK3(K − κ)3(K + κ)3((K5 − 2(κ2 + 4)K3 + κ4K) cosK− (5K4 − 6K2κ2 + κ4) sinK + 8K3 cosκ) (3.22)343.1. Gravitational contribution to relative entropywhere K ≡ R(k1 + k2) and κ ≡ R(k1− k2). The canonical energy splits intoa left-moving part and a right-moving part because in the dual CFT there isno mixing between the holomorphic and antiholomorphic components. Thisis because the conformal group in two dimensions, SO(2, 2), factorizes intoSO(2, 2) ' SL(2,R)× SL(2,R).Taking the inverse Fourier transformation, we obtain the position spacekernelE =ˆBdx+1 dx+2 K(2)(x+1 , x+2 ) 〈T++(x+1 )〉〈T++(x+2 )〉 + {+↔ −} (3.23)where the kernel K(2) is given byK(2)(x±1 , x±2 ) =4pi2GNR2{(R− x±1 )2(R+ x±2 )2, x±1 ≥ x±2(R+ x±1 )2(R− x±2 )2, x±1 < x±2. (3.24)Notice that the kernel is symmetric under exchange of x±1 ,x±2 and vanishesfor |xi| > R.Using c = 3`AdS2GN , we can write (3.23) as the second order relative entropyof a CFT state. Like the leading order result from the entanglement first law,the integrals in (3.23) can be taken over any surface B with boundary ∂B.The fact that we only need the stress tensor on a Cauchy surface for DB isspecial to the stress tensor in two dimensions since the conservation relationsallow us to find the stress tensor expectation value everywhere in DB fromits value on a time slice. For other operators, or in higher dimensions, theresult will involve integrals over the full domain of dependence. This limitswhat we can say about the relative entropy in this case. We will see anexplicit example of such in the next section.As a consistency check, we can plug in the homogeneous BTZ metricwhere k = 0 and h±(0) = 12 . From (3.6) we have that 〈T±±(ki)〉 = λ16piGNand so it only remains to evaluate (3.22) at k1 = k2 = 0. In taking the limit,the kernel evaluates to 512pi2R4GN/90. The total canonical energy is thenE =(λ16piGN)2(512pi2R4GN90)(3.25)= λ2R445GN(3.26)which agrees with the expected result from (2.57).Another important consistency check will be to demonstrate that thecanonical energy in (3.23) is explicitly positive as required by the positivityof relative entropy. In the next section we give an elementary proof of thepositivity.353.1. Gravitational contribution to relative entropy3.1.4 Positivity of canonical energyLet us test the positivity of relative entropy from our result in (3.23). Con-sider the left-moving part of the perturbation h+(x+) ∝ 〈T++(x+)〉 (as anidentical analysis will follow for h−(x−)). The function h+(x) must be realvalued for a perturbation of AdS3 so we can expand h+(x) in a Taylor seriesh+(x) =∑∞n=0 anxn. The canonical energy is then given byE ∼∑n∑manamˆBˆBdx1dx2 xn1xm2 K(2)(x1, x2) . (3.27)Whereas it might seem convenient to use an orthogonal basic such asthe Legendre or Chebyshev polynomials, this integral is only analyticallytractable for the polynomial basis chosen. The result isE ∼∑n∑manamR4+n+mAn,m (3.28)where the proportionality factor is up to a positive constant andAn,m = 1(n+m+ 3)(n+m+ 1)0, if n+m odd1(n+1)(m+1) , if n,m evennm+n+m+3nm(n+2)(m+2) , if n,m odd(3.29)which is clearly non-negative and symmetric in n,m.To show that the canonical energy is positive, we need to show that thematrix M with entries given by An,m = An−1,m−1 4 is positive semidefinite.To do so, we will use proof by induction and Sylvester’s criterion whichstates that a square matrix M is positive semidefinite if and only if it has apositive determinant and all the upper-left sub-matrices also have positivedeterminants.Proof by inductionSuppose that the N ×N matrix MN whose components are given by An,mis positive semidefinite. Then