Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT
PPrepared for submission to JHEP
Beyond toy models:distilling tensor networks in full AdS/CFT
Ning Bao a Geoffrey Penington b Jonathan Sorce b Aron C. Wall c a Berkeley Center for Theoretical Physics, University of California, 366 Le Conte Hall, Berkeley,CA 94720-7300, U.S.A. b Stanford Institute for Theoretical Physics, Stanford University, 382 Via Pueblo Mall, Stanford,CA 94305-4060, U.S.A. c Centre for Mathematical Sciences, Cambridge University, Wilberforce Rd, Cambridge CB3 0WA,U.K.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We present a general procedure for constructing tensor networks that accu-rately reproduce holographic states in conformal field theories (CFTs). Given a state ina large- N CFT with a static, semiclassical gravitational dual, we build a tensor networkby an iterative series of approximations that eliminate redundant degrees of freedom andminimize the bond dimensions of the resulting network. We argue that the bond dimen-sions of the tensor network will match the areas of the corresponding bulk surfaces. For“tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non-intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a versionof one-shot entanglement distillation in the CFT. Using the known quantum error correct-ing properties of AdS/CFT, we show that bulk legs can be added to the tensor networksto create holographic quantum error correcting codes. These codes behave similarly toprevious holographic tensor network toy models, but describe actual bulk excitations incontinuum AdS/CFT.By assuming some natural generalizations of the “holographic entanglement of pu-rification” conjecture, we are able to construct tensor networks for more general bulkdiscretizations, leading to finer-grained networks that partition the information content ofa Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of sucha tensor network must be set larger than the string/Planck scales, we expect that it canbe chosen to lie well below the AdS scale. However, we also prove a no-go theorem whichshows that the bulk-to-boundary maps cannot all be isometries in a tensor network withintersecting Ryu-Takayanagi surfaces.
ArXiv ePrint: a r X i v : . [ h e p - t h ] D ec ontents Introduction
Among the most striking predictions of the AdS/CFT correspondence [1] is that the entan-glement structure of a holographic CFT state is encoded in the geometry of its semiclassicalgravitational dual. This conjectured correspondence is made precise for static spacetimesby the Ryu-Takayanagi (RT) formula [2, 3], and for dynamical spacetimes by the Hubeny-Rangamani-Takayanagi (HRT) formula [4]. These formulas relate the entanglement entropyof a CFT subregion to the area of an extremal surface in the bulk whose boundary coincideswith that of the subregion, and were derived by path integral arguments in [5, 6].This apparent holographic relationship between geometry and entanglement led to theproposal that tensor networks, which were originally developed as tools for the numericalanalysis of condensed matter systems with restricted entanglement structure, might bea good toy model for the AdS/CFT correspondence [7, 8]. Tensor networks represent aquantum state on a D -dimensional lattice as a contraction of tensors lying on a D + 1-dimensional graph with the lattice as its boundary, naturally disentangling the boundarystate into a geometric “bulk” representation.Since entanglement in the boundary state of a tensor network is related to the ge-ometry of its bulk graph, tensor networks display holographic properties that are at leastsuperficially similar to those of AdS/CFT. For example, all tensor networks follow a versionof the Ryu-Takayanagi formula, in the sense that the entanglement entropy of a boundarysubregion is bounded above by the “minimal area” bulk graph cut sharing a boundary withthat subregion [7]. For a large class of tensor networks, this “Swingle bound” is exactly orapproximately saturated [9, 10]. Many tensor networks also display features of quantumerror correction [9, 11, 12], which are expected to appear in the AdS/CFT correspondence[13]. On the other hand, these models typically feature a flat (or almost flat) entangle-ment spectrum, which is at odds with known entanglement features of AdS/CFT (for morediscussion on this point, see [10]).In much of the existing literature, the idea that AdS/CFT can be explained in termsof tensor networks is taken “seriously, but not literally.” Holographic tensor networksare generally regarded as toy models for AdS/CFT that provide some intuition for howthe geometric structure of a spacetime is encoded in the entanglement of its boundarydual. Even when a tensor network is interpreted literally as an approximate holographicdescription of a CFT state, it is often assumed that each tensor in the network mustrepresent a volume of space that is at least of order (cid:96) d − AdS (where (cid:96)
AdS is the characteristicscale of the semiclassical bulk spacetime, and d is its spacetime dimension). This conclusion, however, is fundamentally at odds with the notion that tensor net-works can be used to understand the Ryu-Takayanagi formula. Since the Ryu-Takayanagiformula is believed to hold exactly up to quantum and stringy corrections, it cannot bedescribed adequately by a model that displays only AdS-scale locality. Any true tensor-network explanation of the Ryu-Takayanagi formula must therefore involve tensor networkmodels that describe spacetime accurately at sub-AdS scales (so long as those scales remainabove the string and Planck scales). For a previous effort to construct tensor networks below the AdS scale, see [14]. – 2 –he Ryu-Takayanagi formula is not the only geometric formula for an information-theoretic quantity that is claimed to hold below AdS scales. Progress in understanding theholographic properties of tensor networks has been accompanied in recent years by a seriesof increasingly bold conjectures regarding the formal relationship between entanglementand geometry in AdS/CFT (e.g., the holographic entanglement of purification conjecture[15, 16], which will play a central role in this paper). While many of these conjectures arepartially inspired by tensor network models, they too are supposed to hold up to quantum(and stringy) corrections — well below the scale of existing tensor network models.In this paper, we take literally the idea that there exists an approximate tensor networkdescription for any holographic state, for essentially any discretization of the bulk, so longas the Planck and string scales are small compared to the discretization scale. As opposedto previous tensor network constructions, where a tensor network toy model is first definedand then shown to have properties similar to AdS/CFT, here we start with a holographicstate in full AdS/CFT and construct a tensor network that describes it with high accuracy.Given a holographic CFT state with a static semiclassical dual, and assuming certainconjectures that naturally extend the holographic entanglement of purification conjecture,we provide an explicit procedure to construct a network that is “geometrically appropriate”for the holographic state in the following sense:(i) it approximately reproduces the original CFT state on its boundary, and(ii) it has the same geometric features as the bulk dual to leading order in N .Our networks include subleading fluctuations around a flat entanglement spectrum, whichwe interpret as corresponding to fluctuations of the areas of extremal surfaces in fullAdS/CFT. As a result, our constructions do not suffer from the usual issue of a flat,non-physical entanglement spectrum. If our assumptions hold up to quantum and stringy corrections (as is commonly as-sumed for the Ryu-Takayanagi formula and the holographic entanglement of purificationconjecture), then essentially any discretization of a bulk geometry gives a correspondingtensor network description of the boundary state at sufficiently large N (and strong cou-pling). The tensor network description of the AdS/CFT correspondence would thereforebe valid even at sub-AdS scales, and could be interpreted not as a “toy model” but as agenuine description of the quantum gravitational theory.When the chosen discretization of the bulk geometry is constructed using only non-intersecting minimal surfaces, we are able to prove the existence of a corresponding tensornetwork, whose graph will always form a tree, without resorting to the holographic entan-glement of purification. The construction relies only on the Ryu-Takayanagi formula forthe von Neumann entropy and its extension to more general formulas for holographic R´enyientropies in [17]; it is only when we extend our construction to finer-grained discretizations(and hence tensor networks with loops) that we require the use of entropies of purifica-tion. Regardless of the particular discretization in question, all of our procedures involve Our tensor networks will not have the same R´enyi entropies as the AdS/CFT states from which theyare constructed, nor do they need to in order for condition (i) to be satisfied. As mentioned in Section 3.1,the R´enyi entropies are very sensitive to small changes in a state that do not alter its physical properties. – 3 – process of disentangling and removing as many boundary degrees of freedom as possibleto simplify the entanglement structure of a holographic state without altering its physicalproperties. In this sense, our constructions form a systematic approach for building tensornetworks that describe their corresponding holographic states with maximal efficiency.In Section 2, we review the fundamental principles of tensor networks, introduce theabstract index notation that will be used in the remainder of the paper, and explain thebasics of entanglement distillation from the tensor network perspective. We also sketchan important, but unpublished, result due to Hayden, Swingle, and Walter [18] on one-shot holographic entropies that allows us to apply the general entanglement distillationprocedure in the holographic context. In Section 3, we then identify a large class of tensornetworks, which we call “tree tensor networks,” that can be constructed rigorously fromholographic CFT states using one-shot entanglement distillation. In Section 4, we showhow these constructions can be adapted to produce holographic quantum error correctingcodes. In Section 5, we review the holographic entanglement of purification conjecture,and show that it can be used to improve the granularity of our networks by localizing theinformation contained within a single Ryu-Takayanagi surface. In Section 6, we explainhow natural generalizations of the holographic entanglement of purification conjecture canbe used to extend this procedure to produce even finer-grained tensor networks, wherethe discretization scale of the network may lie well below the AdS scale so long as itexceeds the string and Planck lengths. In Section 7, we discuss quantum fluctuations ofthe spacetime geometry and propose an interpretation in terms of quantum superpositionsof tensor networks, which require significantly fewer degrees of freedom to describe thana full, “fluctuating” geometry. We identify an uncertainty relationship between the areasof intersecting Ryu-Takayanagi surfaces, and prove a related no-go theorem that limitsthe kinds of bulk-to-boundary isometries that can be obtained in a tensor network thataccurately reproduces the geometry of AdS/CFT. Finally, in Section 8, we summarize ouressential results and present several potential avenues for future work.
A pure state | ψ (cid:105) on a multipartite Hilbert space H = H A ⊗ · · · ⊗ H A n may be thought ofas a tensor with n (abstract) up-indices, each one corresponding to a tensor factor of H .In the tensor interpretation, we write such a state as | ψ (cid:105) ↔ ψ A ...A n . (2.1)Such a tensor can generally be written as an outer product and contraction of other tensors,each of which acts only on some subset of the tensor factors A through A n . For n = 2, forexample, a state might be written as ψ A A = P A BC Q A C B , (2.2)– 4 –here the indices B and C correspond to some auxiliary Hilbert spaces H B and H C (andtheir dual spaces H ∗ B and H ∗ C ), defined solely for the purpose of constructing tensors P and Q . Note that up-indices always refer to vector spaces, while down-indices refer to theircorresponding dual spaces.While it is always possible to find some outer product representation of any given ten-sor, there are significant computational advantages to finding one in which the contractedHilbert spaces have small dimension compared to the physical Hilbert space factors. (Thesecontracted Hilbert spaces are often referred to as “bonds,” with their dimensions referredto as the “bond dimensions” of the network.) Even in cases where the bond dimension ischosen to be of the same order as the physical Hilbert space dimension, many physicallyinteresting states have outer product representations with some particular restricted struc-ture that allows them to be simulated efficiently on a classical computer (see, e.g., [19] and[20]).From the perspective of holography, one advantage of such an outer product repre-sentation of a quantum many-body state is that it has a natural geometric interpretationthat shares features with holographic spacetimes. This geometric interpretation is calleda tensor network . A tensor network is constructed from an outer product representationof a quantum state by drawing a vertex for each tensor, with one edge for each of itsindices. In our convention, edges corresponding to up-indices will be labeled with arrowsthat point away from the vertex, while edges corresponding to down-indices will be labeledwith arrows that point toward the vertex. Contractions are denoted by connecting thecorresponding edges. As a simple example, the tensor network corresponding to the statein equation (2.2) is given in Figure 1. P QA A BC Figure 1 : A simple network for the state given in equation (2.2). Outward-pointing ar-rows denote up-indices, inward-pointing arrow denote down-indices, and arrows connectingtensors denote contractions.An important connection between tensor networks and quantum information theoryarises from the fact that each bond in a tensor network can be thought of as a projectiononto a maximally entangled state in the bond Hilbert space. For concreteness, considerthe tensor network representation ψ AB = P Aγ Q Bγ (2.3)of a state on the bipartite Hilbert space H A ⊗ H B . The tensor network is constructed bycontraction over some bond Hilbert space H γ . The inner product on this bond Hilbert space– 5 –elects a preferred maximally entangled state | φ (cid:105) on the product space H γ ⊗ H γ , where H γ is the complex conjugate vector space to H γ . Since the state is maximally entangled,tracing out either of its tensor factors yields the identity operator on the remaining tensorfactor (up to a normalization factor). In index notation, this statement is simply φ γγ φ ∗ γ (cid:48) γ = 1 d δ γγ (cid:48) , (2.4) φ γγ φ ∗ γγ (cid:48) = 1 d δ γγ (cid:48) , (2.5)where φ ∗ denotes the tensor corresponding to the dual state (cid:104) φ | . Equations (2.4) and (2.5)imply that φ and φ ∗ can be used to raise and lower indices between H γ and H γ . Since φ ∗ can be obtained from φ by lowering its indices, we will generally drop the asterisk and referto the tensors as φ γγ and φ γγ , respectively.Using φ to raise and lower indices, the state in equation (2.3) can be rewritten as ψ AB = P Aγ Q Bγ φ γγ , (2.6)which is a slightly different tensor network representation of the same state. In the familiarbra-ket notation, this corresponds to projecting a state | P (cid:105) ∈ H A ⊗ H γ and a state | Q (cid:105) ∈H B ⊗ H γ onto a maximally entangled state | φ (cid:105) ∈ H γ ⊗ H γ , i.e., | ψ (cid:105) = (cid:104) φ | ( | P (cid:105) ⊗ | Q (cid:105) ) . (2.7)A tensor network in which every bond takes this form is sometimes referred to as a projected entangled-pair state (PEPS) network (see, e.g., [10]), though the term PEPS ismore commonly used to refer to a more highly-restricted class of tensor networks on a(usually square) lattice where each tensor has a single uncontracted physical index [21]. Toavoid confusion, we refer to a tensor network with bonds of the form (2.7) as a projectionof entangled pairs (PEP). Note that any tensor network can be rewritten in this form byraising and lowering indices with the appropriate maximally entangled state on each bond.The original tensor network given by equation (2.3) is drawn in Figure 2a, and itsequivalent PEP network in Figure 2b. Since PEP networks manifestly represent bonds asmaximally entangled states, it is generally useful to consider them when drawing connec-tions between tensor networks and AdS/CFT, where geometric features of a semiclassicalspacetime correspond directly to entanglement features of its boundary dual.Particular classes of tensor networks are known to reproduce various features of AdS/CFT[9, 10], such as a version of the Ryu-Takayanagi formula and holographic quantum er-ror correction. In this paper, we work backwards, beginning with these known featuresof AdS/CFT and using them to produce tensor networks with corresponding properties.Much of our protocol is reliant on the procedure of entanglement distillation , which showshow the entanglement between subregions of some physical state can be distilled out ofthe state in the form of a large number of EPR pairs (which will, ultimately, become the Formally, the inner product on H γ is a bilinear map L : H γ × H γ → C , or, equivalently, a tensor in thespace H ∗ γ ⊗ H ∗ γ . Since the inner product is nondegenerate by assumption, it has an inverse tensor L − on H γ ⊗ H γ , which can be shown to be a maximally entangled state on the tensor product Hilbert space. – 6 – QA Bγ (a)
P φ QA Bγ γ (b)
Figure 2 : Tensor networks for equations (2.3) and (2.6). (a) A tensor network on abipartite system with a single contraction. (b) A PEP network for the same state createdby replacing the contraction with a maximally entangled state.maximally entangled bonds of a PEP-style network). This procedure is the subject of thefollowing subsection.
