Coarse-graining holographic states: A semiclassical flowin general spacetimes

Chitraang Murdia,1,2 Yasunori Nomura,1,2,3 and Pratik Rath 1,2

1Berkeley Center for Theoretical Physics, Department of Physics, University of California,Berkeley, California 94720, USA

2Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA3Kavli Institute for the Physics and Mathematics of the Universe (WPI),

The University of Tokyo Institutes for Advanced Study, Kashiwa 277-8583, Japan

(Received 16 August 2020; accepted 27 August 2020; published 1 October 2020)

Motivated by the understanding of holography as realized in tensor networks, we develop a bulkprocedure that can be interpreted as generating a sequence of coarse-grained holographic states. Thecoarse-graining procedure involves identifying degrees of freedom entangled at short distances anddisentangling them. This is manifested in the bulk by a flow equation that generates a codimension-1object, which we refer to as the holographic slice. We generalize the earlier classical construction to includebulk quantum corrections, which naturally involves the generalized entropy as a measure of the number ofrelevant boundary degrees of freedom. The semiclassical coarse graining results in a flow that approachesquantum extremal surfaces such as entanglement islands that have appeared in discussions of the black holeinformation paradox. We also discuss the relation of the present picture to the view that the holographicdictionary works as quantum error correction.

DOI: 10.1103/PhysRevD.102.086001

I. INTRODUCTION

The holographic principle, as embodied by the AdS/CFTcorrespondence, has led to a tremendous amount of pro-gress in our understanding of quantum gravity. In particu-lar, the realization that entanglement plays a crucial rolein generating bulk spacetime has put the holographiccorrespondence on much stronger footing [1–5]. Thishas led to key insights about bulk reconstruction andsubregion duality, culminating in entanglement wedgereconstruction [6–11]. Interestingly, several of theseinsights are quite general and do not seem to require ananti–de Sitter (AdS) setting in particular, and thus they canbe used to understand features of holography in generalspacetimes [12–15].1A particular manifestation of the above ideas can be seen

in tensor networks (TNs) that serve as useful toy models ofholography [5,25–28]. TNs prepare quantum states with alot of structure and via the process of “pushing” the state

generate a sequence of boundary states, each of whichsatisfies the Ryu-Takayanagi (RT) formula [1,2].2 Thisprocedure involves disentangling certain short-distancedegrees of freedom and coarse grains the state by reducingit to one in a smaller effective Hilbert space. Applying thisprocedure to a general smooth classical spacetime leads to aflow equation in the continuum limit, as we shall reviewlater [15]. The flow equation takes the form3

dxμ

dλ¼ 1

2ðθklμ þ θlkμÞ;

where xμ are the embedding coordinates of a codimension-2 surface σ on which the holographic states are defined, andfkμ; lμg are the future-directed null vectors orthogonal to σ,with θk;l being the classical expansions in the correspond-ing directions. This flow satisfies all of the requiredproperties for it to be interpreted as a disentanglingprocedure resulting in a sequence of coarse-grained states.In this work, we go beyond the classical flow equation by

including bulk quantum corrections. In the TN picture, weinclude these effects by modifying the network such that itincludes nonuniversal tensors/bonds as well as bonds

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

1An early work in this direction is the so-called surface/statecorrespondence [16,17], of which the construction of Refs. [12–15] can be viewed as a covariant generalization. For other workon holography beyond AdS/CFT, see, e.g., Refs. [18–24].

2We distinguish this from the Hubeny-Rangamani-Takayanagi(HRT) formula [3] which applies in time-dependent spacetimes.

3The sign convention for the flow parameter λ in this paper isopposite to that in Ref. [15].

PHYSICAL REVIEW D 102, 086001 (2020)

2470-0010=2020=102(8)=086001(16) 086001-1 Published by the American Physical Society

connecting tensors nonlocally. With this picture in mind,we develop a coarse-graining procedure analogous to theclassical flow equation which pleasantly fits in with ourunderstanding of holography. In the continuum limit, theprocedure leads to a flow equation similar to that in theclassical case:

dxμ

dλ¼ 1

2ðΘklμ þ ΘlkμÞ;

where Θk;l now represent quantum expansions [29].4 Thisis our primary result. It is in line with many results in whichquantum corrections are included by replacing the areaA=4GN with the generalized entropy Sgen [29–33]. Thoughmotivated by TNs, which often face issues in describingtime-dependent situations, our procedure can be appliedquite generally. In fact, we obtain consistent descriptions ingeneral time-dependent spacetimes.Another important progress in understanding holo-

graphy is the view that the holographic dictionary worksas quantum error correction [34–37], where a small Hilbertspace of semiclassical bulk states is mapped isometricallyinto a larger boundary Hilbert space. In our framework, thispicture arises after considering a collection of states overwhich we want to build a low-energy bulk description.Choosing such a collection is equivalent to erecting acode subspace. We argue that while there is no invariantchoice of code subspace in a general time-dependentspacetime, our framework gives a natural choice(s) deter-mined by the coarse-graining procedure. This procedureleads to a one-parameter family of “dualities” depending onthe amount of coarse-graining performed, providing animproved understanding of the holographic dictionary ingeneral spacetimes.

A. Overview

In Sec. II, we first establish the framework in which weare working. We explain how quantum corrections affectthe description of holography in general spacetimes and theassociated HRT formula. In Sec. III, we review the classicalflow equation. In Sec. IV, we motivate our coarse-grainingprocedure with a toy model of TNs, elucidating howfeatures of a state relevant for the quantum-level consid-eration are represented there.In Sec. V, we present our main result, i.e., the procedure

of performing the flow in the bulk at the quantum level,which corresponds to moving the holographic boundary.We also discuss properties of this flow indicating that itcorresponds to a coarse-graining of holographic states. Weelucidate that the way the flow ends can be used as anindicator of qualitative features of the boundary state

describing a given spacetime, using the example of acollapse-formed evaporating black hole. In Sec. VI, wediscuss how the picture of quantum error correction may beimplemented in our framework. Finally, conclusions aregiven in Sec. VII.

II. FRAMEWORK

In this work, we follow and further develop the frame-work of holography for general spacetimes proposed inRef. [12]. In this framework, we consider an arbitraryspacetime M and posit the existence of a dual “boundary”theory that lives on a holographic screen [38], which is acodimension-1 hypersurface H embedded in M. Thishypersurface is foliated by marginally trapped/antitrappedcodimension-2 surfaces called leaves, which we denote byσ. A marginally trapped/antitrapped surface σ is defined bythe property that σ has classical expansion θ ¼ 0 in one ofthe orthogonal null directions. The proposal is that theboundary theory describes everything in the “interior” ofH, and states of the theory are naturally defined on theleaves σ, which provide a preferred foliation of H intoconstant time surfaces. Based on the covariant entropybound, it is expected that the boundary theory effectivelypossesses AðσÞ=4GN degrees of freedom, where AðσÞ isthe area of a leaf. The AdS/CFT correspondence can beviewed as a special case of this duality, where the holo-graphic screen is sent to the conformal boundary of AdS.Given this setup, it was shown in Ref. [39] that the HRT

formula for computing entanglement entropy can beapplied consistently using a maximin procedure [7], i.e.,for any subregion A of a leaf

SðAÞ ¼ AðγAÞ4GN

; ð1Þ

where SðAÞ is the von Neumann entropy of the reduceddensity matrix on subregion A, and AðγAÞ is the area of theHRT surface γA of A. The entanglement wedge, denoted byEWðAÞ, is defined as the bulk domain of dependence of anybulk partial Cauchy slice ΣA with ∂ΣA ¼ A ∪ γA, which isoften called the homology surface. The entropies obtainedby the above procedure can be shown to satisfy all of thebasic properties of von Neumann entropy and are consis-tent with more constraining inequalities satisfied by holo-graphic states in AdS/CFT [40–42].Now, in order to generalize this framework to the

quantum level, we can follow the simple guiding principleof replacing A=4GN with the generalized entropy Sgen toinclude quantum corrections in the bulk [43,44],

A4GN

→ Sgen ¼A

4GNþ Sbulk; ð2Þ

where Sbulk is the von Neumann entropy of the bulkreduced density matrix on the homology surface which

4We use a modified version of the quantum expansion whichincludes a bulk entropy contribution from an exterior region, asdescribed in Sec. V.