consider the block matrix constructed asMN+1 =(M˜N B˜B˜T AN+1,N+1)= AN+1,N+1(MN BBT 1)(3.30)4The inelegant notation change is due to conventional matrix notation starting atn = 1, while the Taylor series starts at n = 0.363.1. Gravitational contribution to relative entropywhere B is an N column vector with entries given by Ai,n+1. Since MN ispositive semidefinite, it has a positive determinant and all the upper-left sub-matrices of MN also have positive determinants by Sylvester’s criterion. Toshow thatMN+1 is positive semidefinite, we need only show it has a positivedeterminant since all the upper-left sub-matrices are already known.The determinant of MN+1 may be evaluated using the formuladet(MN+1) = An+1,n+1[2 det(MN )− det(MN +BTB)](3.31)so it is sufficient to showdet(MN +BTB) < 2 det(MN ) . (3.32)We denote the eigenvalues ofMN+BTB by λM+Bi where they are orderedfrom largest to smallest λM+B1 ≥ λM+B2 ≥ ... ≥ λM+BN . Since BTB is arank-one matrix, the sole non-zero eigenvalue is given by β = Tr(BTB) =∑Ni=1Ai,N+1 ≥ 0. Since BTB is positive semidefinite, there exists an upperbound on det(MN + BTB) given by the Weyl inequality λM+Bi ≤ λMi + βiwhere λMi are the eigenvalues of MN in order from largest to smallest λM1 ≥λM2 ≥ ... ≥ λMN [53]. We then expand the determinant asdet(MN +BTB) =N∏i=1λM+Bi≤ λM+B1λMN∏i=1λMi =(1 +βλM1)det(MN ) .So it remains to show that λM1 −βB ≥ 0 to complete the proof. The maximumeigenvalue λM1 is bounded from below by the minimum sum of a column ofMN through the Perron-Frobenius theorem (equivalently Gershgorin circletheorem) [53]. For the matrix MN , the minimum sum of a column vectoris simply the sum of the Nth column∑Ni=1Ai,N . Therefore, it remains toshow thatN∑i=1(Ai,N −A2i,N+1) ≥ 0 . (3.33)We split this sum up into two cases. The first case is if N is even. Thenwe haveN/2∑i=1A2i,N −(N+1)/2∑i=1A22i−1,N+1 =N/2∑i=1(A2i,N −A22i−1,N+1)(3.34)373.2. Scalar field in AdS3since the final term in∑(N−1)/2i=1 A22i−1,N+1 is zero. Explicitly analyzing thecoefficients, we see that(A2i,N −A22i−1,N+1)is always positive for all i ∈{1..N/2}, so clearly the entire sum is positive.In the case of odd N , the sum becomes(N+1)/2∑i=1A2i−1,N −N/2∑i=1A22i,N+1 = AN,N +N/2∑i=1(A2i−1,N −A22i,N+1). (3.35)Each term in this sum is also positive, so we have shown λM1 − βB ≥ 0.The expressions in (3.34) and (3.35) are not obviously positive, but theyreduce to some (tractable, but unattractive) polynomial equations whichcan be shown to be positive. Therefore we have shown det(MN + BTB) <2 det(MN ), and thus MN+1 is positive semidefinite given that MN is. SinceM1 is positive semidefinite by induction, so too is MN for all N . Therefore,the canonical energy is explicitly positive semidefinite as expected by thepositivity of relative entropy.3.2 Scalar field in AdS3As an extension of the previous results, we could consider adding a simplescalar field to the AdS spacetime. The canonical energy in this case is givenby the sum of the gravitation and matter contributionE = Egrav +ˆΣBξaTmatterab dΣb (3.36)where Tmatterab is the stress tensor for the scalar field. Explicitly taking z asthe bulk coordinate, the matter contribution to the canonical energy isEmatter =ˆΣBdzdxxpi(R2 − z2 − x2)RzTmattertt . (3.37)In the holographic context, one considers excited states of a CFT whichare induced by normalizable modes of bulk scalar fields in AdS. The bulkfields correspond to non-zero one-point functions of the dual operator O. Asin the gravitational case, we wish to express the canonical energy in terms ofa product of one-point functions; this time