Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality

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by  Xi Dong

Reconstruction of Bulk Operators within the Entanglement Wedge
in Gauge-Gravity Duality
Xi Dong,1 Daniel Harlow,2 and Aron C. Wall1
1School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
2Center for the Fundamental Laws of Nature, Harvard University, Cambridge MA, 02138 USA
In this Letter we prove a simple theorem in quantum information theory, which implies that
bulk operators in the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence can
be reconstructed as CFT operators in a spatial subregion A, provided that they lie in its entan-
glement wedge. This is an improvement on existing reconstruction methods, which have at most
succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis,
Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier
ideas interpreting the correspondence as a quantum error correcting code.
6
1 INTRODUCTION the nice feature that it can be done quite explicitly, but
0 it has the disadvantage that it relies on solving a rather
2 The AdS/CFT correspondence tells us that certain nonstandard Cauchy problem [11, 15]. In this paper we

l large-N strongly-coupled CFTs define theories of quan- willrefertoitastheHKLLprocedure,after[9],whowere
u
J tum gravity in asymptotically AdS space [1–6]. For this the first to study it in detail.
definition to be complete, we need to know how to use TheHKLLprocedurehastheinterestingpropertythat
1
1 CFTlanguagetoaskanyquestionofinterestinthebulk. it also is sometimes able to reconstruct a bulk operator
The most basic items in this dictionary are: φ(x,r) as a CFT operator with nontrivial support only

] on some spatial subregion A of a boundary Cauchy slice
h
• Quantum states in the Hilbert space of the CFT Σ [9, 14]. This is believed to occur whenever the point
t
- correspond to quantum states in the bulk. (x,r) lies in the causal wedge of A, denoted C .1 C is
p A A
e defined as the intersection in the bulk of the bulk causal
• The conformal and global symmetry generators in
h future and past of the boundary domain of dependence
[ the CFT correspond to the analogous asymptotic
of A [16].
symmetries in the bulk. For example the CFT
3 Causal wedge reconstruction via the HKLL procedure
Hamiltonian maps to the ADM Hamiltonian, and
v gives an explicit illustration of “subregion-subregion du-
a U(1) charge maps to the electric flux at infinity.
6
ality”, which is the notion that a spatial subregion A
1
4 • InthelargeN limit,anysingle-traceprimaryoper- in the boundary theory contains complete information
5 atorO(x)intheCFT,withscalingdimension∆of about some subregion of the bulk [15, 17, 18]. It has
0 orderN0,correspondstoabulkfieldφ(x,r). They been proposed however that the subregion dual to A is
1. arerelatedbythe“extrapolatedictionary”[4,7,8] notjustCA,butratheralargerregion: theentanglement
0 wedge EA [18–20]. Todefinetheentanglementwedge, we
6 must first define the HRT surface χ [21] (the covariant
A
1 O(x)= lim r∆φ(x,r). (1) generalization of the Ryu-Takayanagi proposal [22]). χ
A
: r→∞
v is defined as a codimension-two bulk surface of extremal
i area, which has boundary ∂χ =∂A and is homologous
X These give an excellent starting point for understanding A
to A through the bulk (in the event that multiple such
r thecorrespondence,butthelimitingprocedureintheex-
a surfaces exist, it is the one with least area). The HRT
trapolate dictionary makes it difficult to concretely dis-
formula tells us that at leading order in 1/N, the area of
cuss what is going on deep within the bulk. There is a
χ is proportional to the von Neumann entropy of the
good reason for this: although the CFT side of the du- A
region A in the CFT [21–23]. The entanglement wedge
ality is defined exactly, the bulk side is ultimately some
E is then defined as the bulk domain of dependence of
nonperturbativetheoryofquantumgravity,whichisonly A
any achronal bulk surface whose boundary is A∪χ .
approximatelygivenbyasemiclassicalpathintegralover A
local bulk fields. This means that any attempt to “back
off of the extrapolate dictionary” and formulate a CFT
representationofthebulkoperatorφ(x,r),usuallycalled
a reconstruction, must itself be only an approximate no- 1 Asmentionedabove,thisreconstructionreliesonsolvinganon-
standardCauchyproblem[11,15]. Ingeneralitisnotknownif
tion. Nonetheless, there is a standard algorithm for pro-
thesolutionreallyexists,butitcanbefoundexplicitly(inadis-
ducing such a φ(x,r) perturbatively in 1/N: one con-
tributionalsense[14])forthecaseofsphericalboundaryregions
structs it by solving the bulk equations of motion with in the vicinity of the AdS vacuum; in this case it is called the
boundary conditions (1) [4, 9–14]. This algorithm has AdS-Rindlerreconstruction.