Consider a state | ψ (cid:105) in a CFT with large central charge that is known to have a static,semiclassical gravitational dual. If the domain of the CFT is partitioned into connectedregions A and A c , then the Ryu-Takayanagi formula states that the entanglement entropyof | ψ (cid:105) between A and A c is given to leading order in the gravitational constant G N by thearea of the minimal codimension-2 bulk surface anchored on ∂A and homologous to A , i.e., S ( ψ ( A ) ) = S ( ψ ( A c ) ) = min γ,∂γ = ∂A area( γ )4 G N + O (1) . (2.8)Implicit in this statement is a simultaneous regularization procedure where an ultravioletcutoff is chosen in the CFT alongside a matching radial cutoff in the bulk spacetime [2]. The Ryu-Takayanagi formula encourages us to think of the information encoded inthe entanglement spectrum of A and A c as lying physically on the extremal surface in thebulk that partitions A and A c . For example, the Ryu-Takayanagi formula is sometimesinterpreted as counting the number of “bit-threads” of entanglement, each of which occupya Planckian area 1 / G N of the extremal surface [24]. One way to more concretely justifythis intuition comes from entanglement distillation , which makes precise the statement In fact, it suffices to consider states whose semiclassical gravitational duals contain a moment of time re-flection symmetry around the Cauchy slice being considered. This assumption is crucial to our construction,however, in Section 8, we discuss the possibility of lifting this restriction in future work. That the subleading corrections to the Ryu-Takayanagi formula are O (1) in G N was established in [22]and [23]. – 7 –hat entanglement entropy measures the number of qubits (or, more precisely, “ebits”) ofentanglement shared between a region and its complement.Entanglement distillation is the procedure by which a large number m of copies of somebipartite quantum state | ψ (cid:105) ∈ H A ⊗ H A c can be converted into a large number n of Bellpairs with some fixed asymptotic ratio n/m ≈ S ( A ) / ln (2). For our purposes, the mostuseful formulation of this principle is that for large m , the state | ψ (cid:105) ⊗ m can be expressedwith high fidelity as | ψ (cid:105) ⊗ m ≈ ( V ⊗ W ) √ D e S ( A ) m − O ( √ m ) (cid:88) i =0 | ii (cid:105) ⊗ e O ( √ m ) (cid:88) j =0 √ p j | jj (cid:105) , (2.9)where V and W are isometries that embed Hilbert spaces of size e S ( A ) m − O ( √ m ) and e O ( √ m ) ,along with their complex conjugate Hilbert spaces, back into the physical space H A ⊗H A c . (See, e.g., [25] for further details.)The first factor of the tensor product in equation (2.9) is just a maximally entan-gled Bell state on O ( n ) qubits, while the second term lives in a Hilbert space of subleadingdimension in the asymptotic entanglement entropy n ln (2) = mS ( A ) . The fact that asymp-totically many copies of | ψ (cid:105) can be approximately represented as a number of Bell pairsdetermined by the entanglement entropy gives partial justification for thinking of S ( A ) asa measure of the number of degrees of freedom entangled between A and A c , which in turnencourages us to think of all the information in the entanglement spectrum of a subregionof a holographic state as living physically on its Ryu-Takayanagi surface.Using the notation of Section 2.1, equation (2.9) is a tensor network of the form ψ A (1) A c (1) ...A ( m ) A c ( m ) ≈ V A (1) ...A ( m ) γf W A c (1) ...A c ( m ) γf φ γγ σ ff , (2.10)where φ γγ is the maximally entangled state on O ( n ) qubits and σ ff is the leftover stateon a Hilbert space of subleading dimension.We see from equation (2.10) that entanglement distillation can be used to constructa simple tensor network that reproduces | ψ (cid:105) ⊗ m with high fidelity for large m and anygiven quantum state | ψ (cid:105) . We could easily apply this procedure to a holographic CFT state(or, indeed, a non-holographic CFT state) and obtain a tensor network for the productstate | ψ CFT (cid:105) ⊗ m . In the holographic case, this tensor network will respect the geometry ofAdS/CFT in the sense that its internal bond dimensions are inherited from the areas ofthe Ryu-Takayanagi surfaces.Of course, in quantum gravity, people do not generally consider a large number of copiesof a single holographic state. Any hope of understanding the entanglement structure of a In most of the original literature, “entanglement distillation” is used to refer to the general case of amixed state ρ ∈ S ( H A ⊗ H A c ), while “entanglement concentration” is used in the special case where ρ ispure. Here, we use the terms interchangeably. In the literature, the word “isometry” is commonly used to refer to a Hilbert space map V satisfying V † V = . These maps are not generally isomorphisms in the mathematical sense, i.e., they are not invertible,unless the domain and target spaces have the same dimension. This expression follows from a procedure of smoothing and binning the entanglement spectrum of | ψ (cid:105) ⊗ m in a way that is analogous to the procedure for a holographic state detailed in Section 3. – 8 – ingle holographic state using the tools of quantum information theory is inhibited by thefact that almost all operational interpretations of a state’s von Neumann entropy involvean asymptotic number of copies of the state in question. For our purposes, therefore, itis instead useful to consider “one-shot” or “smooth” entropies, which determine how wellprocedures like entanglement distillation can be carried out using only a single copy of astate. Luckily, the one-shot entropies of holographic states are highly constrained by thelarge central charge of the CFT, which is the focus of the following section. If the von Neumann entropy only has a physically meaningful interpretation in the limitof asymptotically many copies of a state, then it might seem peculiar that the holographicdictionary relates the von Neumann entropy of a single reduced state to the (physicallysignificant) area of a corresponding Ryu-Takayanagi surface. The resolution to this puzzle isquite simple: as we shall see, the semiclassical limit of large N (or, equivalently, G N → ρ are the max-entropy S max = log (rank( ρ )) (2.11)and the min-entropy S min = log( λ − ( ρ )) , (2.12)where λ max ( ρ ) is the largest eigenvalue of ρ . These quantities agree with the von Neumannentropy for a state with a flat probability spectrum, but generically differ for arbitrarydensity matrices while satisfying S max ≥ S ≥ S min . The max- and min-entropies can beinterpreted as R´enyi entropies S α = 11 − α log Tr( ρ α ) (2.13)in the limits α → α → ∞ respectively. (Note that the von Neumann entropy S isgiven by the R´enyi entropy in the limit α → ε , it is more natural for our purposes (and in many similar situations) to considerthe smooth max-entropy S ε max = min (cid:107) ρ − σ (cid:107) <ε log (rank( σ )) (2.14)and the smooth min-entropy S ε min = max (cid:107) ρ − σ (cid:107) <ε log( λ − ( σ )) . (2.15)In other words, we consider the minimum max-entropy (respectively the maximum min-entropy) of any state σ lying within a ball of radius ε around the state ρ , where we haveused the trace norm (cid:107) ρ − σ (cid:107) = Tr (cid:112) ( ρ − σ ) † ( ρ − σ ) as a metric on the space of densitymatrices. – 9 –or holographic theories, the smooth min- and max-entropies are expected to agreewith the von Neumann entropy to leading order in 1 /G N . This was shown in [26] for singleintervals in ground and thermal states of 1 + 1-dimensional holographic CFTs (where thedensity of states of the modular Hamiltonian may be computed explicitly), and in [18] forarbitrary regions in holographic theories of arbitrary dimension. Somewhat counterintu-itively, this generic behavior of the smooth min- and max-entropies follows from the factthat in holographic theories, the R´enyi entropies are given to leading order by S α = s α G N , (2.16)where s α is independent of G N but depends non-trivially on α and is related to the areas ofsurfaces in particular backreacted geometries [17, 28]. Since [18] remains unpublished, weinclude here a simplified version of the derivation of the smooth min- and max-entropiesfor holographic states, which follows on very general grounds from (2.16). Let K = − log( ρ ) be the modular Hamiltonian corresponding to ρ . Let { E i } be theeigenvalues of K , with density of states defined by D ( E ) ≡ (cid:88) i δ ( E − E i ) . (2.17)Then the partition function Z ( α ) = (cid:90) ∞ d E D ( E ) e − αE (2.18)is related to the R´enyi entropies by e (1 − α ) S α = Tr ( ρ α ) = Z ( α ) . (2.19)Hence e (1 − α ) S α is the Laplace transform of D ( E ) and thus the density of states is given by D ( E ) = IL ( e (1 − α ) S α )( E ) = (cid:90) C d α e αE e (1 − α ) S α , (2.20)where IL ( e (1 − α ) S α ) is the inverse Laplace transform of e (1 − α ) S α and C is a contour parallelto the imaginary axis with sufficiently large positive real part. If S α has the form given in(2.16), then D ( E ) can be evaluated by a saddle point approximation for sufficiently small G N . To leading order, we must therefore find D ( E ) = e f ( G N E ) /G N + o (1 /G N ) (2.21)for some function f ( G N E ) that can be found by evaluating the exponent in (2.20) atthe saddle point. If we substitute this expression for the density of states back into the For discussion of smooth max-entropies in general quantum field theories, see [27]. While this article was in preparation, a pair of articles [29, 30] appeared with a similar analysis of theR´enyi entropy spectrum; we will discuss their proposal that tensor networks correspond to area-eigenstatesin more detail in section 7.1. In this case, it suffices to take real part greater than or equal to one, and probably greater than orequal to zero. (Assuming the modular Hamiltonian has no maximum temperature state.) – 10 –xpression given in equation (2.18), and substitute E (cid:48) = EG N , we find that the trace of ρ can be written as Z (1) = (cid:90) d E D ( E ) e − E = (cid:90) d E (cid:48) e ( f ( E (cid:48) ) − E (cid:48) ) /G N )+ o (1 /G N ) . (2.22)Note that the coefficient of G N that would appear from substituting E (cid:48) for E in the measureof the integral has been absorbed into the subleading corrections of order e o (1 /G N ) .By the same argument given above for the density of states, we find that the integralfor Z (1) will be dominated at small G N by the leading saddle point E (cid:48) . We can thereforeapproximate the integral to within any arbitrarily small precision ε by integrating over arestricted range of eigenvalues of the modular Hamiltonian, constraining E to lie in therange E (cid:48) /G N − O ( (cid:112) log (1 /ε ) /G N ) < E < E (cid:48) /G N + O ( (cid:112) log (1 /ε ) /G N ) . (2.23)Since E (cid:48) is the solution to the saddle point equation f (cid:48) ( E (cid:48) ) = 1, it is independent of G N .The error in approximating this integral controls the error induced by shaving off thelargest and smallest eigenvalues of ρ . More precisely, if we define a “smoothed state” σ = P ρP/
Tr(
P ρP ), where P is the projector onto the eigenspaces of K = − log( ρ ) witheigenvalues in the range given in (2.23), then it is clear that (i) σ lies within an O ( ε )-ballof ρ , and (ii) it has maximal and minimal eigenvalues given by λ max ( σ ) = e − E (cid:48) /G N + O (1 / √ G N ) , (2.24) λ min ( σ ) = e − E (cid:48) /G N − O (1 / √ G N ) . (2.25)It follows immediately from the definitions given in equations (2.14) and (2.15) that thesmooth min- and max-entropies of ρ agree with one another to leading order in G N , sincewe have rank( σ ) λ min ( σ ) ≤ Tr( σ ) ≤ rank( σ ) λ max ( σ ) . (2.26)More precisely, since σ is normalized with Tr ( σ ) = 1 , equations (2.24) and (2.25) imply S max ( σ ) = log(rank( σ )) ≤ log (cid:18) λ min ( σ ) (cid:19) = E (cid:48) G N + O (cid:18) √ G N (cid:19) (2.27)and S min ( σ ) = log( λ − ( σ )) = E (cid:48) G N − O (cid:18) √ G N (cid:19) . (2.28)Since S max ( σ ) must be greater than S min ( σ ), equations (2.27) and (2.28) together implythat the min and max entropies of σ agree with one another to leading order in G N . Since σ is within an O ( ε )-ball of ρ , the same statement holds true for the smooth min and maxentropies of ρ , as we previously claimed.The only remaining question is to find the saddle point value E (cid:48) . The von Neumannentropy of ρ is given by S ( ρ ) = (cid:90) d E D ( E ) E e − E = E (cid:48) G N + O (1) , (2.29)– 11 –here we have used the form of D ( E ) given in (2.17). It follows from the Ryu-Takayanagiformula that the saddle point value E (cid:48) is given by A/
4, where A is the area of the cor-responding RT surface. The smooth min- and max-entropies are therefore equal to thevon Neumann entropy up to O (1 / √ G N ) corrections for any fixed nonzero ε . Since thevon Neumann entropy S is of order O (1 /G N ), the corrections grow as the square root ofthe entropy (for fixed values of the UV cutoff). It follows that for holographic states, thesmooth min and max entropies satisfy S min = S − O ( √ S ) , (2.30) S max = S + O ( √ S ) . (2.31)This is exactly the same scaling that is seen when we take the asymptotic limit of a largenumber of copies of a state. The semiclassical holographic limit of large central chargeis therefore replicating, at least partially, the effects of the asymptotic i.i.d. limit of alarge number of identical copies of a single, non-holographic state. When we constructholographic tensor networks in the following section, we will see that these subleadingcorrections to the smooth min- and max-entropies can be related to known subleadingcontributions to holographic entanglement in AdS/CFT.
In order to construct a meaningful tensor network for a state in the AdS/CFT correspon-dence, it is necessary to produce a network that (i) reproduces the correct boundary statewith high fidelity, and (ii) has a bulk geometry that matches the bulk spacetime. Sincetensor networks have discrete geometries, property (ii) must be interpreted in terms ofsome discretization of the bulk spacetime. In this section, we consider tree networks —those constructed by discretizing the bulk with non-intersecting Ryu-Takayanagi surfaces.Given such a discretization, the underlying graph of the corresponding tensor network istaken to be the dual graph of the set of Ryu-Takayanagi surfaces and their correspondingboundary regions. A sample discretization of vacuum
AdS , along with the correspondingdual graph, is shown in Figure 3.Once a network is constructed on this graph, it is straightforward to quantify howwell the resulting state satisfies property (i) by looking at the inner product betweenthe state constructed by the tensor network and the target state. We will say that thenetwork satisfies property (ii) if the bond dimension of each edge in the network matchesthe area of the Ryu-Takayanagi surface through which it passes. More precisely, we willrequire that each bond γ in the network satisfies dim( H γ ) = e A γ / G N + o (1 /G N ) , where A γ Until now, we have assumed that ε is some fixed small number that is independent of G N . However,the range of integration required to approximate (2.22) depends only weakly on the allowed error ε as (cid:112) log(1 /ε ). Hence we can make the error ε non-perturbatively small with respect to G N , at the small costof allowing the smooth min- and max-entropies be separated by O ( (cid:112) Sf ( S )) for some super-logarithmicfunction f ( S ). (We can then obtain ε = e − f ( S ) , which is non-perturbatively small.) A natural choice mightbe f ( S ) = S δ for some small δ , or maybe f ( S ) = (log( S )) . Regardless of the specific function chosen, it iseasy to ensure that O ( (cid:112) Sf ( S )) is subleading compared to the entropy S . – 12 – igure 3 : A bulk discretization of vacuum AdS by non-intersecting RT surfaces. Bluecurves represent extremal surfaces in the bulk, red lines are edges in the dual graph, andblack dots are vertices in the dual graph. In the resulting tensor network, the dangling edgespassing through the boundary of the spacetime will correspond to uncontracted (physical)Hilbert space indices.is the area of the corresponding RT surface. Since the dimension of any bond Hilbertspace could be made arbitrarily large without altering the state by adding zero probabilitystates, we also require that each contraction is full rank in the bond space in the sense thatit contains no trivial contractions. In fact, this condition is not quite strong enough,as the dimension of H γ could still be made arbitrarily large by the addition of stateswith arbitrarily small probability — in our construction, this possibility is avoided byensuring that the eigenvalues of each bond are bounded below by a nontrivial function ofthe bond entropy (cf. the smoothing process of Section 2.3, where each “smoothed state”has bounded minimal eigenvalue). To explain the procedure of constructing tree networks via one-shot entanglement dis-tillation, we restrict temporarily to the case where the bulk is discretized by a singleRyu-Takayanagi surface. For such a discretization, the boundary is partitioned into twoconnected regions A and A c . The generic procedure for arbitrary non-intersecting bulkpartitions is detailed in Section 3.2.As discussed in Section 2.3, the smooth min- and max-entropies of a holographic stateagree with the von Neumann entropy to leading order in G N . It follows that for the reducedCFT density matrix ψ ( A ) of a holographic CFT state | ψ (cid:105) , there exists a normalized state Both sides of this equality are infinite. As usual, equations involving the entanglement entropy or thearea of extremal surfaces should be interpreted in the context of a regularization scheme in which the CFTstate is regulated on a lattice with finite spacing and the bulk spacetime is regulated with a radial cutoff. Formally, for a bond of the form P Aγ Q Bγ , we require that there is no nonzero vector v γ in H γ or dualvector ω γ in H ∗ γ such that P Aγ ω γ or Q Bγ v γ identically vanishes. – 13 – ε ( A ) within ε trace distance of ψ ( A ) satisfyingrank (cid:16) ψ ε ( A ) (cid:17) = e S ( A )+ O ( √ S ) , (3.1) λ max (cid:16) ψ ε ( A ) (cid:17) = e − S ( A )+ O ( √ S ) , (3.2)where λ max (cid:16) ψ ε ( A ) (cid:17) is the largest eigenvalue of ψ ε ( A ) .Given the full, pure CFT state | ψ (cid:105) ∈ H A ⊗ H A c , one can write the Schmidt decom-position | ψ (cid:105) = (cid:88) n (cid:112) λ n | n (cid:105) A | n (cid:105) A c , (3.3)where { λ n } are the eigenvalues of the reduced states ψ ( A ) and ψ ( A c ) . If { (cid:101) λ n } are theeigenvalues of the smoothed state ψ ε ( A ) , then it is easy to verify that the state | ψ ε (cid:105) = (cid:88) n (cid:113)(cid:101) λ n | n (cid:105) A | n (cid:105) A c (3.4)approximates the original state | ψ (cid:105) with very high fidelity. In particular, we have (cid:12)(cid:12)(cid:10) ψ ε (cid:12)(cid:12) ψ (cid:11)(cid:12)(cid:12) = F ( ψ ( A ) , ψ ε ( A ) ) ≥ (cid:18) − (cid:107) ψ ( A ) − ψ ε ( A ) (cid:107) (cid:19) ≥ − ε, (3.5)where F ( ρ, σ ) = (cid:2) Tr (cid:112) √ ρσ √ ρ (cid:3) is the fidelity of two quantum states. The first equality in(3.5) follows from the definition of fidelity and the form of the states (3.3) and (3.4), whilethe subsequent inequality is one of the Fuchs-van de Graaf inequalities [31].If we re-order the probability spectrum (cid:101) λ n such that it is monotonically decreasing,i.e. (cid:101) λ n +1 ≤ (cid:101) λ n , and break the resulting sum into blocks of size ∆, then we may rewrite(3.4) as | ψ ε (cid:105) = rank[ ψ ε ( A ) ] / ∆ − (cid:88) n =0 ∆ − (cid:88) m =0 (cid:113)(cid:101) λ n ∆+ m | n ∆ + m (cid:105) A | n ∆ + m (cid:105) A c (3.6)Now, suppose we discard the m -dependence of the eigenvalues (cid:101) λ n ∆+ m and replace all ofthe eigenvalues in each block with the average value of that block, (cid:101) λ avg n ∆ . The resulting state(which is still correctly normalized) is | Ψ ε (cid:105) = rank[ ψ ε ( A ) ] / ∆ − (cid:88) n =0 ∆ − (cid:88) m =0 (cid:113)(cid:101) λ avg n ∆ | n ∆ + m (cid:105) A | n ∆ + m (cid:105) A c , (3.7)and satisfies (cid:107) Ψ ε − ψ ε ( A ) (cid:107) ≤ λ max [ ψ ε ( A ) ] · ∆ ≡ δ. (3.8) In reality, the Hilbert space of the actual CFT will not factorize in this way, due to ultraviolet issues.However, since we have already regularized the theory, there is no problem splitting the Hilbert space intotensor factors. – 14 –y the same arguments as in (3.5), the overlap between | Ψ ε (cid:105) and the original CFT state | ψ (cid:105) is bounded below by (cid:12)(cid:12)(cid:10) ψ (cid:12)(cid:12) Ψ ε (cid:11)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:10) ψ (cid:12)(cid:12) ψ ε (cid:11)(cid:12)(cid:12) (cid:12)(cid:12)(cid:10) ψ ε (cid:12)(cid:12) Ψ ε (cid:11)(cid:12)(cid:12) ≥ − ε − δ. (3.9)If we choose ∆ = e S − O ( √ S ) , with the O ( √ S ) dependence chosen to approximately cancelthe O ( √ S ) dependence of λ max [ ψ ε ( A ) ] given in equation (3.2), then we can ensure that δ remains small, while the state becomes | Ψ ε (cid:105) = e O ( √ S ) (cid:88) n =0 e S − O ( √ S ) (cid:88) m =0 (cid:113)(cid:101) λ avg n ∆ | n ∆ + m (cid:105) A | n ∆ + m (cid:105) A c , . (3.10)To properly distill the EPR pairs out of this state, we define auxiliary Hilbert spaces H γ and H f with dimensions given bydim H f = e O ( √ S ) , (3.11)dim H γ = e S − O ( √ S ) , (3.12)where the precise values are chosen to match the range of the sums in equation (3.10). Wedefine the isometries H f ⊗ H γ (cid:44) → H A and H f ⊗ H γ (cid:44) → H A c by V | n (cid:105) f | m (cid:105) γ = | n ∆ + m (cid:105) A , (3.13) W | n (cid:105) f | m (cid:105) γ = | n ∆ + m (cid:105) A c (3.14)for some arbitrarily chosen bases of the auxiliary Hilbert spaces and the correspondingbases in their complex conjugate Hilbert spaces. We may then rewrite the state | Ψ ε (cid:105) as | Ψ ε (cid:105) = ( V ⊗ W ) e O ( √ S ) (cid:88) n =0 (cid:113)(cid:101) λ avg n ∆ | nn (cid:105) ff ⊗ e S − O ( √ S ) (cid:88) m =0 | mm (cid:105) γγ . (3.15)This expression is essentially identical to (2.9), except that it approximates