of a general operator O, ratherthan the stress tensor T±±:δ(2)O S =ˆdx1dx2K(2)O (x1,x2)〈O(x1)〉〈O(x2)〉 . (3.38)383.2. Scalar field in AdS3The total change in the entanglement entropy up to second order in theperturbations is then∆S = δ(1)S + δ(2)S + δ(2)O S . (3.39)The δ(1)O S term vanishes for a perturbation around the vacuum so the onlylinear contribution to the entanglement entropy is from the first law.In the remainder of this Chapter we focus on explicitly evaluating thesecond order contribution from a field, δ(2)SO.3.2.1 General scalar fieldConsider the case of a scalar field φ in AdS which is dual to the operator O.The equation of motion of the scalar field φ, in AdS spacetime is1z∂z (z∂zφ) + ∂µ∂µφ− m2z2φ = 0 , (3.40)where µ run over the CFT dimensions.The solution of (3.40) may be written in term of the Fourier transformof φ,φ(z, t, x) =ˆω2−k2>1dωdk ei(kx−ωt)φω,k(z) (3.41)so that the equation of motion becomes[z2(ω2 − k2) + z2∂2z − z∂z −m2]φω,k(z) = 0 . (3.42)The solution for each Fourier component is then given by a sum of Besselfunctionsφω,k(z) = C1zJν(qz) + C2zYν(qz) (3.43)where ν =√1 +m2 = ∆ − 1 and q = √ω2 − k2. Near the boundary, thefield must behave aslimz→0φ(z, t, x) = z∆φ1(t, x) + z2−∆φ0(t, x) (3.44)where φ0 and φ1 are the normalizable and the non-normalizable modes re-spectively. These modes can be related to the Bessel function solutionsthrough the holographic dictionary in (1.16). We thus have〈O(t, x)〉 = (2∆− 2)φ1(t, x) , φ0(t, x) = 0 . (3.45)393.2. Scalar field in AdS3The constants C1, C2, can be determined by taking the z → 0 limit of theBessel function so thatC1 =2∆Γ(∆)∆− 1 q−ν/2〈Oω,k(z)〉 , C2 = 0 . (3.46)Therefore, the scalar field takes the formφ(z, t, x) =2∆Γ(∆)∆− 1 zˆω2>k2dωdk ei(kx−ωt)q−ν/2Jν(qz)〈Oω,k(z)〉 . (3.47)This is a fairly well-known result and can be found in [54, 55]We can now compute the stress tensor fromT(2)ab = ∂aφ∂bφ−12gab(∂aφ∂bφ+m2φ2). (3.48)The evaluation (3.48) is fairly involved, but with the aid of symbolic com-putation, the result isT(2)ab =ˆω21>k21ˆω22>k22dw1dk1dw2dk2 T˜(2)tt (z, w1, w2, k1, k2)ei (k1+k2)x〈Ow1,k2〉〈Ow2,k2〉 .For the canonical energy evaluated on a Cauchy slice ΣB, will only need theT(2)tt component of this equation. This is given byT˜(2)tt (z, w1, w2, k1, k2) =14(2∆Γ(∆)∆− 1)2q−ν1 q−ν2×[ (−z2(w1w2 + k1k2) + 2ν(ν − 1)) Jν(zq1)Jν(zq2)−(ν − 1) (q1zJν−1(q1z)Jν(q2z) + q2zJν−1(q2z)Jν(q1z))+q1q2z2Jν−1(q1z)Jν−1(q2z)].Although this formula is not very transparent, it can be simplified in certaincases. For m = 0, the stress tensor kernel reduces toT˜(2)tt (z, w1, w2, k1, k2)|m=0 =4z2q1q2(q1q2J0 (zq1) J0 (zq2)−(k1k2 + w1w2) J1 (zq1) J1 (zq2)).The entanglement entropy contribution from the scalar field is thenδ(2)SO =ˆΣBˆω21>k21ˆω22>k22dz dx dw1dk1dw2dk2 (3.49)×pi(R2 − z2 − x2)ei(k1+k2)xRzT˜(2)tt (z, w1, w2, k1, k2)〈Ow1,k2〉〈Ow2,k2〉 .403.3. Summary3.2.2 Constant scalar fieldasdasdTo simplify this expression even more, consider the case where 〈O〉 isconstant so that the solution for φ is φ(x, z) = γ〈O〉z∆ with some normal-ization constant γ. In this case, the integrals in (3.49) are calculable and thecontribution to the entanglement entropy is explicitlyδ(2)SO = −pi3/24γ2〈O〉2R2∆Ωd−2 ∆Γ(∆)Γ(∆ + 32). (3.50)It is actually possible to carry out the constant scalar field calculation ind dimensions [1]. In this case, the formula generalizes toδ(2)SO = −pi`d−1AdS4γ2〈O〉2R2∆Ωd−2∆Γ(d2 − 12)Γ(∆− d2 + 1)Γ(∆ + 32). (3.51)This result agrees with previous calculations in [56, 57].3.3 SummaryThis Chapter has covered a lot of material. Firstly, we derived the mostgeneral perturbation to AdS spacetime from