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-2

is appropriately modified at the quantum level. This ismotivated by various examples in which this naturally works[29–33]. Furthermore, Sgen is a natural quantity because it isa quantity that is renormalization scheme independent, andhence is expected to be associated with fundamental degreesof freedom in the UV theory [31,45,46].The generalization to include bulk quantum corrections

requires a refined understanding of the holographic dualitywhich we now turn to. First, we note that the globaldescription of a state involves both the interior and exteriorportions of the holographic screen [38]. Although thegeneralization of the HRT formula we describe appliesto the interior region, it will be important to keep track ofthe exterior as well.Next, the quantum extension of Eq. (2) implies that the

location of the screen on which the holographic theory is

defined needs to be shifted accordingly. Let us consider aspecific global bulk state. We propose that the boundarytheory describing the dynamics of this state lives ona modified version of a Q-screen H0 [30], rather than aclassical holographic screen H. A Q-screen is defined as acodimension-1 hypersurface foliated by quantum margin-ally trapped/antitrapped surfaces, i.e., surfaces that have thequantum expansion Θ ¼ 0 in one of the orthogonal nulldirections. Usually the quantum expansion is defined byincluding a contribution from the von Neumann entropy ofthe interior or exterior region of a leaf which is a simplecodimension-2 surface. In this paper, however, we considerthat leaf σ is given by σint ∪ σext such that σint and σext aresplit by a small regulating region Σϵ on a Cauchy slice Σ,as seen in Fig. 1. This induces a division of the Cauchy sliceas Σ ¼ Σint ∪ Σϵ ∪ Σext. Now, we define the generalizedentropy of σ to be

SgenðσÞ ¼AðσÞ4GN

þ SbulkðΣint ∪ ΣextÞ: ð3Þ

With this definition of generalized entropy, one can definea quantum expansion Θ by the variation of SgenðσÞ underdeformations of σint while holding σext fixed. Using thisdefinition of Θ, we can locate marginally trapped/anti-trapped surfaces self-consistently for any given ϵ. Thelocation H0 of the holographic screen is a Q-screen definedusing this definition of Θ in the limit ϵ → 0.Generalizing the HRT formula of Eq. (1), we postulate

that the von Neumann entropy of a subregion A on the leafσ of a Q-screen can be computed as

SðAÞ ¼ AðA ∪ ΓAÞ4GN

þ SbulkðΣAÞ; ð4Þ

where ΓA is the minimal quantum extremal surface (QES)[44], and ΣA is the homology surface with ∂ΣA ¼ A ∪ ΓA.To find ΓA, we can use a maximin procedure at the quantumlevel [47] applied to general spacelike surfaces containingσ. We treat σext as a single unit that cannot be furtherdivided into subregions, which must be either included inor excluded from A. We assume that the leaf σ is convexwhere σext is treated as an indivisible unit. Thus, SðAÞobtained by Eq. (4) satisfies properties required for it to beinterpreted as the von Neumann entropy of the densitymatrix of subregion A. With this assumption, we will showthat the same applies to any renormalized leaf σðλÞ obtainedfrom σ by our coarse-graining procedure.In AdS/CFT, the regime of validity of the quantum

extremal surface formula [Eq. (4)] has been suggested to beall orders in GN [44]. However, there are subtleties with thedefinition of entanglement entropy for gravitons whichhave not been completely resolved (see, e.g., Ref. [29]). Atthe least, we expect the formula to hold at Oð1Þ, where itcan already lead to a surface different from that obtained by

FIG. 1. The leaf σ is split into σint ∪ σext such that σint and σextare separated by a small regulating region Σϵ on a Cauchy slice Σ.This induces a division of the Cauchy slice as Σ ¼ Σint ∪Σϵ ∪ Σext. We define the location of the holographic screen byrequiring that it is marginally quantum trapped/antitrapped undervariations of σint.

COARSE-GRAINING HOLOGRAPHIC STATES: A … PHYS. REV. D 102, 086001 (2020)

086001-3

using Eq. (1) [48–52]. In order to avoid subtleties withgravitons, one could consider a setup with bulk matterhaving a large central charge c so that the graviton con-tribution is subleading in a 1=c expansion. We expect asimilar regime of validity for Eq. (4) in general spacetimes.Note that in Eq. (4) we have included the area con-

tribution from A in addition to that from ΓA. This is requiredif there is a spacetime region outside the leaf, as is the casein generic spacetimes. In this case, SbulkðΣAÞ receives acontribution from entanglement of bulk fields across A,which is divergent. This divergence is then canceled withthat in the area contribution from A in the first term, makingSðAÞwell defined. The same applies to the AdS/CFT case ifwe impose transparent boundary conditions near theboundary which lead to kinetic terms coupling the interiorand exterior of AdS space [48,49]. In fact, the classicalformula in Eq. (1) must also have the contribution from theboundary subregion area, AðγAÞ → AðA ∪ γAÞ, in thesecases, although this does not affect the minimizationleading to γA and hence the result of Ref. [15].There are special cases in which the area contribu-

tion from A—as well as the corresponding contributionfrom SbulkðΣAÞ—is absent. This occurs when the space-time outside the leaf is “absent,” as is the case if Dirichletboundary conditions are imposed on the Q-screen, orif reflective boundary conditions are imposed in AdS/CFT. Even in this case, however, our coarse-grainingprocedure—which corresponds to moving the leaf portionσint inward—induces the area contribution from A on amoved (i.e., renormalized) leaf σintðλÞ, reflecting the factthat the spacetime continues across σintðλÞ.5

III. REVIEW OF CLASSICAL FLOW

In previous work [15], it was shown that a coarse-graining procedure motivated by TNs can be defined in thebulk at the level of classical geometry. Here we review thisconstruction, which allows us to elucidate a generalizationto include bulk quantum corrections.A key idea is to realize that a TN defines a sequence of

states that can be generated by including fewer tensors,layer by layer, as shown in Fig. 2. For example, one canconsider a state defined on the outermost legs which lives inHilbert spaceHσ. A coarse-grained version of this state canthen be given by a smaller TN that is obtained by peelingoff the outermost layer. This state lives in a smaller HilbertspaceHσ1 , and the TN provides an isometric map fromHσ1to Hσ . The sequence can then continue, giving a series ofHilbert spaces Hσ2 , Hσ3 , and so on.In fact, this peeling-off procedure can be decomposed

further into smaller steps. For any subregion A of a given

boundary, there is an isometric map from the in-planelegs at the RT surface γA to the boundary legs in A [25,26].This implies that a particular subspace of the boundarysubregion legs, corresponding to the in-plane legs at γA,is maximally entangled with the complementary sub-region A via γA, which acts as an entanglement “bottle-neck”; see Fig. 3. All of the other subregion legs can bedisentangled by applying a unitary UA that acts locallywithin A.Therefore, if one is to preserve only long-range entan-

glement while getting rid of short-range entanglement, onecould compress the state down to that defined at the legsof the surface σ0 ¼ γA ∪ A. This reduces the size of theeffective Hilbert space, mapping a pure state in the largerboundary Hilbert space Hσ to a pure state in a smallerboundary Hilbert space Hσ0 . This can be done by consid-ering small subregions of σ and truncating the TN to end atσ0. One useful way to interpret this step is that we areretaining the complementary entanglement wedge EWðAÞ.This step can then be repeated multiple times to generate asequence of states, all of which are increasingly coarsegrained.6

Now, we apply this idea to a general spacetime M byconsidering infinitesimal subregions A of size δ (→ 0) onthe boundary leaf σ. In order to coarse grain, we findthe HRT surface γA and reduce the accessible spacetimeregion to the complementary entanglement wedge, EWðAÞ.Repeating this multiple times involves shrinking the

FIG. 2. ATN defines a boundary state in the Hilbert spaceHσ atthe outer legs. One can, however, also consider “coarse-grained”states defined at inner layers, e.g., states defined in Hilbert spacesHσ1 and Hσ2 .

5This is different from what has been done in the AdS/CFTliterature in the context of TT deformations [53–58], whichcorresponds to (re)imposing Dirichlet boundary conditions ateach step in the coarse graining, i.e., at σintðλÞ for all λ.