a single copyof the original CFT state | ψ (cid:105) . Approximate states such as those given in equation (3.15) are considerably more com-mon in the quantum information literature than in discussions of quantum gravity, so wepause briefly to discuss their physical relevance. The first and most important point tonote is that expectation value of any bounded operator ˆ O on the Hilbert space of | ψ (cid:105) iswell approximated by its expectation value in the approximate state. In particular, sincethe overlap between the original CFT state | ψ (cid:105) and the new state | Ψ ε (cid:105) takes the form givenin (3.9), we can guarantee that the expectation values of bounded operators between thetwo states differ at most by (cid:12)(cid:12)(cid:12)(cid:68) ψ (cid:12)(cid:12) ˆ O | ψ (cid:69) − (cid:68) Ψ ε (cid:12)(cid:12) ˆ O | Ψ ε (cid:69)(cid:12)(cid:12)(cid:12) ≤ √ ε + δ (cid:107) ˆ O (cid:107) . (3.16)If both ε and δ can be made sufficiently small, then this bound is quite narrow. Since thetime evolution operator e iHt is bounded above by (cid:107) e iHt (cid:107) ≤ t that is either real or– 15 –n the upper half-plane, a similar bound holds for correlation functions at arbitrary times.In particular, correlation functions of bounded operators satisfy (cid:12)(cid:12)(cid:12)(cid:68) ψ (cid:12)(cid:12) ˆ O ( t ) . . . ˆ O n ( t n ) | ψ (cid:69) − (cid:68) Ψ ε (cid:12)(cid:12) ˆ O ( t ) . . . ˆ O n ( t n ) | Ψ ε (cid:69)(cid:12)(cid:12)(cid:12) ≤ √ ε + δ (cid:107) ˆ O (cid:107) . . . (cid:107) ˆ O n (cid:107) . (3.17)The states | ψ (cid:105) and | Ψ ε (cid:105) therefore generally produce approximately the same values for anyEuclidean or Lorentzian correlation function with arbitrarily large time gaps, includingout-of-time order Lorentzian correlation functions.There are a few scenarios in which | ψ (cid:105) and | Ψ ε (cid:105) can display qualitatively differentbehavior; we argue that none of these scenarios are actually physically important. If (cid:68) ψ (cid:12)(cid:12) ˆ O | ψ (cid:69) is itself very small compared to (cid:107) ˆ O (cid:107) , then there may O (1) differences in the relative size of (cid:68) ψ (cid:12)(cid:12) ˆ O | ψ (cid:69) and (cid:68) Ψ ε (cid:12)(cid:12) ˆ O | Ψ ε (cid:69) . However, in this case, since ε can be arbitrarilysmall, both the expectation values would have to be zero at leading order; they would stillagree up to O ( ε ) corrections.Secondly, non-observable quantities such as R´enyi entropies for α (cid:54) = 1 may (and will)look very different for | ψ (cid:105) and | Ψ ε (cid:105) . This is unsurprising, as the R´enyi entropies are verysensitive to small and (physically) insignificant perturbations, and can vary drasticallywithout significantly altering the expectation values of bounded operators.Finally, we are often interested in unbounded operators. In this case we do not haveany bound on the error in expectation values for the approximate state. However, in suchcircumstances we can generally replace an unbounded self-adjoint operator ˆ O by somebounded function f ( ˆ O ) of the operator without affecting the important physics. We willthen obtain a tight bound on the error of the expectation value (cid:104) f ( ˆ O ) (cid:105) when approximatingit in the distilled state | Ψ ε (cid:105) .So long as we originally chose ε small and chose ∆ correctly to ensure that δ is small,then (3.9) ensures that the distilled state | Ψ ε (cid:105) is a good approximation of the original CFTstate | ψ (cid:105) . As discussed above, this implies that the physics of the two states should be thesame up to non-perturbatively small corrections. Moreover, the expression given in (3.15)is a tensor network with a geometry matching the semiclassical dual of | ψ (cid:105) . In the abstractindex notation of section 2.1, the state in (3.15) is written asΨ AA c = V Afγ W A c fγ φ γγ σ ff , (3.18)where | φ (cid:105) = e S − O ( √ S ) (cid:88) m =0 | mm (cid:105) γγ , (3.19) | σ (cid:105) = e O ( √ S ) (cid:88) n =0 (cid:113)(cid:101) λ avg n ∆ | nn (cid:105) ff . (3.20) Here, we mean that the R´enyi entropies are non-observable when provided with only a single copy ofthe state — they can be computed from matrix moments of a state, which are observable given multiplereplicas of the state in question. – 16 –he network corresponding to (3.18) is sketched in Figure 4, superposed over the corre-sponding discretization of vacuum
AdS . The tensors V and W correspond to the entan-glement wedges of regions A and A c , respectively, as isometries that embed the states φ and σ into the boundary. Since φ is a maximally entangled state on a Hilbert space ofdimension e O ( S ( A )) , it has the right entanglement to reproduce the Ryu-Takayanagi surfacethat separates A from A c . Because the bond dimension of the legs of the state | σ (cid:105) is verysmall compared to the large bond dimensions of the φ -legs, we can think of the σ -legs asthin “cobwebs”, attached to the thick “girders” of the φ -legs. V WφσA A c γ γf f Figure 4 : A tensor network for a bipartite discretization of
AdS by a single Ryu-Takayanagi surface. φ γγ is a maximally entangled state on a Hilbert space of dimension e O ( S ( A )) , while σ ff is a (generally not maximally entangled) state on a Hilbert space ofdimension e O (cid:16) √ S ( A ) (cid:17) . The tensors V and W embed these states isometrically into theboundary.It is important to be clear about our motivation in including a second approximationstep, where we average (cid:101) λ i within each block and hence extract the dependence of (cid:101) λ i on i into the state | σ (cid:105) in the (relatively) small auxiliary Hilbert space H f ⊗ H f . After all, wecould already have constructed a tensor network with the correct discretized geometry byusing the smoothed, but unflattened state (cid:88) i (cid:113)(cid:101) λ i | i (cid:105) γ | i (cid:105) γ . (3.21)Our purpose in flattening the (cid:101) λ i spectrum is not to claim that the state produced is in anysense “better” or “more holographic” than the smooth state that was constructed priorto this flattening procedure. In fact, the degrees of freedom extracted into | σ (cid:105) are highlynon-unique, as they depend a great deal on the exact block size ∆ chosen while flatteningthe spectrum. However, by showing that the original state | ψ (cid:105) can be approximated in thisway, we are showing explicitly that the entanglement spectrum is so flat (up to smooth-ing) that the effective number of degrees of freedom in | ψ (cid:105) that describe the gradient ofthe entanglement spectrum is subleading compared to the effective number of degrees offreedom that are simply maximally entangled between the two sides.– 17 –ote that to obtain a tensor network description with the correct leading order bonddimensions, we only needed the smooth max-entropy to be sufficiently small. The addi-tional requirement that the smooth min-entropy agree with smooth max-entropy to leadingorder was imposed to ensure that the auxiliary Hilbert space H f on which the state is notmaximally entangled has subleading dimension e O ( √ S ) .Before moving on to more general tree tensor networks, we pause to consider the roleof the “cobweb” state | σ (cid:105) . This state does not have an immediate interpretation in theusual Ryu-Takayanagi picture of bulk entanglement, at least at leading order in G N . Sincethe state | σ (cid:105) is of subleading size in the entanglement entropy and hence in the centralcharge, the most natural interpretation is that it arises from quantum fluctuations in thebulk geometry (e.g., graviton fluctuations) that alter the areas of Ryu-Takayanagi surfacesat subleading order in G N . Generally, such fluctuations are expected to be suppressedby a factor of O ( √ G N ), meaning that the resulting fluctuations in the entropy are also oforder O (1 / √ G N ). Since this matches the rank of | σ (cid:105) , it seems natural to associate thesubleading state with these geometric fluctuations. We discuss this proposal in more detailin Section 7.1.It is worth commenting that we are also at liberty to absorb the state | σ (cid:105) into one of(or a combination of) the isometries V and W . This simplifies the picture of the tensornetwork, but comes at the cost of at least one of the operators V and W no longer being anisometry. In fact, they will not even be approximate isometries, although they will remainisometries if interpreted as operators V : H γ (cid:44) → H f ⊗ H A and W : H γ (cid:44) → H f ⊗ H A c .The question of whether tensors in the network are (at least approximate) isometries isimportant for various reasons, both in ensuring that the boundary state of the networkcorrectly approximates the original CFT state and in understanding the error correctingproperties of the network. As such we shall always keep the state | σ (cid:105) , and its generalizationsin more complicated networks, explicit. The argument given above can be extended to construct a tree tensor network for anarbitrary discretization of the bulk by (non-intersecting) Ryu-Takayanagi surfaces. Thisgeneralization works roughly as one would expect: one simply localizes degrees of freedomto each RT surface in turn, each time creating an additional link and tensor in the network.However, some important subtleties arise during this process. It is easy to construct a An alternative approach [32, 33] describes the fluctuations of the areas of extremal surfaces as “edgemodes,” i.e. superselection sectors that commute with the algebra of observables on both sides of thesurface. In our approach, however, these fluctuations are described explicitly by the states of the | σ (cid:105) tensorthat lies on the Ryu-Takayanagi surface. To see that this is the correct scaling of metric fluctuations, note that a one graviton state with order-unity frequency has an O (1 /G N ) energy, but the energy is the square of the amplitude of the metric strain.Note however that in contexts where we are only interested in the average area, the O (1 / √ G N ) term doesnot appear because, for linearized gravitons, positive fluctuations are just as likely as negative fluctuations.That is why the quantum corrections in the holographic von Neumann entropy (2.8), which traces over thewhole probability distribution, are merely O (1). In our tensor network contexts, however, we need to keeptrack of the Hilbert space dimensions, which do not average out. – 18 –uperficially-reasonable procedure that will not actually approximate the original statewith high fidelity. We will therefore describe the procedure for constructing generic treenetworks in some detail. Our argument is inductive: we assume that we have successfully constructed a treetensor network for a simpler discretization with one fewer RT surface, and then show thatwe can always add an additional RT surface while (approximately) preserving the bulkbond dimensions and the boundary CFT state. After arbitrarily many inductive steps,the final network will still approximate the original “target” CFT state on its boundary.However, to obtain rigorous bounds on our final error in approximating the original CFTstate, we must be somewhat careful in the order in which we choose to add RT surfaces.Specifically, there must exist some choice of boundary node, which we shall label the “root”node, such that each “parent” tensor was added after all of its “children.” (In practice, itseems likely that one will obtain a correct approximation of the original state even whenthe RT surfaces are added in an arbitrary order. Without adding them according to aparticularly nice orientation, however, it is hard to guarantee that one couldn’t obtain alarge boundary error by sheer bad luck.)We begin by choosing a “target” discretization of the bulk by non-intersecting RTsurfaces, such as the one sketched above in Figure 3. This will be the graph of our finaltree tensor network. Designating one of its nodes as a root picks out a preferred orientationfor the tree by flowing away from the root, as sketched in Figure 5. In general, we willchoose the root node to lie on the boundary, although our construction works even if theroot is chosen to lie in the bulk. All boundary nodes that are not the root are now leaves ofthis oriented graph. Note that this orientation is defined only for the purpose of orderingthe RT surfaces that make up the discretization, and is independent of the orientationimposed on the tensor network to denote up- and down-indices (cf. Section 2.1).To construct a tree tensor network for this graph, edges will be added to the networkinductively from leaves up to the root. To preserve the isometry properties of the tensornetwork, no RT surface can be added to the network before all of its children have beenadded (according to the orientation induced by choosing a boundary root). Different choicesof root node on the boundary, and even different orderings of RT surfaces that are consistentwith a single root-leaf orientation, will in general produce different tree tensor networks.However, all such networks are geometrically appropriate for the AdS/CFT correspondencein the sense that they have bond dimensions that match the holographic geometry andboundary states that approximately reproduce the original CFT state.To define the isometry properties of a tree tensor network precisely, we first define thestate associated to any bulk region bounded by a mixture of Ryu-Takayanagi surfaces andsubregions of the boundary to be the state produced by a truncation of the tree tensornetwork to that region. This is sketched in Figure 6 for a subregion of the tree tensornetwork that was introduced in Figures 3 and 5. Importantly, the edge states | φ (cid:105) and | σ (cid:105) associated to each RT surface are included in the state assigned to the region. There isa natural map from the state associated to a bulk region to the state associated to any For similar work in other contexts, see [34] and [35]. – 19 – igure 5 : The bulk discretization of vacuum
AdS that was originally sketched in Figure3 has here been given a root-leaf orientation on its dual graph by choosing an arbitraryboundary edge as the “root” (represented here by a white circle).larger region which contains the smaller region. This is essentially the “inclusion” mapof the tensor network, which consists of all tensors that are included in the state of thelarger region but not included in the smaller one; we call this the extension map . Theextension map does not include the edge states associated to its “input” RT surfaces, asthose are already included in the state of the smaller subregion; however, it will include theedge states associated to any “output” RT surfaces that bound the larger bulk subregion.This convention is chosen so that an extension map “beginning” on a given RT surface canalways be composed with an extension map that “ends” on the same surface.In the bipartite construction of Section 3.1, the bulk tensors V and W each servedas the extension map from the state in the complementary bulk region out to the globalboundary. In this construction, the maps V and W were exact isometries. For a treenetwork constructed from a boundary root orientation (as sketched in Figure 5), we willshow inductively that extension maps flowing entirely along the direction of the orientationare always exact isometries. An extension map which flows partially against the orientationof the graph will not in general be an exact isometry; however, it will be an approximateisometry with respect to a particular state-dependent metric. (For example, in Figure 6a,the extension map from the shaded region through the network out to the “right half”of the boundary is an exact isometry, as it flows along the orientation of the graph. Theextension map from the shaded region out to the global boundary, however, is only anapproximate isometry, as it must flow against the orientation of the graph to reach theroot node at the boundary.)We now present the inductive argument for constructing a tree tensor network foran arbitrary (non-intersecting) bulk discretization. For clarity, this whole procedure issketched in Figure 7 for the final step of the oriented discretization sketched in Figure 5.After a boundary edge has been designated as the root and a root-leaf orientation has– 20 – a) φσφ σφσ (b) Figure 6 : (a) The bulk discretization of vacuum
AdS shown in Figure 5 can be dividedinto bulk states by selecting regions of the bulk that are bounded by Ryu-Takayanagisurfaces and boundary subregions. Such a region is shaded here. (b) The bulk stateobtained from truncating a tree tensor network to the shaded region. Each edge of the treetensor network is composed of a maximally entangled state | φ (cid:105) and a subleading state | σ (cid:105) ,just as in the bipartite construction of Section 3.1. The bulk state on the shaded region isdefined by removing all tensors outside of the shaded region while keeping the edge states | φ (cid:105) and | σ (cid:105) that define the edges at the boundary of the shaded region. The result is atruncated state on the tensor product Hilbert space of the edges that cross the boundaryof the shaded region.been imposed on the dual graph, we pick one of the “uppermost” Ryu-Takayanagi surfaces(i.e., one of the surfaces that neighbors the root node) to be the last surface added tothe network, and assume that we have already constructed a tree tensor network for thediscretization that includes all but this final surface. To make an inductive argument, weassume that the tensor network for the “all-but-one” discretization has been constructedso that it:(a) approximately reproduces the original “target” CFT state on the boundary,(b) has internal bond dimensions that match the areas of the discretization surfaces, and(c) has the isometry properties detailed above (i.e., extension maps that follow the flow ofthe root-leaf orientation are exact isometries). The choice of “uppermost” RT surface is in general non-unique, and adding surfaces to the networkin different orders will produce different tensor networks that all approximate the original CFT state withhigh fidelity. – 21 –e now consider the smallest bulk subregion a containing the new RT surface in itsinterior (cf. Figure 7a, where the bulk subregion a is shaded). In the tensor network forthe reduced discretization, the state associated to this region is formed by a single tensortogether with | φ (cid:105) and | σ (cid:105) states on each already-constructed RT surface on the boundaryof a . Because we chose the RT surface to neighbor the root node, the extension map fromthis state to entire boundary state flows entirely along the orientation of the tree and so isan exact isometry by assumption. Furthermore, we have assumed that the entire boundarystate is approximately equal to the target CFT state.Since the smooth min- and max-entropies of the target state depend only weakly onthe error ε , and since the extension map from a to the global boundary is an isometry, thesmooth min- and max-entropies of the subregion state on a will agree with those of thetarget state to leading order. We can therefore apply the exact same bipartite distillationprocedure to the subregion state on a that we used to construct the global bipartite tensornetwork in Section 3.1. This will produce a bipartite tensor network ( V ⊗ W ) | φ (cid:105)| σ (cid:105) of theform given in (3.15) that approximates the state of the subregion a . We label our isometriesso that W is the “upwards” isometry that includes the root node in its image, while V isthe “downwards” isometry that maps away from the root node of the discretization (cf.Figure 7c).This newly-distilled state approximately reproduces the subregion state on a , and couldbe substituted directly into the network as in Figure 7c. Doing so, however, would requireerasing all of the | φ (cid:105) and | σ (cid:105) states associated to each of the neighboring, previously-addedRT surfaces, replacing them with the outward-pointing legs of V . Instead, we wish only toreplace the central tensor associated to a . Fortunately, as discussed in Section 2.1, thereis a canonical isomorphism between the states | φ (cid:105) ∈ H γ ⊗ H γ and | σ (cid:105) ∈ H f ⊗ H f andoperators φ : H γ → H γ and σ : H f → H f . Since | φ (cid:105) and | σ (cid:105) are full-rank, these operatorsare invertible. We can therefore simply replace the central tensor in the subregion by V (cid:48) W | φ (cid:105)| σ (cid:105) , where V (cid:48) = (cid:32)(cid:89) i σ − i φ − i (cid:33) V (3.22)and the product is taken over all RT surfaces on the boundary of a . In the case wherethere is a unique “uppermost” RT surface for our discretization, the isometry W will mapdirectly to the boundary and will not require modification. In the case of a discretizationwhere two or more RT surfaces both neighbor the root node, the isometry W will need tobe modified on any of its outgoing legs that pass through already-constructed RT surfaces,as in equation (3.22).By construction, the new state associated to the bulk subregion will approximate theold state, and since by assumption the rest of the tensor network forms an isometry from V (cid:48) W | φ (cid:105)| σ (cid:105) (or V (cid:48) W (cid:48) | φ (cid:105)| σ (cid:105) in the case of multiple “uppermost” RT surfaces) to the globalboundary, the new state constructed by the entire network will continue to approximatethe target CFT state. The map V , which maps the new bulk subregion across the newRT surface into the rest of the network, is an exact isometry even though V (cid:48) is not.This validates our assumption that extension maps that follow the root-leaf orientation– 22 –f the network should always be isometries. By induction, we can therefore construct atensor network state whose geometry agrees with an arbitrary tree discretization and whichapproximately reproduces the original CFT state.The only claim that we have made but are yet to prove is that extension maps whichflow partially against the root-leaf orientation of the underlying graph are still approximate,if not exact, isometries. This claim was not required as part of our inductive procedure forconstructing tree networks, but will be useful for constructing more general networks infollowing sections. Consider an arbitrary RT surface in some tree network discretization.We have two extension maps, one in each direction, that map the edge state | φ (cid:105)| σ (cid:105) associ-ated with this surface to the entire boundary. One map V : H γ ⊗ H f (cid:44) → H A c flows entirelywith the orientation of the graph and so is an exact isometry. When the RT surface wasfirst added to the network, the other extension map W : H γ ⊗ H f (cid:44) → H A was also an exactisometry for exactly the same reasons. However, because this extension map flows par-tially against the root-leaf orientation of the graph, it will continue to change as additionalsurfaces are added to the network. This is in contrast to the “downwards” isometry V ,which remains unaltered because of the order in which we chose to add the RT surfaces.In the final network, once all RT surfaces have been added, we call the extension map thatflows partially against the root-leaf orientation X : H γ ⊗ H f → H A . This map will generalnot be an exact isometry; we will show that it is still an approximately isometry in anappropriate sense.Because our construction is designed so that the tensor network approximately repro-duces the boundary state at every stage in its construction, we find V ⊗ W | φ (cid:105)| σ (cid:105) ≈ V ⊗ X | φ (cid:105)| σ (cid:105) . (3.23)The left-hand side of (3.23) describes the state produced by the entire tensor network whenthe RT surface is first added to the network, while the right-hand side is the state of thefinal tensor network. Since | φ (cid:105)| σ (cid:105) is fully entangled (i.e., its reduced density matrices oneither side of the RT surface are full-rank), exactness of (3.23) would imply that X = W and so the extension map that flows partially against the network orientation would haveto remain an exact isometry. As such, we can interpret (3.23) as showing that X is anapproximate isometry with respect to a particular metric that is adapted to the state | φ (cid:105)| σ (cid:105) .It can be equivalently written as (cid:107) ( X − W ) φ ⊗ σ (cid:107) = (cid:107) ( X − W ) ρ / φ ρ / σ (cid:107) ≤ ε, (3.24)where (cid:107) A (cid:107) ≡ (cid:112) Tr( A † A ) is the Hilbert-Schmidt norm, ρ φ ρ σ = φ σ is the reduced densitymatrix of | φ (cid:105)| σ (cid:105) , and ε > V from (3.23), and so givesa distance that depends only on X , W and the reduced density matrix ρ φ ρ σ .Finally, we can look at the partial trace of (3.23) over H f ⊗ H γ . Using the fact that thetrace norm (cid:107) ρ (cid:107) = Tr (cid:112) ρ † ρ is monotonically decreasing under the partial trace, it followsfrom (3.23) that (cid:13)(cid:13)(cid:13) Tr fγ (cid:16) X | φ (cid:105)| σ (cid:105)(cid:104) φ |(cid:104) σ | X † (cid:17) − ρ φ ρ σ (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ρ / φ ρ / σ ( X † X − ) ρ / φ ρ / σ (cid:13)(cid:13)(cid:13) ≤ ε (3.25)– 23 – a) φσ φσ (b) VWφ σ (c) V (cid:48) Wφ σφσ φσ (d)