the CFT requirements on Tµν .The result agreed with the expectation that the two-dimensional CFT dualfactorizes into holomorphic and antiholomorphic parts.We then derived the canonical energy for such a spacetime with the mainresult in (3.23). This result reduces to the BTZ black hole case discussed inChapter 2 as well as [50]. Furthermore, we proved the positivity directly, byusing some powerful theorems from linear algebra.Lastly, we found the contrition to the entanglement entropy from the ad-dition of an operator O in the CFT. Our calculation makes use of the mattercontribution canonical energy via (3.37). Although the formal expression isalmost unwieldy, in the simple case of constant 〈O〉, the expression reducesnicely to (3.51).Overall, these methods contribute to the understanding how to recon-struct bulk geometry from the entanglement structure of a CFT. In partic-ular, we find that (to second order), the pure gravitational components maybe written as the product of CFT one-point functions. This remains possiblefor scalar perturbations, albeit more complicated.41Chapter 4Emergent de Sitter spacetimefrom entanglementIn a recent paper by de Boer et al. [24] it was recognized that, to firstorder, the entanglement entropy of a spatial ball-shaped region obeys theKlein-Gordon equation in de Sitter (dS) spacetime. In this construction,the size of the ball directly determines how far into the future/past a fieldpropagates. Causality in the bulk dS is then equivalent to an ordering ofthe spheres based on their size. This result holds for an arbitrary number ofdimensions and is independent of the standard AdS/CFT correspondence.In this Chapter, we first review the results of [24] for the first order entan-glement entropy. We then show how the second order entanglement entropycalculated from the canonical energy formalism is related to dynamical fieldsin dS. We find that the total entanglement entropy looks like a scalar fieldtheory with a cubic interaction in dS spacetime. Although the original workin [24] was independent of the number of dimension, our result is only ford = 2.4.1 Emergent de Sitter dynamics fromentanglementIn this section we review the argument of de Boer et al. [24] regarding theentanglement structure in a CFT and it’s implications for dynamics in deSitter spacetime.In the case of a flat d-dimensional CFT the vacuum subtracted entangle-ment entropy of a spherical region B, which is centred at x, with radius R,isδ(1)S(R, x) = 2piˆBdd−1x′R2 − |x− x′|22R〈Ttt(x′)〉 (4.1)where δ(1)S denotes the entanglement entropy to first order in Ttt, the energy424.1. Emergent de Sitter dynamics from entanglementdensity operator. As discussed in Chapter 2 this comes from the first law ofentanglement δ(1)S = δ(1)〈HB〉.It was noticed by the authors of [24] that the integration kernel in (4.1) isthe bulk-to-boundary propagator in d-dimensional de Sitter spacetime withthe metricds2 =L2R2(−dR2 + dx2). (4.2)The scale L is a free parameter in this theory.The entropy δ(1)S(R, x) may be interpreted as a solution to the free scalarwave equation in dS spacetime(∇2dS −m2)δ(1)S(R, x) = 0 , (4.3)with the mass parameter set by m2 = −d/L2. Notice the mass is tachyonic.Fortunately, we will show that the boundary conditions imposed will removethe unstable modes associated with the tachyon.The CFT can be interpreted as living on either future or past infinityI±. In this case, we will take I+ so that a future direction light cone ofa point (R, x) in dS intersects I+on the region with entanglement entropyδ(1)(R, x). The diagrammatic interpretation is