6We note that this is similar to the construction suggested inRefs. [59,60], although here we directly use the TN descriptionand its fine structure, as opposed to constructing the TN usinginformation about the boundary state, such as entanglement ofpurification [61,62] or reflected entropy [63].

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-4

domain of dependence at each step by finding new HRTsurfaces anchored to infinitesimal subregions, as seenin Fig. 4.In the continuum limit, this reduces to the original

construction of Ref. [15]; see Fig. 5. Here, we considerthe intersection of the complementary entanglementwedges EWðApÞ for infinitesimal subregions Ap, centeredaround arbitrary points on the leaf, denoted by p. This leadsto a new domain of dependence RðσÞ,

RðσÞ ¼ ∩p EWðApÞ; ð5Þ

which can be interpreted as defining the state on a new“renormalized” leaf σ1 such that the domain of dependence

of σ1 is RðσÞ, i.e., Dðσ1Þ ¼ RðσÞ.7 The HRT prescriptioncan be shown to consistently apply for subregions on thisrenormalized leaf as well, owing to the fact that it is still aconvex surface [15]. Thus, we may interpret this as thespacetime continuum version of the procedure yielding thesequence of states described above using TNs.

FIG. 4. A sequence of coarse-graining steps. At each step, weconsider infinitesimal subregions of size δ (→ 0) and reduce thespacetime region to their respective complementary entanglementwedges.

FIG. 5. Coarse graining over infinitesimal subregions on σ canbe performed by considering the intersection of complementaryentanglement wedges. This leads to a domain of dependenceRðσÞ which corresponds to a new renormalized leaf σ1.

(a)

(b)

FIG. 3. (a) The von Neumann entropy of subregion A iscomputed by the minimal cut γA that splits the TN into twoparts containing A and A, respectively. (b) By applying a localunitary on A, we can find maximally entangled legs across γA,which serve as a bottleneck for the entanglement betweenA and A.

7The domain of dependence of a closed codimension-2 surfaceis defined as the domain of dependence of a spacelike hyper-surface enclosed by the surface.

COARSE-GRAINING HOLOGRAPHIC STATES: A … PHYS. REV. D 102, 086001 (2020)

086001-5

This continuum procedure can be written in terms of aflow equation for the leaf σðλÞ, which is interpreted as aLorentzian mean curvature flow:

dxμ

dλ¼ 1

2ðθklμ þ θlkμÞ; ð6Þ

where xμ are the embedding coordinates of σðλÞ, fkμ; lμgare the future-directed null vectors orthogonal to σðλÞ,normalized such that k · l ¼ −2, and θk;l are the classicalexpansions in the k, l directions, respectively. The sequenceof renormalized leaves spans a codimension-1 hypersur-face, which was termed the holographic slice. In particular,it is a partial Cauchy slice of the bulk domain ofdependence, DðσÞ, of the original leaf σ.It was shown that the flow described above satisfies

various interesting properties that are consistent with thecoarse-graining interpretation. These include the fact thatthe area of the leaf σðλÞ decreases monotonically with λ,implying that the number of degrees of freedom in theeffective Hilbert spaceHeffðσðλÞÞ decreases as we flow. Bychoosing statistically isotropic subregions with randomshapes, one can obtain a preferred holographic slice thatpreserves the symmetries of the system. Alternatively, byvarying flow rates along the transverse directions, onecould get a range of different, but gauge equivalent, slicesof DðσÞ.

IV. MOTIVATION FROM TENSOR NETWORKS

Having the classical construction in hand, we nowdescribe how to generalize it to include bulk quantumcorrections. Let us take a specific state defined on a leaf of aQ-screen. We want to understand how coarse graining ofthis state works using the TN picture.We expect that the state is still modeled by a TN at the

quantum level. In order to represent the effect of bulkquantum corrections appropriately, this TN must includetwo additional features compared with the classical case.First, tensors and bonds used in the network should ingeneral not all be “featureless,” i.e., all of the tensors are thesame and connected by maximally entangling bonds, aswas the case in simple perfect tensor [25] or random tensor[26] networks. Reflecting the existence of excitations ofbulk low-energy fields, tensors and/or bonds must have anonuniversal structure representing such excitations. Thisgenerally makes the network not fully isometric. Second,since bulk low-energy quantum fields can have long-rangeentanglement, corresponding to Sbulk in Sgen, there shouldbe longer bonds connecting non-nearest-neighbor tensors,although the number of such nonlocal bonds (or moreprecisely, the total dimension associated with them) isgenerally suppressed as the bonds become longer. A typicalTN of this sort is depicted in Fig. 6.Once we have a TN representation of the state, the

scenario in Sec. III can be generalized in a relatively simple

manner. To do so, we must first establish how to computethe entropy of a boundary subregion, following the formulain Eq. (4). In general, the boundary legs consist of both theshortest, local bonds and longer, nonlocal bonds cut by theboundary. When we compute the entropy of subregion A ofσint, i.e., a subset of these legs, we must minimize

SgenðA;XAÞ ¼AðA ∪ XAÞ

4GNþ SbulkðΞAÞ ð7Þ

over all surfaces XA anchored to the boundary of A, whereΞA is the homology surface with ∂ΞA ¼ A ∪ XA. In thisexpression, the area term represents the contribution fromthe shortest bonds, while SbulkðΞAÞ from longer bonds, cutby ∂ΞA. In short, SgenðA;XAÞ is given by the entropy of allof the bonds connecting tensors inside and outside ∂ΞA,regardless of their lengths; see Fig. 7. (This reflects the factthat the precise way to separate the contributions from localand nonlocal bonds is arbitrary and does not have aninvariant meaning.)With this prescription, we can follow the analysis in

Sec. III and coarse grain the region A of the boundary stateby removing the bulk regions that are entangled with A, i.e.,by reducing the TN to a smaller one giving a state on

σ0 ¼ A ∪ ΓA: ð8Þ

Note that here the complement A of A is defined as that onthe entire leaf σ ¼ σint ∪ σext, not on σint. The QES ΓA isgiven by the surface XA minimizing

SgenðA; XAÞ ¼AðA ∪ XAÞ

4GNþ SbulkðΞAÞ; ð9Þ

where

FIG. 6. A TN representing a state at the quantum level hastensors that are not universal (yellow) and bonds that connecttensors nonlocally (pink).

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-6

∂XA ¼ ∂A ¼ ∂Aint; ð10Þ

where Aint is the complement of A on σint (i.e.,A ¼ Aint ∪ σext) and ∂ΞA ¼ A ∪ XA, meaning that

ΞA ¼ ΞA;int ∪ Σext: ð11Þ

Here, ∂ΞA;int ¼ Aint ∪ XA, and Σext is a spacelike hyper-surface exterior to σext. Note that for σext, we have definedthe surface “enclosed” by σext to be the exterior of σext:

∂Σext ¼ σext: ð12Þ

This procedure gives the interior portion of the newleaf as

σ0int ¼ Aint ∪ ΓA: ð13Þ

We assume that for a TN representing a state with asemiclassical bulk, the RT formula with quantum correc-tions can be applied to the state on this new surface aswell. The process described here can be repeated multipletimes, leading to a similar sequence of coarse-grained statesas before.Note that the assumption that the RT formula continues

to hold is nontrivial given that generic bulk states break theisometric property of the TN. However, we will only needto assume that the RT formula holds for infinitesimalsubregions and their complements, which gives results thatare consistent with the coarse-graining interpretation sug-gested here. As discussed in Sec. II, one could also considerbulk matter with a large central charge so that the non-isometric behavior appears only at subleading order in 1=cfor reasonable bulk states [59,60].