Figure 7 : The final step of a tree tensor network construction for the bulk discretizationof vacuum
AdS with orientation given in Figure 5. (a) It is assumed that a tree networkhas been constructed for the discretization that consists of all but the final (dashed) RTsurface. The bulk subregion a , shaded in gray, is the smallest bulk subregion containingthis surface in its interior. (b) The state of the network on a is sketched explicitly. Sincethe final RT surface has not yet been added to the network, the bulk subregion a containsonly one tensor. As all extension maps away from a follow the root-leaf orientation of thenetwork, they are all exact isometries. (c) The subregion state on a is distilled across thenew RT surface into a bipartite tensor network such as the one given in (3.15). (d) Topreserve the structure of the states | φ i (cid:105) and | σ i (cid:105) on the already-constructed RT surfaces,the bipartite isometry V is replaced by the map V (cid:48) given in equation (3.22). Note that allextension maps that begin on the new smallest bulk subregion (shaded here) are still exactisometries. – 24 –his is a strictly weaker condition than (3.23) and (3.24): inefficiencies in the Fuchs-vande Graaf inequalities [31] mean that we cannot recover (3.23) and (3.24) without some lossof precision. However, it is perhaps the easiest condition to interpret.Let { x i } be the eigenvalues of X † X . If | σ (cid:105) were maximally mixed, we could rewrite(3.25) as 1 d (cid:88) i | x i − | ≤ ε. (3.26)In other words, we would understand (3.25) as saying that on average the eigenvalues of X † X is close to one. Hence X † X (cid:39) and X is an approximate isometry. Since | σ (cid:105) is fullrank but is not maximally mixed, we should instead think of (3.25) as a weighted averageof | x i − | . Of course, since in general X † X and ρ σ will not commute, this interpretationis not quite literal. It does, however, provide the correct intuition.Thus far, we have only considered the approximate isometry condition for extensionmaps that go from an RT surface all the way to the boundary. What about an extensionmap X (cid:48) that flows only partially through the network, whose image lies on RT surfaces inaddition to, or instead of, the boundary? In this case, the “output” RT surfaces may nothave been added to the network when the “input” RT surface was added, so it may notnecessarily be possible to compare the final extension map X (cid:48) to some intermediary exten-sion map W (cid:48) as we did in equation (3.23). However, it will always be possible to compose X (cid:48) with some exact isometry W (cid:48) so that the resulting operator W (cid:48) X (cid:48) is an extension mapfrom the “input” RT surfaces of X (cid:48) out to the global boundary. Since W (cid:48) is an exact isometry, applying (3.25) to the map W (cid:48) X (cid:48) yields the followinginequality: (cid:13)(cid:13)(cid:13) ρ / φ ρ / σ ( X (cid:48)† X (cid:48) − ) ρ / φ ρ / σ (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ρ / φ ρ / σ ( X (cid:48)† W (cid:48)† W (cid:48) X (cid:48) − ) ρ / φ ρ / σ (cid:13)(cid:13)(cid:13) ≤ ε (3.27)We conclude that extension maps that do not flow all the way to the boundary, such as X (cid:48) ,satisfy the same approximate isometry condition given in (3.25) for extension maps that do flow all the way to the boundary.Unfortunately, the approximate isometry conditions (3.23), (3.24), (3.25), and (3.27)are not quite as powerful as one might want. They tell us that X and X (cid:48) are close toisometries “on average”, but, because of the large dimensions of the Hilbert spaces involved,they do not say much about the “worst-case” error. For example, a single eigenvalue x i of X † X can be very far from one without making a large contribution to the averaged error(3.26).More formally, to bound the “worst-case” error, we would want to bound the operatornorm (cid:107) X − W (cid:107) ∞ = sup | ψ (cid:105) (cid:107) ( X − W ) | ψ (cid:105)(cid:107)(cid:107)| ψ (cid:105)(cid:107) . (3.28) In the final network, in general, the extension map W (cid:48) from the output surfaces of X (cid:48) to the globalboundary will not be an exact isometry. However, it will always be possible to find some intermediary net-work, constructed as part of the inductive procedure, where W (cid:48) is exact. Since both final and intermediarynetworks reproduce the boundary state to within tolerance ε , either can be used to prove equation (3.27). – 25 –owever, the tightest bound that we can place on this norm is (cid:107) X − W (cid:107) ∞ ≤ (cid:107) ( X − W φ ⊗ σ ) (cid:107) ∞ (cid:107) φ − ⊗ σ − (cid:107) ∞ ≤ ε (cid:107) φ − ⊗ σ − (cid:107) ∞ , (3.29)where the first inequality follows from the submultiplicativity of the operator norm and thesecond follows from the monotonicity of the Schatten norms and (3.24). Since the operatornorm of φ − ⊗ σ − is quite large, na¨ıvely satisfying (cid:107) φ − ⊗ σ − (cid:107) ∞ = e O ( S ) , (3.30)we cannot make ε small enough to make this a tight bound.If the RT surfaces of a tree tensor network are added in an arbitrary order, ratherthan adding child surfaces before parent surfaces, then the potentially large error in (3.29)prevents us from guaranteeing that the final network correctly approximates the originalboundary state. In this section, we avoided this possibility by imposing a root-leaf orien-tation on our construction; however, when constructing more complicated sub-AdS scaletensor networks in Section 6, the issue of errors in our approximate isometries will ariseonce again without the guarantee of a root-leaf orientation to stop them from blowing up.However, in practice, (3.23) and (3.25) should ensure that “generic” small perturba-tions in the subregion states of our network that occur during an unoriented inductionshould only lead to a small error in the final state. Large errors would only occur if theseperturbations were somehow finely tuned to blow up when mapped out to the boundaryvia extension maps. We therefore think it very likely that our distillation procedure willproduce tensor networks that approximate the original holographic state correctly regard-less of the order in which Ryu-Takayanagi surfaces are added to the network, and moregenerally will produce accurate sub-AdS scale tensor networks even when we no longerhave the luxury of a root-leaf orientation to constrain accumulated errors precisely. Thus far, we have focused on constructing a tensor network for a single, arbitrary holo-graphic CFT state. In the literature, however, it is common to consider tensor networksthat describe not only a single holographic state, but an entire code subspace of holographicstates. These tensor networks have some dangling bulk legs that turn the entire tensor net-work into a bulk-to-boundary map that encodes bulk excitations into a subspace of theboundary (see, e.g., [9] and [10]). Ideally, the map from the bulk to the boundary will havesome appropriate error correcting properties that allow bulk operator reconstruction to beinterpreted in the language of quantum error correction.We discuss here, briefly, how our construction can be extended to create such codespaces. We shall focus on a code space that consists of a single bulk qubit, localized withina single bulk region a of the tree network discretization shown in Figure 8a. The generaliza-tion to more complicated bulk code spaces in tree tensor networks is straightforward, andwe would optimistically expect similar results to hold for the more general constructionsof Sections 5 and 6. – 26 –n much of the existing literature on holographic quantum error correcting codes (see,e.g., [9] and [10]), one begins with a particular tensor network construction and then showsthat it has quantum error correcting properties analogous to those of AdS/CFT. As inthe rest of this paper, we reverse this script. We begin with the known quantum errorcorrecting properties of AdS/CFT [13], and especially entanglement wedge reconstruction[36–39], and show that they imply the existence of tensor network constructions for codespaces of actual holographic states, which behave in exactly the same way as the toy modelsthat have previously appeared in the literature. We emphasize that this should not be takenas an independent proof of entanglement wedge reconstruction. Instead it demonstratesthat the error-correction properties of AdS/CFT itself and of holographic tensor networkstoy models are not merely analogous to one another, but are in fact different examples ofexactly the same phenomenon.Consider, for example, a 2-dimensional “code subspace” of states in AdS/CFT thatcan be built from a single starting state by applying low-energy unitary bulk operatorsin only a single bulk region a of some tree network discretization. Any Ryu-Takayanagisurface neighboring the bulk region a splits the boundary into two regions: region A , theentanglement wedge of which contains the bulk region a , and its complement A c . Thisis sketched for a particular choice of bulk region and Ryu-Takayanagi surface in Figure8b. Because of the usual error correcting properties of AdS/CFT [13, 33, 36, 37], thereduced density matrix on A c is approximately the same for every state in the code space.Moreover, any two orthogonal states in the code space will have reduced density matriceson A that are approximately orthogonal; the trace distance between them will be close tomaximal. This is in fact equivalent to the fact that the reduced density matrix on A c isthe same for every state, including superpositions, in the code space (see, e.g., the weakdecoupling duality described in [40, 41]).Let us suppose we have successfully constructed a tree tensor network for a singlestate | ψ (cid:105) in the code space using the techniques of Section 3. Because the reduced densitymatrices on A c are the same for every RT surface bounding a and every state in the codespace, any state in the code subspace can be represented by a tensor network that isidentical to the one constructed for | ψ (cid:105) except that it differs in the tensor associated to thebulk subregion a . It follows by linearity that we can describe the entire two-dimensionalcode space adding a two-dimensional bulk leg to the tensor in region a (cf. Figure 8c). Theresulting tensor network can be interpreted as a map T : H bulk (cid:44) → H CFT from the bulk tothe boundary. Furthermore, by using our freedom to choose an inner product on H bulk , wecan ensure that T is an exact isometry.Showing that our tensor networks are quantum error correcting in the sense of [9]requires showing that for any choice of RT surface bounding a and corresponding bound-ary subregion A , any bulk operator on H bulk has an equivalent boundary operator whosesupport lies only on A . By “equivalent,” we mean that acting with one of these boundary If the energy of the bulk matter excitations is O (1) in a G N expansion, then the backreaction onthe geometry is O ( G N ), resulting in an O (1) change in the Bekenstein-Hawking entropy of the surfacesbounding the region a . This is smaller than the O ( G − / N ) fluctuations already present in our network, andcan therefore be neglected without changing our error estimates on the bond dimensions of the network. – 27 – (a) A A c a (b) a φσ (c) Figure 8 : (a) A tree tensor network for a particular discretization of vacuum
AdS . A bulksubregion a of the discretization (shaded here) has been chosen to create a code subspace ofexcitations around the vacuum generated by operators whose support lies entirely within a . Note that the | φ (cid:105) and | σ (cid:105) edge states have been suppressed in this sketch for the sakeof visual clarity. (b) Choosing a particular RT surface that bounds a , denoted here witha dashed line, partitions the global boundary into two regions: A , which contains a in itsentanglement wedge, and its complement A c . (c) As explained in the text, a code subspaceof excitations localized in a can be represented by adding a single bulk leg to the tensorin region a . For a particular choice of RT surface bounding a , the “controlled extensionmap”, which maps the bulk leg in a plus the edge states on the chosen RT surface into theboundary region A , is sketched with red arrows. In order to clarify where the extensionmap begins, the edge states | φ (cid:105) and | σ (cid:105) are shown explicitly on the chosen RT surface, andsuppressed on all other RT surfaces in the diagram.– 28 –perators on any state in the code subspace produces approximately the same state thatone would obtain by acting with the original, bulk operator on H bulk . Such boundaryrepresentations of bulk operators are obtained in [9] by using the tensor network to “pushthe bulk operator through the network” and into the boundary.To be precise, the map that will be used to push bulk operators to the boundary region A is the “controlled extension map” X : H f ⊗ H γ ⊗ H bulk → A, sketched in Figure 8c. Wecall this a controlled extension map because once a state | ψ (cid:105) is fixed on the bulk leg H bulk ,the resulting map X | ψ (cid:105) is an extension map in the corresponding tensor network in thesense of Section 3.2. We will show first that X is an approximate isometry in the sense ofequation (3.25), and second that this condition allows us to use X to push bulk operatorsto equivalent operators on the boundary.Let | (cid:105) , | (cid:105) form a basis for H bulk . The controlled extension map X can then berepresented in this basis as X = X (cid:104) | + X (cid:104) | , (4.1)where X and X are the extension maps from the RT surface to A for the tree tensornetworks defined by specifying bulk leg states | (cid:105) and | (cid:105) respectively. We showed inSection 3.2 that extension maps in tree tensor networks are approximate isometries, andso X and X satisfy X † X (cid:39) X † X (cid:39) (4.2)in the sense of equation (3.25). Because orthogonal bulk states are almost orthogonal inregion A , these extension maps satisfy X † X (cid:39) X † X (cid:39) X † X (cid:39) and so X is an approximateisometry in the sense of (3.25). By the same arguments as in Section 3.2, we can also showthat controlled extension maps that end on RT surfaces are also approximate isometries.Given an operator ˆ O b acting on H bulk , we wish to use the fact that X is an approximateisometry to produce a boundary operator ˆ O A supported on A whose action on the codesubspace is approximately the same as ˆ O b . More precisely, for a state | ψ (cid:105) ∈ H bulk , we wishto show ˆ O A T | ψ (cid:105) ≈ T ˆ O b | ψ (cid:105) . (4.4)This is exactly the same sense of bulk reconstruction through quantum error correctionthat was developed for exact tensor network toy models in [9]. In terms of the controlledextension map X and the bulk tensor network state | T A c (cid:105) associated to the entanglementwedge of A c , the tensor network map T can be decomposed as T | ψ (cid:105) = X | T A c (cid:105)| ψ (cid:105) . (4.5)Equation (4.4) therefore becomesˆ O A X | T A c (cid:105)| ψ (cid:105) ≈ X ˆ O b | T A c (cid:105)| ψ (cid:105) . (4.6)– 29 –s in [9], we can find a boundary representation of a bulk operator by simply conju-gating with X , i.e., ˆ O A ≡ X ˆ O b X † . (4.7)Using the approximate isometry condition that X † X acts approximately as the identityon | T A c (cid:105)| ψ (cid:105) for any bulk state | ψ (cid:105) , we see that ˆ O A satisfies equation (4.4). If X were anexact isometry, as in [9], the map from bulk operators to boundary operators given in (4.7)would be unital on the image of X (i.e., it maps the identity on H bulk to the boundaryprojector onto the image of X ). This condition is desirable from the perspective of quantuminformation theory, as it ensures that the map given in (4.7) is a quantum channel in theHeisenberg picture.In order to ensure that this condition holds for our approximate isometries, thus makingthe bulk-to-boundary operator map into a genuine quantum channel, we instead define theboundary representation of a bulk operator asˆ O A ≡ X ( X † X ) − / ˆ O b ( X † X ) − / X † . (4.8)From the isometry condition (3.25), we see that ˆ O A still satisfies equation (4.4), and thusthis an approximate boundary representation of the bulk operator ˆ O b . We can also seeplainly that the bulk-to-boundary operator map given by equation (4.8) maps the identityon H bulk to the identity on the image of X , and is thus a quantum channel on operatorsin the code subspace (i.e., a completely positive, unital map on operators). We concludethat every bulk operator ˆ O b has an equivalent boundary operator representation ˆ O A whosesupport lies only on A and whose action on the code subspace is approximately the sameas the action of ˆ O b , as desired.The boundary operator obtained via the quantum channel given in equation (4.8)may be very different from the boundary operator on the same region obtained throughthe extrapolate dictionary and the HKLL reconstruction procedure [42]; however, theseoperators will agree on the boundary code subspace obtained by taking the image of H bulk under the map defined by the tensor network. Our tensor network description of a statein the AdS/CFT correspondence therefore displays exactly the same bulk-to-boundaryoperator mapping properties of the celebrated HaPPY class of holographic codes [9]. Weconclude that tensor networks models of bulk operator reconstruction may be taken quiteliterally. They are not just toy models! The tree networks constructed in Section 3 already constitute a large class of tensornetworks for the AdS/CFT correspondence, but we might still hope for a more generalconstruction. A single bond in a tree tensor network corresponds to a complete Ryu-Takayanagi surface in the bulk spacetime from which the network was constructed, so theinformation contained in that bond is a priori distributed nonlocally across the entire cor-responding surface. In order to localize degrees of freedom at sub-AdS scales within thebulk geometry, we must therefore find a way to divide each bond geometrically along its– 30 –orresponding Ryu-Takayanagi surface. In the bulk discretization picture of Section 3, thiscorresponds to constructing a holographic tensor network with loops .To construct such a tensor network, we will need to understand the holographic en-tanglement of purification [15, 16], a geometric quantity in the bulk spacetime that isconjectured to correspond to an information-theoretic quantity involving a tensor factor-ization of the information on an RT surface. In this section, we will assume the holographicentanglement of purification conjecture, and then use it to construct a geometrically accu-rate tensor network with a single loop for an arbitrary holographic CFT state. With somefurther assumptions introduced in Section 6, we will then be able to extend this procedureto construct tensor networks with arbitrarily many loops, and therefore to localize infor-mation in the bulk arbitrarily well within the regime of validity of the Ryu-Takayanagiformula (i.e., above the string/Planck length scales).
