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Tensor networks for dynamic spacetimes May, Alex 2017

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Tensor networks for dynamic spacetimesbyAlex MayBsc Joint Honours Physics and Mathematics, McGill University, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)August 2017c© Alex May, 2017AbstractTensor networks give simple representations of complex quantum states. Theyhave proven useful in the study of condensed matter systems and conformal fields,and recently have provided toy models of AdS/CFT. Underlying the tensor net-work - AdS/CFT connection is the association of a graph geometry with the tensornetwork. This geometry is most easily understood as containing only spatial di-rections. In the context of the AdS/CFT correspondence this limits tensor networktoy models to describing static spacetimes. Here we look to extend tensor net-work models of AdS/CFT by capturing the geometry of a dynamic spacetime in anetwork description. We review the role of tensor networks in our understandingof AdS/CFT to motivate this extension, before proposing a network picture thatcaptures key features of AdS/CFT.iiLay AbstractQuantum mechanics has a number of puzzling features. For instance, two parti-cles can be entangled, which indicates their behaviour is tied together over longdistances in a way that challenges our notion of causality. A tensor network is adescription of a quantum system that captures this entanglement pictorially. In thisthesis we apply tensor networks to one of the key problems in theoretical physics,quantum gravity, which seeks to put gravity within the language of quantum me-chanics.The connection between tensor networks and quantum gravity rests on the factthat in both entanglement is described using geometry. The quantum gravity set-ting where the entanglement geometry connection is best understood is known asAdS/CFT, but it is expected that this connection exists more generally. By under-standing this connection in AdS/CFT in the tensor network language we hope tofind new avenues for generalizing our understanding of entanglement and geome-try.iiiPrefaceThe work presented in this thesis is primarily the work of the author, but discus-sions and input from several parties were formative in its development. Mark VanRaamsdonk initiated the project by posing to the author the initial topic of under-standing dynamics in tensor networks. Charles Rabideau suggested a useful nar-rowing of the research goals, focusing on the maximin formula. Discussions withMark Van Raamsdonk continued throughout the project, where he gave valuableguiding suggestions.The content of chapters 4 and 5 of this manuscript is published by the Journalof High Energy Physics[33]. This content also appears in the online preprint serverarXiv under identifier 1611.06220.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Holography and the AdS/CFT correspondence . . . . . . . . . . . . 11.1 The holographic principle . . . . . . . . . . . . . . . . . . . . . . 21.2 Quantum information theory . . . . . . . . . . . . . . . . . . . . 31.3 The AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . 62 Tensor networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Tensor network basics . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Maps defined from tensor networks . . . . . . . . . . . . . . . . 172.3 MERA and entanglement renormalization . . . . . . . . . . . . . 203 Tensor networks and holography . . . . . . . . . . . . . . . . . . . . 233.1 Entanglement and geometry . . . . . . . . . . . . . . . . . . . . 233.2 Early efforts and constraints on holographic tensor networks . . . 253.3 The subregion isometry property . . . . . . . . . . . . . . . . . . 27v4 Examples of real space holographic tensor networks . . . . . . . . . 304.1 HaPPY networks . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Random tensor networks . . . . . . . . . . . . . . . . . . . . . . 355 Tensor networks for dynamic spacetimes . . . . . . . . . . . . . . . 405.1 Length and extremal curves in tensor networks . . . . . . . . . . 415.1.1 Lessons from the maximin formula . . . . . . . . . . . . 415.1.2 A definition of length in tensor networks . . . . . . . . . 425.1.3 A static example . . . . . . . . . . . . . . . . . . . . . . 445.2 Dynamic tensor network states . . . . . . . . . . . . . . . . . . . 476 Holographic tensor networks away from AdS/CFT . . . . . . . . . . 536.1 Building a tensor network for flat space . . . . . . . . . . . . . . 537 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60viList of FiguresFigure 1.1 The cylinder representing AdS2+1. The boundary of the cylin-der is at ρ = ∞. . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.2 Illustration of the error correcting property of the bulk to bound-ary mapping. No single boundary region A, B, or C is sufficientto reconstruct a bulk operator living at O, but any two regionsAB, BC, or CA is. This is because the minimal surface of A(for example), which is labelled as a, does not enclose pointO. However the minimal surface for AB, which is c, does.This is analogous to the three qutrit code introduced in section1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.1 (a) A maximally entangled state in the Hilbert space, repre-sented in the graphical notation. (b) A maximally entangledstate in the dual Hilbert space, represented in the graphical no-tation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.2 A basic example of a contraction of two quantum states intoa tensor network. Algebraically, the objects at left are writtenTabcd and Se f gh. The object at right is Tabcdδ ceδ d f Se f gh. . . . . 15Figure 2.3 (a) An operator M⊗ I applied to a quantum state. (b) In thecase where the vertex represents a maximally entangled state,the operator M can be moved to the other subspace by takingthe transpose. . . . . . . . . . . . . . . . . . . . . . . . . . . 15viiFigure 2.4 (a) Representation of the density matrix ρAB shown in eq. 2.6in the graphical notation. (b) Representation of the reduceddensity matrix ρA in the graphical notation. . . . . . . . . . . 16Figure 2.5 (a) Graphical description of 2.15, which gives a tensor networkstate in terms of the two block tensors C and C¯ defined by a cutγ . (b) Graphical description of eq 2.16, which computes thestate on a cut γ . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.6 Illustration of the tensor network which prepares a matrix prod-uct state. Matrix product states are the output of the DMRGalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.7 Illustration of the MERA network, thought of as a quantum cir-cuit which prepares a state. The blue squares represent unitarymatrices and black triangles represent isometries. Free legs at-tached to blue squares are wrapped to the opposite side of thefigure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 3.1 As the entanglement between the A and B regions in the CFTis removed, the bulk geometry pinches and then is pulled apart. 24Figure 3.2 A tensor network which includes bulk legs. The state on a cut|Ψ〉BB¯ is associated to any path through the dual graph γ . . . . 27Figure 4.1 Illustration of the defining condition for perfect tensors. Thesame equality must hold when any subset consisting of half thelegs is contracted. . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 4.2 Illustration of the operator pushing operation. An operator Oacting on a subset of size n of a perfect tensor with 2n legs isequivalent to an operator O ′ = T †OT acting on the complement. 32Figure 4.3 A tree graph on which the 4 legged perfect tensor can be placedto build a tensor network. The resulting network seems to havethe subregion isometry property, as can be argued by checkingcases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35viiiFigure 4.4 a) Structure of the state |Ψ〉. The beige box at left representsthe state |φ〉A¯C¯E¯ . The curved black line represents the max-imally entangled state (|Ψ+〉⊗n)CE . b) Structure of the state|ΨM〉. The blue box represents the isometry V : EE¯ → A dis-cussed in text. . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 5.1 (a) A HaPPY network. Six legged vertices are perfect ten-sors and two legged projecting pairs are maximally entangledstates. All boundary entropies are given by the graph lengthof a minimal cut in the network. (b) A network which satis-fies the Ryu-Takayanagi formula using the mutual informationbased definition of length, but does not satisfy Ryu-Takayanagiwhen using the graph length. The blue dots and legs representthe projecting state given in eq. 5.11. The dashed line repre-sents the isometric cut for the boundary region A. . . . . . . . 45Figure 5.2 The basic simplification used to compute the density matrix ofregion A in figure 5.1b. The six leg tensors are from the edge ofthe network shown in figure 5.1b. The effect of the state givenin equation 5.11 is to add a normalization factor, representedas the blue loop at right. . . . . . . . . . . . . . . . . . . . . 46Figure 5.3 Graphical representation of the density matrix of the region Afrom figure 5.1 b. . . . . . . . . . . . . . . . . . . . . . . . 47ixFigure 5.4 The four networks included in the set F considered in text. Theprojecting states shown in blue in (a) are defined by an oper-ator O acting on the maximally entangled state, these can bepushed to any subset of three legs, as seen in (b-d). In gen-eral the pushed through operator will not have a tensor productstructure, so the corresponding projecting state will be entan-gled across three legs. This is indicated here by the thick blueline in (b-d). All four networks shown here have the sameboundary state. The entropies of subsets of boundary legs iscalculated by choosing the network which contains an isomet-ric cut enclosing those boundary legs, and applying the isomet-ric cut formula. No one network contains an isometric cut forevery boundary region, but the set of four networks together do. 48Figure 5.5 The identity used to show a cut containing one interior legwhich is blue in 5.4 a is isometric up to a normalization factor.This identity is easily derived from that in figure 5.6. . . . . . 49Figure 5.6 The identity used to show the cut γ1 in figure 5.4a is isometricup to a normalization factor. . . . . . . . . . . . . . . . . . . 50Figure 5.7 Illustration of how to choose an isometric cut for the subregionconsisting of legs adjacent to the d and c vertices. The upperdashed line crosses legs included in the B¯ Hilbert space, whilethe lower dashed line crosses legs included in the B Hilbertspace. It is straightforward to check that both the operatorsabove and below the dashed lines are isometries; it followsthat S(A) = 12 I(B¯ : B). . . . . . . . . . . . . . . . . . . . . . 51xFigure 6.1 Illustration of our procedure for constructing a network whoseminimal lengths approximate those of a given geometry. Theexample shown constructs a network with four boundary legswhich approximates a disk shaped region of R2. In the firststep, four boundary points are chosen and all of the minimalcuts anchored on those points are drawn. The minimal cutsform a planar graph, in this example the graph has verticesA,B,C,D and O. . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 6.2 Illustration of the second step in our procedure for construct-ing a network which approximates a given geometry. In thisstep, the dual of the graph formed in step one is drawn. Theedges of the dual graph are assigned a weight based on the R2length of the edges in the direct graph they cut. For example, aweight of Floor( ¯AB/δ ) is assigned to the edge TY , where δxis a parameter with units of length controlling how closely thegraph approximates lengths in the disk. . . . . . . . . . . . . 