V. COARSE GRAINING AND QUANTUM FLOW

A. Definition

Having found the procedure in TNs, we can now lookfor a continuum version in semiclassical gravity. Giventhe framework established in Sec. II, we can locate theQ-screen for a given state and start a coarse-grainingprocedure analogous to that discussed in Sec. IV.As described in Sec. III, we consider an infinitesimal

subregion on the interior portion σint of the original leaf σand reduce the accessible spacetime region to the com-plementary quantum entanglement wedge QEWðAÞ, whichis determined by the minimal QES ΓA of A such thatQEWðAÞ ¼ DðΣAÞ, where ∂ΣA ¼ A ∪ ΓA. Note that in ageneral spacetime, the global description includes anexterior portion outside σext. Thus, the complement ofan infinitesimal subregion A ⊂ σint on the leaf is A ¼Aint ∪ σext, and the bulk entropy term Sbulk of the gener-alized entropy is given by the von Neumann entropy ofΣA ¼ ΣA;int ∪ Σext. The necessity of including the regionexterior to σext can be argued from complementary recoveryin pure states.Considering many such infinitesimal subregions Ap

on σint centered around points p as in Eq. (5), we cansequentially reduce the accessible spacetime region to

RðσÞ ¼ ∩p QEWðApÞ; ð14Þwhich leads to a renormalized leaf σ1 such thatDðΣ1Þ ¼ RðσÞ, where ∂Σ1 ¼ σ1. This yields a new boun-dary state in a smaller effective Hilbert space definedon σ1. As we show in the Appendix A, the convexity of theoriginal leaf implies that the corresponding renormalizedleaf is also convex, which ensures that the coarse-grainingprescription can be repeatedly applied. The preservation ofconvexity also means that SðAÞ of a subregion A of arenormalized leaf obtained using Eq. (4) satisfies propertiesneeded for it to be interpreted as the von Neumann entropyof the density matrix of the region.In the continuum limit, the behavior of QESs anchored to

small subregions can be studied analytically. While the vonNeumann entropy can in general show complicated behav-iors as the subregion is varied, for an infinitesimal sub-region we may expect that such behaviors arise only fromphysics at scales much larger than the size of the subregion.It is then reasonable to assume that the change of theentropy of the subregion, as well as that of the complement,can be approximated by the volume integral of somedensity function. With this assumption, and reasonablesmoothness assumptions about the spacetime and subre-gions, we show in Appendix B that the resulting QESs aresuch that the deepest point lies in a universal normaldirection to the leaf given by

s ¼ 1

2ðΘklþ ΘlkÞ; ð15Þ

FIG. 7. The minimal QES ΓA of subregion A is determined byminimizing the entropy of all bonds connecting tensors inside andoutside A ∪ ΓA (consisting of 12 black and 2 pink bonds in thefigure).

COARSE-GRAINING HOLOGRAPHIC STATES: A … PHYS. REV. D 102, 086001 (2020)

086001-7

as long as the relevant QESs exist. Here, fkμ; lμg are thefuture-directed null vectors orthogonal to σint, normalizedsuch that k · l ¼ −2, and Θk;l are the correspondingquantum expansions. Here, Θk;l are computed by varyingSgen as defined in Eq. (3). This is sufficient to find thelocation of σ1;int, and hence of σ1, after fixing relativenormalizations for the size of subregions considered ondifferent portions of the leaf. For convenience, we willchoose the normalizations such that the resulting flowequation takes the simplest form. Other possibilities will bediscussed in Sec. V B 3.Following the procedure described above, we can derive

a flow equation, which generalizes the Lorentzian meancurvature flow in Eq. (6) to include bulk quantum correc-tions:

dxμ

dλ¼ 1

2ðΘklμ þ ΘlkμÞ; ð16Þ

where xμ are the embedding coordinates of the interior por-tion σintðλÞ of the renormalized leaves σðλÞ¼σintðλÞ∪σext,parametrized by λ, and Θk;l represent the quantum expan-sions of σðλÞ at xμ. The resulting sequence of σintðλÞ spans acodimension-1 quantum-corrected holographic slice, asshown in Fig. 8.

1. Possibility of appearance of disconnectedleaf portions

While performing the coarse graining as describedabove, it may occur that the minimal QES ΓAp

for Ap inEq. (14) becomes noninfinitesimal. In particular, if thereis a bulk region surrounded by a surface X of area AðXÞand whose entanglement with the exterior of σext exceeds

AðXÞ=4GN on a given spacelike slice containing σi ¼σint;i ∪ σext, then the quantum minimal surface χðApÞ on ithas a disconnected component surrounding the region. Ifsuch a disconnected component remains after the maximi-zation over all of the spacelike slices, then the minimal QESΓðApÞ does have a disconnected component, and as aconsequence RðσÞ will have a “hole” such that ∂RðσÞ ⊃ X.This makes the new leaf σint;iþ1 have a disconnectedcomponent X (⊂ σint;iþ1), in addition to the portion infini-tesimally close to σint;i. The region surrounded by X is thusexcluded from the flow afterward.When it first appears, an excluded region, and hence

the disconnected component of the leaf associated withit, is small. This appearance cannot be seen just by solvingthe flow equation, although our coarse-graining procedureitself captures the occurrence of this phenomenon. After itsappearance, the disconnected component of the leaf alsoflows generally, following the flow equation. This makesthe hole of the spacetime larger, which may eventuallycollide with the component arising from the continuousinward motion of the original leaf portion σint.

B. Properties

We now illustrate some of the salient properties of theflow, showing the consistency of it being interpreted ascoarse graining.

1. Monotonicity of generalized entropyof renormalized leaves

In order to interpret our procedure as coarse graining, thenumber of degrees of freedom must decrease monotoni-cally with λ. The dimension of the effective Hilbert spaceHeffðσðλÞÞ associated with leaf σðλÞ can be defined as theamount of entropy the boundary legs carry in the TNpicture, implying

ln jHeffðσðλÞÞj ¼AðσðλÞÞ4GN

þ SbulkðΣðλÞÞ

¼ SgenðσðλÞÞ; ð17Þ

where jHj represents the dimension of H, and ΣðλÞ is abulk codimension-1 spacelike surface bounded by σðλÞ ¼σintðλÞ ∪ σext, i.e., ΣðλÞ ¼ ΣintðλÞ ∪ Σext. We thus find thatthe condition for the decrease of the degrees of freedom isthe same as the statement that the generalized entropy of therenormalized leaf σðλÞ decreases monotonically with λ. Wenow prove this in a manner similar to Ref. [15].We have defined the evolution vector s, which is tangent

to the holographic slice ϒ and radially evolves the interiorleaf portion inward:

s ¼ 1

2ðΘklþ ΘlkÞ; ð18Þ

FIG. 8. A sequence of renormalized leaves σintðλÞ obtained bysolving the flow equation in Eq. (16) spans a codimension-1quantum-corrected holographic slice. Each leaf represents thedomain of dependence of a spacelike surface ΣintðλÞ with∂ΣintðλÞ ¼ σintðλÞ.

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-8

where the associated quantum expansions satisfy

Θs ¼ ΘkΘl ≤ 0; ð19Þ

as shown in Appendix A.Consider a point p on the leaf portion σintðλÞ and the s

vector orthogonal to σintðλÞ at p. Next consider aninfinitesimal patch of area δA around p. As we flow alongs by a small amount, the rate at which SgenðσðλÞÞ changes isdetermined by the quantum expansion Θs as

δSgen ∝ ΘsδA ≤ 0: ð20Þ

This implies that the contribution to SgenðσðλÞÞ from theinward flow of any infinitesimal patch is negative, andhence SgenðσðλÞÞ must decrease with λ.The argument above relies on the flow equation.

However, as shown in Sec. VA 1, it is possible that oncoarse graining, we obtain a new disconnected componentof σintðλÞ. While the appearance of such a componentcannot be described by the flow equation, it comes with anegative contribution to the generalized entropy of therenormalized leaf. Thus, even on including this effect, wefind that SgenðσðλÞÞ decreases with λ.

2. Containment of subregion flow

Consider the situation where we apply the holographicslice construction only to a finite subregion A of the leafportion σint. This yields a sequence of renormalized leavesgiven by σðλÞ ¼ AðλÞ ∪ A. Here,AðλÞ represents a sequenceof subregions that arise from the radial evolution of A.Now, because of entanglement wedge nesting,

QEWðAÞ ⊂ QEWðσðλÞÞ ð21Þ

for arbitrary λ, since A ⊂ σðλÞ for all λ. Here, QEWðAÞ andQEWðσðλÞÞ are determined by the corresponding minimalQESs. This implies that the boundary of QEWðAÞ acts asan extremal surface barrier for the flow of AðλÞ. Inparticular, AðλÞ remains outside QEWðAÞ for all λ.Incidentally, if there is another QES anchored to ∂A

which lies outside QEWðAÞ, then AðλÞwould not be able togo beyond this nonminimal extremal surface.