For a quantum state ρ ( AB ) on a bipartite Hilbert space H A ⊗ H B , the entanglement ofpurification [43] between subsystems A and B is defined as E P ( A : B ) = inf | Ψ (cid:105) AA (cid:48) BB (cid:48) S ( AA (cid:48) ) , (5.1)where | Ψ (cid:105) AA (cid:48) BB (cid:48) ∈ H A ⊗H A (cid:48) ⊗H B ⊗H B (cid:48) is a purification of ρ ( AB ) . The infimum in equation(5.1) is taken over all possible purifications of ρ ( AB ) in all possible auxiliary Hilbert spaces H A (cid:48) and H B (cid:48) . If ρ ( AB ) is a mixed state, then the von Neumann entropy S ( A ) no longer measures theentanglement (or even the correlation) between A and B in any meaningful sense, sincesome portion of S ( A ) is inherited from the nonzero von Neumann entropy of ρ ( AB ) . Thevon Neumann entropy S ( A ) may be non-zero for a product state ρ ( A ) ⊗ ρ ( B ) , which has nocorrelation between the two subsystems. The entanglement of purification E P is a some-what better measure of the degree to which A and B are entangled (or at least correlated),as it measures the minimal entanglement between AA (cid:48) and BB (cid:48) for any purification. As aresult, the entanglement of purification is zero for product states, and is non-increasing un-der local operations. It is not a true entanglement monotone in the sense of [44], however,since it may be non-zero even for separable states (which only have classical correlation),and it may be increased by classical communication.As with other information-theoretic quantities, one might hope that the entanglementof purification has a geometric dual in the context of AdS/CFT. For two subregions A and A of a holographic CFT state, it has recently been conjectured in [15, 16] that E P ( A : A ) is given to leading order in G N by the area of the entanglement wedge cross-section , E W ( A : A ) . Formally, E W ( A : A ) is defined as E W ( A : A ) = area(Σ A : A )4 G N , (5.2)where Σ A : A is the minimal surface anchored to the Ryu-Takayanagi surface of A ∪ A that partitions the entanglement wedge into a portion whose boundary contains all of A – 31 –nd a disjoint portion whose boundary contains all of A . This surface is sketched in Figure9 in vacuum
AdS for two different typical configurations of boundary regions A and A . A A A c Σ A : A A (cid:48) A (cid:48) (a) A A A c A c Σ A : A A (cid:48) A (cid:48) A (cid:48) A (cid:48) (b) Figure 9 : The entanglement wedge cross-section for two different configurations of bound-ary subregions A and A in vacuum AdS . In both (a) and (b), the entanglement wedgecross-section divides the entanglement wedge (the bulk region bounded by A , A , and RTsurface of A ∪ A ) into two disjoint regions, each of which contains only one of either A or A in its boundary. The subregions of each RT surface inherited from this partition havebeen instructively labeled A (cid:48) and A (cid:48) for reasons explained in Section 5.2.If A and A are connected subregions of some larger connected region A = A ∪ A (i.e., if A and A form a connected partition of A as in Figure 9a, then we call thestate | Ψ (cid:105) A A (cid:48) A A (cid:48) that saturates the infimum in equation (5.1) the minimally entangledpurification (MEP) of the partition A : A . The holographic entanglement of purification conjecture has been generalized to mul-tipartite and conditional entropies (and their geometric duals) in [46–49]. For the moment,however, we will concern ourselves only with the minimally entangled purification of aconnected partition A : A , and the associated entanglement of purification E P ( A : A ). Consider a holographic CFT state | ψ (cid:105) with some subregion A that is further divided intoa connected partition A : A , as sketched in Figure 9a. The reduced state ρ ( A A ) has Of course, since (5.1) contains an infimum rather than a minimum, it is not necessarily saturated byany state | Ψ (cid:105) A A (cid:48) A A (cid:48) . In this case, we take the MEP to be a fixed state that saturates the infimum towithin tolerance ε . Since one might expect that the infimum in (5.1) could be saturated arbitrarily wellby taking the purifying spaces A (cid:48) and A (cid:48) to be arbitrarily large, allowing this finite tolerance in the MEPprevents the dimensions of A (cid:48) and A (cid:48) from blowing up. A similar object, defined by a procedure in which one is also permitted to minimize over all possiblepartitions A : A , has previously been studied in [45]. – 32 – minimally entangled purification | Ψ (cid:105) A A (cid:48) A A (cid:48) chosen so that S ( A A (cid:48) ) = E P ( A : A ) . According to the holographic entanglement of purification conjecture, this implies S ( A A (cid:48) ) = E W ( A : A ) = area(Σ A : A )4 G N . (5.3)Examining Figure 9a, it is easy to see that Σ A : A is the minimal bulk surface anchoredto the boundary region A and a subregion of the Ryu-Takayanagi surface of A ∪ A , whichwe have instructively labeled A (cid:48) . Equation (5.3) looks just like the Ryu-Takayanagi formula(2.8) if one supposes that | Ψ (cid:105) A A (cid:48) A A (cid:48) is a holographic state in some boundary theory witha domain corresponding to the codimension-2 bulk surface made up of A and A alongwith the Ryu-Takayanagi surface of their union.Indeed, as explained in [15, 16], it makes sense to interpret the MEP | Ψ (cid:105) A A (cid:48) A A (cid:48) asa geometric state on some subregion of the bulk, where the auxiliary Hilbert spaces H A (cid:48) and H A (cid:48) are identified with the bulk surfaces A (cid:48) and A (cid:48) , respectively. The conjecturedsurface-state correspondence [50] of holographic CFTs suggests that any codimension-2surface in a holographic spacetime corresponds to some state in a boundary theory thatencodes the physics in the entanglement wedge of that surface. If this is to believed, thenit is natural to assume that the MEP of the partition A : A corresponds to a state onthe codimension-2 bulk surface Σ A A (cid:48) A A (cid:48) = A ∪ A ∪ A (cid:48) ∪ A (cid:48) . To make this proposal precise, we assume that the entanglement entropies of subsys-tems of the MEP are given at leading order by the Ryu-Takayanagi formula applied tothe codimension-2 surface Σ A A (cid:48) A A (cid:48) . In particular, since A (cid:48) and A (cid:48) are subregions of aminimal surface and hence minimal themselves, this assumption tells us that A (cid:48) and A (cid:48) are themselves the Ryu-Takayanagi surfaces for the Hilbert Space factors H A (cid:48) and H A (cid:48) .In other words, their von Neumann entropies satisfy: S ( A (cid:48) ) = area( A (cid:48) )4 G N + o (cid:18) G N (cid:19) , (5.4) S ( A (cid:48) ) = area( A (cid:48) )4 G N + o (cid:18) G N (cid:19) . (5.5)We require one more assumption to build a tensor network using the holographic entangle-ment of purification, which is that the MEP has the same smooth min- and max-entropyproperties (2.30) and (2.31) as holographic CFT states. In Section 2.3, we argued thatthis will be true for any state with von Neumann entropies given by equations of the form(5.4) and (5.5), and with extensive R´enyi entropies that take the geometrical form givenin equation (2.16). If the MEP is indeed a holographic state for the portion of the bulkbounded by Σ A A (cid:48) A A (cid:48) , then it should satisfy both of these properties.If the MEP has entropies given to leading order by the Ryu-Takayanagi formula onsubregions of Σ A A (cid:48) A A (cid:48) , and if the smooth min- and max-entropies of those subregionsare also given by (2.30) and (2.31), then the MEP satisfies all the conditions required to The “entanglement wedge” of a codimension-2 bulk surface Σ is defined in analogy with the entangle-ment wedge of a boundary interval as the subregion of the bulk bounded by Σ and by the minimal surfacehomologous to Σ that shares its boundary. – 33 –uild a tree tensor network (cf. Section 3). This means that we can build a tree networkfor the MEP that matches the bulk geometry contained within Σ A A (cid:48) A A (cid:48) . Specifically,there exists some state | Ψ ( ε ) (cid:105) A A (cid:48) A A (cid:48) that approximates the MEP with high fidelity witha tensor network representationΨ A A (cid:48) A A (cid:48) ( ε ) = T A (cid:48) γ f U A γ f γ f V A γ f γ f W A (cid:48) γ f × φ γ γ (1) φ γ γ (2) φ γ γ (3) σ f f (1) σ f f (2) σ f f (3) (5.6)with bond dimensions given by:dim H γ = e S ( A (cid:48) ) − O ( √ S ( A (cid:48) )) , (5.7)dim H f = e O ( √ S ( A (cid:48) )) , (5.8)dim H γ = e S ( A A (cid:48) ) − O ( √ S ( A A (cid:48) )) , (5.9)dim H f = e O ( √ S ( A A (cid:48) )) , (5.10)dim H γ = e S ( A (cid:48) ) − O ( √ S ( A (cid:48) )) , (5.11)dim H f = e O ( √ S ( A (cid:48) )) . (5.12)The outer product expression given in (5.6) is not particularly illuminating on its own,but has a natural interpretation in the geometric picture of tensor networks. In Figure10a, this tensor network is shown superposed over the geometric picture of the minimallyentangled purification, with each network bond passing through its corresponding bulksurface. Equations (5.3), (5.4), and (5.5) imply that the bond dimensions of this tensornetwork match the areas of the surfaces A (cid:48) , A (cid:48) , and Σ A : A in Figure 9a, justifying ourinterpretation of expression (5.6) as an approximate tensor network for the minimallyentangled purification that matches the geometric properties of its holographic dual.The tree network for the minimally entangled purification constitutes a tensor networkfor the “top half” of the bulk discretization shown in Figures 9a and Figure 10a. To find ageometric tensor network for the full boundary state, we need to find a tensor correspondingto the bulk region that lies between A (cid:48) ∪ A (cid:48) and the complementary boundary region A c .Since the MEP | Ψ (cid:105) A A (cid:48) A A (cid:48) and the original global boundary state | ψ (cid:105) A A A c are bothpurifications of the reduced boundary state ρ ( A A ) , they are related by an isometry on thepurifying space. That is, there exists an isometry X : H A (cid:48) ⊗ H A (cid:48) (cid:44) → H A c (5.13)such that | ψ (cid:105) A A A c = ( I A A ⊗ X ) | Ψ (cid:105) A A (cid:48) A A (cid:48) . (5.14) As in Section 3.2, one must choose a root-leaf orientation for the MEP network in order to determinethe order in which its subregions should be distilled. The choice of ordering will not matter for our purposes,though as usual it will determine which isometries are exact and which are only approximate. Formally, this isometry only exists if dim( H A c ) ≥ dim( H A (cid:48) ⊗ H A (cid:48) ) . Since the dimension of the latterspace is given to leading order by e S ( A c ) , and S ( A c ) is much smaller than log dim( H A c ), this bound issatisfied here and the isometry exists. – 34 – A φ σ φσ φσT U V W A (cid:48) A (cid:48) (a) A A A c φ σ φσ φσU VXT W (b) Figure 10 : (a) A tensor network for the minimally entangled purification of two neighbor-ing boundary regions in vacuum
AdS , as given in equation (5.6). (b) The one-loop tensornetwork for the full boundary state given by equation (5.16), obtained by embedding theMEP isometrically into the global boundary. Note that in both figures, subscripts on the φ ( i ) and σ ( i ) edge states have been suppressed.In tensor notation, this is ψ A A A c = X A c A (cid:48) A (cid:48) Ψ A A (cid:48) A A (cid:48) . (5.15)Since X is an isometry, the minimally entangled purification | Ψ (cid:105) A A (cid:48) A A (cid:48) can be re-placed with the nearby state | Ψ ( ε ) (cid:105) A A (cid:48) A A (cid:48) without any additional loss of precision. Thatis, the state ψ A A A c ( ε ) = X A c A (cid:48) A (cid:48) Ψ A A (cid:48) A A (cid:48) ( ε ) . (5.16)approximates ψ as well as Ψ ( ε ) approximates Ψ . Plugging the tensor network expressionfor Ψ ( ε ) from equation (5.6) into equation (5.16) (and contracting the tensors X , T , and W into a single bulk tensor XT W ) yields a tensor network description for the full (ap-proximate) boundary state ψ ( ε ) with bond dimensions matching the areas of the surfaces A (cid:48) , A (cid:48) , and Σ A : A . This complete tensor network is sketched in Figure 10b. By usingthe natural properties of the minimally entangled purification arising from the holographicentanglement of purification conjecture, we have managed to localize degrees of freedomwithin the Ryu-Takayanagi surface and hence to construct a non-tree tensor network for ageneric holographic state with extremely high fidelity.With this tensor network now fully constructed, we must now ask an important ques-tion about its extension maps: in what sense can they be shown to be isometries? Arethey exact isometries, as in Section 3.1? Are they merely approximate isometries in the– 35 –ense of Section 3.2? Or are they in fact neither of these things? Because the answer tothis question will prove important both in constructing sub-AdS scale networks in Section6 and in formulating the no-go theorem that we prove in Section 7.3, we shall answer thisquestion systematically for each extension map of the network shown in Figure 10b.Most of the extension maps in Figure 10b (e.g. the maps outwards from Σ A : A to A ∪ A (cid:48) and A ∪ A (cid:48) and the maps upwards from A (cid:48) to A ∪ Σ A : A and from A (cid:48) to A ∪ Σ A : A ) were also extension maps in the tree tensor network for the minimally entangledpurification. As such, they are all at least approximate isometries in the sense of Section3.2; depending on the order in which RT surfaces were added to the tree network forthe MEP, several of them will be exact isometries. The extension map XT W flowingdownwards from the horizontal RT surface A (cid:48) ∪ A (cid:48) is not an extension map from the treetensor network for the MEP; however, since it is a composition of an exact isometry X withtwo MEP extension maps T and W , and since T and W are exact isometries regardless ofthe order in which the edges in the tree tensor network for the MEP were distilled, XT W is still an exact isometry in the final network.There is one remaining extension map that one might also hope would be an isometry:the map upwards from A (cid:48) ∪ A (cid:48) to A ∪ A . In this case, we have no good argument that itshould be an exact, or even an approximate, isometry. In particular, the | φ (cid:105) and | σ (cid:105) edgestates on A (cid:48) and A (cid:48) were constructed to approximate the reduced density matrices of theMEP on H A (cid:48) and H A (cid:48) individually . The upwards extension map will be an approximateisometry if (and more importantly only if) the product of these reduced density matricesapproximates the reduced density matrix of the entire MEP on H A (cid:48) ⊗ H A (cid:48) . Unfortunately,the assumptions we have made about the MEP thus far are only sufficient to show that themutual information I ( A (cid:48) : A (cid:48) ) is subleading in G N ; they do not imply that it is zero. Wehave no solid reason to believe that the reduced density matrix on H A (cid:48) ⊗ H A (cid:48) is actuallyclose to a product state with respect to the trace norm. Furthermore, we will see in Section7.3 that there is good reason to think that this cannot be the case. Henceforth, we shall refer to an extension map of this kind as a “moral” isometry.In using this terminology, we mean that — even though the extension map from A (cid:48) ∪ A (cid:48) to A ∪ A in Figure 10b is unlikely to be an exact or even approximate isometry — weexpect that small errors introduced in the “bottom-half” tensor XT W will not blow updramatically when mapped through the moral isometry to alter the “top-half” boundarystate on A ∪ A . Since the moral isometry preserves the normalization of the full-rank state | φ (1) (cid:105)| σ (1) (cid:105)| φ (2) (cid:105)| σ (2) (cid:105) , and also preserves the entanglement entropy of this state to leadingorder, it is tempting to think of the moral isometry as a combination of an exact isometryand some other, non-isometric operator that acts only on a subleading number of degreesof freedom. We will revisit the issue of moral isometries in Section 6, where we invoke themoral isometry condition to argue that distilling the bottom-half tensor XT W into a tree The fact that this map is not even an approximate isometry is somewhat problematic for interpretingthe bottom-half state of the tensor network in Figure 10b in terms of the surface-state correspondence. Itimplies that the bottom half state is not an approximate purification of the reduced density matrix of theoriginal holographic state on H A c . However, we still hope that the relevant smooth entropies will behavecorrectly. – 36 –ensor network of its own will result in a global tensor network that still approximatelyreproduces the original boundary state of the CFT. Of course, nothing in our construction thus far has limited us to considering the case ofa bipartite boundary partition A : A . One might equally well wish to consider a moregeneral multipartite partition, where a boundary region A is partitioned into n connectedsubregions as A : A : · · · : A n . For simplicity, we assume that each subregion A i only shares a boundary with at most two neighbors, A i − and A i +1 . In 2 + 1 spacetimedimensions, any connected partition can be ordered such that this is true. By analogy withthe holographic entanglement of purification conjecture, one would expect that minimizingthe quantity S ( A A (cid:48) ) + S ( A A (cid:48) A A (cid:48) ) + · · · + S ( A A (cid:48) . . . A n − A (cid:48) n − ) (5.17)over all possible purifications into spaces A (cid:48) ⊗ · · · ⊗ A (cid:48) n would correspond to minimizing theareas of surfaces that partition the entanglement wedge of A = A ∪ · · · ∪ A n into n distinctsubregions. Each term in (5.17) should correspond to the area of one entanglement wedgecross-section, and since minimal surfaces cannot cross one another, the minimization ofeach area in the bulk can be performed independently up to subleading corrections. If ourextended conjecture is correct, this implies that each term in the entropy sum (5.17) canalso be minimized independently, at leading order.In the special case of two subregions (i.e., n = 2), the quantity given in (5.17) reducesto the one minimized in defining the entanglement of purification (5.1). If the MEP ofthis partition has entropies given by the areas of extremal surfaces contained within theentanglement wedge, then one can readily construct an “ n -to-one” network for the CFTstate | ψ (cid:105) by repeating the procedure detailed above. This is sketched for n = 3 in Figure11. Note that for n ≥