illustrated in figure 4.1.The general solution to (4.3) has two independent asymptotic solutions,limR→0δ(1)S(R, x) =F (x)R+Rd f(x) . (4.4)Through the holographic dictionary, the first term corresponds to a statewith conformal weight ∆ = −1, and the second to a weight ∆ = d. Requiringthe solution to be regular at R = 0, we haveF (x) = 0, f(x) =pid+12Γ(d+32 )〈Ttt(x)〉 . (4.5)The absence of the F (x) precisely corresponds to removing the unstable(non-normalizable) modes of the tachyon [24].It can be interpreted that the energy density specifies the entanglemententropy at small scales (R → 0), while at larger scales the entanglemententropy is determined through the Lorentzian propagation into the bulkde Sitter geometry. In other words, the entanglement entropy for smallregions is determined by (4.1) and inputting this as an initial condition, theentanglement entropy for large regions is determined by (4.3). In contrast434.2. Interactions in de Sitterη=∞I−I+B(R, x)dSdδ(1)SFigure 4.1: Penrose diagram for dSd spacetime. The entangling region Bis in blue with the light cone reaching out to the unique bulk point (R, xi).The wavy line represents the field δ(1)S propagating from the bulk to theboundary.to the usual route in holography of computing holographic entanglemententropies, this method demonstrates that entanglement entropies themselvesgives hints at constructing dS spacetime holographically.In this picture, the radius of the ball in the CFT is the time coordinatein dS. There exists a specific map between the time in de Sitter spacetimeand the size of the ball in a CFT. A ball that is contained in another ball isconsidered timelike separated, while two disjoint balls are spacelike separatedas shown in figure 4.2. This gives a natural ordering to ball-shaped regionsas was suggested in [58].4.2 Interactions in de SitterThe view presented above is remarkably simple and begs the question; whatdo higher order terms of the entanglement entropy terms correspond to inde Sitter? Since in Chapter 3 we computed the second order correction tothe entanglement entropy in a 2d CFT, we can at least answer this questionfor δ(2)S and d = 2.Recall from Section 2.2 that the second order entanglement entropy is444.2. Interactions in de SitterI+ = {x|R = 0}x2Bx2x3Bx3x4Bx4x5Bx5Figure 4.2: The size of the entangling region determines the bulk depth inthe dS geometry. There is a one-to-one mapping between bulk points andspherical regions on the asymptotic future boundary I+. The light cone froma bulk point reaches the boundary on a spherical region.given by the sum of two termsδ(2)S(R, x) = δ(2)S+ + δ(2)S− , (4.6)where we suppress the (R, x) dependence of δ(2)S± for convenience. Actingwith the Klein-Gordon equation for de Sitter spacetime on δ(2)S± gives(∇2dS −m2)δ(2)S± =ˆdx1ˆdx2 (∇2dS −m2)K(2)(R, x;x1, x2)〈T±±(x1)〉〈T±±(x2)〉 .Since only K(2) depends on x, we are only concerned with the action of thewave equation of K(2). Evaluating this explicitly gives(∇2dS −m2)K(2)(R, x;x1;x2) =12cL2K(1)(R, x;x1)K(1)(R, x;x2) (4.7)where K(1)(R, x;x1) = 2piR2−|x−x1|22R as per (4.1) and K(2) is given by (3.24).Reverting back to entropies, the equation of motion reads(∇2dS −m2)δ(2)S±(R, x) =12cL2(δ(1)S±(R, x))2. (4.8)Remarkably, the second order term has factored into an interaction termbetween two bulk-to-boundary propagators. If the first order term δ(1)Scorresponds to a free scalar field in de Sitter spacetime, then the secondorder term δ(2)S± corresponds to adding an cubic interaction for this field.Equation (4.8) can be derived from the LagrangianL± = 12(∇aδS±)2 + 12m2δS2± +4cL2(δS±)3 (4.9)454.2. Interactions in de Sitterx−R x+RδS+ δS+δS+x1 x2g3(R, x)I+I+Figure 4.3: A visual interpretation of (4.7) as a Feynman diagram. The δS±field propagates from a point on asymptotic past infinity (I−), interacts withanother δS± with a vertex given by g3 = 12cL2 and produces another δS± fieldwhich reaches the bulk point (R, x).where the ± fields are independent and the total scalar field in dS is δS± =δ(1)S± + δ(2)S± + O(〈T 〉3). In this way the cubic interaction term in theLagrangian precisely produces the diagram in figure 4.3. It is interestingthat static entanglement entropy of regions produce dynamics (a classicalfield theory) in dS spacetime.The past R evolution of δS± can be described by a past-directed dSGreen’s functionδS±(R, x) =ˆdx′ G(1)dS (R, x;x′)〈T±±(x′)〉 . (4.10)This function propagates the field from a location on future infinity I+ toa point inside the bulk dS space. Likewise, the full bulk-to-bulk Green’sfunction is required for the second order contributionδ(2)S±(R, x) =12cˆdSdR′dx′√|gdS|GdS(R, x;R′, x′)(δS±(R′, x′))2.(4.11)The bulk-to-boundary function G(1)dS (R, x0;x) can be obtained by takingthe R→ 0 limit of the bulk-to-bulk propagatorGdS(R, x;R′, x′) = −4piR2 +R′2 − (x− x′)24RR′(4.12)so thatG(1)dS (R, x;x′) = 2piR2 − (x− x′)22R, (4.13)464.2. Interactions in de Sitterwhich matches (4.1) from the first law of entanglement.These propagators are defined to be non-zero only within the past di-rected light-cone, which is important in reproducing both the support andthe exact form of K(2)(x1, x2) from (3.24).Positivity checkThe emergent de Sitter realization provides a very quick way to check thepositivity of Fisher information. By directly inserting the propagator (4.12)into expression (4.11), we haveδ(2)S±(R, x0) = − 12cL2ˆdSdR′dx′√|gdS|R2 +R′2 − (x− x′)2RR′(δS±(R′, x′))2.(4.14)The determinant of the metric is positive, along with the square of δS±. Themiddle term is also positive for (x′ − x)2 ≤ (R −R′)2 which lies completelyin the range of integration. Thus it is evident that δ(2)S± is strictly non-negative as required by the positivity of relative entropy.Thermal stateThe analysis thus far has assumed the entropy is that of a state perturbedfrom the vacuum. Using the CFT method in [1], it is straightforward togeneralize to the entropy of a state perturbed from a thermal state. Theresult is given by the propagation of a scalar field in a spacetime with themetricds2thermal =4pi2L2dSβ sinh2(2piRβ) (−dR2 + dx2) . (4.15)The bulk-to-boundary propagator in this spacetime isK(1)β (R, x;x′) =2βsinh(2piRβ) sinh(pi (R− x+ x′)β)sinh(pi (R+ x− x′)β),with the entanglement entropyδ(1)Sβ(R, x) = 2piˆdx′ K(1)β (R, x;x′)〈Ttt(x′)〉 . (4.16)The kernel for the second order entanglement entropy, or the bulk-to-bulk474.3. Summarypropagator, isK(2)β (R, x;x1, x2) =24β2c sinh2(2piRβ)×sinh2(pi(R−x1+x)β)sinh2(pi(R+x2−x′)β), x2 ≤ x1sinh2(pi(R+x1−x′)β)sinh2(pi(R−x+x′)β), x1 ≤ x20 , |xi| > R.From the entropy, we see that states perturbed from a background state(thermal or vacuum) correspond to a dynamical scalar field in a backgroundspacetime. Whether this spacetime could be dynamical itself is an interestingopen question.4.3 SummaryIn this Chapter we have reviewed the previous work of [24] which relates theentanglement entropy of a CFT to tachyonic fields in an auxiliary de Sitterspacetime. Using the results from Chapter 3 we extended this interpreta-tion to include interacting fields in de Sitter. This construction provides asimple way to confirm the positivity of the canonical energy in Section 3.1.4.Additionally, we confirmed