3. Remaining freedom

In general, the proof in Appendix B allows us to fix thedirection of the flow at each point of σintðλÞ to be the vectors, as discussed in Eq. (15). However, there is no canonicalchoice of normalization, reflecting the arbitrariness ofchoosing relative sizes of subregions for different pointson σintðλÞ. The ratio of these sizes must stay finite in thecontinuum limit, and yet it can still lead to inequivalentflow equations parametrized as

dxμ

dλ¼ αðyi; λÞðΘklμ þ ΘlkμÞ; ð22Þ

where yi represents the tangential coordinates on σintðλÞ,and αðyi; λÞ > 0. These flow equations in general result indifferent holographic slices, which are all gauge equivalentby the equations of motion. By choosing subregions of thesame characteristic size at all p, we can fix the preferrednormalization that leads to Eq. (16).We note that this provides a natural gauge choice

motivated by holography; the spacetime inside the holo-graphic screen, which is now the Q-screen H0, is para-metrized by λ, yi, and t, where t is a time parameter on theholographic screen giving a sequence of leaves at differenttimes. [If disconnected components of σintðλÞ appear duringthe flow, then we must extend yi to incorporate thosecomponents.]An alternative choice for the normalization is to take λ to

be the proper length along the trajectory pðλÞ of a point onσintðλÞ. Here, the trajectory is defined such that if a point onσintðλþ dλÞ is located on the two-dimensional surfaceorthogonal to σintðλÞ at pðλÞ, then it is regarded as the“same” point as pðλÞ, i.e., pðλþ dλÞ. This provides anothernatural gauge choice motivated by holography.

C. End of the flow

The quantum flow procedure described above provides away to probe a spacetime inside the holographic screen byfollowing the holographic slice inward. A key qualitativefeature of the spacetime is given by how and where theholographic slice ends.The holographic slice can end in one of three pos-

sible ways:(i) The slice ends at an empty surface. This can simply

occur such that σintðλÞ keeps moving inward, and theslice is capped off at a point, as shown in Fig. 9(a).Alternatively, as discussed in Sec. VA 1, discon-nected components of σintðλÞ may appear during theflow, which then grow outward and coalesce withthe original component of σintðλÞ moving inward,ending up with an empty surface.

(ii) The slice asymptotes to a QES [as shown inFig. 9(b)]. As the interior portion σintðλÞ of renor-malized leaves approaches the QES, the flow slowsdown becauseΘk;Θl → 0. Note that a QES homolo-gous to the initial leaf portion σintð0Þ—even if it isnonminimal—acts as a barrier which cannot becrossed as we flow in.

(iii) The slice terminates abruptly [as shown in Fig. 9(c)].This occurs when the minimal QES associated withA, the complement of an infinitesimal subregion A,becomes noninfinitesimal. At this point, we need toterminate the flow.

It is worth mentioning that while some of these cases haveclassical analogues, the second possibility of case (i) andthe case (iii) are exclusive to the quantum flow.

COARSE-GRAINING HOLOGRAPHIC STATES: A … PHYS. REV. D 102, 086001 (2020)

086001-9

D. Example

Various examples for the classical flow equation ofEq. (6) were discussed in Ref. [15]. In many situations,the minimal QES is a small perturbation to the classicalHRT surface, and accordingly the quantum-corrected flowequation in Eq. (16) results in a holographic slice that isperturbatively close to the classical holographic slice. Thereare, however, cases in which the two flows are significantlydifferent. Here we illustrate an example of these: a blackhole formed from collapse.In the classical case, it was found that the holographic

slice stays close to the horizon for a long time until itreaches the matter forming the black hole [15]. It is thencapped off to form a complete Cauchy slice of thespacetime, as seen in Figs. 5 and 6 of Ref. [15]. How isthis modified at the quantum level?Far from the black hole horizon, the flow is largely

unaffected. It is, however, significantly modified once weapproach the horizon. As the black hole evaporates, thereare Hawking modes that escape to the region exterior toσext, denoted R, leaving behind their interior partners

entangled with them. As the leaf portion σint is movedinward by the flow, its classical area decreases but theentropy contribution from the Hawking partners increases.About a Planck distance inside the horizon, the two effectscompete with each other, resulting in a QES where the flowends. The mechanism by which the QES emerges here isidentical to the one that appeared in a specific example inRef. [64].8 Thus, after including bulk quantum corrections,the holographic slice becomes a partial Cauchy slice of thespacetime that excludes a large portion of the interior.9 Thesame feature can be found in the case of an AdS black hole,where one could allow the black hole to evaporate bycoupling the conformal field theory to a bath. Our coarse-graining procedure then leads to a flow that stops at thesame QES as that found in Refs. [48,49].Another mechanism excising the black hole interior

comes from the phenomenon discussed in Sec. VA 1.As the black hole evaporates, there are a large numberof interior partners of Hawking radiation that accumulate

FIG. 10. Eddington-Finkelstein diagram representing blackhole formation and evaporation with quantum holographic slicesdepicted for three characteristic times.

FIG. 9. Three possible ways in which the holographic slicecan end: (a) ends at an empty surface, (b) asymptotes to a QES,and (c) terminates abruptly.

8Recently, Ref. [65] appeared which found such a nontrivialQES in cosmological spacetimes. The coarse-graining flowwould end at the QES in these situations as well.

9This does not necessarily mean that the interior of the blackhole is absent. It is possible that the semiclassical interior pictureemerges through approximately state-independent operators act-ing on modes (the hard modes [66,67]) whose characteristicfrequencies are larger than the local Hawking temperature [68].

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-10

behind the horizon, which eventually exceed the area of thehorizon at the Page time. Hence the interior of such an oldblack hole would not be swept by renormalized leaves(even if the flow did not halt as described above).The phenomenon that the holographic slice does not

penetrate deep into the black hole horizon was already seenin the classical case. There is, however, an important differ-ence in the quantum case. As shown in Figs. 10 and 11,holographic slices become partial Cauchy slices during themiddle of the evolution of a black hole. (These can be con-trasted with Figs. 5 and 6 of Ref. [15].) This implies, inparticular, that with a given time parametrization on aboundary, e.g., on the holographic screen, the concept ofblack hole formation and evaporation can be rigorouslydefined through the behavior of the flow discussed inSec. V C.

VI. RELATION TO QUANTUM ERRORCORRECTION

In this section, we discuss the relation between ourcoarse-graining procedure and the picture that the holo-graphic dictionary works as quantum error correction[34–37], in which a small Hilbert space of semiclassicalbulk states is mapped isometrically into a larger boundaryHilbert space. In our framework, this picture arises afterconsidering a collection of states over which we want tobuild a low-energy bulk description.

In the context of quantum error correction, one choosesthe set of semiclassical bulk states that can be representedas a code subspace in the boundary theory. In a generaltime-dependent spacetime, however, there is no naturalchoice of code subspace fixed by the bulk effective theory.This is because degrees of freedom that appear natural onone time slice need not be in bijection with those thatappear natural on a different time slice. For example, if asingle heavy particle decays into a large number ofradiation particles within the causal domain of σint, thenwe may naturally choose a code subspace associated withthe degrees of freedom of the parent particle, e.g., its spin,or a larger subspace determined by the coarse-grainedentropy of the final-state radiation.Our framework addresses this issue by providing a

specific gauge choice given by the coarse-graining pro-cedure. Suppose there are a set of states giving similargeometries on their holographic slices. Then, we canrepresent all of these states approximately at once by asingle TN, which has “dangling” legs so that different statesin these legs correspond to different elements in the set; seeFig. 12. This is a choice of code subspace motivated by thecoarse-graining procedure.The introduction of dangling legs amounts to dividing

bulk degrees of freedom into two classes: those representedby a code subspace and the rest. Let us denote the asso-ciated Hilbert space factors by Hcode and Hfrozen, respec-tively. States in Hcode correspond to the bulk degrees offreedom that we aim to reconstruct, whileHfrozen is viewedas “frozen.” Namely, the degrees of freedom correspondingto Hfrozen are treated essentially as part of the background,despite the fact that they are associated with quantum statesin the conventional bulk effective field theory.We can now define the coarse-graining in this setup,

namely, on a continuum analogue of a TN with danglinglegs. Specifically, we take the maximally mixed state inHcode, while picking a fixed state in Hfrozen determined by

FIG. 11. Penrose diagram version of Fig. 10. The region Routside σext is depicted by orange lines.