3, the n -to-one network contains bonds that correspond to extremalsurface subregions with areas that remain finite even when the ultraviolet CFT regulatorand corresponding bulk radius regulator are removed. If the holographic entanglement ofpurification conjecture and its extensions hold down to the string and Planck scales, asassumed, then the areas of these subregions can be chosen to be arbitrarily small relativeto the AdS scale by choosing suitably small boundary subregions (so long as we remainabove the string and Planck scales). The tensor network would therefore be capturingthe bulk geometry at sub-AdS scales. In [47], a slightly different notion of multipartite entanglement of purification was shown to satisfyconstraining inequalities involving linear combinations of entanglement entropies, which are also satisfiedby the corresponding geometric quantity. Similar proof techniques, both holographic and informationtheoretic, should suffice to place analogous constraints on the quantity given in equation (5.17). Since theholographic entanglement of purification conjecture was originally motivated in [15] by showing that thegeometric bulk quantities and the information-theoretic boundary quantities satisfy the same constraints,our construction is equally well-motivated. One important exception occurs when two neighboring entanglement wedge cross-sections Σ A : A andΣ A : A undergo a “phase transition” in the sense that they “jump” discontinuously across the RT surface – 37 – A : A Σ A : A A A A A c A (cid:48) A (cid:48) A (cid:48) (a) A A A φσ φσ φ σφσφ σV (cid:48) V V V (cid:48) V V (cid:48) A (cid:48) A (cid:48) A (cid:48) (b) Figure 11 : A rough sketch of the procedure for constructing a “3-to-one” network for aholographic CFT state. The boundary subregion A is given a connected partition A : A : A , and a minimally entangled purification for this partition is found by minimizingthe sum S ( A A (cid:48) ) + S ( A A (cid:48) A A (cid:48) ) . In (a), this partition is shown along with the surfacesΣ A : A and Σ A : A whose minimal areas should correspond to the minimization of this sum.If the MEP has entropies corresponding to areas of surfaces in the entanglement wedge of A ∪ A ∪ A , then a “top-half” tree tensor network can be constructed as in (b), whichcan then be completed by an isometry on the purifying space to obtain a tensor networkfor the global CFT state.Thus far, we have implicitly assumed that we are working in 2+1 spacetime dimensions.While the construction detailed above will certainly work in higher-dimensional spacetimes,it would no longer be completely accurate to claim that the n -to-one network captures thegeometry at sub-AdS scales; while RT surface subregions can be chosen with finite, sub-AdS width, they also have at least one transverse direction that extends all the way to theboundary of the spacetime. Localizing the information on a single Ryu-Takayanagi surfaceto bounded, sub-AdS bulk regions in higher-dimensional spacetimes is a subtle procedure,and requires techniques from Section 6. We will therefore comment on this generalizationbriefly in Section 6.2. even when the middle boundary region A is made arbitrarily small. In this case, the region of the RTsurface that is “skipped over” by this phase transition cannot be directly probed by a single applicationof the holographic entanglement of purification, and may in fact have an area well above the AdS scale.Nevertheless, techniques in Section 6 should still allow us to construct a sub-AdS tensor network within sucha region by using multiple, iterated applications of the holographic entanglement of purification conjecture.(This complication does not arise for bulk geometries that are close enough to a 2+1 AdS vacuum.) – 38 – Iteration and Sub-AdS Locality
In Section 3, we showed that the Ryu-Takayanagi formula, together with constraints onthe smooth min- and max-entropies that follow from the extensive growth of the ordinaryR´enyi entropies, is sufficient to construct geometrically appropriate tensor networks corre-sponding to an arbitrary discretization of the bulk by non-intersecting extremal surfaces.The resulting tensor networks are always tree tensor networks, where each bond of thenetwork is associated to an entire extremal surface. In tree tensor networks, informationis never localized within a single Ryu-Takayanagi surface.In Section 5, however, we were able to show that the holographic entanglement ofpurification conjecture can be used to associate network bonds to subregions of a singleRyu-Takayanagi surface. By assuming a natural extension of the holographic entanglementof purification conjecture to multipartite partitions of the boundary, these subregions couldbe made to have finite size even when the CFT and bulk regulators are removed. If theholographic entanglement of purification conjecture for multipartite boundary partitionsholds up to stringy and quantum corrections, then these extremal surface subregions canbe made arbitrarily small compared to the AdS scale in the semiclassical limit G N → λ → ∞ . We would like to go further by achieving some form of sub-AdS locality not only inthe sense of dividing bonds along a single Ryu-Takayanagi surface, but in the generalgranularity of the network. More precisely, we would like to construct tensor networkswhere each tensor is associated to a bulk subregion that occupies a volume well below (cid:96) d − AdS . We approach this problem by proposing a procedure to construct a tensor networkfor discretizations of the bulk whose discretization scale lies well below (cid:96)
AdS . We begin by considering the simplest tensor network that our prior techniques were unableto address, namely the four-tensor square network shown in Figure 12a that correspondsto a discretization of the bulk by two complete, intersecting extremal surfaces. One way toconstruct such a network involves a process of iteration: begin by constructing a one-loopnetwork like the one shown in Figure 10b, where the “bottom half” of the discretization in12a is represented by a single tensor, then divide this tensor into two tensors that representthe bulk subregions in the discretization. We reproduce the one-loop network in Figure 12bwith a relabeling of the tensors that is slightly more convenient for our current purposes.This process of iteration is functionally almost identical to the inductive procedure forconstructing tree networks detailed in Section 3.2.The bulk state assigned to the “bottom half” of the one-loop network in Figure 12b,as defined in Section 3, is the state comprised of the bulk tensor W along with the edgestates φ and σ that correspond to neighboring extremal surfaces. As in 3, this state can beapproximated by a tree tensor network in which the tensor W is replaced by an expression Note that since the bond dimension of the corresponding network edge goes like e area / G N , the bonddimension will still diverge in the semiclassical limit. – 39 –f the form W A A A (cid:48) A (cid:48) ≈ V A A (cid:48) γf V A A (cid:48) γf φ γγ σ ff , (6.1)where φ is a maximally entangled state on a space of dimension e S ε max ( A (cid:48) A ) , and S ε max ( A (cid:48) A )is the smooth max-entropy of the bottom-half bulk state in the subregion A (cid:48) A . The opera-tors V and V are not themselves isometries. However, when combined with the edge state φ and σ operators on the horizontal edges of the network, they become exact isometriesflowing outwards from the newly created vertical edge.Such a network can always be constructed for any state by entanglement distillation;however, the resulting network is only geometrically appropriate for the discretization givenin Figure 12a if the smooth min- and max-entropies satisfy S ε max ( A (cid:48) A ) = Area(Σ A : A )4 G N + O (cid:18) √ G N (cid:19) and S ε min ( A (cid:48) A ) = Area(Σ A : A )4 G N + O (cid:18) √ G N (cid:19) . (6.2) A A A A (a) A A A c φσ φσ φ σV V W (b) A A A A φσ φσ φ σφσV V V V (c) Figure 12 : Constructing a tensor network for the four-tensor discretization by an iter-ative procedure. (a) A discretization of the bulk into four regions by two intersectingRyu-Takayanagi surfaces, along with the corresponding dual graph. (b) The one-loopnetwork for the boundary partition A : A obtained from the minimally entangled purifi-cation of this partition. (c) The full four-tensor network, obtained from (b) by distillingentanglement out of the bottom tensor W .Arguments given in Section 2.3 would imply equations (6.2) if the tensor network statefor the bulk subregion represented by W has entropies given by the areas of extremal sur-faces in the entanglement wedge of A ∪ A . However, unlike the minimally entangledpurification, this state has not previously been conjectured to have this property. Never-theless, such a conjecture is closely analogous to the conjectures used in Section 5, andfollows intuitively from the surface-state conjecture [50]. The state defined by W and itsneighboring edge states is naturally associated to the “bottom half” bulk subregion, and– 40 –o the area of the extremal surface dividing A and A is unquestionably the “natural”geometric quantity that would be associated to the smooth min- and max-entropies of thisstate.If this is indeed the case, and the bulk tensor W can be distilled into a tree networkfor the bottom half of the four-tensor discretization from Figure 12a, then the resultingexpression is a tensor network for the four-tensor discretization whose bond dimensionsmatch the areas of extremal surfaces in the bulk. This completed network is sketched inFigure 12c. The only remaining question to ask is whether the boundary state of thisnetwork accurately reproduces the original “target” CFT state.Arguments given in Section 5 imply that the one-loop network of Figure 12b wellapproximates the original CFT state. Since the tree network for the bottom half statewell approximates the original bottom half state in the one-loop network of Figure 12b,the final network should still well approximate the boundary CFT state, so long as theerror induced in the bottom half state from its approximation as a tree network doesn’tdramatically increase in size when the rest of the network is added.Of course, here we run into something of a problem. The extension map upwards fromthe bottom half state is not an exact isometry. Indeed, as we discussed in Section 5.2, it isnot even an approximate isometry in the sense of Section 3.2; instead we proposed that itshould be called a “moral” isometry. We therefore will not have good control over the totalaccumulated error between the state produced by the four-tensor network and the targetstate. This is in contrast to the networks in Section 3 (and Section 5), where we couldprecisely control the total error that could be accumulated, so long as the RT surfaces wereadded in an appropriate order.On the other hand, the edge states on the horizontal RT surface and the top half stateare both normalized quantum states. This means that the upwards flowing map preservesthe norm of the edge states on the horizontal RT surface, which are fully entangled. Hencewe can be relatively hopeful that a generic perturbation to the bottom half state will notbe dramatically blown up in size by the upwards flowing map, and the final state producedby the network in Figure 12c should approximately reproduce the original CFT state.Another way of seeing that the four-tensor network should approximately reproducethe correct boundary state is to compare the exact isometry W in Figure 12b to the finaldownwards-flowing extension map made up of tensors V and V and bottom-half edgestates | φ (cid:105) and | σ (cid:105) in Figure 12c. For the sake of this argument, let the full downwards-flowing extension map be denoted by B . Since the network in Figure 12c was obtained fromthe network in Figure 12b by tree network distillation, B and W must have approximatelythe same action on the reduced density matrices of the | φ (cid:105) and | σ (cid:105) states on the horizontalRT surface, i.e., (cid:107) ( B − W ) φ σ φ σ (cid:107) (cid:28) . (6.3)The condition for the four-tensor network to approximately reproduce the original holo-graphic state is for B and W to approximately agree on the full state for the top half ofthe four-tensor network, i.e., (cid:107) ( B − W ) ρ / T (cid:107) (cid:28) , (6.4)– 41 –here ρ T is the truncated tensor network state for the top half of Figures 12b and 12c.These conditions are inequivalent, as we do not expect ρ / T to be a product state acrossthe two halves of the horizontal RT surface, because the upwards-flowing extension map isonly a moral isometry. However, our version of the holographic entanglement of purificationconjecture implies that the correlations between these two halves are subleading in G N inthe sense of the mutual information. Equation (6.3) can be interpreted as taking a weightedaverage of ( B − W ) over the product of the marginal distributions on each half of ρ T , whileequation (6.4) can be interpreted as taking a weighted average over the joint distribution.We expect that, barring unlikely disasters, equation (6.3), which follows from conjecturesgiven in Section 5, should imply equation (6.4).We will revisit the question of exact, approximate, and moral isometries in Section 7.3,where it is shown that any geometrically appropriate tensor network of the form shown inFigure 12c must have some moral isometries, as constructing such a tensor network withstronger bulk-to-boundary isometry conditions is inconsistent with the dynamics of theoriginal CFT state on the boundary. It is fairly easy to extend this construction to tensor networks on arbitrarily fine griddiscretizations of the bulk. To iterate the one-loop (now n -loop) network, a grid is chosenlike the one in Figure 13, where the horizontal surfaces are extremal surfaces, and eachvertical segment is an extremal surface linking the horizontal RT surfaces on either of itsendpoints. The vertical surfaces are chosen via a “top-to-bottom” inductive procedure,where the top point of each segment is fixed at the bottom of the previous segment, whilethe bottom point is chosen to minimize the total area of the surface. In other words, eachvertical segment is the minimal surface connecting neighboring horizontal RT surfacessubject to the constraint that it must continue the vertical segment above it.To extend the one-loop iteration procedure to this discretization, we begin by con-structing an n -to-one network for the top “row” of the grid, the bottom tensor of which isthen distilled into a grid network for the remainder of the bulk by induction. By assumingthat the bulk state represented by the bottom tensor satisfies the surface-state correspon-dence, and hence the holographic entanglement of purification conjecture, this top-rowdistillation procedure can be repeated until a network is produced for the entire grid. Notethat because the bulk spacetime is curved, the grid on which we discretize cannot have rightangles everywhere: in a negatively-curved spacetime, there cannot exist a quadrilateral,bounded by geodesics, with each corner having an angle of π/
2. The iterative procedurefor constructing a tensor network on this grid will generally produce vertical surfaces thathave kinks as they pass through each horizontal RT surface.It is worth noting that this procedure can be used to generalize the n -to-one networksof Section 5 to higher-dimensional spacetimes. In Section 5, the holographic entanglementof purification conjecture was used to construct a tree tensor network for the entanglementwedge of a boundary region that was partitioned as A : · · · : A n . This tree tensor networkwas then extended to the global spacetime by an isometry on the purifying space. Gen-eralizing this procedure to higher dimensions requires constructing a tensor network for a– 42 – a) (b)
Figure 13 : (a) A grid discretization for vacuum
AdS , for which one can find a tensornetwork by iterating an “ n -to-one” loop network from the top to the bottom of the grid.As explained in the text, not all surfaces in this discretization will meet at right angles. Ifthe network is constructed “top-to-bottom”, then the bottom of each vertical segment willmeet the corresponding horizontal RT surface at a right angle. (b) The dual graph of thisdiscretization, which forms the underlying geometry for a holographic tensor network.boundary partition that is a grid , not simply a one-dimensional chain. Such a partitioncannot be represented by a tree tensor network, and will generally require a grid networkthat looks more like the one sketched in Figure 13. Since we now know how to constructgrid networks in 2 + 1 dimensions by iteration, however, we are able to construct n -to-onenetworks in higher-dimensional spacetimes; one simply constructs the minimally entangledpurification for a “chain” partition A : · · · : A n , then distills the other grid directionsusing the iterative techniques explained above. By repeating this procedure inductively, itis possible to construct a tensor network for a grid discretization in spacetimes of arbitrarydimension.All our constructions in this section, as well as those in Section 5, are built on theholographic entanglement of purification conjecture, which itself has yet to be proven andmay not be exactly true as stated. Moreover, even if the holographic entanglement ofpurification conjecture is itself valid, the various generalizations of it that we used inthis section could be one step too far. However, we believe that the second possibilityis considerably less likely than the first. All of our constructions follow from the samebasic guiding principle as the holographic entanglement of purification conjecture itself:that there should exist a state associated to any convex bulk surface (the “surface-statecorrespondence” [50]) and that by minimizing an entropy or a sum of entropies over allpossible purifications of a reduced density matrix, we can obtain something close to thestate associated to the surface that minimizes the corresponding area or sum of areas.– 43 –aving completed our description of the details of our general construction, we offer twofinal motivations for believing that it is the most natural way to construct a geometricallyappropriate tensor network, assuming one exists. The first is that radial flow in AdS/CFThas long been understood to be a form of renormalization group flow for the boundary CFTstate. Since renormalization of a state is best understood as a process of disentangling andremoving redundant degrees of freedom [7, 8, 19], the radial flow of a boundary subregionto a Ryu-Takayanagi surface should correspond to performing some position-dependent RGflow on the boundary. Our procedure for constructing holographic tensor networks consistsof disentangling and discarding as many degrees of freedom as possible on a boundarysubregion without changing the reduced density matrix on the complementary subregionof the boundary. If renormalization group flow is a philosophically correct approach todescribing the bulk in AdS/CFT, then our constructions should also be valid, with theadditional benefit that they have the potential to work at sub-AdS scales.A second supporting motivation for our construction is that when we construct tensornetworks through entanglement minimization and distillation, the resulting networks havebulk legs that are “as small as possible” while still approximately preserving the boundarystate. The usual Ryu-Takayanagi inequalities for tensor networks [7] imply that the bonddimensions of a tensor network have lower bounds that are determined by the bulk geom-etry. If any geometrically accurate tensor network exists for a given CFT boundary state,then it should be found by a maximally efficient minimization procedure; we believe thatour minimization procedure is the most obvious one to consider. In this section we consider the effects of quantum fluctuations of the spacetime geometry,which we mentioned briefly in Section 3.1 when discussing subleading entanglement in ourtensor networks. We first argue that these fluctuations are best understood as quantum su-perpositions of tensor networks. We then point out the existence of a quantum uncertaintyrelation between the areas of intersecting holographic entropy surfaces, whereby a veryprecise measurement of the area of one such surface causes the area of the other surface togrow dramatically in size. This poses some issues for the interpretation of tensor networkswhose underlying bulk discretizations contain intersecting Ryu-Takayanagi surfaces, suchas those constructed in Section 6. We close the section by formalizing these issues in theform of a no-go theorem that limits the isometry conditions that can be imposed on thebulk-to-boundary maps of such networks.
