Locality of Gravitational Systems from Entanglement of Conformal Field Theories.pdf

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Locality of Gravitational Systems from Entanglement of Conformal Field Theories

Jennifer Lin,1

Matilde Marcolli,2

Hirosi Ooguri,3,4

and Bogdan Stoica3

1Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA

2Department of Mathematics, California Institute of Technology, 253-37, Pasadena, California 91125, USA

3Walter Burke Institute for Theoretical Physics, California Institute of Technology, 452-48, Pasadena, California 91125, USA

4Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan

(Received 27 December 2014; published 2 June 2015)The Ryu-Takayanagi formula relates the entanglement entropy in a conformal field theory to the area of

a minimal surface in its holographic dual. We show that this relation can be inverted for any state in the

conformal field theory to compute the bulk stress-energy tensor near the boundary of the bulk spacetime,

reconstructing the local data in the bulk from the entanglement on the boundary. We also show that

positivity, monotonicity, and convexity of the relative entropy for small spherical domains between the

reduced density matrices of any state and of the ground state of the conformal field theory are guaranteed by

positivity conditions on the bulk matter energy density. As positivity and monotonicity of the relative

entropy are general properties of quantum systems, this can be interpreted as a derivation of bulk energy

conditions in any holographic system for which the Ryu-Takayanagi prescription applies. We discuss an

information theoretical interpretation of the convexity in terms of the Fisher metric.

DOI: 10.1103/PhysRevLett.114.221601 PACS numbers: 11.25.Tq

Introduction.Gauge-gravity duality posits an exactequivalence between certain conformal field theories

(CFTs) with many degrees of freedom and higher-

dimensional theories of gravity. It is of obvious interest

to understand how bulk spacetime geometry and gravita-

tional dynamics emerge from a nongravitating theory. In

recent years, there have appeared hints that quantum

entanglement plays a key role. One important development

in this direction was the Ryu-Takayanagi proposal [1,2]

that the entanglement entropy (EE) between a spatial

domain D of a CFT and its complement equals the areaof the bulk extremal surface homologous to it,

SEE minD

area4GN

: 1

Using Eq. (1), Refs. [39]studied the connection between

linearized gravity and entanglement physics of the CFT.

In this Letter, we continue this program. We develop

tomographic techniques to diagnose local energy density

in the bulk by the Radon transform of quantum entanglement

data on the boundary. Moreover, we show that properties of

entanglement on the boundary such as the positivity,

monotonicity, and convexity of the relative entropy are

guaranteed by energy conditions in the bulk. As the

positivity and monotonicity of the relative entropy are

general properties of quantum systems, we can interpret

this as a derivation of energy conditions in the bulk for any

holographic system, assuming the Ryu-Takayanagi formula.Relative entropy (see, e.g., Ref. [4]) is a measure of

distinguishability between two quantum states in the same

Hilbert space, for two density matrices 0 and 1 being

defined as

S1j0 tr1log1 tr1log0: 2

It is positive and monotonic:

S1j0 0; SW1jW0 SV1jV0; 3

whereWV. When0and 1are reduced density matriceson a spatial domain D for two states of a quantum fieldtheory (QFT), which is the case we restrict to from this

point on, monotonicity implies thatS1j0increases withthe size ofD. That is, over a family of scalable domainswith characteristic size R,

RS1j0 0: 4

Defining the modular HamiltonianHmodof0implicitlythrough

0 eHmod

treHmod ; 5

Eq.(3)is equivalent to

S1j0 hHmodi SEE 0; 6

with hHmodi tr1Hmod tr0Hmod the change inthe expectation value of the operatorHmod [Eq. (5)] andSEEtr1log1 tr0log0 the change in theentanglement entropy across D as one goes between thestates.

When the states under comparison are close, the pos-

itivity [Eq. (6)] is saturated to leading order[4,6,8]:

PRL 114, 221601 (2015) P H Y S I C A L R E V I E W L E T T E R S week ending

5 JUNE 2015

0031-9007=15=114(22)=221601(5) 221601-1 2015 American Physical Society
http://dx.doi.org/10.1103/PhysRevLett.114.221601http://dx.doi.org/10.1103/PhysRevLett.114.221601http://dx.doi.org/10.1103/PhysRevLett.114.221601http://dx.doi.org/10.1103/PhysRevLett.114.221601http://dx.doi.org/10.1103/PhysRevLett.114.221601


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S1j0 hHmodi SEE0: 7

This is the entanglement first law for its resemblance to the

first law of thermodynamics. Indeed, when 0 is a thermal

density matrix 0eH=treH, Eq. (7) reduces tohHi TS, an exact quantum version of the thermalfirst law.