56xiAcknowledgmentsThis work was carried out under the supervision of Mark Van Raamsdonk, whowas involved in discussions throughout this project and reviewed drafts of thismanuscript. Charles Rabideau made useful contributions in the early stages ofthe project. We are also indebted to Michael Walter, Grant Salton, Zhao Yang,David Stephen, Jordan Cotler, and Adam Levine for helpful discussions. DominikNeuenfeld and Jaehoon Lee provided feedback on early versions of the manuscript.AM was partially supported by the It from Qubit Collaboration, which is spon-sored by the Simons Foundation. AM was also supported by a CGSM award givenby the National Research Council of Canada. This research benefited from the Itfrom Qubit summer school held at the Perimeter Institute for Theoretical Physics.Research at Perimeter Institute is supported by the Government of Canada throughIndustry Canada and by the Province of Ontario through the Ministry of EconomicDevelopment & Innovation.AM also wishes to acknowledge the personal support of Mark Van Raamsdonk,whose flexible style of direction allowed the author to pursue varied directions ofresearch, Jessica Allanach, for her patience and personal support, the Lertzman-Lepofsky’s, whose welcoming home provided the backdrop for finding many of theresults presented here, to Patrick Hayden for early guidance and encouragement,and to Fang Xi Lin, Eric Hanson, Joelle and Bernie Ducharme, and both the UBCand McGill physics communities.xiiChapter 1Holography and the AdS/CFTcorrespondenceGravity is well described by general relativity, which can be understood as an ef-fective field theory. Leading order quantum corrections to general relativity canbe calculated by standard effective field theory methods [11]. The effective fieldsapproach however requires a high energy cutoff, and this description of gravity isexpected to fail at energies near to and above this cutoff.One approach to finding a UV complete theory of gravity is to search for anappropriate field theory which is renormalizable and whose low energy descriptionlooks like the (non-renormalizable) terms in the Einstein-Hilbert action. Indeed,the description of the weak force was initially in terms of a non-renormalizableLagrangian, which was later found to have a high energy renormalizable descrip-tion.There are basic reasons however to expect this approach to fail and a moreradical proposal to be necessary. One such reason is the understanding that a UVcomplete theory of quantum gravity must be holographic [27, 41], which a standardfield theory description is not. To understand what a holographic theory is we willbriefly review some facts about black holes and quantum mechanics below.11.1 The holographic principleBlack holes provide a key low energy probe of high energy physics, and for thisreason have long been at the focus of research in quantum gravity. An early im-portant line of work on black holes concerned black hole thermodynamics, whichshowed that the quantities which describe a black hole - mass, area, charge, andangular momentum - obey relations identical to those of ordinary thermodynamics[48, 49]. In particular, the area of the black hole plays the role of the thermody-namic entropy in this analogy.Black hole thermodynamics lead Bekenstein to argue that the similarity be-tween the black hole relations and thermodynamic relations is more than an anal-ogy, and that in fact the area of the black hole really was the entropy of the blackhole[6, 7]. A key argument of Bekensteins is that making the identificationSBH =A4G(1.1)would lead toδStotal = δ (SBH +Smatter)≥ 0. (1.2)Thus, identifying the black hole area as the entropy of the black hole correctly leadsto the second law of thermodynamics being maintained in a black hole spacetime.Bekensteins claim was put on firm ground by Hawking, who did a calculationin quantum field theory in a black hole background [24] to determine that the blackhole would radiate at a temperatureT =18piGM. (1.3)This temperature is consistent with the thermodynamic relation dE = T dS whenthe entropy is identified as S = A/4G, as argued by Bekenstein.The simple statements above about black holes are already enough to arguefor the holographic principle. Suppose we start with a physical system which iscontained in a spherical spatial region Γ. Call the thermal entropy of this system2S(Γ). Then it is straightforward to show thatlogN ≥ S(Γ), (1.4)where N is the number of states available to the system Γ. We will suppose thatthe system has nearly achieved this bound, and has a maximal entropy so thatS(γ)≈ logN.Now suppose that we add matter to our spatial region Γ until a black holeforms which occupies all of Γ. Then the regions entropy is A/4G, as given by theBekenstein-Hawking formula. The second law of thermodynamics says that theentropy after matter is added should be greater than before, so thatS(Γ)≤ A4G. (1.5)Combining this with the relation S(Γ)≈ logN,logN ≤ A4G. (1.6)That is, the number of degrees of freedom in a region Γ is bounded above by thearea of the region.This should come as a surprise. A field theory description of a typical systeminvolves degrees of freedom at each spatial point, and thus the total number ofdegrees of freedom required scales with the volume. The field theory is only alow energy effective theory however, and our black hole arguments indicate thatin fact the complete theory should have far fewer degrees of freedom. Motivatedby this, we can state the holographic principle as follows: a UV complete theoryof quantum gravity in 3+ 1 dimensions should have an equivalent description in2+1 dimensions.1.2 Quantum information theoryThe holographic principle tells us that a quantum theory of gravity can be writtenin one fewer spatial dimensions than expected. Because of the success of non-holographic physics, we know that at least low energy processes do have a simple,3local Lagrangian description in 3+ 1 dimensions. These processes should haveequivalent descriptions in the lower dimensional theory, which already raises basicquestions. Since naively a local quantum field theory in 3+ 1 dimensions is de-scribed by a larger dimensional Hilbert space than a 2+1 dimensional one, it mustbe some special sector of the QFT which is mapped into the lower dimensionaltheory. We can ask what the structure of this sector is, and how the mapping to 2+1dimensions works precisely.These considerations motivate the study of quantum information theory in thecontext of holography and quantum gravity. Information theorists have alreadyconsidered similar questions to the above. Indeed, a primary goal of quantum in-formation theory is to understand how one Hilbert space may be encoded into an-other. This area of study has been directed more specifically at storing informationin a way which is safe against errors or losses, and goes by the name of quantumerror correction.For later use, we give a very brief introduction to quantum error correctingcodes here. A detailed exposition can be found in Neilson 2002 [34]. Constructionof an error correcting code begins with the notion of a logical Hilbert space, Hlogicaland a physical Hilbert space Hphys. The error correcting code is defined by anencoding map E : Hlogical → Hphys and a decoding map D : Hphys→ Hlogical . Thedimension of Hphys will be larger than Hlogical , and the image of the encoding mapwill form a subspace Hcode. In standard terminology, it is the subspace Hcode whichis known as the error correcting code.We can say that an error correcting code protects against an error A if we havethat, for ρ ∈ Hlogical ,(D◦A◦E)(ρ) = ρ. (1.7)There are known necessary and sufficient conditions known as the error correctionconditions for when there exists a decoding map D which corrects a certain errorA, given the subspace Hcode.A particular class of error correcting codes which will be of interest are thestabilizer codes. A stabilizer code is defined using a stabilizer group, which isitself a subgroup of the Pauli group on n qubits. The Pauli group Gn consists of4all n-fold tensor products of Pauli operators, for instance G2 is generated by theelementsG2 = 〈I⊗X ,X⊗ I, I⊗Y,Y ⊗ I〉. (1.8)A stabilizer group is typically defined by a set of generators drawn from the Pauligroup. The stabilizer code is defined as the joint +1 eigenspace of the generatorsof the stabilizer group.We can illustrate the construction of a stabilizer code using a simple example.Our physical Hilbert space will consist of three qutrits, Hphys = H1⊗H2⊗H3 andthe stabilizer group S is generated by the elements1S1 = XXXS2 = ZZZ (1.9)with X and Z the generalized Pauli operators acting on qutrits2. We look for thespace of states which have the property Si|ψ〉= |ψ〉 for all i. One can confirm thatthe following states all share this property:|0L〉= 1√3(|000〉+ |111〉+ |222〉)|1L〉= 1√3(|012〉+ |201〉+ |120〉)|2L〉= 1√3(|021〉+ |102〉+ |210〉). (1.10)Since each of the Si|ψ〉 = |ψ〉 divide the Hilbert space into thirds, there shouldindeed be exactly 3 linearly independent states in the code subspace.The subspace defined by the span of the three states |0L〉, |1L〉, |2L〉 forms anerror correcting code which can correct against the erasure of any one of the qutrits.To see this, it is sufficient to show that there is a unitary which acts on only twoof the qubits which will output the logical qutrit. That is that there exists U12 such1As is convention in discussions of stabilizer codes, we have omitted the ⊗ between the Paulioperators. Thus ZZZ should be read as Z⊗Z⊗Z.2These have the property that X |n〉= |n−1〉 and Z|n〉= e2piin/3|n〉5thatU12⊗ I|i〉123 = |i〉1⊗|χ〉23. (1.11)with |χ〉 = 1√3(|00〉+ |11〉+ |22〉). One can readily but tediously find the matrixelements of U12 by substituting the states |0L〉, |1L〉, |2L〉 into the above relation. Bysymmetry of the stabilizers 1.9, we know that there must also exist such a U23 andU31 which recover |i〉 from the H2⊗H3 or H3⊗H1 Hilbert spaces.A key lesson to be taken from the theory of quantum error correction is thatone Hilbert space may be encoded into another larger one with redundancy, inthe sense that loss of part of the larger Hilbert space may still allow recovery of theencoded state. This is surprising, as it seems in tension with the no-cloning theoremof quantum mechanics. It is important though that despite the recoverability ofthe state after loss of part of the physical Hilbert space it is only ever possible toconstruct one copy of the logical state by acting on the physical Hilbert space. Inthe example above this plays out as the need to have a majority of the qutrits, 2 of3, to perform the reconstruction.1.3 The AdS/CFT correspondenceAt the time of writing there is only one well established theory of quantum gravitywhich realizes the holographic principle. This is the AdS/CFT correspondence;it describes quantum gravity in d + 1 dimensions in an asymptotically AdS back-ground in terms of a d dimensional conformal field theory living in Minkowskispace. We will give an outline of the correspondence here, and refer the reader tothe literature for a more complete discussion. Some pedagogical introductions canbe found in Ammon [2] and Polchinksi [37].In its strongest form, the AdS/CFT correspondence asserts the equivalence ofstring theory in an AdS background with certain conformal field theories. We willbe concerned however with a limit where the string theory is approximated byquantum fields on a classical gravity background. To be precise, the equivalence6gives thatZCFT [J] = ZAdS[Φ→ J], (1.12)where the CFT partition function is computed in the presence of some externalsources denoted J, and the AdS partition function is computed with fields subjectedto boundary conditions set by those sources. In the remainder of this section wewill unpack some of the ingredients of the equation above, and outline some neededfeatures of this correspondence.We will first briefly describe AdS space. In the context of tensor networks it isAdS2+1 which is most relevant, so to simplify our discussion we will focus on thatcase. A metric for AdS2+1 isds2 = l2AdS(−cosh2ρdt2+dρ2+ sinh2ρ dθ 2), (1.13)with ρ ∈R+, t ∈R+, θ ∈ [0,2pi], and lAdS a length scale known as the AdS radius.Any constant time slice has metric ds2 = l2AdS(dρ2+ sinh2ρ dθ 2) which is just thePoincare plane, H2. We can visualize AdS2+1 as a cylinder, as shown in figure 1.1.A second useful coordinate system for describing AdS is Poincare coordinates.These do not cover the entire AdS space, but a restricted region known as thePoincare patch. The metric isds2 =l2AdSz2(−dt2+dz2+dxµdxµ) (1.14)with 0≤ z < ∞, and t,x ∈ R.AdS space has a boundary, which we can visualize as living on the surface ofthe cylinder. In global coordinates this boundary is at ρ → ∞, while in Poincarecoordinates it is at z = 0. It is often useful to think of the boundary as the result ofa limiting procedure, where a surface is drawn at z = ε , this surface is treated asthe boundary, and then later we take ε → 0.We should also very briefly introduce the other component of the AdS/CFTcorrespondence, conformal field theories. Most relevant for us are 1+1 dimensionalconformal field theory, since these are dual according to the correspondence tostring theory in AdS2+1. A conformal field theory is a quantum field theory that7ρtθFigure 1.1: The cylinder representing AdS2+1. The boundary of the cylinderis at ρ = ∞.has a larger symmetry group than the Lorentz group. In particular, conformal fieldshave no intrinsic length scale and are left invariant under scale transformations.We will be interested in spacelike geodesics of this AdS spacetime which areanchored on two boundary points. Starting with the metric 1.13 it is not