In this paper, we have generally aimed to describe a holographic boundary state with a single tensor network that was expected to capture the bulk geometry. In a full theory ofquantum gravity, however, the bulk geometry itself is expected to be quantum mechanical,and therefore subject to quantum fluctuations around some semiclassical background. InSection 3.1, we proposed that the necessity of including in our networks some subleadingedge states | σ (cid:105) , which are not maximally entangled, is intimately related to the existence– 44 –f these fluctuations. Specifically, we argued that the O (1 / √ G N ) log rank of these statessuggests that they correspond to fluctuations in the areas of extremal surfaces in AdS/CFT.Another approach would be to have a single tensor network encode a single, non-fluctuating bulk geometry. In this interpretation, a holographic boundary state should bedescribed not as a single tensor network that encodes the quantum fluctuations in the geom-etry, but as a weighted quantum superposition of networks, each of which describes a (veryslightly different) non-fluctuating bulk geometry. The idea that fluctuations over differentgeometries correspond to taking a quantum superposition of different tensor networks hasbeen previously discussed in [29, 30, 51, 52].We can replace our single network with a superposition of tensor networks by reinter-preting its subleading tensors. In Section 3.1, we suggested that because of their relativelysmall dimension, the σ -legs could be thought of as thin “cobwebs” adhering to the thick“girders” of the main network of φ -legs. Instead of thinking of the cobwebs as part ofthe tensor network, however, we can choose to interpret them as determining the weightswith which the many different “girder-only” networks are superposed against one another.In doing so, one would eliminate the cobwebs associated with geometric fluctuations inany single network, instead using them to weight a superposition of “fixed” geometries.One major advantage of this approach, as we will see, is that the holographic state can bedescribed, with high accuracy, by a superposition of only O (1 / √ G N ) fixed-geometry tensornetworks; this is a huge improvement over the e O (1 / √ G N ) -rank Hilbert space of fluctuationsthat was necessary in Section 3.1.To be more precise, consider an edge state | σ (cid:105) in a holographic tensor network. Bymeasuring this state in its Schmidt basis, we obtain a tensor network with no cobwebson the corresponding edge. If we measure all cobwebs in the network according to thisprocedure, then the resulting network for any measurement outcome has no cobwebs onany edge. To write the original holographic state in terms of these measured networks, weneed to use a superposition of the networks associated with every possible measurementoutcome, with each network weighted by the corresponding Schmidt coefficient in eachmeasured cobweb state | σ (cid:105) . Given the absence of the cobwebs that we associated withfluctuations in geometry, one might reasonably suggest that each of these tensor networkcorresponds to a fixed, non-fluctuating geometry. The original, holographic state can thenbe written as a superposition of appropriately weighted geometries.So far, this “superpositions” framework is merely a different way of interpreting thetensor networks we already constructed in Section 3. However, interpreting the full, semi-classical tensor network as an ensemble of “girder-only” networks with slightly differentgeometries suggests that we should allow the dimensions of the girders to vary for differ-ent networks in the superposition. This can be accomplished with a slight adjustment tothe block-averaging procedure from Section 3.1, in which we allow different blocks to havedifferent widths. In fact, we will now show that for any fixed error ε , the “target” holo-graphic state can be approximated to within tolerance ε by choosing the block widths to In such an approach it would be natural to represent long range entanglement of geometry fluctuationsusing cobwebs that have legs extending nonlocally to many different girders. – 45 –e proportional to εe E O ( √ G N ), where E is the block-averaged eigenvalue of the modularHamiltonian K = − log ρ . This makes the total number of blocks, and hence the totalnumber of girder-only networks in the superposition, order O (1 / √ G N ) . To define a general block-averaging prescription, let n be an index that labels theeigenvalues of K as in Section 3.1, and let w n be the width of the block containing the n th eigenvalue. If p n = e − E are the eigenvalues of the density matrix ρ , then the one-normerror induced by replacing each p n with the average eigenvalue within its block is given by ε = (cid:90) | p n − p εn | d n ≈ − (cid:90) dp n dn w n d n, (7.1)where we have approximated p n as being roughly linear within each block and approximatedthe index n by a smooth function. To find the optimal block-averaging procedure forrepresenting ρ , we want to minimize the number of blocks—and hence the total numberof tensor networks that must be superposed to describe ρ —subject to the constraint of afixed error ε . The total number of blocks is given by N blocks = (cid:90) w n d n, (7.2)and so the function w n that minimizes the total number of blocks subject to the constraintgiven by (7.1) satisfies 1 w n = λ dp n dn , (7.3)where λ is a Lagrange multiplier that does not depend on n . We may solve for the valueof λ by plugging this expression back into (7.1), yielding √− λ = 12 ε (cid:90) (cid:114) − dp n dn d n. (7.4)With respect to the smooth index n , the density of states is given by D ( E ) = dndE , (7.5)and so the eigenvalues p n = − e − E of the density matrix satisfy dp n dn = − e − E D ( E ) . (7.6)It follows from (7.4), then, that the Lagrange multiplier λ is given by √− λ = 12 ε (cid:90) (cid:113) D ( E ) e − E d E. (7.7)Using expression (7.3) for the optimal block widths w n , we find that the optimal numberof blocks is given by N blocks = (cid:90) w n d n = 14 ε (cid:20)(cid:90) (cid:113) D ( E ) e − E d E (cid:21) . (7.8)– 46 –n Section 2.3, we argued that the spectrum of K is tightly constrained around theleading saddle point in D ( E ) e − E . Near this saddle point, the function can be approximatedby a normalized Gaussian of width O (1 / √ G N ), i.e., D ( E ) e − E = O ( (cid:112) G N ) e − O ( G N )( E − S ) . (7.9)The optimal number of blocks (7.8) may therefore be computed as N blocks = 1 ε O ( √ G N ) , (7.10)as we claimed above. The optimal block widths, w n , satisfy w n = 2 √− λ (cid:115) − dndp n = εO ( G / N ) (cid:113) D ( E ) e E . (7.11)Near the saddle point, the variation in the right hand side of (7.9) is subleading and so wehave D ( E ) ≈ O ( (cid:112) G N ) e E . (7.12)It follows that the block widths in the optimal block-averaging procedure are proportionalto e E . Assuming that the modular energy E has a holographic interpretation as A/ G N ,where A is the area of the minimal surface, this is the right size for the tensor networkgeometry to match the semiclassical geometry of the holographic state.Unlike our construction in Section 3, however, the optimal block-averaging procedurerepresents the semiclassical holographic geometry as a superposition of O (1 / √ G N ) ratherthan e O (1 / √ G N ) “fixed-geometry” networks. We take this as a suggestion that the super-position framework, in which one allows the different tensor networks in the ensemble tohave slightly different “fixed” geometries, is a more efficient and informative description ofthe holographic state.In this superposition framework, it is natural to interpret a single tensor networkin the superposition as corresponding to an (approximate) eigenstate of the bulk areaoperator [29, 30, 53, 54]. If the usual bulk state generated by a path integral correspondsto a canonical ensemble of the area operator, then the state of a single tensor networkcorresponds to a microcanonical ensemble which takes values over a tiny range of areas. This interpretation is compelling because it makes the flat entanglement spectrum of manytensor network models (in which all R´enyi entropies are equal) into a feature rather thana bug. In a single tensor network, the R´enyi entropy is flat; to get a state with a non-flatspectrum, one must take superpositions of different geometries. Having said all this, we will now identify a serious issue with the approach of [29, 30].Namely, area-eigenvalue states with a flat or nearly-flat entanglement spectrum do not We do not expect that the area spectrum will contain large exact degeneracies, but because of the (cid:15) -smoothing, we can approximate an area spectrum with exponentially small gaps with a degenerate spectrum. – 47 –orrespond to a bulk geometry that is similar to the original state. As a result, it is notpossible to construct geometrically appropriate tensor networks that are in a microcanonicalarea ensemble for multiple directions simultaneously.The problem is that in general relativity, the area and boost angle are canonicallyconjugate quantities obeying a Heisenberg uncertainty relation [55, 56]:∆ E ∆ t ≥ , (7.13)where the modular Hamiltonian is E = area / G N + O (1) [57] and the conjugate time t is the boost angle (in hyperbolic radians). Obtaining a flat entanglement spectrum in aholographic state requires constraining ∆ E to a small, O (1) number, and hence measuringthe area of the corresponding surface to within a tolerance of O ( G N ) . Since a single bin in our construction measures E to an accuracy of O ( (cid:15) ), (7.13) impliesthat the uncertainty in the corresponding boost angle is O (1 /(cid:15) ). This is quite large, and infact it is large enough to take us out of the validity of the static slice regime. In particular,introducing a large crease of extrinsic curvature at the horizontal Ryu-Takayanagi surfaceof Figure 12c will make it so that the area of the vertical HRT surface (which follows a spacetime geodesic and therefore no longer lies on the creased slice) will have a significantlygreater area than the minimal surface on the original static slice, as shown in Figure 14. Since this is true for either sign of the boost angle, the expectation value of the area givenan uncertain boost is also larger.This uncertainty principle implies that if we start by distilling information on thehorizontal surface, consider just one term of the resulting superposition, and then attempta “vertical” distillation, then the leading O (1 /G N ) part of the vertical entropies will belarger than on the original static slice. In other words, it will not be possible to constructa geometrically appropriate tensor network for a single term of the superposition. Thisproblem is related to the issues for dynamical tensor networks that we discuss in section8.4.
The uncertainty relationship discussed in the previous section shows that we cannot si-multaneously measure the areas of horizontal and vertical Ryu-Takayanagi surfaces withhigh accuracy. This might make one wonder whether it is really possible, in a single ten-sor network, to localize information on both the vertical and horizontal Ryu-Takayanagisurfaces simultaneously, as required for our iterative constructions in 6 to be geometricallyaccurate.In this section, we will prove that there is indeed such an obstruction preventingthe construction of certain kinds of geometrically accurate tensor networks with crossing In evaluating the entropy of the HRT surface, it is helpful to use a boost-invariant UV cutoff surface,so that the area of surfaces on the slice is independent of the boost angle. In the exact AdS/CFT bulk state, this spuriously large entropy must disappear when we take all termsin the superposition, due to destructive interference. However, it can be difficult to keep approximationsunder control when there is destructive interference among a large number of terms, since small errors canaccumulate and leave a substantial remainder. – 48 – igure 14 : A spacelike slice of vacuum
AdS formed by boosting one half of the t = 0static slice. This boost introduces a nontrivial extrinsic curvature at the dashed line inthe surface. The original RT surface for a particular boundary region on the t = 0 slice,sketched by a purple line, is no longer the entangling surface for the corresponding boundaryregion in the half-boosted state. Instead, one must consider the HRT surface, sketched inblue, which has strictly larger area.Ryu-Takayanagi surfaces, even in the simplest case of a four-tensor network (section 6.1).However, our no-go theorem is only valid for tensor networks having approximate isometryproperties that are stronger than those of our actual network. Thus, we can optimisticallyhope that our tensor network constructions are still valid. When constructing sub-AdS scale tensor networks in Sections 5 and 6, we found thatthe bulk-to-boundary “extension maps” associated with a particular network were not gen-erally exact isometries. In the n -to-one loop networks of Section 5, at least the “downward-flowing” map could be shown to be an exact isometry — or, depending on the order in whichthe “top-half” RT surfaces were distilled in constructing a tree network for the MEP, atleast an approximate isometry in the sense of equation (3.25). In the full sub-AdS networkof Section 6, however, we found that the bulk-to-boundary maps in our network could not generally be proven to be exact or approximate isometries, and were in fact only “morally”isometric in the sense that they preserved the normalization of the state and preserved theentanglement entropy to leading order. The non-exactness of these isometries stands incontrast to the holographic tensor network toy models of AdS/CFT introduced in [9]. If not, then we believe the construction in 5, where we localize information on a single Ryu-Takayanagisurface, will still be valid. – 49 –n this section, we prove that geometrically appropriate tensor networks for genericbulk discretizations of static states in AdS/CFT cannot , in fact, have bulk-to-boundaryextension maps which are all exact or even approximate isometries. We formalize this forthe four-tensor network for vacuum
AdS of section 6 in the following theorem: Theorem 1.