In general, the modular Hamiltonian(5) associated to a

density matrix is nonlocal. However, there are a few simplecases where it is explicitly known. When 0 is the reduced

density matrix of a CFT vacuum state on a disk of radiusR,which we take to be centered at~x00 [10],

HmodZ

Ddd1x

R2 j~xj2R

Tttx; 8

whereTtt is the energy density of the CFT.Our goal in this Letter is to use the entanglement in

the CFT, in particular, the relative entropy, to elucidate

local physics in the bulk (for related recent work, see also

Refs.[3,7,1116]).Our starting point is ad-dimensional CFTwhose vacuum

state is dual to antide Sitter space (AdSd1). We consideran arbitrary excited state of the CFT which has a semi-

classical holographic bulk dual, whose metric can be

parametrized as

gAdSl

2

AdS

z2 dz2 hdxdx: 9

Spacetime indicesa;b; run overz;t;xi while ; ;run overt; xi, and i 1;; d 1 are boundary spatialdirections. We assume that the Ryu-Takayanagi formula

holds in the excited state, and the relative entropy betweenthe reduced density matrix1 of the excited state and0 of

the ground state for the entangling diskD of radius R iscomputable using the formulas discussed previously.

To apply a perturbative analysis in the bulk, we assume

that the radius R of the entangling domain is smallcompared to the typical energy scale E hTi1=d of thestate measured by the boundary stress tensor T as inER 1.

In this limit, to order less than E2dR2d, we will show thatthe relative entropy is expressed as the integral of the local

bulk energy density ,

S1j0 82GNZ

V

R2 z2 x2R

ffiffiffiffiffi

gVp ; 10

where GN is Newtons constant, V is a d-dimensionalregion on a constant-time slice bounded by the domain Don the boundary and the Ryu-Takayanagi surface in the

bulk, and ffiffiffiffiffi

gVp

is the volume form in the bulk (including the

time direction). In Ref. [8], it was shown that the first law

S1j0 0 in the linear approximation is equivalent tothe linearized Einstein equation. This holds to order

OEdRd. Our result (10) improves the approximation tothe order less than E2dR2d by taking into account thebackreaction from the bulk stress tensor.

Taking one derivative with respect to R, we find

RS1j0 82GNZ

V

1z

2 x2R2

ffiffiffiffiffi

gVp

: 11

Both positivity S 0 and monotonicity RS 0 areuniversal properties of the relative entropy. We find that,

in the gravitational dual, they are translated to positivity of

the integrals of the bulk energy density weighted byR2 z2 x2 ffiffiffiffiffigVp ( 0 in V). In particular, the weak energycondition 0 guarantees these properties. Though the

weak energy condition is not necessarily satisfied in AdS, it

holds near the boundary of AdS, where we are evaluating

Eqs.(10)and(11).

One more derivative relates the relative entropy to the

integral of the energy density on the boundary ofV,

2RR1R R2S1j0 162GNZ

ffiffiffiffiffi

gp ; 12

where ffiffiffiffiffi

gp

is the volume form on the Ryu-Takayanagi

surface. We will show that Eq. (12)can be inverted using

the Radon transform to express the bulk stress tensor point

by point in the near-AdS region using the entanglement

information of the CFT.In the holographic setup, it is generally believed that bulk

locality emerges from the entanglement information in the

boundary CFT and that the relation between bulk and

boundary observables is nonlocal. In this Letter, we give

an explicit example in which the bulk stress tensor is

expressed in terms of the boundary relative entropy, showing

how these general expectations are realized in a specific

setup.Hmod andSEE in holography.We first review how

each quantity appearing in the relative entropy definition(6)

is mapped holographically. The modular HamiltonianHmodfor the reduced density matrix of a CFT vacuum state on the

entangling diskD of radius R, centered at a point on theboundary, is expressed in terms of the CFT stress tensorTttas in Eq.(8). It vanishes in the CFT vacuum. We can also use