difficult toshow that these geodesics form semicircles. For later reference, we record that thelength of such a geodesic isA =lAdS2GNln(Lε)(1.15)where L is the size of the boundary interval on which the geodesic is anchored, andz = ε defines the cutoff surface we have used.It is known [12] that the entanglement entropy of a single interval in a 1+1dimensional conformal field theory is given byS(A) =c3logLε(1.16)where c is known as the central charge, which is determined by the conformal fieldwe are working with. ε is a UV cutoff indicating that modes of wavelength shorter8than ε have been excluded from contributing to the entanglement entropy.Ryu and Takayanagi [39] pointed out that the length of the boundary anchoredgeodesics in AdS space and the entanglement entropy of the same boundary inter-val agree if one takesc =3lAdS2GN(1.17)and identifies the UV cutoff in the CFT with the long distance z cutoff in AdS.Indeed, both of these identifications had already been established via other linesof reasoning, which made compelling evidence for the identification of entangle-ment entropies in CFTs with lengths in AdS. More precisely, Ryu and Takayanagiconjectured the formulaS(A) = minγAL(γA)4G, (1.18)where {γA} are all the spacelike curves anchored on region A. This was conjecturedto compute entanglement entropies in conformal fields dual to asymptotically AdSgeometries.Importantly, the Ryu-Takayanagi formula relies on their being a preferred timeslicing of the AdS spacetime. The minimal surfaces γA considered in the minimiza-tion must lie in a well chosen time slice. This must be the case, as its clear that ifthey could be chosen in any Cauchy slice of AdS with the region A on its boundaryone could always take the minimal length to be arbitrarily close to zero by makingthe slice close to lightlike. It turns out that for a spacetime with a timelike Killingvector (a static spacetime) and boundary regions which are at a constant time t0,the bulk Cauchy slice defined by t = t0 will contain the appropriate minimal curve.For spacetimes without such a timelike Killing vector, which we call dynamicspacetimes, a more involved prescription for calculating boundary entanglemententropies from bulk geometry is required. In particular the Ryu-Takayanagi (RT)formula needs to be replaced by the Hubeny-Ranganmani-Takayanagi (HRT) for-9mula, which readsS(A) = extremalγAL(γA)4G. (1.19)That is, the minimization procedure is simply replaced by an extremization. Anequivalent3 formula for computing entanglement entropies using AdS geometry isthe maximin formula, which statesS(A) = maxΣ(minγAL(γA)4G). (1.20)That is, for each bulk Cauchy slice Σ search through all possible curves γA, pickthe shortest, and call it γΣ. The maximin formula states that the length of longestof the γΣ computes the entanglement entropy of A.Finally, we return to the idea of AdS/CFT as an encoding of one Hilbert spaceinto another. From the equality of partition functions 1.12 we can understand howthe encoding of the boundary Hilbert space into the bulk Hilbert space works. Wespecify a boundary state, then solve the bulk equations of motion to determine thebulk fields. To reverse the mapping, we take limits of the bulk fields as they go tothe boundary and recover boundary data.The above gives a global prescription for determining bulk data from the bound-ary or vice versa. It is also interesting to ask, given a boundary region, what bulkregion is determined by that boundary data? In fact, we have already learned howto determine some bulk data from a subregion of the boundary. The RT formulagives us the length of a certain bulk curve γA from the data of a boundary intervalA. Restricting attention to subintervals of A, we can determine the length of a in-finite family of curves that sweep out the entire region enclosed by γA. Thus, theregion enclosed by the minimal curve γA, which we will label D(A), is a reasonablecandidate for the bulk region that can be reconstructed from A. More covariantly,the candidate bulk region is the bulk domain of dependence of D(A), which we callW (A)4. The region W (A) is known as the entanglement wedge of A.3Actually the equivalence of HRT and maximin require the assumption of the null energy condi-tion, but this is a very weak assumption.4The domain of dependence of D(A) is defined as W (A) = D(A)∪ J+(A)∪ J−(A) where J+(A)is the set of points for which all backward directed timelike curves pass though D(A) and J−(A) is10OABCabcFigure 1.2: Illustration of the error correcting property of the bulk to bound-ary mapping. No single boundary region A, B, or C is sufficient to re-construct a bulk operator living at O, but any two regions AB, BC, orCA is. This is because the minimal surface of A (for example), whichis labelled as a, does not enclose point O. However the minimal sur-face for AB, which is c, does. This is analogous to the three qutrit codeintroduced in section 1.2.To define the error correcting property more concretely, we can consider anarbitrary operator OW (A) that lives at point x inside the entanglement wedge of A,W (A). Acting with such an operator allows us to probe the bulk data, thus, ourreconstruction idea implies that there should be a dual operator OA that lives in theboundary theory and has the property thattr(ρW (A)OW (A)) = tr(ρAOA). (1.21)We refer to the above as the error correction property.It has recently been understood that the RT formula actually implies that it ispossible to reconstruct data inside the entanglement wedge [23]. Because a singlebulk point lies in the entanglement wedge of many different boundary regions,this means the bulk to boundary encoding in AdS/CFT has the structure of anerror correcting code, see 1.2 for an illustration. In the context of AdS/CFT theencoding of bulk entanglement wedge data into the boundary has recently beenunderstood [15, 18, 28]. Tensor network models were used early on to explore thedefined as the set of points for which all forward directed timelike curves pass through D(A).11entanglement wedge reconstruction idea [1, 36], and consequently played a rolein the development of the general understanding of the relation between the RTformula and quantum error correction.12Chapter 2Tensor networksIn this section we develop the basics of the tensor network formalism. We empha-size two perspectives on networks which will later have connections to holography:the idea of a tensor network as a map, and the connection between the graph struc-ture of a network and the entanglement of the state it prepares. We leave makingthe connection to holography to chapter 3.2.1 Tensor network basicsThe tensor network formalism describes graphically the pattern of contraction ofa set of simple objects to form a complex quantum state. The basic objects in thegraphical formalism are vertices with some number of lines attached. Each vertexcorresponds to a quantum state, and the lines each correspond to a ket or bra index.For instance|Ψ+〉=D∑m=1|m〉|m〉 (2.1)is represented by figure 2.1 a. We attach a direction (inward or outward) to eachline in the diagram, with inward arrows indicating ket indices and outward arrowsindicating bra indices. Thus 〈Ψ+| is represented as in figure 2.1 b.It will be convenient to write down quantum states without explicitly includingtheir basis vectors. For example, the maximally entangled state |Ψ+〉 is written13(a) (b)Figure 2.1: (a) A maximally entangled state in the Hilbert space, representedin the graphical notation. (b) A maximally entangled state in the dualHilbert space, represented in the graphical notation.δab, leaving the choice of basis implicit. In this notation ket indices are loweredand bra indices are raised, so 〈Ψ+| becomes δ ab. More generally a quantum state|φ〉=∑a,bTab|a〉|b〉 (2.2)is specified as Tab. The upper and lower indices carry transformation rules withthem. Since|φ〉= ∑a,b,cTa(U†)abUbc |c〉=∑bT ′b|b〉, (2.3)we see that lower indices transform according to Ta→ Ta(U†)ab under a change ofbasis described by |b〉 = Ubc |c〉. Similarly under the same change of basis upperindices transform according to T a→ T aUba .The basic operation of the tensor network formalism is the composition of twoquantum states. Composition of two ket states, say Tabcd and Se f gh is performed byintroducing maximally entangled bra states,Tabcd ◦Se f gh→ Tabcdδ ceδ d f Se f gh. (2.4)This is illustrated in figure 2.2. The contraction performed above was not the14ST STFigure 2.2: A basic example of a contraction of two quantum states into atensor network. Algebraically, the objects at left are written Tabcd andSe f gh. The object at right is Tabcdδ ceδ d f Se f gh.M(a)MT(b)Figure 2.3: (a) An operator M⊗ I applied to a quantum state. (b) In the casewhere the vertex represents a maximally entangled state, the operator Mcan be moved to the other subspace by taking the transpose.unique choice, since different pairs of indices could have been contracted. In gen-eral, to describe a pattern of contraction of two or more quantum states a graph isspecified. States are associated with vertices, with each line attached to a vertexrepresenting a particular index. The contraction is performed by placing maximallyentangled pairs on the edges and connecting in going and out going lines.Operators acting on quantum states we represent as vertices having both inwarddirected and outward directed lines attached, and are written as tensors with upperand lower indices, for example Mab. Diagrammatically, applying an operator to astate is given by connecting lines. The algebraic equivalent is performing the ap-propriate sum. Thus the operator Mab applied to a state Tab is represented by figure2.3 a or by MacTcb. In the case of maximally entangled states, it is straightforwardto show the identityM⊗ I|Ψ+〉= I⊗MT |Ψ+〉. (2.5)We will refer to this as the transpose rule below.Indices can be raised and lowered by contracting with maximally entangled15(a) (b)T ∗TT ∗TFigure 2.4: (a) Representation of the density matrix ρAB shown in eq. 2.6 inthe graphical notation. (b) Representation of the reduced density matrixρA in the graphical notation.pairs. In particular an operator Mab can be mapped to a state by Mab → Mab =Macδcb, and states to operators by Mab→Mab = Macδ cb. In the simplest case of astate with two indices this is also known as the Choi-Jamiołkowski mapping [29]between pure bipartite states and operators. More generally, we can raise and lowerindices on objects with arbitrary numbers of indices by appropriate contractionswith maximally entangled pairs.It is also possible to represent density matrices in a tensor network diagram.The density matrix corresponding to the state in eq. 2.2, given byρAB =∑abTab(T ∗)cd |a〉〈c|A⊗|b〉〈d|B, (2.6)is drawn as the network shown in figure 2.4 a. Contracting corresponding inwardor outward indices in the diagram performs the partial trace. We show the diagramfor ρA in figure 2.4 b.In general the contraction of two properly normalized quantum states resultsin an unnormalized output, meaning it is necessary to add a final normalizationfactor after all the contractions have been performed. For this reason we frequentlydrop any normalization factors on our initial states, for instance writing |Ψ+〉 =∑ |m〉|m〉, since this has no effect on the final state after contraction and addingproper normalization.16We recall an important bound on the von Neumann entropy of a subsystem ofa tensor network state. Suppose we have a quantum state which is written|ψ〉=∑Ti1...in j1... jn |i1〉A¯...|in〉A¯| j1〉A...| jn〉A=∑IJTIJ|I〉A¯|J〉A, (2.7)where the capital indices stand in for a set of lower case indices, and we are in-terested in the entropy of the A subsystem. If this state is described by a tensornetwork we can consider a cut γ passing through the network and separating offthe region A. Such a cut is specified by a path in the dual graph, which passesthrough a sequence of maximally entangled pairs. For each cut, there is a corre-sponding decomposition of the TIJ given by|ψ〉= ∑IJKLAIJδ JKBKL|I〉A¯|L〉A. (2.8)Now define states by ∑J BKL|L〉A = |Kˆ〉A and ∑I AIK |I〉= |Kˆ〉A¯. This gives|ψ〉=|K|∑K=1|Kˆ〉A¯|Kˆ〉A. (2.9)From this we have that rank(ρA)≤ |K|. Since the von Neumann entropy is boundedabove by the log of the rank, we haveS(ρA)≤ logdimγ, (2.10)where we define the dimension of the cut by dimγ ≡ |K|. Equality occurs when|Kˆ〉A and |Kˆ〉A¯ are orthonormal bases.2.2 Maps defined from tensor networksAny cut which partitions the network defines two tensors, call them C and C¯, whichcontract to give the boundary state. That is we can write|ψ〉AA¯ =∑IJKCIJδ JKC¯KL|IA〉|LA¯〉. (2.11)17This state can be formed by acting with the operators1C =∑IJCI j1... jn |IA〉〈 jB1 |...〈 jBn |,C¯ =∑KJC¯K j1... jn |KA¯〉〈 jB¯1 |...