FIG. 12. A TN representing a collection of states has danglinglegs as well as nonuniversal tensors and nonlocal bonds.

COARSE-GRAINING HOLOGRAPHIC STATES: A … PHYS. REV. D 102, 086001 (2020)

086001-11

the network structure, i.e., the background geometry. Thiscan be thought of as considering a coarse-grained versionof a generic state within the code subspace. Indeed, themaximally mixed state plays a crucial role in AdS/CFT,in which reconstruction of an operator in the maximallymixed state is sufficient for the operator to be reconstructedon arbitrary states in the code subspace [69,70].The coarse-graining procedure then follows that in

Secs. IV and V, but this time Sbulk receives contributionsfrom both dangling legs, Scode, and nonlocal legs, Sfrozen.Since the QESs for A and A need not agree (see Fig. 13),the region between the two surfaces—often termed theno-man’s land—is partially entangled with both A and A.Nevertheless, using Eq. (14) we can obtain a pictureanalogous to that in Secs. IV and V.Once we perform the flow to obtain a renormalized leaf

σðλÞ appropriate to deal with the problem, e.g., by makingσintðλÞ a surface surrounding the region we are interested in,then we can consider the set of all states in Hcode, ratherthan the maximally mixed state, to analyze the system inmore detail. As in the corresponding TN case, we can theninterpret such coarse-grained states in two ways.One way is to view a state in the set as defining an

entangled state in the combined bulk-boundary Hilbertspace,

jψi ∈ Hbulk ⊗ Hboundary; ð23Þwhere Hbulk represents the space of bulk states in the codesubspace.Another way is to regard the set as giving an isomorphic

map between the bulk Hilbert space (i.e., the space ofdangling legs) and a subspace of the much larger boundaryHilbert space:

fjψig∶Hbulk ↔ Hcode ⊂ Hboundary: ð24Þ

Note that in the TN picture,Hboundary consists of both localand nonlocal bonds cut by the boundary surface obtainedby the flow, as well as the part associated with σext. Thisimplies that the dimension of the boundary effective Hilbertspace is given by

ln jHboundaryðσðλÞÞj ¼AðσðλÞÞ4GN

þ SfrozenðΣðλÞÞ: ð25Þ

This can be compared with Eq. (17).In this way, holographic properties such as the HRT

formula and entanglement wedge reconstruction can benaturally interpreted [11,35]. The interpretation is consis-tent with the analysis in Ref. [69] that the region recon-structible by state-independent operators—termed thereconstruction wedge in Ref. [70]—can be computed byconsidering QEWðAÞ of the maximally mixed state inHcode. In this picture, our coarse-graining procedure isinterpreted to produce a sequence of holographic encodingmaps parametrized by λ, each of which can be viewed as aholographic duality of the form in Eq. (24).

VII. CONCLUSIONS

In this paper, we have generalized the holographiccoarse-graining procedure described in Ref. [15] to includebulk quantum corrections. Interestingly, the generalizationinvolves promoting classical expansions θ to quantumexpansions Θ, as has been found in many other examples[29–33]. We have demonstrated that the flow equationobtained in the bulk has all of the properties consistent withan interpretation as a coarse-graining process in the holo-graphic theory. Our procedure also gives a way in which theregion exterior to the holographic screen is treated at thequantum level. It would be interesting to explicitly under-stand the detailed coarse-graining procedure from a boun-dary theory perspective.

ACKNOWLEDGMENTS

We thank Nico Salzetta and Arvin Shahbazi-Moghaddam for discussions. This work was supportedin part by the Department of Energy, Office of Science,Office of High Energy Physics under Contract No. DE-AC02-05CH11231 and Award No. DE-SC0019380 and inpart by World Premier International Research CenterInitiative (WPI Initiative), MEXT, Japan.

APPENDIX A: CONVEXITY OFRENORMALIZED LEAVES

Definition 1. On a spacelike slice Σ, a compact set S isdefined to be convex if the quantum minimal surface χAanchored to the boundary ∂A of a codimension-2 regionA ⊂ S is such that ∀A, χA ⊂ S. Here, the quantum minimalsurface χA is defined as the surface on Σ which minimizes

FIG. 13. With the state in Hcode being maximally mixed, theQESs ΓA and ΓA can differ.

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-12

the generalized entropy Sgen for the region on Σ boundedby χA ∪ A.Definition 2. A codimension-2 compact surface σ is

called a convex boundary if on every codimension-1spacelike slice Σ such that σ ⊂ Σ, the closure of the interiorof σ is a convex set.Theorem 1. If σ is a convex boundary, then for any

subregion A ⊂ σ, A ∪ ΓA is also a convex boundary.Proof.—Suppose that for some spacelike slice Σ, which

contains A ∪ ΓA, there is B in the closure of the interior ofA ∪ ΓA such that the unique quantum minimal surface χBgoes outside A ∪ ΓA.Since σ is assumed to be a convex boundary, χB cannot

go outside σ, i.e., it cannot cross A. Thus, χB must cross ΓA,as shown in Fig. 14.As ΓA ¼ X1 ∪ X2 is the quantum minimal surface for A,

AðX1Þ4G

þAðX2Þ4G

þAðAÞ4G

þ SbulkðabÞ

≤AðX2Þ4G

þAðX3Þ4G

þAðAÞ4G

þ SbulkðabcÞ; ðA1Þ

where the right-hand side corresponds to the surfaceXA ¼ X2 ∪ X3.Also, χB ¼ X3 ∪ X4 is the unique quantum minimal

surface for B, so

AðX3Þ4G

þAðX4Þ4G

þAðBÞ4G

þ SbulkðbcÞ

<AðX1Þ4G

þAðX4Þ4G

þAðBÞ4G

þ SbulkðbÞ; ðA2Þ

where the right-hand side corresponds to the surfaceXB ¼ X1 ∪ X4.

Combining Eqs. (A1) and (A2), we have

SbulkðabÞ þ SbulkðbcÞ < SbulkðbÞ þ SbulkðabcÞ; ðA3Þwhich contradicts strong subadditivity. Thus, the quantumminimal surface χB crossing the quantum minimal surfaceΓA results in a contradiction. ▪By repeatedly applying this theorem, we can show that if

the initial leaf has the interior portion σint that is convex,then a sequential quantum flow procedure results in aconvex interior portion at each step. In the continuum limit,this sequential procedure gives us the same renormalizedleaves that were obtained using the flow equation. Thus, weconclude that if the initial leaf portion σintð0Þ is a convexboundary, then the renormalized leaf portion σintðλÞ is alsoa convex boundary for all λ > 0.Lemma 1. Consider a slice Σ and a compact set S ⊂ Σ.

If S is convex then ΘΣð∂SÞ ≤ 0. Here, ΘΣð∂SÞ is the traceof the quantum extrinsic curvature of ∂S embedded in Σ forthe normal pointing inward.Proof.—Suppose that ΘΣð∂SÞ > 0 somewhere on ∂S.

One can explicitly construct minimal surfaces that areoutside S by considering small enough subregions anch-ored to this portion of ∂S. ▪Let the future-directed null vectors orthogonal to a

codimension-2 spacelike surface σ be k and l, which wenormalize as k · l ¼ −2.Theorem 2. If σ is a convex boundary, then the quantum

expansions in the inward direction Θk and Θ−l are bothnonpositive.Proof.—Consider some spacelike slice Σ. The inward

normal n on Σ is given by

n ¼ αk − βl; ðA4Þ

with α, β ≥ 0.Suppose Θk > 0. Now we can choose a slice Σ such

that ΘΣðσÞ > 0 by taking α ≫ β. Thus, σ is not a convexboundary because the closure of its interior is notconvex on Σ due to Lemma 1. A similar argument holdsif Θl < 0. ▪When we discuss the convexity of the leaf σðλÞ ¼

σintðλÞ ∪ σext, we treat σext as a single unit on σ whichcannot be further divided into subregions. Thus, σext isincluded or excluded as a whole when we consider anyboundary subregion A.