2
Recently, plausibility arguments for entanglement
wedgereconstructionhavebeendevelopingalongtwodif-
ferent lines. [24] proposed a re-interpretation of causal
wedgereconstructionasquantumerrorcorrection,which
is a structure that very naturally allows an extension to
entanglement wedge reconstruction. This was explicitly
realized in toy models [25, 26]. At the same time it has
gradually been understood that a proposal [27] for the
first 1/N correction to the HRT formula is indicative FIG. 1. Factorizing the bulk and boundary on a time slice.
thatentanglementwedgereconstructionshouldbepossi- TheentanglementwedgeEAisshaded. Forsimplicitywehave
ble [24, 28, 29]. In particular Jafferis, Lewkowycz, Mal- shown a connected boundary region A, although this might
not be the case.
dacena, and Suh (JLMS) have given a remarkable bulk
formula for the modular Hamiltonian associated to any
CFT region, which implies that the relative entropy of
twostatesinaboundaryspatialregionAissimplyequal Yang-Mills theory with gauge group SU(N), and with
to the relative entropy of the two bulk states in EA [29]. gauge coupling g ∼ 1 to equate the string and Planck
The purpose of this paper is to tie all of these ideas to- scales, this corresponds to the set of primary operators
gether into a proof that entanglement wedge reconstruc- whose dimensions are (cid:46) N1/4, together with their con-
tion is in fact possible in AdS/CFT: given a boundary formal descendants. If the duality is correct, meaning
subregion A, all bulk operators in EA have CFT recon- that correlation functions of local boundary operators in
structions as operators in A. thesestatescanbereproducedbybulkFeynman-Witten
diagramswithsomeeffectiveaction,3 thenthemachinery
of [4, 9–14] enables us to explicitly reconstruct all low-
ENTANGLEMENT WEDGE RECONSTRUCTION energy bulk operators φ(x,r) in such a way that their
AS QUANTUM ERROR CORRECTION
correlation functions in any state in the code subspace
agree with those computed in bulk effective field theory
We mentioned above that the holographic nature of withthateffectiveactiontoallordersin1/N. Forthisto
AdS/CFT prevents bulk operator reconstruction from workindetailwemustchoosesomesortofcovariantUV
being a precise notion. This reveals itself not just in cutoff in the bulk, but we will not discuss this explicitly
the perturbative nature of the the HKLL algorithm, but in what follows.
also in its regime of validity. As one acts with more We can now describe entanglement wedge reconstruc-
andmorereconstructedoperatorsinaregion,thethresh- tion. Say that we split a Cauchy slice Σ of the boundary
old for black hole formation is eventually crossed. This CFT into a region A and its complement A. The CFT
leads to a breakdown of the construction. In [24] this Hilbert space has a tensor factorization as
was formalized into the notion that one should think of
reconstructed bulk operators as only making sense in a H =H ⊗H . (2)
CFT A A
codesubspace oftheCFTHilbertspace,whichwedenote
H . The choice of this subspace is not unique, since Similarlywecanthinkofthecodesubspaceasfactorizing
code
in general we can choose to define it based on whatever as
observables we are interested in studying, but the sim-
plest thing to do is choose a state which we know has a Hcode =Ha⊗Ha, (3)
geometricinterpretation,andthenconsiderthesubspace
where H denotes the Hilbert space of bulk excitations
of all states where the backreaction of the metric about a
in E , and H denotes the Hilbert space of bulk excita-
that geometry is perturbatively small.2 A a
tions in E . We illustrate this in Figure 1. Both tensor
As a concrete example, we can consider the subspace A
factorizationsarecomplicatedbythefactthatgaugecon-
of states of the CFT on a sphere whose energy is less
straintsmightbepresent[31–38],butlikelythisproblem
than that of a Planck-sized black hole in the center of
can be absorbed into the choice of UV cutoff (as shown
the bulk, together with the image of this subspace under
for U(1) gauge fields by [39]). We could dispense with
the conformal group. For example in the N = 4 super
this issue by formulating our arguments in terms of sub-
algebras instead of subfactors, see [32] for the relevant
2 This is an overly conservative definition of the code subspace,
since a certain amount of backreaction (such as that present in
the solar system) can be included by resumming subclasses of 3 See [6] for a discussion of which CFTs have this property, and
diagrams in the perturbative expansion [11, 30]. We adopt it also for how to determine the effective action in terms of CFT
nonethelesstosimplifyourargumentsbelow. dataiftheydo.