〈 jB¯n |, (2.12)on a collection of maximally entangled pairs. That is|ψ〉AA¯ = (C⊗C¯)n⊗i=1|Ψ+〉B¯iB¯i . (2.13)In this picture C and C¯ act as maps from an interior Hilbert space onto theboundary. There is freedom in how we choose the operators C and C¯. For example,we could form the same state by contraction with a different choice of entangledstates |Ψi〉 by writing|ψ〉AA¯ = (CΛ−1⊗C¯Λ¯−1)n⊗i=1(λ i⊗ λ¯ i)|Ψ+〉BiB¯i= (C′⊗C¯′)n⊗i=1|Ψi〉BiB¯i , (2.14)where Λ =⊗iλ i and Λ¯ =⊗i λ¯ i. Additionally, we can move an operator λ i ontothe B¯i Hilbert space using the transpose rule. We could also choose operators Λand Λ¯ which are not product. In this case we can no longer write⊗i |Ψi〉 for thestate acted on by C and C¯.The general expression for |ψ〉 without placing assumptions on the form of theprojecting state is|ψ〉AA¯ = (C⊗C¯)|Ψ〉BB¯, (2.15)where we label the Hilbert space⊗i Bi by B and⊗i B¯i by B¯. We give the graphicaldescription of this expression in figure 2.5 a. Eq. 2.15 expresses the boundary state1If the boundary state is thought of as formed by contracting two states |C〉 = ∑CIJ |I〉|J〉 and|C¯〉 = ∑C¯KJ |K〉|J〉, these are just the operators that |C〉 and |C¯〉 are brought to under the Choi-Jamiołkowski mapping.18|Ψ〉C¯C(a)|Ψ〉C¯CC¯†C†(b)Figure 2.5: (a) Graphical description of 2.15, which gives a tensor networkstate in terms of the two block tensors C and C¯ defined by a cut γ . (b)Graphical description of eq 2.16, which computes the state on a cut γ .as the output of two operators acting on a state localized to the cut γ . This suggestsa natural mapping to the cut,|γ〉BB¯ =C†C⊗C¯†C¯|Ψ〉BB¯. (2.16)This expression for the state on a cut is given graphically as figure 2.5 b.In addition to thinking of tensor networks as maps in the sense described above,earlier literature [25, 36, 38, 50] also considers tensor networks as maps from aset of “bulk” uncontracted legs to the boundary legs. To build a network of thistype, we place a tensor with n+ 1 legs on a vertex with n edges. Contraction19is performed according to the pattern of the graph as before, but now there is anextra leg associated with each vertex that remains uncontracted. It is these extrauncontracted legs that we refer to as bulk legs. A cut γ through the network nowdefines a map from the cut legs plus the bulk legs to the boundary. In the presenceof bulk legs we can think of the full tensor network as defining a state on the bulkand boundary legs, or as defining a mapping from bulk to boundary legs. We willhave use of both perspectives.2.3 MERA and entanglement renormalizationOutside of the holographic context the most prominent applications of tensor net-works are in condensed matter theory. Tensor networks have proven central to thedevelopment of efficient numerical approximation of ground states as well as toreal space renormalization techniques. These applications highlight some key fea-tures of tensor networks, which has provided and may continue to provide insightinto how tensor networks may be usefully applied in holography. For this reasonwe outline some of these applications here.An early tensor network application relates to the density matrix renormal-ization group (DMRG)2 [40]. This is a numerical technique for determining theground state of 1D systems. Very roughly, the procedure is as follows. We aregiven a Hamiltonian H which acts of a lattice with some finite number of sites.After generating some initial ansatz |Ψ〉 for the ground state, we iteratively im-prove the accuracy of this ansatz by a repeated cutting and varying procedure. Wefirst imagine splitting |Ψ〉 into two Hilbert spaces and writing it in the Schmidtdecomposition,|Ψ〉= ΣiΨi|ψi〉A|ψi〉B. (2.17)We then optimize the choice of Ψi by minimizing 〈Ψ|H|Ψ〉. Next, we move thecut that divides A and B one step over, and repeat. We continue this until reachingone end of the lattice, then turn around and sweep through the sites in the otherdirection. This procedure continues until the state |Ψ〉 is left unchanged by the2Although similar ideas can be used to do a real space renormalization procedure in 1D, the“DMRG” algorithm described here is not a renormalization procedure and the naming is unfortunate.20Figure 2.6: Illustration of the tensor network which prepares a matrix productstate. Matrix product states are the output of the DMRG algorithm.sweeping procedure.The DMRG proved highly successful at describing ground states of gappedHamiltonians. Central to this success is the DMRGs tracking of entanglement. Ateach step, the algorithm varies the Schmidt coefficients between two subsystemsA and B in order to best approximate the ground state. It was later realized thatDMRG could be usefully understood in terms of a tensor network representationtermed the matrix product state (MPS), shown in figure 2.6. The MPS containslinks between nearest neighbour sites, accounting for its suitability in describingstates with nearest neighbour interactions.At a critical point correlation lengths diverge and we expect long range entan-glement to be present in ground states. To describe a critical system efficientlywe should expect to need a tensor network with a graph structure with links be-tween sites at all distances. The MERA network[45, 46] 3 provides just such arepresentation. We illustrate the network in figure 2.7.At the uppermost layer of figure 2.7 a qubit is fed into a three index tensor,which is required to be an isometry when thought of as a map from its two upper toits two lower indices. At the next layer the outputs from the previous layer are fedinto two unitary gates. The subsequent layer is then fed into a layer of isometries,and so on. The purpose of the three legged tensors is to increase the total numberof legs, eventually building a large system. These legs also add entanglement, butsome nearby legs will be more entangled than others if a network of only thesetensors are used4 . The four index tensors add local entanglement and ensure atranslationally invariant state is prepared. In a typical application of MERA the3MERA stands for Multiscale Entanglement Renormalization Ansatz, which is again sometimesa misnomer due to the wide range of uses and perspectives taken with regard to MERA.4The reader may draw such a network and convince themselves that some neighbouring legs areconnected at the first level of the network, while others are connected many layers up.21|0i1Figure 2.7: Illustration of the MERA network, thought of as a quantum cir-cuit which prepares a state. The blue squares represent unitary matri-ces and black triangles represent isometries. Free legs attached to bluesquares are wrapped to the opposite side of the figure.choice of unitary and isometry is optimized to minimize the energy of the preparedstate.The MERA network may also be thought of as acting on some already knownstate on N sites, with each layer of the MERA reducing the number of sites to N/2.From this perspective MERA acts as a real space renormalization procedure. Thestate output after one layer of MERA is a coarse grained version of the earlier layer,with the unitary-isometry structure chosen so that entanglement at the decimatedlength scale is removed while larger scale entanglement is preserved. The Hamil-tonian also flows under this transformation, and critical points are those where theHamiltonian is invariant under this transformation. A useful review of MERA isprovided in Vidal [47].22Chapter 3Tensor networks and holography3.1 Entanglement and geometryThe Ryu-Takayanagi formula, aside from being a powerful computational tool,reveals a startling connection between entanglement in holographic CFTs and theirbulk dual gravity theories. This was emphasized early on by Van Raamsdonk [44]who considered the thermofield double state,|Ψ〉=∑ie−βEi/2|Ei〉A⊗|Ei〉B, (3.1)where A and B are the Hilbert spaces for two CFTs. Van Raamsdonk recalled thatthis state had been understood to correspond to a wormhole geometry in the bulk[32]. This is already surprising as the A and B CFTs are non-interacting and theironly relation is that they have been put in this entangled state. It seems that theentanglement between the A and B subsystems is somehow responsible for thebulk wormhole connection between the spacetimes.To get a more quantitative handle on this one can consider tuning the parame-ter β to decrease the entanglement between A and B. We can measure the entan-glement using the Von Neumann entropy of A, and relate this to the area of thewormhole neck via the RT formula. When this is done it is found that as the en-tanglement decreases the wormhole neck narrows before finally pinching off as theentanglement vanishes, leaving two disconnected spacetimes. Following on this23Figure 3.1: As the entanglement between the A and B regions in the CFT isremoved, the bulk geometry pinches and then is pulled apart.perspective of entanglement builds geometry various authors have pursued a line ofwork which takes as starting point the RT formula and tries to extract gravitationalphysics from properties of entanglement [30, 43]. This program has successfullyrecovered Einsteins equations to first and second order [20, 21], and proven severalpositive energy theorems [31].Tensor networks realize the entanglement-geometry connection in an immedi-ate way. Consider for example a quantum state which contains two unentangledsubsystems. It will be possible to prepare such a quantum state using two discon-nected networks, and interpreting the networks graph as a discretized geometry weimmediately find that the spacetime dual to the state contains two disconnectedregions, just as in the AdS/CFT example. More quantitatively, the entropy bound2.10 which readsS(A)≤ logdimγ (3.2)for tensor networks tells us that given some amount of entanglement between twosubsystems any cut through the network which divides the two regions must con-tain some minimal number of legs.It is interesting that in the tensor network picture entanglement only specifiesa minimal number of legs required. This implies the graph geometry is not totally24specified by the entanglement properties of the quantum state it prepares, rather,there are many networks with potentially distinct graph structures which preparethe same state. This has lead to the introduction of various additional requirementson the tensor network in cases where one would like to interpret it geometrically.Typically some requirement is added which ensures the bound 3.2 is saturated,allowing entanglement entropies to be determined from the graph structure alone.In section 4.1 and section 4.2 we will see two examples of this.3.2 Early efforts and constraints on holographic tensornetworksTensor network states display a connection between their graph geometry and en-tanglement properties, and so it is natural to wonder if tensor networks can be usedto build toy models of AdS/CFT, or even to define more general notions of holog-raphy than AdS/CFT. This idea was first pointed out and pursued by Swingle [42]and has been an area of active interest since.Early efforts at establishing contact between tensor networks and AdS/CFTfocused on the MERA network. This is natural as MERA had already proven itsusefulness in approximating states in a CFT. Further, Swingle noted the minimalsurfaces in MERA are similar to the AdS minimal surfaces, and bound 2.10 onthe entropy then connects these minimal surfaces to boundary entanglement. Qideveloped this proposal by including a set of bulk legs in the MERA network andconsidering the tensor network as a map between bulk and boundary Hilbert spaces[38].Another perspective taken early on was to look for general conditions con-straining all holographic tensor networks, rather than to construct a specific model.Such arguments lead Bao et al. [4] to the realization that MERA could only hopeto describe AdS geometry at lengths greater than the AdS radius. It was also foundthat there was no dimension for the bulk Hilbert space which would satisfy boththe RT formula and the Bousso bound1 for a MERA network.There were two developments which side stepped these limitations. First, some1The Bousso bound is a bound on the entropy of a certain light-sheet which generalizes theS≤ A/4G bound coming from avoiding black hole formation.25authors pursued other choices of tensor network, beginning with the networks builtfrom perfect tensors [36]. These networks have the advantage of realizing theAdS geometry in a more direct way - minimal cuts in perfect networks realizethe RT formula exactly. These also have the advantage of being translationallysymmetric and isotropic, whereas a MERA network has a directionality. However,they have the significant disadvantage that the boundary state they prepare does notapproximate a CFT state. Nonetheless these