APPENDIX B: DERIVATION OF THEFLOW EQUATION

Consider a codimension-2, closed, achronal surface σ inan arbitrary (dþ 1)-dimensional spacetime M. Suppose σis a convex boundary. We assume that both M and σ aresufficiently smooth so that variations in the spacetimemetric gμν and induced metric on σ, denoted by hij, occuron characteristic length scales Lg and Lσ, respectively.

FIG. 14. The quantum minimal surface χB crossing the quan-tum minimal surface ΓA.

COARSE-GRAINING HOLOGRAPHIC STATES: A … PHYS. REV. D 102, 086001 (2020)

086001-13

We also assume that the changes of the variational entropycurrent density JμðxÞ (discussed below) occur on a char-acteristic length scale LS.Theorem 3. Consider subregion A of characteristic

length δ ≪ Lg; Lσ; LS on the surface σ. This subregionA is chosen to be a (d − 1)-dimensional ellipsoid on σ atorderOðδÞ. Then, at the leading order, the QES anchored to∂A lives on the hypersurface generated by the evolutionvector10

s ¼ 1

2ðΘklþ ΘlkÞ ¼ Θtt − Θzz ðB1Þ

normal to σ. Here, k and l are future-directed null vectorsorthogonal to σ normalized as k · l ¼ −2, and t and z arevectors related to these by

k ¼ ðtþ zÞ; l ¼ ðt − zÞ: ðB2ÞProof.—Since the subregion A is assumed to be an

ellipsoid, we label its center point as p. We can then setup Riemann normal coordinates in the local neighborhoodof p:

gμνðxÞ ¼ ημν −1

3Rμνρσxρxσ þOðx3Þ: ðB3Þ

In these coordinates, we are considering a patch of sizeOðδÞ around the origin p with Rμνρσ ∼Oð1=L2

gÞ, so at anypoint in this patch

gμνðxÞ ¼ ημν þO

�δ2

L2g

�: ðB4Þ

Since there is still a remaining SOðd; 1Þ symmetry thatpreserves the Riemann normal coordinate form of themetric, we can use these local Lorentz boosts and rotationsto set t and z as the coordinates in the normal direction to σat p, while yi parametrize the tangential directions. At orderOðδÞ, the subregion A is then an ellipsoid in the yi

coordinates centered at the origin p.In a small region around p, we can define a variational

entropy current density that measures how Sbulk changes.More formally, let XA be a surface anchored to theboundary of A, or equivalently of A: XA ¼ XA. Let ΞA

be the homology surface with ∂ΞA ¼ A ∪ XA; then,

SbulkðΞAÞ ¼ S0 −ZSJμðxÞdaμ; ðB5Þ

where S0 is the Sbulk associated with the full σ, so it isindependent of the choice of subregion A or the surface XA.

JμðxÞ is the aforementioned variational entropy currentdensity which upon integrating over S, a homology surfacewith boundary ∂S ¼ A ∪ XA, determines how SbulkðΞAÞdiffers from S0.We now Taylor expand the entropy current density about

p, so for any point within OðδÞ distance of p

JμðxÞ ¼ J μ

�1þO

�δ

LS

��; ðB6Þ

where J μ ¼ Jμð0Þ. Recall that LS is the length scale of thevariations of corresponding entropy variations.Let Kt

ij, Kzij denote the extrinsic curvature tensors of σ

for the t and z normals, respectively. It follows thatKt

ij; Kzij ∼Oð1=LσÞ. Since t and z are normal to σ, the

equations for the surface σ, described by tLðyiÞ and zLðyiÞ,can be Taylor expanded in the region A as

tLðyiÞ ¼ −1

2Kt

ijyiyj þO

�δ3

L2σ

�; ðB7Þ

zLðyiÞ ¼1

2Kz

ijyiyj þO

�δ3

L2σ

�; ðB8Þ

where the negative sign in the first line is due to the timelikesignature of the t normal.From Eqs. (B7) and (B8), it follows that at the leading

order in δ,

∇2tL ¼ −ηijKtij; ðB9Þ

∇2zL ¼ ηijKzij; ðB10Þ

where ∇2 ¼ ∂i∂i. Note that hij ¼ ηij at this order.It follows that the ratio of quantum null expansions on

the surface σ is

Θk

Θl¼ ηijðKt

ij þ KzijÞ þ 4GNðJ t − J zÞ

ηijðKtij − Kz

ijÞ þ 4GNð−J t − J zÞ; ðB11Þ

or, equivalently,

Θt

Θz¼ Θk þ Θl

Θk − Θl¼ ηijKt

ij − 4GNJ z

ηijKzij þ 4GNJ t

: ðB12Þ

Here, we have used that the bulk entropy is given byEq. (B5) along with the Taylor expansion in Eq. (B6).The QES ΓA can be parametrized in a similar way using

tEðyiÞ and zEðyiÞ. The boundary conditions satisfied by theQES are

tEð∂AÞ ¼ tLð∂AÞ; ðB13Þ

zEð∂AÞ ¼ zLð∂AÞ: ðB14Þ10In this Appendix, we ignore the possibility that there is no

QES infinitesimally close to A as δ → 0, i.e., the possibility(iii) discussed in Sec. V C.

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-14

Since the region A is chosen to be an ellipsoidal region inthe yi coordinates at OðδÞ, we have symmetry under yi →−yi at this order. Consequently, the t and z directions arenormal to the QES at the center point ðtEð0Þ; zEð0Þ; 0Þ.Let Kt

ij, Kzij denote the extrinsic curvature tensors of the

QES for the t and z normals, respectively. We assume thatthe QES is approximately flat at length scale δ, i.e.,Kt

ij; Kzij ≪ 1=δ. We will show that this assumption is

self-consistent as long as the entropy current density isnot too large.We can Taylor expand tEðyiÞ and zEðyiÞ as

tEðyiÞ ¼ tEð0Þ −1

2Kt

ijyiyj; ðB15Þ

zEðyiÞ ¼ zEð0Þ þ1

2Kz

ijyiyj: ðB16Þ

Since the QES has vanishing quantum null expansion, wehave at the leading order

ηijðKtij þ Kz

ijÞ þ 4GNðJ t − J zÞ ¼ 0; ðB17Þ

ηijðKtij − Kz

ijÞ þ 4GNð−J t − J zÞ ¼ 0: ðB18Þ

Here, we have used the expansion in Eq. (B6) because anypoint on the QES is at most OðδÞ distant from the origin p.These equations, along with Eqs. (B15) and (B16), result

in the following differential equations for tEðyiÞ and zEðyiÞat the leading order:

∇2tE ¼ −ηijKtij ¼ −4GNJ z; ðB19Þ

∇2zE ¼ ηijKzij ¼ −4GNJ t: ðB20Þ

Earlier, we assumed that Ktij; K

zij ≪ 1=δ, which is justified

as long as J t;J z ≪ 1=ð4GNδÞ, which is the case if J t, J zdo not diverge as δ → 0.Let us consider the quantities δt ¼ tE − tL and δz ¼

zE − zL. These satisfy the following differential equationsat the leading order:

∇2δt ¼ ηijKtij − 4GNJ z; ðB21Þ

∇2δz ¼ −ηijKzij − 4GNJ t: ðB22Þ

The boundary conditions are given by

δtð∂AÞ ¼ δzð∂AÞ ¼ 0: ðB23Þ

It is now clear that at the leading order, δt=Θt and−δz=Θz satisfy the same differential equation with the sameboundary conditions, since Θt and Θz can be regarded asconstant at this order. Thus,

δtΘt

¼ −δzΘz

ð1þOðδÞÞ ðB24Þ

for all points on the extremal surface. Rewritten, theextremal surface lives on the hypersurface generated bythe evolution vector s ¼ Θtt − Θzz normal to σ. ▪In Theorem 3 above, we have assumed that the sub-

region A is a (d − 1)-dimensional ellipsoid on the sur-face σ. Nonetheless, the proof goes through if thesubregion A has a reflection symmetry ðy1; y2;…; yd−1Þ →ð−y1;−y2;…;−yd−1Þ about the center point p atorder OðδÞ.In fact, we expect this theorem to hold for a more general

subregion A because the above proof works if we can findany point p ∈ A such that the normal vectors to σ at pmatch with the normal vectors to the QES at the pointcorresponding to p at order OðδÞ. Under the condition thatδ ≪ Lg; Lσ; LS, such a point lies at the “center,” in the sensethat the above leading-order treatment works; for example,the QES of the form of Eqs. (B15) and (B16) is correctly“anchored” to ∂A at the leading order in δ.Finally, our discussion in this Appendix applies in the

context of the main text to the interior portion of the leaf,σint. The existence of the exterior portion σext does notchange the fact that the QES of A lies on the hypersurfacegiven by Eq. (B24).