Eq. (8) to evaluate the expectation value ofHmod for anyexcited state in the same Hilbert space, by computing the

expectation valuehTtti of the CFT stress tensor usingholographic renormalization (see, e.g., Refs. [1719]) orthe shortcut of Ref. [8] to exploit the fact that the relative

entropy in the CFT vanishes in the limit that the entangling

domain shrinks to zero. As long as the bulk matter fields

contributing tohTi are dual to operators with scalingdimension > d=2, both methods give

hHmodi limz0

dld1AdS16GN

ZD

dd1xR2 j~xj2

R zdijhij: 13


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In general, the right-hand side is modified by boundary

counter terms if it involves operators with d=2.The holographic EE in Einstein gravity is given by the

Ryu-Takayanagi area formula(1). On a constant time slice

of pure AdS, the codimension-2 bulk extremal surface

ending on a boundary sphere of radius R is the half-sphere

z0r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 r2p :

14

The EE of the entangling disk of radius R in the CFTvacuum is equal to the area functional of pure AdS

evaluated on the surface [Eq. (14)]. Suppose we perturb

the bulk metric away from pure AdS by hab, which isparametrically small. Because the original surface was

extremal, the leading variation in the holographic EE

comes from evaluating z0r (14) on the perturbed areafunctional. One finds[6]

SEE l

d1AdS

8GNR

Z

dd1xR2ij xixjzdhij: 15

At orderh2, one must account for corrections to the shapeof the Ryu-Takayanagi surface; see, e.g., Ref. [4].

Linearized Einstein equations.We now summarize the

derivation of the linearized gravitational equations of

motion from the entanglement first law (7), as presented

in Ref. [8]. The idea of Ref. [8] was to apply the Stokes

theorem to the bulkd-dimensional regionVon a constant-time slice bounded by the entangling disk D on theboundary and the extremal surface in the bulk. One

can write Hmod and SEE as integrals over D and ,respectively, of a local (d 1)-formthat is a functional ofthe metric fluctuation hab. Within Einstein gravity,

Refs. [8,20,21] explicitly construct a hab that givesEqs.(13)and(15) when integrated overD and ,

ZD hHmodi;

Z

SEE: 16

Moreover, the exterior derivative of this is given by

d 2tEgtthgtt ffiffiffiffiffi

gVp

dz dxi1 dxid1 ; 17

with ffiffiffiffiffi

gVp

the natural volume form onVinduced from thebulk spacetime metric, and

R

fR2 z2 t t02 x2t 2t t0zzxiig

the Killing vector associated with (14), which is a bifurcate

Killing horizon in pure AdS. The linear gravitational

equations of motion in vacuum are expressed asEgabh 0.By the Stokes theorem, the relative entropy is

S1j0 hHmodi SEEZ

Vd: 18

Considering Eq. (18) for every disk on a spatial slice at

fixed time t 0, the entanglement first law S1j0 0can be shown to be equivalent toEgtth 0. Considering itfor Lorentz-boosted frames gives vanishing of the other

boundary components, Egh 0. Finally, an argumentappealing to the initial-value formulation gives vanishing of

the remaining components of the linearized Einstein tensor

that carry z indices.

To summarize, Ref.[8]proved the existence of a (d 1)-formas a functional of a metric fluctuation hab, for whichEq.(16)holds off shell and Eq.(17)holds withEgabhthelinearized gravity equations of motion in vacuum.

By accounting for the 1=N correction to the Ryu-Takayanagi formula[22], Ref. [9]showed that the entan-

glement first law (7) implies the bulk linearized Einstein

equations sourced by the difference in the quantum expect-

ation value of the bulk stress-energy tensor in the quantum

state of bulk fields relative to their vacuum state, htabi.Assuming that the source of the linearized Einstein equa-

tion is a local QFT operator, one can then argue thathtabiin the derivation of Ref. [9] can be uplifted to the bulk

operator tab. In contrast, in this Letter, we remain in thelarge Nclassical gravity limit, but assume the linearizedEinstein equations sourced by the classical value of the

bulk stress tensor.Effects due to bulk stress tensor.We now evaluate the

(d 1)-formof Ref.[8]on the bulk metric fluctuation habof the dual to an arbitrary excited state of a CFT, but in the

interior of the Ryu-Takayanagi surface for the entangling

disk(14), whose radius satisfies ER 1. As the deviationof the bulk metric in the enclosed volume V is parametri-cally small, all results of the above discussion carry over

and we can still use Eq. (18). However,Egabhin Eq.(17)should be evaluated on thehabwhich is reconstructed fromCFT data at the nonlinear level and is not identically zero.