networks realize both the RT formulaand the error correction properties of AdS/CFT and have consequently proven tobe interesting toy models. We discuss them at some length beginning in section4.1.The second direction which side steps the constraints of Bao et al. [4] is toreinterpret the MERA network geometry not as the spatial geometry of AdS, butrather as de Sitter. This proposal first appeared in Beny [8], where the causalstructure of the MERA was identified as a discrete version of that appearing in deSitter space.A second development by Czech et al. [16, 17] also associated MERA with deSitter space, but beginning with a very different starting point. There, the authorsconsidered the mathematical space known as kinematic space. For any space with ameasure, we can define the associated kinematic space by parameterizing the set ofall geodesics and can equip this new space with a natural measure. Further, for thecase of AdS3 it was possible to determine a metric for the new space and identifyit as dS2. This new kinematic space turns out to encode entanglement properties ofthe CFT is a direct way. In particular, the volume of a region in kinematic space iscomputed as the conditional mutual information of a set of three boundary regions.The interpretation of the MERA network as geometric is much more natural if oneidentifies the MERA as approximating kinematic space. In fact, conditional mutualinformation in MERA is computed approximately by counting vertices in a certainregion, just as in kinematic space. Further, the causal structure identified earlier inthe MERA network can be matched to the causal structure of kinematic space.We will focus on networks which approximate spatial slices of AdS in thisthesis. However, our lack of discussion of MERA and kinematic space shouldnot be taken as representative of the importance of this work. Both the real spaceand kinematic space networks are interesting. Typically, real space models do26(a)A(b)| iC¯C|i1Figure 3.2: A tensor network which includes bulk legs. The state on a cut|Ψ〉BB¯ is associated to any path through the dual graph γ .not effectively approximate CFT ground states but can realize the error correctionproperty, while kinematic space models do the reverse. A possible exception tothis is the recent hyper-invariant models [19].3.3 The subregion isometry propertyThere are various aspect of AdS/CFT that we might want to capture in a toy model,but most tensor network literature has focused on two: the entanglement-geometryconnection, as made precise in the Ryu-Takayanagi formula, and the error correc-tion property. In fact, we can show fairly easily that both of these properties can berealized in a tensor network given that the tensor network has what we will call thesubregion isometry property.We illustrate a tensor network schematically in figure 3.2. A cut γ has beenchosen which divides the network into two regions, labelled C,C¯ and defines astate on the cut |Ψ〉BB¯ as discussed in section 2.2. There are bulk legs associatedwith the regions C and C¯, and projected into these is the bulk state |Φ〉W (A)W (A¯).Given a network with these basic components we define the subregion isometryproperty as follows:27Definition 1 A tensor network is said to have the subregion isometry propertywhen, given a cut of minimal length γA anchored on a boundary region A, thetensor network defines maps C : HW (A)B→ A and C¯ : HW (A¯)B¯→ HA¯ which are bothisometries.Much of the tensor network literature consists of methods for constructing net-works which have this property. We discuss HaPPY networks and random tensornetworks from this perspective in sections 4.1 and 4.2, respectively. For now, let usassume it is possible to construct such networks and investigate the consequences.Algebraically, the state prepared by figure 3.2 is|ψ〉=C⊗C¯(|Ψ〉BB¯⊗|Φ〉W (A)W (A¯)). (3.3)We will be interested in the reduced density matrix of a subregion A,ρA = trA¯(C⊗C¯(|Ψ〉〈Ψ|BB¯⊗|Φ〉〈Φ|W (A)W (A¯))C†⊗C¯†). (3.4)If we now use the cyclic property of the trace and that C¯†C = I we findρA =C[trB¯(|Ψ〉〈Ψ|BB¯)⊗ trW (A¯)(|Φ〉〈Φ|W (A)W (A¯))]C†. (3.5)From here we can straightforwardly see the Ryu-Takayanagi property. Sincethe von Neumann entropy is unchanged under conjugation by an isometry, we haveS(ρA) = S(trB¯(|Ψ〉〈Ψ|BB¯)+S(ρW (A))= |γA| logD+S(ρW (A)). (3.6)where |γA| represents the number of legs cut by γ , and D is the dimension of asingle leg. Typically we identify |γA| logD as the length of the cut, so thatS(ρA) = minγAL(γA)+S(ρW (A)). (3.7)Which is exactly the Ryu-Takayanagi formula, including the bulk entropy term.The subregion isometry property also leads immediately to the error correctionproperty. Recall that the error correction property can be stated precisely by the28requirement 1.21, which requires there be a boundary operator OA living on theregion A for every bulk operator OW (A) living in the entanglement wedge and suchthattr(ρAOA) = tr(ρW (A)OW (A)). (3.8)To construct the operator OA from OW (A) in a tensor network, we defineOA ≡C(OW (A)⊗ IB)C†. (3.9)Then it is a simple calculation to check 1.21,tr(ρAOA) = tr(ρAC(OW (A)⊗ IB)C†) = tr(C†ρAC(OW (A)⊗ IB)) (3.10)Now from 3.4 we have thatC†ρAC = ρW (A), (3.11)which leads totr(ρAOA) = tr((ρW (A)⊗ρB)(OW (A)⊗ IB)) = tr(ρW (A)OW (A)). (3.12)We should also note that expression 3.4 has already made use of C¯ being an isome-try, so both C and C¯ being isometries is used in the proof of both the error correctionand RT formulas.29Chapter 4Examples of real spaceholographic tensor networks4.1 HaPPY networksDue to the shortcomings of the MERA, some authors began pursuing other classesof tensor network as possible toy models of AdS/CFT. HaPPY networks [36] areone model which has been introduced, they have the property that cuts which crossa minimal number of legs saturate the bound 2.10. This gives HaPPY networks aprecise connection between entanglement and graph geometry, as they satisfy theRT formula exactly. We outline some facts about HaPPY networks in this section.The basic building block of a HaPPY network is a perfect tensor. We remindthe reader of the definition of a perfect tensor below.Definition 2 A perfect tensor is a tensor Ta1a2...a2n with an even number of indicesand having the property thatTa1...anan+1...a2n(T∗)b1...bnan+1...a2n = δ b1a1 ...δbnan , (4.1)where the an+1...a2n can be chosen to be any of the 2n legs of the tensor.30We can also raise and lower legs on the left side of 4.1, givingTa1...anan+1...a2n(T ∗)b1...bn an+1...a2n = δb1a1 ...δbnan . (4.2)This shows we can think of the perfection condition as the statement that the ten-sor defines a unitary transformation from any set of n legs to the complement.It follows by contracting indices on both sides of 4.2 that perfect tensors defineisometries from any subset of legs of size k < n to the complement. We illustratethe perfection condition in figure 4.1.To explicitly construct a perfect tensor, we can begin by thinking about thethree qutrit error correcting code discussed in section 1.2. Recall the error correct-ing code consisted of a code subspace Hcode spanned by three states |0L〉, |1L〉, |2L〉which were written explicitly in equation 1.10. To construct a perfect tensor, wetake the state|Ψ〉b123 = |0〉b⊗|0L〉123+ |1〉b⊗|1L〉123+ |2〉b⊗|2L〉123 (4.3)which one can check explicitly defines a perfect tensor. The four index tensordefined by 4.3 actually acts as the encoding map |i〉 → |iL〉. The mapping is givenbyb〈i|Ψ〉b123 = |iL〉123. (4.4)A useful operation involving perfect tensors is operator pushing. Suppose wehave a perfect tensor T and an operator O which acts on three legs. Then we canrewrite the tensor OT as TO ′ by defining O ′ = T †OT . We illustrate this in figure4.2. An operator acting on a single leg of a 2n leg perfect tensor can be pushedthrough to any n legs, but in general the operator O ′ will not act as a tensor productacross those legs.To construct a HaPPY network, perfect tensors are placed on the vertices of agraph with a non-positive curvature condition1. Reference [36] which introducedHaPPY networks does not keep track of the distinction between upper and lower1By non-positive curvature it is meant that distance (measured in number of legs cut) betweenpoints in the dual graph has no maximum away from the boundary.31T † T =Figure 4.1: Illustration of the defining condition for perfect tensors. Thesame equality must hold when any subset consisting of half the legsis contracted.=O T O ′TFigure 4.2: Illustration of the operator pushing operation. An operator O act-ing on a subset of size n of a perfect tensor with 2n legs is equivalent toan operator O ′ = T †OT acting on the complement.indices in their construction, so to translate their construction to the language usedhere we must consider a maximally entangled state being placed along every edgeof this non-positively curved graph. This done, we may perform the contraction,leaving a boundary state whose entanglement entropies saturate 2.10.In the case of HaPPY networks the length of curves through the graph is definedbyLG(γ) = log(dimγ). (4.5)Herein we will refer to this as the graph length. In section 5.1.2 we will discuss analternative notion of length in the network.The key result regarding HaPPY networks which gives them a precise entanglement-geometry connection isS(A) = minγALG(γA), (4.6)where A is a single boundary interval and the minimization is taken over cuts γA32enclosing A. To show this, the authors show that both sides of a minimal cut can beinterpreted as a unitary circuit from the cut legs and a subset of the boundary legsto the remainder of the boundary legs. We restate this result in a slightly changedlanguage as follows.Theorem 3 In a HaPPY network, a cut which is anchored on a boundary intervalA and crosses a minimal number of legs defines a map from the cut legs to theinterval A which is an isometry.We will refer to cuts of a network that define isometries on both sides as iso-metric cuts. This result allows us to calculate the state defined on a cut as givenin eq. 2.16 whenever the cut is minimal. Recalling that all of the contractions in aHaPPY network are performed with maximally entangled pairs, 2.16 becomes|γ〉BB¯ = (C†C⊗C¯†C¯)n⊗i=1|Ψ+〉BiB¯i . (4.7)When the cut is minimal theorem 3 gives C†C = I and C¯†C¯ = I, so the state on thecut is just a collection of maximally entangled pairs, with one pair for each edgecut by γ .In fact, we can straightforwardly extend theorem 3 to an if and only if statementas follows.Theorem 4 In a HaPPY network, a cut γ enclosing a single boundary intervaldefines an isometry on both sides if and only if logdimγ is minimal.Proof. That a cut being minimal implies the maps it defines are isometries is givenas theorem 3.Next we show that an isometric cut is minimal. Consider the boundary stateas written in 2.15, which corresponds to the diagram in figure 2.5a. To form thedensity matrix on a region A we draw an arrow reversed duplicate of 2.5a, andcontract the A¯ legs. Then since C¯†C¯ = I and |Ψ〉=⊗i |Ψ+〉 we are left withρA =CC†. (4.8)33To get the normalization factor note that tr(CC†) = tr(C†C) = logdimγ . Further,since CC† is a projector its non-zero eigenvalues are equal to one. Using these twofacts we have thatS(ρA) = logdimγ. (4.9)At the same time, the bound 2.10 gives that S(ρA)≤ logdimγ ′ for any cut γ ′ in thenetwork. Combining this with 4.9 we havelogdimγ ≤ logdimγ ′ (4.10)for any cut in the network. Thus any cut which defines an isometry on both sidesis minimal.As a consequence of theorem 4 the RT formula for HaPPY networks can berestated asS(A) = L(γAiso). (4.11)Here γ is any isometric cut enclosing A. We will refer to this as the isometric cutformula2.It would be useful to extend theorem 3 to the case where the network includesbulk legs so as to establish the isometric subregion property for HaPPY networks.One HaPPY network with bulk legs which appears to have this property is shownin figure 4.3. Each vertex has three planar legs and one bulk leg. The network isa tree in the computer science sense, and we can assign the root node a label of0, the vertices one edge away from the root 1, and so on. The edges may then beassigned a directionality by having them point from lower to higher label numbers.By checking cases, the reader can convince themselves that minimal cuts defineisometric mappings from bulk plus cut legs to the boundary.2As with theorems 3 and 4 the isometric cut formula is proven only for boundary regions consist-ing of a single interval.34Figure 4.3: A tree graph on which the 4 legged perfect tensor can be placedto build a tensor network. The resulting network seems to have thesubregion isometry property, as can be argued by checking cases.4.2 Random tensor networksThe