3
definitions, but we have opted for the latter for familiar- HereA denotesabulkoperatorthatisalocalintegral
loc
ity. over the HRT surface χ . At leading order in 1/N, or
A
Entanglement wedge reconstruction is then the state- equivalentlyatleadingorderinthegravitationalcoupling
ment that any bulk operator O acting within H can G, we have
a a
always be represented in the CFT with an operator O
A
that has support only on HA. In fact this is the precise A = Area(χA), (9)
definition of the idea that the operator O can be “cor- loc 4G
a
rected”fortheerasureoftheregionA[40],andtheobser-
asrequiredbytheHRTformula,butA receivescorrec-
vation that a given O can be reconstructed on different loc
a tions at higher orders in 1/N [27, 44], or in the presence
choices of A reflects the ability of the code to correct for
of more general gravitational interactions [45–55].
a variety of erasures [24]. We will establish the existence
In fact [27] only established (8) to order N0. In [56] it
of O below, but first we need to review a recent result
A
was suggested that (8) continues to hold to all orders in
that will be essential for the proof.
1/N,providedthatonealwaysdefinesχ tobeextremal
A
with respect to the sum on the right hand side of (8)
REVIEW OF THE JLMS ARGUMENT (sometimes called the “generalized entropy”); such a χA
isknownasaquantumextremalsurface. Wewillassume
thistobethecasethroughoutthecodesubspaceinwhat
In [29], JLMS argued for an equivalence of relative en-
follows: if it is not then our arguments only establish
tropy between bulk and boundary. In this section we
entanglement wedge reconstruction to order N0.
give a more detailed derivation of this result, which also
To establish the JLMS result, we now observe that
extendsittohigherordersinthesemiclassicalexpansion
linearizing (8) about σ and using the first law (7), we
given an assumption we state precisely below. Let us
have
first recall that the von Neumann entropy of any state is
defined as (cid:16) (cid:16) (cid:17)(cid:17)
Tr(δσ K )=Tr δσ A{σ}+K . (10)
A σA a loc σa
S(ρ)≡−Tr(ρlogρ) , (4)
Here we have taken δσ to be an arbitrary perturbation
its modular Hamiltonian is defined as
that acts within the code subspace H . We have writ-
code
Kρ ≡−logρ, (5) ten A{loσc} to emphasize that Aloc is still located at the
surface defined by extremizing S(σ )+A . (10) is lin-
a loc
and the relative entropy of a state ρ to a state σ is ear in δσ, so we can integrate it to obtain
S(ρ|σ)≡Tr(ρlogρ)−Tr(ρlogσ)=−S(ρ)+Tr(ρK ) . (cid:16) (cid:16) (cid:17)(cid:17)
σ Tr(ρ K )=Tr ρ{σ} A{σ}+K , (11)
(6) A σA a loc σa
Relativeentropyisnon-negative,andvanishesifandonly
where now ρ and σ are arbitrary states acting within
if ρ=σ. Under small perturbations of the state, the en-
tropy and modular Hamiltonian are related by the “first Hcode. We have written ρ{aσ} to clarify that we are still
law of entanglement”:4 factorizing the bulk Hilbert space at the quantum ex-
tremalsurfaceforσ,eventhoughwearenowconsidering
S(ρ+δρ)−S(ρ)=Tr(δρK )+O(δρ2). (7) another state ρ as well. Since (11) holds for any ρ acting
ρ
within H , it implies that
code
Now consider the setup of the previous section, where
in some holographic CFT we pick a code subspace Π K Π =K +A{σ}, (12)
H = H ⊗ H in which effective field theory per- c σA c σa loc
code a a
turbatively coupled to gravity is valid, and in which all
where Π is the projection operator onto the code sub-
c
states can be derived from path integral constructions
space, and where we have defined the operators on the
in this effective theory. Following JLMS, we recall that right hand side to annihilate H⊥ . This is one of the
Faulkner et al. [27] showed that such states5 ρ obey code
main results of [29].