the result also holds for states perturbed from athermal background.48Chapter 5ConclusionFirstly, let us comment on the canonical energy procedure from Chapter3. While the Ryu-Takayangi formula gives a simple means to compute theexact entanglement entropy of a CFT, due to the difficulty of a direct CFTcomputation, it is impossible to verify in complex systems. In general, thefirst order contribution to the Ryu-Takayanagi entropy is given by the firstlaw of entanglement, a feature that is universal to all CFTs. At what orderdoes the Ryu-Takayangi formula differentiate holographic theories from othertheories?In this thesis we have suggested that the second order contribution mightbe appropriate to answer such a question. Unfortunately, in the case studiedin this thesis, the result remained universal for all two-dimensional CFTs, andhence did not restrict the number of possible holographic theories. We didfind however, that this contribution manifests itself as a cubic interactionterm within the emergent de Sitter formalism of [24]. This is an excitingresult that may yield insight into the structure of holographic theories andhow more realistic models of our universe (such as de Sitter spacetime) mayemerge.There are many possible future directions for the line of inquiry pursuedin Chapter 4 as summarized in figure 5.1. Firstly, it is possible to extend thiswork to arbitrary order in 〈T 〉 which has recently been done in [59]. In thiscase, the cubic potential in (4.9) is replaced with an exponential of δS. Itis doubtful that this expansion could be attained in higher dimensions sincethe result depends on the infinite Virasoro symmetry in 2d CFTs. It is alsodoubtful that these results could be derived from the gravitational approachsince it would require new quantities beyond the canonical energy.Another reason the de Sitter interpration might be unique to d = 2CFTs is the fact that the conformal group in two-dimensions factorizes asSO(2, 2) ' SL(2,R) × SL(2,R). It is clear from this why we obtainedtwo copies of de Sitter space, one for each of the left- and right-movingcomponents. Such a factorization does not hold in higher dimensions, so theemergent spacetime might be radically more complicated.We could also imagine adding additional operators Oi in the CFT to49Chapter 5. ConclusionAuxiliary dS HolographydS2 dSdδ(1)S δ(2)S δ(n)S δ(2)O S δ(1)S δ(2)S δ(n)S δ(2)O SFigure 5.1: A summary of progress and future directions for the auxiliaryde Sitter approach. Green boxes indicate solved problems, and red boxesindicate present challenges.see what they correspond to within the auxiliary dS space. However, unlikethe contribution from the stress tensor, the entropy from these operatorsinvolves integration of one-point functions over the full domain of dependenceDB. Rather than the space of entanglement entropies of spheres, one mightconsider the space of causal diamonds, to adequately generalize the auxiliaryde Sitter interpretation.It would also be interesting to see how this prescription works in higherdimensions. The first order term is already known to hold for arbitrary di-mensions. Unfortunately, pure gravity in higher dimensions faces the sameproblem as additional operators in the CFT. The entanglement entropy de-pends on the entire causal region DB, requiring a more advanced version ofde Sitter holography.Since the future boundary of the auxiliary de Sitter space does not includethe time direction of the CFT, any extension of these results to dynamicalentropies will also be problematic. 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