key property of a HaPPY network which gave it the RT formula was thata minimal cut through the network defined isometries on both sides. However,HaPPY networks with bulk legs have not been proven to have the subregion isom-etry property, a key feature we would like in a holographic toy model. There arenow other classes of network which are known to have this property however, themost general and versatile among them being the random tensor networks whichwe now discuss.The reduced density matrix of any subsystem of a state defined by a perfecttensor is maximally mixed. In fact, we can characterize a perfect tensor as one withmaximal entanglement across any division of the Hilbert space. This is actually ageneric property: a state drawn at random according to the Harr measure will withhigh probability have nearly maximal entanglement across its subsystems. Thisexpectation becomes more and more precise as the dimension D of each subsystembecomes large.There is a tradition in quantum information theory of studying random quantumstates [14] and there are well developed techniques available for their study. Be-35cause of this, and because the closely related perfect tensors have already provenuseful, it is natural to consider tensor networks built from random tensors in theholographic context.To build a random tensor network one begins with a graph. We will give thevertices labels x, and place states |Vx〉 at each vertex. |Vx〉 is defined by a tensor withnx+1 legs, where nx is the number of edges connected to vertex x. The states |Vx〉are drawn independently at random according to the Harr measure by beginningwith a reference state |0x〉 and acting with a random unitary U .Next, maximally entangled pairs are placed on the edges of the graph and usedto contract legs. One uncontracted leg is left on each vertex which will serve asa bulk leg. We will then project some bulk state into these legs, leaving us witha network that defines a boundary state. We can write the uncontracted state as adensity matrix,ρ =⊗x|Vx〉〈Vx| (4.12)and the contraction can be expressed using the partial trace,ρ = tr(ρP(⊗x|Vx〉〈Vx|))(4.13)withρP = ρb⊗(⊗x|xy〉〈xy|). (4.14)The dimension of a leg connecting vertices x and y will be referred to as the bonddimension Dxy.It is possible to make use of random matrix techniques to calculate entangle-ment properties of the state eq. 4.12. We will only outline a few of the key stepsin doing this here and refer the reader to Hayden [25] for details. First, one canrealize that it is easier to calculate the 2nd Renyi entropy than the Von Neumannentropy when using random tensors. Later it can be argued that the Renyi andVon Neumann entropies agree for these particular states. The 2nd Renyi entropy is36defined bye−S2(ρA) =tr[(ρ⊗ρ)FA]tr[ρ⊗ρ] (4.15)whereFA is known as a swap operator and is defined byFA(|n〉A1⊗|m〉A¯1⊗|n′〉A2⊗|m′〉A¯2) = |n′〉A1⊗|m〉A¯1⊗|n〉A2⊗|m′〉A¯2. (4.16)The convenience of the Renyi entropy lies in the fact that we can average over alltensors |Vx〉 at each site before doing the projection and the trace, so thattr[(ρ⊗ρ)FA] = tr[(ρP⊗ρP)FA⊗x|Vx〉〈Vx|⊗ |Vx〉〈Vx|]. (4.17)The average over tensors can now be done explicitly,|Vx〉〈Vx|⊗ |Vx〉〈Vx|=∫dU (U⊗U)|0〉〈0|⊗ |0〉〈0|(U†⊗U†)=Ix+FxD2x +Dx. (4.18)Inserting this into 4.17 and using the result to take the average of equation 4.15,one finds that evaluating the 2nd Renyi entropy becomes equivalent to calculatingthe partition function of an Ising model with spin 1/2 variables. In the limit of largebond dimension Dxy this partition function can be evaluated by approximating it byits minimal energy configuration.Recall our notation in which HAA¯ is the boundary Hilbert space, with HA theHilbert space enclosed by some cut γ which we are considering. The bulk legs en-closed by γ form the Hilbert space HC, and those outside the cut form HC¯. Thus theentire network can be thought of as a map M : CC¯→ AA¯ or, by attaching maximallyentangled pairs to the bulk legs, as a state |ΨM〉AA¯CC¯.We would like to understand when a random tensor network will have the iso-metric subregion property, which we saw in section 3.3 leads to both the RT anderror correction properties. To do this, it is first useful to understand when the map-ping M : CC¯→ AA¯ from the full bulk Hilbert space to the boundary is an isometry.37This mapping being an isometry is actually equivalent to the state |ΨM〉AA¯CC¯ beingmaximally entangled across the CC¯ and AA¯ subsystems. The random matrix tech-niques outlined above can be used to calculate the entropy S(CC¯). Doing so, it isfound that a necessary and sufficient condition for M : CC¯→ AA¯ to be an isometryis that|Ω| logDb < |∂Ω| logD (4.19)where Ω is any region in the bulk, Db is the dimension of the bulk legs, and D isthe dimension of the planar legs.We now have the background we need to understand the isometric subregionproperty in random tensor networks. We will again look at entanglement propertiesof |ΨM〉AA¯CC¯, in particular, we can show that ifI(C : A¯C¯) = 0andS(C) = logdimC (4.20)then it follows that the map given by cutting the network along γ is an isometryfrom the cut legs and bulk legs to the boundary. Indeed, the random matrix tech-niques can be used to show the above statements are true assuming 4.19 and takinga large bond dimension limit.To see that 4.20 implies the subregion isometry property, note that I(C : A¯C¯)= 0gives that ρCC¯A¯ = ρA⊗ ρA¯C¯ and that S(C) = logdimC implies ρA is maximallymixed. ThusρCC¯A =ICdimC⊗ρC¯A. (4.21)One purification of this state is |ΨM〉CC¯AA¯, but another purification is|Ψ〉= (|Ψ+〉⊗n)CE ⊗|φ〉C¯A¯E¯ (4.22)where |Ψ+〉 are maximally entangled pairs.Now, we make use of the fact that different purifications of the same state are38(a) (b)Figure 4.4: a) Structure of the state |Ψ〉. The beige box at left represents thestate |φ〉A¯C¯E¯ . The curved black line represents the maximally entangledstate (|Ψ+〉⊗n)CE . b) Structure of the state |ΨM〉. The blue box repre-sents the isometry V : EE¯→ A discussed in text.related by isometries, in this case there must exist a map V : E¯E→ A such that|ΨM〉=V |Ψ〉=V ((|Ψ+〉⊗n)CE ⊗|φ〉C¯A¯E¯). (4.23)Figure 4.4 illustrates the structure of this state diagrammatically. We see that themap V can be identified with the map defined by γ from a set of interior legs, whichhere form the system E¯, and the bulk legs C into the boundary A. Similarly, if wecan establish that I(C¯ : AC) = 0 then it follows that the map defined by the exteriorof γ is also an isometry from cut legs and bulk legs to the boundary region A¯. Oncewe have that the two sides of the cut γ are isometries, the error correction propertyand RT formula follow immediately by the same arguments as given in section 3.3.39Chapter 5Tensor networks for dynamicspacetimesThe HaPPY networks of section 4.1 display the RT formula. In this sense thesenetworks are a toy model with features analogous to AdS/CFT. In this analogy, theboundary legs play the role of CFT degrees of freedom, and the tensor networksgraph geometry plays the role of the geometry of a bulk Cauchy slice. It is inter-esting to understand the limitations of this toy model. In particular, we know thatthe RT formula applies only to static spacetimes. For a dynamic spacetime RT isreplaced by the HRT or maximin formula, which so far our tensor network modelcontains no analogue of.In this chapter we will extend the HaPPY network models to include a net-work analogue of the maximin formula. Performing this extension forces on us achanged perspective, in particular highlighting 4.11 as the most productive way tounderstand the RT formula and connecting the tensor network picture more closelywith the perspective developed in [23].We will argue in section 5.1.1 that the definition of length used previously indiscussions of tensor networks, equation 4.5, is incompatible with any descriptionof a dynamic spacetime. From there, we go on to develop a new definition of lengthin networks based on the notion of a state on a cut developed in section 2.1, anddiscuss our tensor network model for a dynamic spacetime in 5.405.1 Length and extremal curves in tensor networks5.1.1 Lessons from the maximin formulaRecall that the maximin formula states that the entropy of a boundary region A canbe calculated asS(A) = maxΣ(minγAL(γA)). (5.1)That is consider a spacelike slice of the boundary and a subset A of this slice. Oneach spacelike surface Σ which has the chosen boundary, calculate the length ofthe minimal surface homologous to A. From the set of all those lengths choose thelargest element. The resulting length will give the entropy S(A).To translate this statement into tensor network language it is clear that we needto think of a boundary state as associated with a set of networks. Indeed, as men-tioned preceding equation 2.14, there are various ways we can modify a networkwhile preserving the boundary state. Beginning with a defining network thesetransformations give a set of networks corresponding to a single boundary state.We will explore the possibility that a set of networks generated in this way canbe searched over to calculate boundary entropies, analogous to optimization overspacelike slices in the maximin formula.However, suppose that we have decided on a set of networks and that the max-imin formula is true for these networks and their boundary state. Then the maxi-mization step of the maximin formula gives that the minimal lengths in each net-work are bounded above by the entropy,minγAL(γA)≤ S(A). (5.2)This is a key inequality restricting the possible definitions of the length L(γA) inthe network. Indeed, suppose that we took L(γ) = LG(γ), the graph length. Thenthe rank bound on the entanglement entropy given in 2.10 says thatS(A)≤minγA(log(dimγA)) = minγALG(γ). (5.3)41This is the opposite inequality to 5.2, so we have that S(A) = minγALG(γ) for allnetworks in the set optimized over. This means that every network in the set mustcontain the extremal curve anchored on A. Repeating this for each of the possibleboundary regions, we would conclude that every network in the set must containthe extremal curves for each boundary region. In a dynamic spacetime howeverno one slice should contain all of the extremal curves. Taking the graph lengththen prevents any description of the geometry of slices other than the constant timeslices of static spacetimes.We see that a requirement for describing spacelike slices of dynamic geome-tries using tensor networks is a new definition of length in networks. With a defi-nition of length in hand, one approach is to determine the extremal cut by variationover the set of networks. However, it turns out to be simpler to generalize theisometric cut formula 4.11 than to try and generalize the statement 4.6 of RT interms of minimal cuts. Indeed, a possible generalization of 4.11 is just 4.11 again,with the modification that the isometric cut can now be chosen from within a set ofnetworks. We find in the next section that there is a simple way to define L(γ) thathas reasonable geometric properties and which extends the isometric cut formulato the dynamic setting.5.1.2 A definition of length in tensor networksFrom our analysis of the maximin formula we know that the definition of lengthwill need to be changed. We claim that there is a simple way to define L(γ) whichguarantees S(A) = L(γAiso) whenever such an isometric cut exists in the set of net-works associated with the boundary state. To see this first calculate the boundarystate on A in terms of the operators defined by an isometric cut γAiso. This is mosteasily done by looking again at figure 2.5 and considering an arrow reversed du-plicate of the network in figure 2.5a. We then contract the A¯ legs and use thatD†D = I, which yieldsρA =C trB¯(|Ψ〉〈Ψ|)C†. (5.4)Since C is an isometry, we have that S(ρA) = S(trB¯|Ψ〉〈Ψ|). Next, consider thelength L(γAiso). As discussed in section 2.2 any cut has a state associated with it,42given by 2.16. In particular since γAiso is an isometric cut we have|γiso〉= |Ψ〉BB¯. (5.5)From this it is clear that defining the length as the entropy of one side of |Ψ〉 wouldcorrectly compute S(A). We prefer to write this more symmetrically as the mutualinformationL(γAiso) =12I|γ〉(B : B¯). (5.6)For a HaPPY network, the state |Ψ〉 is a product of maximally entangled pairs andthe length of a minimal cut reduces to the graph length. What about an arbitrarycut in a HaPPY network? In this case we make use of the fact that when |Ψ〉 isproduct, the length of a minimal cut becomes a sum over each leg in the cutL(γAiso) =∑i12I|γ〉(Bi : B¯i). (5.7)Thus it is natural to associate a length to each leg individually,L(γi) =12I|γ〉(Bi : B¯i). (5.8)For an arbitrary curve we can define its length to be the sum of the lengths ofeach