S(ρ )=S(ρ )+Tr(ρ A ) . (8)
A a a loc
lyticcontinuationofcertain“Renyientropies”Tr(ρn),andthat
4 This is derived in e.g. [41, 42] by linearizing the entropy. thebulkpathintegralforcalculatingthen-thRenyientropydoes
To deal with possible non-commutativity of ρ and δρ one not spontaneously break the associated Zn symmetry. There is
can use the Baker-Campbell-Hausdorff formula and recall that alsosomesubtletyinapplyingtheargumenttostatesthatdonot
Tr(A[B,C])=0ifAandB aresimultaneouslydiagonalizable. possessamomentoftime-reflectionsymmetry,butthisseemsto
5 The argument of [27], like the earlier classical argument [23], justbeamatterofconvenience,andanargumentthatdispenses
assumesthatthevonNeumannentropycanbefoundbyanana- withthiscriterionwillappearelsewheresoon[43].
4
Moreover combining (6), (8), and (11), we find that 1. For any X acting on H and any state |φ(cid:105) ∈
A A
H , we have
S(ρ |σ )=S(ρ{σ}|σ ) code
A A a a
(cid:104) (cid:16) (cid:17) (cid:16) (cid:17) (cid:104)φ|[O,X ]|φ(cid:105)=0. (16)
+ Tr ρ{σ}A{σ} −Tr ρ{ρ}A{ρ} A
a loc a loc
(cid:16) (cid:17) (cid:16) (cid:17)(cid:105) 2. ThereexistsanoperatorO actingjustonH such
+S ρ{σ} −S ρ{ρ} . (13) A A
a a that O and O have the same action on H , i.e.
A code
To order N0 the terms in square brackets cancel due to
the extremality of χ{ρ} with respect to S(ρ )+A , so O |φ(cid:105)=O|φ(cid:105), O†|φ(cid:105)=O†|φ(cid:105), (17)
A a loc A A
we find the equivalence of relative entropies which is the
for any state |φ(cid:105)∈H .
other main result of [29].6 code
More generally, we find that the relative entropies dif- Proof. Firstwenotethatthetwostatementsareguaran-
fer by the difference of generalized entropies of ρ on the teedtobeequivalentbythetheoremprovedinAppendix
two quantum extremal surfaces χ{ρ} and χ{σ}. In either B of [24]; for the convenience of the reader we sketch the
A A
case, we observe that if ρ {σ} = σ , then (13) implies logic of that proof in an appendix below. Therefore we
a a
S(ρ |σ )=0, and thus ρ =σ . This is the result that will only need to prove the first statement. Furthermore
A A A A
we will use in the proof below. it is sufficient to prove the theorem when O is a Hermi-
By symmetry, the results (13),(12) apply also for the tian operator which we assume now.8
complement regions A and a, although for a mixed state Consider any state |φ(cid:105) ∈ H , and let λ be an arbi-
code
one must use S(ρ ) rather than S(ρ ) when extremizing trary real number. For the two states |φ(cid:105) and
a a
to find χ .7
A |ψ(cid:105)≡eiλO|φ(cid:105) (18)
AtorderN0itisintuitivelyclearthat(13)saysthatwe
must be able to reconstruct in the entanglement wedge. which are both in H , the assumption of the theorem
code
Relative entropy is a measure of the distinguishability of guaranteestheexistenceofatensorfactorizationH =
code
two quantum states, so we can only have (13) if ρA,σA Ha⊗Ha such that O acts only on Ha, which implies
have just as much information about the bulk as ρ ,σ .
a a
We now prove a theorem that makes this precise, and ρa =σa (19)
extends it to higher orders in 1/N given the proposal of
because the two states only differ by the action of a uni-
[56].
tary operator eiλO on H . Using (15) we find
a
ρ =σ ⇒ (cid:104)ψ|X |ψ(cid:105)=(cid:104)φ|X |φ(cid:105). (20)
A A A A
A RECONSTRUCTION THEOREM
Using (18) we may rewrite (20) as
Theorem. Let H be a finite-dimensional Hilbert space, (cid:104)φ|e−iλOX eiλO|φ(cid:105)−(cid:104)φ|X |φ(cid:105)=0. (21)
H=H ⊗H be a tensor factorization, and H be a A A
A A code
subspace of H. Let O be an operator that, together with Expanding this equation to linear order in λ, we find
its Hermitian conjugate, acts within H . If for any
code (cid:104)φ|[O,X ]|φ(cid:105)=0, (22)
two pure states |φ(cid:105),|ψ(cid:105) ∈ H , there exists a tensor A
code
factorization Hcode =Ha⊗Ha such that O acts only on which proves (16), and hence also (17).