No four-tensor network of the form shown in Figure 12c can simultaneouslysatisfy the following four properties:(i) Each leading edge state | φ (cid:105) is maximally entangled on a Hilbert space of dimension e S ± o ( S ) , where S is proportional to the area of the corresponding bulk surface, satis-fying S = area4 G N + o (cid:18) G N (cid:19) . (7.14) (ii) Each subleading edge state | σ (cid:105) is submaximally entangled on a Hilbert space of di-mension e o ( S ) . (iii) The four extension maps that map either side of either RT surface to the boundaryare all approximate isometries in the sense that V † V ≈ (7.15) for any such map V , where this approximation means that V † V is close to the identityin the operator norm.(iv) The boundary state of the tensor network approximately reproduces the boundary stateof the AdS vacuum with high fidelity. Suppose that the four-tensor network shown in Figure 12c does satisfy all conditionsgiven in Theorem 1. We denote the “upwards-pointing” extension map from the horizontalRT surface as T , and the “downwards-pointing” map as B . In other words, if we were tocollapse all the tensors in the top and bottom halves of Figure 12c, excluding the tensorson the horizontal RT surface itself, then the resulting tensor network would have only T as its top-half tensor and B as its bottom-half tensor (sketched in Figure 15a). We denotethe actual CFT state as | ψ CFT (cid:105) , and the tensor network state as | ψ TN (cid:105) . Assumption (iv)of Theorem 1 ensures | ψ CFT (cid:105) ≈ | ψ TN (cid:105) . (7.16)From the form of the tensor network shown in Figure 15a, we see that | ψ TN (cid:105) has the form | ψ TN (cid:105) = ( T ⊗ B ) | φσ (cid:105) , (7.17)where | φσ (cid:105) represents the combined pure state of all edge states | φ (cid:105) and | σ (cid:105) on the horizontalRT surface.Let K CFT B be the modular Hamiltonian of the bottom-half boundary state ψ CFT A A , and e iK CFT B the corresponding boost operator. Since this operator is unitary, we have e iK CFT B | ψ CFT (cid:105) ≈ e iK CFT B | ψ TN (cid:105) . (7.18)– 50 – A A A φ σ φσTB (a) A A A A φ σ φσ φσφσV V V V e iK σ ζ (b) Figure 15 : Figures to accompany the no-go theorem for the four-tensor network. (a)The extension maps for the top and bottom halves of the horizontal RT surfaces are drawnexplicitly as tensors T and B . (b) The boost operator on the “bottom half” of the boundary, A A , can be represented on the tensor network by a unitary operator e iK σ that acts onlyon the subleading edge states | σ (cid:105) . The graph cut ζ , sketched here, has dimension given toleading order by the combined size of the | φ (cid:105) Hilbert spaces on the vertical RT surface.Since the reduced state of the tensor network on A A approximately reproduces thereduced state of the CFT on the same region, their modular Hamiltonians approximatelyagree. It follows that the action of the modular Hamiltonian on the CFT state can berepresented in the tensor network as e iK CFT B | ψ CFT (cid:105) ≈ e iK TN B | ψ TN (cid:105) , (7.19)where K TN B is the modular Hamiltonian of the bottom-half boundary state ψ CFT A A in thetensor network.We see immediately from the form of the tensor network state given in equation (7.17)that this modular Hamiltonian takes the explicit form K TN B = − log ψ TN A A = − log Tr A A ( T BφσB † T † ) . (7.20)Condition (iii) of Theorem 1 ensures that T † T is close to the identity, ensuring that thepartial trace over A A in the above expression can be replaced by a partial trace over The modular Hamiltonians may not actually be close in the sense of the operator norm; however, theiraction on the global states | ψ TN (cid:105) and | ψ CFT (cid:105) are approximately the same. – 51 –he domain of T , which we call H f ⊗ H γ . Here, as in Section 3, H f corresponds to thesubleading states | σ (cid:105) while H γ corresponds to the maximally entangled states | φ (cid:105) . Themodular Hamiltonian of the reduced tensor network state on A A therefore satisfies K TN B ≈ − B (log Tr fγ φσ ) B † . (7.21)In other words, the modular Hamiltonian of the tensor network on the bottom-half bound-ary region A A can be approximately represented by K TN B ≈ BK φσ B † , (7.22)where K φσ is the modular Hamiltonian of the reduced RT surface-state Tr fγ φσ. Since the edge states | φ (cid:105) are maximally entangled, their modular Hamiltonian whenrestricted to either side of the RT surface is simply a multiple of the identity. This con-tributes only a normalization factor to the overall network. We therefore write the modularHamiltonian of the tensor network on A A as K TN B ≈ BK σ B † , (7.23)up to an additive constant coming from the normalization, which we ignore. Here K σ is anoperator that acts only on the subleading boundary states | σ (cid:105) , and only acts on one sideof the RT surface (in this case, the bottom half).Returning to the tensor network expression for the modular flow of the CFT given inequation (7.19), we see that the boost operator on the bottom half of the boundary CFTstate can be represented on the tensor network as e iK CFT B | ψ CFT (cid:105) ≈ Be iK σ B † | ψ TN (cid:105) . (7.24)From the expression for the tensor network state given in (7.17), we may rewrite thisexpression as e iK CFT B | ψ CFT (cid:105) ≈ ( T ⊗ B ) e iK σ | φσ (cid:105) , (7.25)where we have used condition (iii) of Theorem 1 to ensure that B † B is close to the identity.This final tensor network representation for the boosted CFT state is sketched in Figure15b.Equation 7.25 essentially tells us that the modular flow of the CFT on the boundaryregion A A can be represented by a unitary operator that acts only on the subleadingstates | σ (cid:105) on the horizontal RT surface. This conclusion, however, contradicts assumptions(i) and (ii) of Theorem 1, which restrict the bond dimensions of the network. To see thiscontradiction, consider the entanglement entropy of the boundary region A A in the CFTstate e iK CFT B | ψ CFT (cid:105) . In the bulk, this operator acts as a boost on the entanglement wedgeof A A . In vacuum
AdS , it is easy to show that the entangling surface of A A in theboosted state has greater area in the boosted state than in the unboosted state. By theHRT formula [4], it follows that the entanglement entropy of A A in the boosted statemust be greater than the corresponding entropy in the unboosted state at leading order,– 52 –.e., S ( A A ) > area(vert) / G N , where area(vert) represents the area of the vertical RTsurface (see Fig. 14). However , this is in direct contradiction a bound derived by Swingle in [7]! There it wasshown that the entanglement entropy of a boundary subregion A in any tensor network isbounded above by log dim ζ for any graph cut ζ that partitions A from its complement.From the tensor network representation of the boosted state given in equation (7.25) andthe graph cut sketched in Figure 15b, conditions (i) and (ii) of Theorem 1 imply that theentanglement entropy of A A in the boosted state satisfies S ( A A ) ≤ area(vert)4 G N (7.26)to leading order in G N . The bound given in (7.26) clearly contradicts the HRT formula forthe boosted state.We conclude that no geometrically appropriate four-tensor network can be constructedfor the
AdS vacuum with approximate isometries from each RT surface to the bound-ary. This provides partial justification for our relatively weak isometry conditions: boththe state-dependent approximate isometry condition of Section 3 and, in particular, the“moral” isometries of Sections 5 and 6. While one might initially think that stronger isom-etry conditions should be possible in a tensor network construction of a holographic state,it turns out that such conditions are incompatible with the dynamics of AdS/CFT. Thegeometrically appropriate four-tensor network constructed in Section 6 avoids our no-gotheorem precisely because some of its extension maps are only moral isometries. A A A A φ σφ φφT T B B Figure 16 : A four-tensor network with explicit entanglement between the right and leftsides of the horizontal RT surface, where all four tensors share a common cobweb state.Adding subleading entanglement like this cannot be used to subvert our no-go theorem.Note that while our proof of the no-go theorem required the existence of a subleadingstate on the horizontal RT surface to absorb the action of the modular Hamiltonian, it A more pessimistic interpretation would be that either the entropy of purification conjecture, or one ofthe additional conjectures that we made in Section 6, fails to hold in the form necessary for the constructionsin Section 6 to go through. – 53 –id not necessarily require that the subleading states take the exact form shown in Figure12c. In fact, the same proof would apply for many different kinds of cobweb entanglement(using the terminology of Section 7.1, where we referred to maximally entangled states as“girders” and subleading states as “cobwebs”), so long as the cobweb states are able toabsorb the modular Hamiltonian on the horizontal RT surface and do not interfere with thesize of the girders. In particular, one could consider a four-tensor network with a genericcobweb state that is entangled among all four quadrants of the network, sketched in Figure16. Such a tensor network still satisfies all the necessary conditions of our no-go theorem,and so cannot have approximate bulk-to-boundary isometries while accurately reproducingthe dynamics of the boundary CFT state.
In this article, we have given a constructive procedure for distilling the spatial geometry of astatic spacetime from the quantum entanglement of a boundary CFT, including structureat sub-AdS scales. We described a general procedure for constructing large-scale “treetensor networks” (Sec. 3), explained how the holographic entanglement of purificationconjecture could be used to partition the information on a single Ryu-Takayanagi surfacebelow the AdS scale (Sec 5), and proposed an iterative approach for constructing a sub-AdS tensor network by distilling the geometry incrementally (Sec. 6). We showed that treetensor networks are always quantum error correcting in the appropriate sense (Sec. 4), andsuggested that our more general, sub-AdS constructions should have similar properties.We also proved an important no-go theorem (Sec. 7) showing that the bulk-to-boundarymaps of any geometrically appropriate tensor network for AdS/CFT cannot be exact orapproximate isometries in the usual sense.While our constructive procedures can always be performed so long as the smooth min-and max-entropies of various surface subregions agree at leading order, our holographic con-jectures ensure that the resulting tensor network geometry also matches the geometry ofthe corresponding AdS/CFT state in the sense that the Hilbert space dimensions of itslegs match the areas of corresponding spacetime surfaces. This constitutes significant dataabout the spacetime geometry, in the sense that a sufficiently large set of bulk area observ-ables can be used to reconstruct the metric, as has been shown explicitly in four spacetimedimensions [58] and likely holds more generally. Furthermore, our holographic conjecturesensure that any two choices of iterative construction for a single bulk discretization shouldproduce tensor networks that differ only at subleading orders in 1 /G N ; in the limit as thediscretization scale is taken arbitrarily small, any two tensor networks corresponding to different discretization schemes should converge up to subleading corrections. So if ourconjectures hold, then for the first time we have used tensor networks to obtain the “it” ofcontinuous spatial geometry from the “qubit” of quantum entanglement for states in fullAdS/CFT.In fact, our dictionary for constructing a tensor network uses only quantum informa-tion properties of the boundary state. This implies that, if we regulate the CFT on alattice, acting on any lattice points with local unitaries does not change the resulting bulk– 54 –eometry. This raises some philosophical puzzles given that entanglement is not a linearquantum observable and therefore cannot be measured by a normal quantum observation[59]. These puzzles may be related to the AMPS firewalls paradox [60–63] and claims thatthe construction of geometry is necessarily state-dependent [64, 65], but in this article wemake no claims about which geometrical features are truly measurable. In describing our constructions, we usually had in mind either the vacuum state or smallperturbations around it. As a result, we have mostly ignored some subtleties than canappear in more complicated states. However, we anticipate that our results can be extendedto essentially arbitrary spacetimes. In particular, it should be possible to use our resultsto probe the geometry of the so-called “entanglement shadow,” i.e., a bulk region thatcannot be probed by normal Ryu-Takayanagi surfaces [66–69]. Such entanglement shadowregions appear, for example, in the region surrounding a massive star or black hole. We canprobe such regions using our iterative construction in Section 6, beginning by dividing RTsurfaces that pass just outside the entanglement shadow into pieces using the holographicentanglement of purification. By iteratively constructing new surfaces anchored to theseRT surfaces, it is clear that we can get farther into the bulk than we could get by startingon the boundary. This raises the question of how deep into a general bulk we can probe. It is particularlyinteresting to ask this question in the context of wormholes extended between multipleasymptotic CFT regions [49, 71–73]. Because our construction relies on taking consecutivebipartite divisions of the system, our construction always requires starting with the fullentangled state of all the asymptotic CFT regions. We can then easily construct a Hilbertspace representing the compact RT surface of an entire connected boundary component.It is not clear, however, that there will always be nontrivial entanglement wedge cross-sections extending to the global RT surface, because it may be the case that the minimalcross-section of the entanglement wedge will close off on the boundary rather than travelingdown the wormhole throat.In light of the above reflections, we believe that the only possible obstruction to con-tinuing deeper into the bulk using our iterative procedure would be a locally minimal areasurface Σ for an entire connected component of the boundary with the property that forany bipartite division of Σ = Σ ∪ Σ , the minimal area surface separating Σ and Σ isalways whichever of Σ or Σ has the least area. Such a surface Σ would be analogous toa Haar-random pure state, for which the entanglement entropy of any bipartite divisionscales as the volume of the smaller subsystem. Even in these cases, however, one mightsometimes be able to go deeper if there exists a nontrivial entanglement wedge cross-sectionfrom Σ to the true global minimum. It would be interesting to confirm this intuition usinggeometrical proofs similar to [68]. For a similar discussion of how entanglement shadows can be probed by extremal surfaces anchored topoints in the bulk, see [70]. – 55 – .2 Bit Threads
Another potential avenue for future work is the apparent connection between our construc-tion and the bit-thread formalism for holographic entanglement [24]. Bit threads describeboundary entanglement in terms of smooth flows on the bulk geometry, with intuitioninherited from the machinery of maximal flows on discrete graphs. In some sense, then,the bit thread formalism is inspired by the notion of discretizing the bulk geometry of aholographic state; discretizing the bulk geometry of a holographic state while preserving itsentanglement structure is exactly what we have done in this paper by constructing tensornetworks for AdS/CFT. Since our construction is largely based on the Ryu-Takayanagi for-mula and its generalizations, and bit threads interpretations of holographic entanglementare formally equivalent to RT, our work could be reframed in the language of bit threads.The bit threads formalism suffers from the fact that known holographic entropy in-equalities [74, 75] seem generally more difficult to prove in the language of bit threads thanin the usual Ryu-Takayanagi picture [76, 77]. It is possible, however, that the bit threadformalism is more useful than the Ryu-Takayanagi formula for describing localization ofboundary entanglement to small “cells” of the bulk. Many times in this work, especially inSection 3.2, we have appealed to notions of “information flow” toward or away from bulksubregions to describe our construction intuitively. This notion is best expressed in thelanguage of bit threads, and one might expect that reframing our work in the bit threadformalism could yield new insight into the information content of bulk subregions in tensornetworks with sub-AdS locality. In particular, one might ask the following: what happenswhen one tries to define a spacetime flow that maximizes the flow through a single bulksubregion in our tensor network construction, or a combination of bulk subregions? Doesthis maximal bulk flow have a natural interpretation in terms of boundary information?Flows of this type were recently discussed in [78]. We leave further analysis of this questionfor future work.
An important aspect of the holographic bulk that cannot be decomposed in our constructionis the geometry of the Kaluza-Klein fiber directions. For example, the ABJM model vacuumis dual to
AdS × S [79], while the N = 4 Super Yang-Mills vacuum is dual to AdS × S [1]. In such states, we cannot possibly use entropy of purification to subdivide the S n factor into smaller pieces, because spherical symmetry implies that all entanglement cross-sections will be symmetrical. In our construction, these fiber directions simply go alongfor the ride, without being subdivided, even though their radius of curvature is of the sameorder as that of the AdS factor, which we do subdivide. Note that, as in all theories with large extra dimensions, the 10 or 11 dimensionalPlanck length will be parametrically larger than the effective Planck scale of the Kaluza-Klein reduced theory on AdS. This raises the question of whether ignoring the fiber direc- Interestingly, it might be possible to do better in excited geometries which are not spherically symmetric. See [80, 81] for a possible idea for how to understand the fiber directions in terms of entanglementbetween different field degrees of freedom. If this idea is correct it would be interesting to combine it withour construction to obtain a discretization of the full space. – 56 –ions might actually allow us to subdivide the AdS factor at a finer scale than we couldif we were also subdividing the sphere. After all, our construction depends on the Hilbertspace dimension being large, and including all the modes of the sphere makes the Hilbertspace larger than it would be otherwise. It seems likely that higher-curvature/stringy cor-rections to the Ryu-Takayanagi formula would prevent this from working, but this subjectbears further investigation.
Another important unanswered question is how to extend our construction to dynamicalsettings. The main inhibition to extending the construction to holographic states withdynamical bulk spacetimes is that while boundary entropies are still given by the areas ofextremal surfaces in the bulk, those extremal surfaces can generally not all be chosen to liein a single spacelike bulk slice [4]. In a static spacetime with Killing time parameter t , bycontrast, all Ryu-Takayanagi surfaces of the t = 0 boundary slice can be made to lie in the t = 0 slice of the bulk. Such a proposal would presumably require either (i) discretizingtime to produce a d -dimensional tensor network for a d -dimensional spacetime [52, 82], orelse (ii) constructing different tensor networks for different Cauchy slices of the bulk [83],which nevertheless give rise to the same boundary state. In the latter case, it is temptingto identify the gauge equivalence of tensor networks with the Hamiltonian constraint of thecontinuum bulk general relativity, since both involve an equivalence of states at differenttimes.Unfortunately, it is impossible for tensor networks on dynamical Cauchy slices to begeometrically appropriate in the same sense as on static slices. Let us consider any bound-ary region R , and attempt to construct a tensor network on a Cauchy slice Σ which does not include the HRT surface X R . By the maximin construction [84], the minimal area cut γ always has less area than the HRT surface: area( γ ) < area( X R ) (for an example, see Figure14). On the other hand, the Swingle bound [7] requires that area( γ R ) ≥ S ( R ) = area( X R ).This is a contradiction, unless we allow the log of the bond dimension to exceed the areaeven at leading order.Perhaps, then, dynamical tensor networks are described by tensor networks that in-clude long, nonlocal links connecting different parts of the network, which carry O (1 /G N )amounts of information. Presumably we could still construct a tensor network by ourminimization procedures. But for a general boundary slice, only a single family of non-intersecting HRT surfaces could be simultaneously placed on the same Cauchy slice andrepresented by edges with geometrically appropriate bond dimensions. The other directionswould have to have information flow exceeding their area.It may still be possible to construct such tensor networks in a compelling way by usingthe modular flow techniques of [85, 86]. In this picture, when two flat slices meet at an HRTsurface with a nonzero boost angle, there is a nonlocal exchange of information along theRT surface due to the modular flow associated with the boost. (Otherwise, the maximinprinciple would be violated.) This clarifies that, although our construction localizes theinformation of the RT surface at sub-AdS scales, this localization must be understood asonly being valid in a particular Lorentz frame of reference.– 57 –n the other hand, since the modular flow relates the dynamical and static cases, thedynamical problems posed by the Swingle bound are to some degree already present in thestatic case. This observation led to our no-go theorem in Section 7.3, which shows thatgeometrically appropriate tensor networks for the AdS/CFT correspondence cannot gener-ally be expected to have approximate bulk-to-boundary isometries when Ryu-Takayanagisurfaces intersect in the bulk. The fact that our distillation procedure depends only on the entanglement structure ofthe state suggests that it may be more broadly applicable to other kinds of entangledstates, perhaps e.g. those that are hypothesized to live on so-called holographic screens[87, 88] in cosmology. Given a quantum state on a lattice, one need only check thatits smooth min- and max-entropies of purification have favorable properties; if so, it willproject a holographic state onto its interior. The surface-state conjecture implies that thequadrilaterals of our tensor network grid (see e.g. Figure 13a) have suitable holographicstates living on their boundaries. In principle, we could use our construction to determinethese boundary states explicitly.We do not expect our quadrilaterals to be associated with “perfect tensors” [9], sincecutting a quadrilateral along the diagonal results in a surface with less area than the twoother sides of the triangle, allowing a nontrivial distillation to be performed along thediagonal. But we do expect that there will be an approximate isometry (in the sense ofSection 3.2) mapping any one of the edges to the other three edges. In general, it will beimportant to prove as many isometry-like relations as possible (subject to the constraintsof our no-go theorem), both for the purposes of quantum error correction and to determinehow sensitive the tensor network is to the precise order in which distillations are performed.This article goes in the direction of starting with a boundary state and analyzing whatthe holographic tensor network must be. A complementary approach would be to startwith the tensors associated with different kinds of geometries, and then synthesize themback together into an arbitrary geometry. In doing so it would be important to check thatall expected isometries continue to hold. It would also be critical to show that, to highaccuracy, the tensor network associated with a geometrical region does not significantlydepend on either its external spatial context, or the methodology used to construct it.If this can be done, then the holographic principle would finally be freed from thestraitjacket of asymptotically AdS boundary conditions. It could be applied equally wellto universes with other asymptotic structures, or even to closed cosmologies! In the lattercase, the tensor network could be evaluated to give some complex number for each possiblechoice of spatial geometry. Such a “tensor network partition function” would in effectdefine a special cosmological state over the space of 3-metrics. It would be interesting todetermine what relationship this special state might have to other proposals for specialinitial conditions, e.g. the Hartle-Hawking state.– 58 – cknowledgments
We would like to acknowledge useful conversations with Ahmed Almheiri, Chris Akers, RaphaelBousso, Xi Dong, William Donnelly, Dan Harlow, Patrick Hayden, Ted Jacobson, Isaac Kim, AitorLewkowycz, Juan Maldacena, Don Marolf, Masamichi Miyaji, Ali Mollabashi, Xiao-Liang Qi, DanRanard, Eva Silverstein, Steve Shenker, Brian Swingle, Tadashi Takayanagi, and Guifre Vidal.We especially thank Patrick Hayden, Brian Swingle, and Michael Walter for sharing with us theirunpublished work on smooth min- and max-entropies in quantum field theory. GP, JS, and AWwere supported by the Simons Foundation (“It from Qubit”), AFOSR grant number FA9550-16-1-0082, and the Stanford Institute for Theoretical Physics. AW was also supported in part bythe John Templeton Foundation, Grant ID
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