leg, where we calculate the length of a single leg by finding an isometriccut containing that leg. In the HaPPY network this assigns all legs a length oflogdimγi.For a non-HaPPY network we can attempt to assign lengths to every leg by thesame procedure of looking at the product factors of the isometric cuts which arein that network. In general however not every leg will be part of an isometric cut,and it may not be possible to assign a length to every leg. This manner of buildingup the lengths of arbitrary cuts using the lengths of isometric cuts is reminiscent ofthe differential entropy formula [3, 26]. Extremal cuts in the differential entropyformula play the role of isometric cuts in the procedure outlined here. This isconsistent with our interpretation of 4.11 as applying to dynamic spacetimes, sincethe isometric cuts of 4.11 are playing the role of extremal curves.43As a basic check on this definition of length, we should confirm that all iso-metric cuts passing a single leg will assign the same value of length to that leg.Indeed, for any isometric cut crossing a segment γi the length of that segment isgiven by the mutual information I(B : B¯)/2 computed in the projecting state |Ψi〉,and is independent of the operators C and C¯ defined by whichever isometric cut hasbeen chosen. Further, the projecting state |Ψi〉 is fixed for a given network.We can also notice that contracting the network in a different basis has no effecton the length of a cut. This follows from the invariance of the mutual informationunder local unitaries. Finally, it would be nice to see that if a cut γ is composed oftwo segments γ1 and γ2 thenL(γ) = L(γ1)+L(γ2). (5.9)We’ll refer to this as the additivity property. The additivity property is not guaran-teed by our definition of length, but rather depends on the structure of the state |Ψ〉appearing in 5.5. For example if this state is product across each leg, that is if|γiso〉=n⊗i=1|Ψi〉, (5.10)then the length is additive at the level of individual legs. However, in other casesit may happen that entanglement is present across the Bi, in which case the lengthwill not be additive across individual legs. In the dynamic example given in sec-tion 5.2 we will allow legs which are contracted with a common vertex to shareentanglement, meaning additivity may fail at the level of a small number (in thecase there, three) legs.5.1.3 A static exampleAs an illustration of this assignment of length we look at a network which doesnot satisfy RT when the graph length is used, but does when using the mutualinformation based definition. Our example is based on the network shown in figure5.1a. The six legged tensors are perfect tensors, and the two legged tensors shownas solid black dots are maximally entangled pairs used to form the contraction.This is a HaPPY network and the boundary entropies are all given by the minimal44(a)A(b)| iC¯C|i1Figure 5.1: (a) A HaPPY network. Six legged vertices are perfect tensorsand two legged projecting pairs are maximally entangled states. Allboundary entropies are given by the graph length of a minimal cut inthe network. (b) A network which satisfies the Ryu-Takayanagi formulausing the mutual information based definition of length, but does notsatisfy Ryu-Takayanagi when using the graph length. The blue dots andlegs represent the projecting state given in eq. 5.11. The dashed linerepresents the isometric cut for the boundary region A.number of legs cut to separate off a boundary region.For our example we replace the maximally entangled pairs around the edge ofthe network with another state, |Ψi〉, which for convenience we write in the form1|Ψi〉= (O⊗ I)|Ψ+〉. (5.11)The modified network is shown in figure 5.1b. We claim that this network satisfiesthe RT formula using our new definition of length in terms of mutual information,but not using the graph length. To see this we begin by computing the entropy ofthe three boundary legs marked region A.We can go a long ways towards computing this entropy using the graphicalnotation. To do this we draw an arrow reversed copy of figure 5.1b, then contract1It is always possible to write |Ψ〉 in this way because we can write |Ψi〉 = A⊗B|Ψ+〉 and thenuse the transpose rule to move B to the other subspace.45=Figure 5.2: The basic simplification used to compute the density matrix ofregion A in figure 5.1b. The six leg tensors are from the edge of thenetwork shown in figure 5.1b. The effect of the state given in equation5.11 is to add a normalization factor, represented as the blue loop atright.all the legs in A¯. To understand what happens when this is done consider thediagram in figure 5.2. The simplification shown there gives that all the insertionsof O that are not adjacent to region A turn into pure normalization factors. Aftercontinuing the contraction we are left with the density matrix illustrated in figure5.3. As an operator expression, this isρA = T (OO†⊗ I⊗OO†)T †. (5.12)The entropy is given by:S(ρA) = S(T †ρAT ) = 2 ·S(OO†)+1 (5.13)By choosing O to be non-unitary we find an entropy less than 3 = LG(γmin), so wehave that the RT formula using the graph length fails.What is the minimal length when computed using eq. 5.6? The cut with theminimal number of legs actually defines an isometry on both sides2. This is infact what we used when showing that the reduced density matrix ρA was given by2To be precise, the A¯ side of this cut has the property C¯†C¯ = αI for a scalar α . This scalar isdivided out when the normalization is added to the state.46Figure 5.3: Graphical representation of the density matrix of the region Afrom figure 5.1 b.expression 5.12. Since both sides of the cut are isometries the state on the cut isjust given by the projecting state, which in this case is|Ψ〉= |Ψi〉B1B¯1⊗|Ψ+〉B2B¯2⊗|Ψi〉B3B¯3 . (5.14)There are two legs with operator insertions, which have lengthL =12IΨi(B : B¯) = S(OO†), (5.15)while the leg with no insertion has length IΨ+(B : B¯)/2= 1, giving L= 2·S(OO†)+1 = S(ρA). It is straightforward to check the minimal lengths and boundary en-tropies of any other region A in the network shown in figure 5.1 b agree.5.2 Dynamic tensor network statesConsider an AdS spacetime and a spacelike slice of its boundary. The maximin for-mula shows that the entanglement entropy of a given boundary region can be foundby determining the area of an extremal surface extending into the AdS spacetime.For a dynamic spacetime these extremal surfaces may lie in many different slicesof the interior. In the tensor network picture we have identified isometric cuts asthe network analogue of extremal surfaces. Further, we have suggested a set ofnetworks contracting to a single boundary state is the analogue of the set of space-like slices of the bulk spacetime. A tensor network state which is analogous to anevolving spacetime then should have isometric cuts for different boundary regions47abcfegd12(a)2abcfegd(b)abcfegd(c)abcfegd(d)1Figure 5.4: The four networks included in the set F considered in text. Theprojecting states shown in blue in (a) are defined by an operatorO actingon the maximally entangled state, these can be pushed to any subset ofthree legs, as seen in (b-d). In general the pushed through operator willnot have a tensor product structure, so the corresponding projecting statewill be entangled across three legs. This is indicated here by the thickblue line in (b-d). All four networks shown here have the same bound-ary state. The entropies of subsets of boundary legs is calculated bychoosing the network which contains an isometric cut enclosing thoseboundary legs, and applying the isometric cut formula. No one networkcontains an isometric cut for every boundary region, but the set of fournetworks together do.48=Figure 5.5: The identity used to show a cut containing one interior leg whichis blue in 5.4 a is isometric up to a normalization factor. This identity iseasily derived from that in figure 5.6.living in different networks drawn from this set.We will call this set of networks F , and specify the networks it contains bygiving a defining network N0 along with a set of allowed transformation rules.Continuing our analogy, we view these transformations as corresponding to defor-mations of the interior spacelike slices. Importantly, these transformations mustpreserve the boundary state. An example of such a transformation was given asequation 2.14.To construct examples of boundary states with a geometry corresponding to adynamic spacetime we begin with the example network of figure 5.1a and replacethree of the maximally entangled pairs which are projected into the central vertexwith the state|Ψi〉= (O⊗ I)|Ψ+〉. (5.16)This is our defining network, shown in figure 5.4a. The allowed transformationswe take to be the operator pushing operation discussed in section 2.1. This resultsin the four networks shown in figure 5.4b-d being included in the set F .It is not difficult to see that no one of these networks contains isometric cuts forevery boundary region, but the set of four together do. Consider for example thedefining network, figure 5.4 a, with O sitting on three of the interior legs (markedas blue legs). Consider first the subregion consisting of exterior legs attached tovertex f. The cut enclosing f and crossing three legs is isometric, this follows fromthe identity shown in figure 5.5. Similarly, the cut γ2 which encloses f and g canbe shown to be isometric by use of the identity in figure 5.6. Further, the boundary49=Figure 5.6: The identity used to show the cut γ1 in figure 5.4a is isometric upto a normalization factor.legs adjacent to vertices f,g,b or c,d,e are also enclosed by isometric cuts containedin the network of figure 5.4a, since a minimal cut which crosses three interior bluelegs is isometric.The need for additional networks to be included in the set F arises when weconsider subregions adjacent to two or fewer black interior legs. Take for examplethe subregion containing vertices b, c and d. The cut γ2 which crosses one blue andtwo black interior legs is not isometric, nor is the other possible cut which crossesone black and two blue interior legs. To find an isometric cut enclosing b-c-d weconsider network 5.4 b. Since all the operator insertions now live on the cut γ2 andγ2 crosses a minimal number of legs, it is isometric. Similarly, an isometric cutfor enclosing d-e-f can be found in network 5.4 and an isometric cut for e-f-g innetwork 5.4.A complication arises in considering the region containing only c, d or e, orregions f-e, e-d, d-c, c-b. Consider region d-c, the remaining possibilities are han-dled similarly. In this case an isometric cut can be found in network 5.4 b. To seethis, we must return to the notion of a cut through the network. Recall that a cut γcorresponds to a specification of projecting state, on which operators C and C¯ actto prepare the boundary state. Implicitly, choosing a cut involves specifying whichlegs of the projecting state are acted on by the operator C and which by the oper-ator C¯, corresponding to our breakdown of the projecting state Hilbert space intoHB andHB¯. To specify this in the graphical notation we can use a double line tospecify a cut through the network. One cut γ crosses the legs which are associatedwith the B Hilbert space, with a second cut γ¯ denoting the legs in the B¯ Hilbertspace. We have adopted this notation in figure 5.7 to specify an isometric cut for50Figure 5.7: Illustration of how to choose an isometric cut for the subregionconsisting of legs adjacent to the d and c vertices. The upper dashedline crosses legs included in the B¯ Hilbert space, while the lower dashedline crosses legs included in the B Hilbert space. It is straightforwardto check that both the operators above and below the dashed lines areisometries; it follows that S(A) = 12 I(B¯ : B).the region c-d. It is straightforward to show that the operators defined by this cutare isometric, from which we can conclude that the mutual information across theB and B¯ subsystems of the projecting state is equal to the entropy of the boundaryregion c-d, as needed.One advantage to the isometric cut formula is that it is not necessary to limitthe networks which are included in the set F searched over. Indeed, it is a con-sequence of our definitions that any cut which is isometric will have as its lengththe entropy of the enclosed boundary region. Thus although we specified only thefour networks given in figure 5.4 we could add arbitrary networks to the set F .Those without isometric cuts would not disturb the isometric cut formula at all,and any additional networks having isometric cuts would give unchanged valuesfor the boundary entropies.It is not difficult to construct further examples of sets of networks satisfying51the