[1] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602(2006).

[2] S. Ryu and T. Takayanagi, J. High Energy Phys. 08 (2006)045.

[3] V. E. Hubeny, M. Rangamani, and T. Takayanagi, J. HighEnergy Phys. 07 (2007) 062.

[4] M. Van Raamsdonk, Gen. Relativ. Gravit. 42, 2323 (2010).[5] B. Swingle, Phys. Rev. D 86, 065007 (2012).

[6] B. Czech, J. L. Karczmarek, F. Nogueira, and M. VanRaamsdonk, Classical Quantum Gravity 29, 155009(2012).

[7] A. C. Wall, Classical Quantum Gravity 31, 225007 (2014).[8] M. Headrick, V. E. Hubeny, A. Lawrence, and M.

Rangamani, J. High Energy Phys. 12 (2014) 162.[9] D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. J. Suh,

J. High Energy Phys. 06 (2016) 004.

COARSE-GRAINING HOLOGRAPHIC STATES: A … PHYS. REV. D 102, 086001 (2020)

086001-15

[10] X. Dong, D. Harlow, and A. C. Wall, Phys. Rev. Lett. 117,021601 (2016).

[11] J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle,and M. Walter, Phys. Rev. X 9, 031011 (2019).

[12] Y. Nomura, N. Salzetta, F. Sanches, and S. J. Weinberg,Phys. Rev. D 95, 086002 (2017).

[13] Y. Nomura, P. Rath, and N. Salzetta, Phys. Rev. D 97,106025 (2018).

[14] Y. Nomura, P. Rath, and N. Salzetta, Phys. Rev. D 97,106010 (2018).

[15] Y. Nomura, P. Rath, and N. Salzetta, Phys. Rev. D 98,026010 (2018).

[16] M. Miyaji and T. Takayanagi, Prog. Theor. Exp. Phys. 2015,073B03 (2015).

[17] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi, andK. Watanabe, Phys. Rev. Lett. 115, 171602 (2015).

[18] M. Alishahiha, A. Karch, E. Silverstein, and D. Tong, AIPConf. Proc. 743, 393 (2004).

[19] X. Dong, B. Horn, S. Matsuura, E. Silverstein, and G.Torroba, Phys. Rev. D 85, 104035 (2012).

[20] S. Leuven, E. Verlinde, and M. Visser, J. High Energy Phys.06 (2018) 097.

[21] X. Dong, E. Silverstein, and G. Torroba, J. High EnergyPhys. 07 (2018) 050.

[22] S. Cooper, M. Rozali, B. Swingle, M. Van Raamsdonk,C. Waddell, and D. Wakeham, J. High Energy Phys. 07(2019) 065.

[23] V. Gorbenko, E. Silverstein, and G. Torroba, J. High EnergyPhys. 03 (2019) 085.

[24] H. Geng, arXiv:2005.00021.[25] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, J. High

Energy Phys. 06 (2015) 149.[26] P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and

Z. Yang, J. High Energy Phys. 11 (2016) 009.[27] W. Donnelly, B. Michel, D. Marolf, and J. Wien, J. High

Energy Phys. 04 (2017) 093.[28] X.-L. Qi, Z. Yang, and Y.-Z. You, J. High Energy Phys. 08

(2017) 060.[29] R. Bousso, Z. Fisher, S. Leichenauer, and A. C. Wall, Phys.

Rev. D 93, 064044 (2016).[30] R. Bousso and N. Engelhardt, Phys. Rev. D 93, 024025

(2016).[31] A. C. Wall, arXiv:1804.10610.[32] R.Bousso,V.Chandrasekaran, andA.Shahbazi-Moghaddam,

Phys. Rev. D 101, 046001 (2020).[33] R. Bousso, A. Shahbazi-Moghaddam, and M. Tomasevic,

Phys. Rev. Lett. 123, 241301 (2019).[34] A. Almheiri, X. Dong, and D. Harlow, J. High Energy Phys.

04 (2015) 163.[35] D. Harlow, Commun. Math. Phys. 354, 865 (2017).[36] C. Akers and P. Rath, J. High Energy Phys. 05 (2019) 052.[37] X. Dong, D. Harlow, and D. Marolf, J. High Energy Phys.

10 (2019) 240.[38] R. Bousso, Rev. Mod. Phys. 74, 825 (2002).[39] F. Sanches and S. J. Weinberg, Phys. Rev. D 94, 084034

(2016).

[40] M. Headrick and T. Takayanagi, Phys. Rev. D 76, 106013(2007).

[41] P. Hayden, M. Headrick, and A. Maloney, Phys. Rev. D 87,046003 (2013).

[42] N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully, and M.Walter, J. High Energy Phys. 09 (2015) 130.

[43] T. Faulkner, A. Lewkowycz, and J. Maldacena, J. HighEnergy Phys. 11 (2013) 074.

[44] N. Engelhardt and A. C. Wall, J. High Energy Phys. 01(2015) 073.

[45] L. Susskind and J. Uglum, Phys. Rev. D 50, 2700(1994).

[46] S. N. Solodukhin, Living Rev. Relativity 14, 8 (2011).[47] C. Akers, N. Engelhardt, G. Penington, and M. Usatyuk,

arXiv:1912.02799.[48] G. Penington, arXiv:1905.08255.[49] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield,

J. High Energy Phys. 12 (2019) 063.[50] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao,

J. High Energy Phys. 03 (2020) 149.[51] G. Penington, S. H. Shenker, D. Stanford, and Z. Yang,

arXiv:1911.11977.[52] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and

A. Tajdini, J. High Energy Phys. 05 (2020) 013.[53] L. McGough, M. Mezei, and H. Verlinde, J. High Energy

Phys. 04 (2018) 010.[54] W. Donnelly and V. Shyam, Phys. Rev. Lett. 121, 131602

(2018).[55] P. Caputa, S. Datta, and V. Shyam, J. High Energy Phys. 05

(2019) 112.[56] A. Banerjee, A. Bhattacharyya, and S. Chakraborty, Nucl.

Phys. B948, 114775 (2019).[57] C. Murdia, Y. Nomura, P. Rath, and N. Salzetta, Phys. Rev.

D 100, 026011 (2019).[58] A. Lewkowycz, J. Liu, E. Silverstein, and G. Torroba,

J. High Energy Phys. 04 (2020) 152.[59] N. Bao, G. Penington, J. Sorce, and A. C. Wall, J. High

Energy Phys. 11 (2019) 069.[60] N. Bao, G. Penington, J. Sorce, and A. C. Wall, arXiv:

1902.10157.[61] T. Takayanagi and K. Umemoto, Nat. Phys. 14, 573

(2018).[62] P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel, and

B. Swingle, J. High Energy Phys. 01 (2018) 098.[63] S. Dutta and T. Faulkner, arXiv:1905.00577.[64] T. Hartman, E. Shaghoulian, and A. Strominger, J. High

Energy Phys. 07 (2020) 022.[65] T. Hartman, Y. Jiang, and E. Shaghoulian, arXiv:

2008.01022.[66] Y. Nomura, Phys. Rev. D 99, 086004 (2019).[67] Y. Nomura, Phys. Rev. D 101, 066024 (2020).[68] Y. Nomura, Phys. Rev. D 102, 026001 (2020).[69] P. Hayden and G. Penington, J. High Energy Phys. 12

(2019) 007.[70] C. Akers, A. Levine, and S. Leichenauer, Phys. Rev. D 100,

126006 (2019).

MURDIA, NOMURA, and RATH PHYS. REV. D 102, 086001 (2020)

086001-16