H , and the reduced density matrices
a
In the above theorem we assumed that H is finite-
ρA ≡TrA|φ(cid:105)(cid:104)φ|, σA ≡TrA|ψ(cid:105)(cid:104)ψ|, dimensionaltoavoidanysubtletieswiththeproofofAp-
ρ ≡Tr |φ(cid:105)(cid:104)φ|, σ ≡Tr |ψ(cid:105)(cid:104)ψ| (14) pendix B of [24]. In AdS/CFT we can accomplish this
a a a a
byintroducingaUVcutoffintheCFT.Thisonlyaffects
satisfy physicsneartheasymptoticboundaryofAdS,andthere-
fore is not essential to our discussion. In bulk language
ρ =σ ⇒ ρ =σ , (15)
a a A A the assumptions for the theorem follow from picking an
then both of the following statements are true: OthatliesintheentanglementwedgeE foranystatein
A
H , with the choice of factorization coming from the
code
quantumextremalsurfaceforthestate|φ(cid:105). Thisensures
that we can apply eq. (13) for thecomplement region A,
6 This equivalence is√complicated by gravitons, which are metric
variationsoforder Gandthuscanproducesecond-ordervaria- which then implies (15).
tionsoforderN0. Thesecanbedispensedwithbyintroducinga
gauge(e.g. theonein[29])wherethequantumextremalsurface
χA doesnotmoveuntilorderN0.
7 In this case, we expect that the two entanglement wedges can 8 To see this, we recall that any operator is a (complex) linear
neveroverlap. combinationoftwoHermitianoperators.
5
DISCUSSION tween (16) and (17). More details can be found in the
full proof provided in Appendix B of [24].
We view the theorem of the previous section as estab- SupposethatHcodeisspannedbyanorthonormalbasis
lishing the existence of entanglement wedge reconstruc- |i(cid:105) . Consider the state
AA
tion. Several comments are in order: (cid:88)
|Φ(cid:105)= |i(cid:105) ⊗|i(cid:105) , (23)
Our argument ultimately rests on the validity of the R AA
quantumversion(8)oftheHRTformula,derivedtoorder i
N0 by [27], and conjecturally extended to higher orders where R is a reference system whose Hilbert space is
in1/N by[56]. Understandingthoseresultsbetteristhus spannedbyanorthonormalbasis|i(cid:105) . Using(23)wemay
R
clearlyofinterestforbulkreconstruction. Anydiscussion mirroranyoperatorOactingwithinH toanoperator
code
ofbulkreconstructionwillultimatelybeapproximate,so O on R such that O|Φ(cid:105)=O |Φ(cid:105), O†|Φ(cid:105)=O†|Φ(cid:105).
R R R
it would be interesting to prove an approximate version The task is then to instead view O as O ⊗I and
R R A
of our theorem and use it to clarify the stability of our mirror it back to an operator O on A. To do this we
A
results under small perturbations. need the Schmidt decomposition
Note that our proof is constructive. After establishing (cid:88)
|Φ(cid:105)= c |α(cid:105) ⊗|α(cid:105) , (24)
the assumptions of the theorem of Appendix B in [24] α A RA
hold, that theorem provides an explicit formula for the α
reconstructionO . Ourconstructionisthusanimprove- where the states |α(cid:105) with c (cid:54)= 0 generally span a
A RA α
ment on the HKLL procedure even within the causal subspace of H ⊗H . We can mirror O ⊗I onto A if
R A R A
wedge, since the nonstandard Cauchy problem involved it acts within this subspace which is guaranteed by
in causal wedge reconstruction has only been solved in
[O ⊗I ,ρ (Φ)]=0. (25)
special cases. However, it does not give much insight R A RA
into how to think about the reconstruction from a bulk This statement is implied by (16). Therefore we obtain
point of view. This is to be contrasted with the HKLL (17).
algorithm, which (at least when it works) proceeds by
solvingbulkequationsofmotion. Webelievethatabulk
interpretation should exist, and finding it could be quite
illuminating.
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perconformal field theories and supergravity,” Int. J.
Theor. Phys. 38, 1113–1133 (1999), [Adv. Theor. Math.
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