isometric cut formula based on the construction used here. Indeed, we maycontinue the pattern of contraction given by figure 5.1a and construct networkswith an arbitrary number of layers. We can then proceed to replace a subset of theprojecting maximally entangled pairs by non-maximally entangled states; allowingthe same freedom of pushing operators through adjacent tensors then gives a set ofnetworks associated with a fixed boundary state. We have not yet systematicallystudied which networks defined in this way contain isometric cuts for all possibleboundary regions; doing so remains a direction for future work.It is interesting to reconsider the dynamic picture developed earlier in the pres-ence of bulk legs. If we assume we have a tensor network with the isometric sub-region property, a nice picture emerges. We can consider evolving the bulk stateforward in time by applying an operator to a portion of its Hilbert space. When wemap this bulk state to the boundary by projecting it into the tensor network, we canpush through this time evolution operator to the interior legs of the tensor network.The time evolution operator can then be treated as the same type of deformationsto the network as were considered in this section, showing that local time evolutionleads to deforming the Cauchy slice. Since we do not have examples of HaPPYnetworks which have the subregion isometry property, one could instead use therandom tensor networks of section 4.2.52Chapter 6Holographic tensor networksaway from AdS/CFTIn this chapter we consider starting with a continuous geometry and then buildinga network which approximates it. This allows us to construct networks whichapproximate spacetimes of interest other than AdS, for instance flat space. It ispossible to do this by making use of the random tensor construction.6.1 Building a tensor network for flat spaceIn the random tensor construction it is possible to prove the RT formula when theirare no bulk legs very generally [25]. In particular, there is no condition analogousto 4.19 when no bulk legs are present - a network constructed on any graph willdisplay the RT formula. However, we should keep in mind that as with the otherentropy calculations performed using random tensor techniques the RT formula isproven for random tensors in a limit of large bond dimension.Since RT holds without restrictions on the graph geometry, we can try to con-struct networks that approximate flat space. This was already claimed in ref. [25],however, we argue that the construction given there is problematic and give an al-ternative construction. We also wish to acknowledge that our construction borrowsa technique from Bao et al. [5].We consider a disk M = {(x,y) : 0 ≤ x2 + y2 ≤ 1} which is endowed with53OABCDFigure 6.1: Illustration of our procedure for constructing a network whoseminimal lengths approximate those of a given geometry. The exampleshown constructs a network with four boundary legs which approxi-mates a disk shaped region of R2. In the first step, four boundary pointsare chosen and all of the minimal cuts anchored on those points aredrawn. The minimal cuts form a planar graph, in this example the graphhas vertices A,B,C,D and O.a distance function d(u,v). Our goal is to fill in the disk with a planar tensornetwork which satisfies the Ryu-Takayanagi formula and whose minimal surfaceshave lengths approximating the function d(u,v). We introduce a parameter δ whichrepresents a unit of length in the continuous geometry. In particular we say thenetwork approximates the geometry of the disk to a resolution of ε if∣∣∣∣d(u,v)− δlogD ·L(u,v)∣∣∣∣≤ ε, (6.1)where L(u,v) is the graph length in the network, D is the dimension of the legsin the network. The δ/ logD should be understood as a conversion factor fromgraph length (unitless) to physical length, so that δ is the length associated withcutting one leg of dimension D. Below we construct a network and show that, inthis network, given any choice of resolution ε there is a choice of δ such that 6.1is satisfied.54Our strategy is to construct a graph whose minimal cuts satisfy 6.1 and thenpopulate that graph with random tensors of large bond dimension. With randomtensors placed on the vertices, the results of ref. [25] then guarantee the Ryu-Takayanagi formula is satisfied. Let us consider as an example a disk which isa section of flat space, so d(u,v) =√(u1− v1)2+(u2− v2)2. A reasonable firstapproach is to tile the disk with a regular polygon. This can be done with triangles,squares, or hexagons. However, none of these tilings correctly reproduce lengths inthe disk in the sense of 6.1. For example in a tiling with squares, the graph lengthfunction isL(u,v) = |u1− v1|+ |u2− v2|, (6.2)which doesn’t approximate the Euclidean distance. Additionally, such a distancefunction gives highly degenerate minimal surfaces - for example a staircase shapedpath gives the same distance between two boundary points as a path which turnsonly once. Regular tilings using triangles or hexagons produce similar graph dis-tance functions and also have degenerate minimal surfaces.Our construction begins by specifying a set of points on the edge of the disk.The closeness of our approximation is set in part by the number of points on theboundary chosen, which we will denote by N. The construction proceeds by draw-ing every minimal surface between pairs of these points; this is illustrated in figure6.1. The resulting surfaces define a graph which we take to be the dual graph ofthe tensor network being constructed. Importantly, the edges in the direct graphare assigned a weighting wi set bywi = Floor(d(u j,uk)/δ ). (6.3)Up to the rounding implemented by the floor function, the weight of the edges inthe direct graph is given by the length of the edges in the dual graph which theycut, measured in units of δ . Forming the direct graph from the dual graph andassigning the weightings is illustrated in figure 6.2.Finally, random tensors are placed on the vertices of the direct graph and thenumber of legs along an edge is chosen to be equal to the weighting wi associated55OABCDSTUVWXYZFigure 6.2: Illustration of the second step in our procedure for constructinga network which approximates a given geometry. In this step, the dualof the graph formed in step one is drawn. The edges of the dual graphare assigned a weight based on the R2 length of the edges in the directgraph they cut. For example, a weight of Floor( ¯AB/δ ) is assigned to theedge TY , where δx is a parameter with units of length controlling howclosely the graph approximates lengths in the disk.with that leg. We can then show that the resulting tensor network has lengths whichsatisfy 6.1. To prove this, note that a cut in the tensor network is also a path in thedual graph. We will consider a minimal cut passing from u0→ uN where the ui arevertices in the dual graph. Consider one segment of that path which passes from uito u j. The length of this segment is given byL(ui,u j) = (number of legs crossed) logD. (6.4)The number of legs crossed is just wi, which is the length of that segment of the56path given in units of δ ,L(ui,u j) = Floor(d(ui,u j)δ)· logD (6.5)Inserting this into 6.1 gives that∣∣∣∣d(ui,u j)− δlogD ·L(ui,u j)∣∣∣∣≤ δ . (6.6)The number of boundary points chosen, N, sets the maximal number of segments ina minimal cut through the disk, which we call f (N)1. Then the triangle inequalitygives that for a minimal path through the networkd(u0,un)≤ δ f (N). (6.7)A network with resolution ε then can be constructed by choosing δ = ε/ f (N).It would be interesting to explore further the properties of the flat space networkconstructed here, and to understand if other flat space network constructions arepossible. For example, the network here is far from being translationally invariant,and it is interesting to ask if a translationally invariant flat space network can beconstructed. This seems unlikely, as translational invariance requires we use one ofthe regular tilings with triangles, squares, or hexagons, which produce non-uniqueminimal surfaces and distance measures which do not approximate flat space. Onepossibility however is to use a random tiling. In this case translational invaraince isrestored at large enough distance scales, but minimal surfaces in the random tilinghave some hope of being unique.It would also be interesting to understand if the flat space network constructedhere, or a possible random tiling, could be upgraded to have bulk legs and to havethe isometric subregion property. More generally, the construction here allows us totake any geometry and use its minimal surfaces to construct a corresponding tensornetwork. We can thus ask for any geometry about its properties as a mapping.1A chord AB divides the points C,D... on the circles edge into two sets of size n1 and n2 wheren1 + n2 ≤ N− 2. Since every pairing of such points gives a chord which crosses AB once we canbound the number of cuts through AB by ((N−2)/2)257Chapter 7Final remarksOne of the fundamental puzzles of quantum gravity concerns the exact relationshipbetween geometry and entanglement. The tensor network has the basic advantageof giving an immediate relationship between these two apparently distinct ideas,at least at a discrete level. The work presented in this thesis regarding dynamicspacetimes strengthens this connection by showing tensor networks can capture notjust the geometry of a special class of spacelike slices (constant time slices in static,asymptotically AdS spacetimes), but actually capture features of the geometry ofarbitrary Cauchy slices in any asymptotically AdS spacetime.From this perspective, a natural question poses itself: how can timelike direc-tions be represented in a tensor network or similar formalism? Since in the dynamictensor network picture it is the mutual information which naturally defines lengths,this seems connected to a standing question in quantum information theory of howto define a mutual information between a Hilbert space at a particular time and thesame Hilbert space at a later time.Another direction the tensor network-geometry-entanglement connection mightbe pursued is towards a continuum picture. It would be interesting to under-stand how to take the continuum limit of a tensor network, or to use tensor net-work ideas to inspire the development of microscopic models that incorporate theentanglement-geometry connection naturally. Two active directions in this area arethe development of continuous-MERA (cMERA) [22] and a new perspective onviewing Euclidean path integrals as continuum limits of tensor networks [13].58As we began to address in section 6, the tensor network gives a way to thinkabout entanglement and geometry outside of the context of AdS/CFT. As seen per-haps most explicitly in the random tensor models their can be bulk to boundarymappings where the bulk is not hyperbolic. This presents the possibility of toymodels for flat space holography, a direction we feel has not yet been satisfacto-rily explored. It would be interesting to understand if the mapping from bulk toboundary defined by a flat space network can ever be an isometry.Finally, we mention a few other directions which are being pursued in recentholographic tensor network literature. First, an alternative approach to understand-ing the dynamics of tensor networks has appeared [35]. The approach taken therefocuses on extending the symmetries of a HaPPY network to include time trans-lation. Nicely, they are able to show that the symmetry group of their HaPPYnetwork forms a group which had been previously understood to approximate theconformal group. The approach given in this thesis has the advantage of displayinga network analogue of the maximin formula. It would be interesting to understandbetter the connection between these two approaches.Suggestions have been made that complexity of a boundary CFT state be dualto the volume [9] or action [10] of a certain bulk region. This was initially inpart inspired by the idea of a tensor network, which can also be understood as aquantum circuit, as building up spacetime. Indeed, the correct tensor network todescribe a geometry may be related to the efficiency of that network in building theboundary state, as suggested by the MERA, which makes the complexity-volumeproposal natural. A clear picture of how time evolution works in tensor networkswould be useful in elucidating this potential connection.59Bibliography[1] A. Almheiri, X. Dong, and D. Harlow. Bulk locality and quantum errorcorrection in ads/cft. Journal of High Energy Physics, 4(2015):1–34, 2015.→ pages 12[2] M. Ammon and J. Erdmenger. Gauge/gravity duality: foundations andapplications. 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