Rather, the linearized Einstein tensor couples to bulk matter

in the form of the bulk stress tensor tab. Our main result[Eq.(10)] can now be derived by using

Egabhab 8GNtab: 19

The energy density appearing on the right-hand side of

Eq. (10) corresponds to the tt component of the stress-energy tensor, ttt.

For example, a massive scalar field in the bulk can

contribute to the metric perturbation habas

hO

i2z2, with

the scaling dimension of the corresponding operator on theboundary andhOi its expectation value, leading to anOhOi2R2 effect in Eq. (10). On the other hand,corrections to the relative entropy by nonlinear gravity

effects are of the order OE2dR2d or higher, which weignore. Thus, effects due to relevant operators with < dare visible in our approximation.

By taking a derivative of Eq. (10) with respect to the

radius R of the entangling domain, we find Eq. (11).Though the derivative also generates an integral over the


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Ryu-Takayanagi surface , it vanishes because t vanishes

on the surface. Thus, we have shown that the positivity and

the monotonicity of the relative entropy are translated to

positivity of the integrals of the energy density weighted

byR2 z2 x2 ffiffiffiffiffigVp . In other words, we derive thepositivity of bulk energy conditions (10) and (11) from

entropy inequalities on the boundary that hold for

any CFT.

Inverting the bulk integral.

We found thatRS1j0is given by the integral of the energy density over the

regionVinside the Ryu-Takayanagi surface. We can invertthis relation to compute point by point in the bulk by

using the relative entropy S1j0.To show this, note that

RR1S1j0 162GNZ

V ffiffiffiffiffi

gVp

; 20

so differentiating again,

2RR1R R2S1j0 162GNZ

ffiffiffiffiffi

gp ; 21

where ffiffiffiffiffi

gp

is the natural volume form on the Ryu-

Takayanagi surface induced from the bulk spacetime

metric. The right-hand side is still non-negative if we

assume the positivity of the bulk energy density. Thus,

2RR1R R2S1j0 0: 22

Here, the bulk geometry is the unperturbed AdS, and its

spacelike section is the d-dimensional hyperbolic space.The surface is then totally geodesic. In this case, the

integral (21) is the Radon transform and its inverse is

known. For a smooth function fond-dimensional hyper-bolic space, the Radon transformRfis an integral offoverann-dimensional geodesically complete submanifold withn < d. This gives a function on the space of geodesicallycomplete submanifolds. The dualRadon transformRRfgives back a function on the original hyperbolic space in the

following way: pick a point in the hyperbolic space,

consider all geodesically complete submanifolds passing

through the point, and integrate Rf over such submani-folds. It was shown by Helgason [23]that ifdis odd,f isobtained by applying an appropriate differential operator on

RRf. We are interested in the case nd 1 for whichf 4d1=2d=21d=21QRRf; 23

where Q is constructed from the Laplace-Beltramioperator on the hyperbolic space as

Q 1d 2 2d 3 d 2 1: 24

Applying this to Eq. (21), we find

4d3=2d=21d=2GN1QR2RR1R R2S1j0; 25

when d is odd. There exists a similar formula for deven[24].

Note that, even if we are evaluating at

z;t;x

in the

near-AdS region, there are totally geodesic surfaces that

pass through this point and go deep into the bulk, where thegeometry can depart significantly from AdS. However,

contributions from these surfaces are negligible when

Ez 1, where E is the typical energy scale of the CFTstate. In this case, we can choose anotherz0 so thatz z0and the geometry underz0is still approximately AdS. Sincemost totally geodesic surfaces passing throughz;t;xstayunder z0, an integral over such surfaces is well approxi-mated by the inverse Radon transform in the hyperbolic

space.The energy density is the time-time component of the

stress-energy tensortab. By computing the relative entropyin other Lorentz frames, we can also derive components talong the boundary. Finally, we can use the conservation

law,atab0, to obtainthe remainingcomponents,tz; t.Thus, we can use the entanglement data on the boundary to

reconstruct all components of the bulk stress tensor.

Since the Radon transform preserves positivity, thepositivity of the energy density implies the positivity of

2RR1R R2S1j0. Conversely, the positivity ofthe latter implies the positivity of its dual Radon transform.

It is interesting to note thatQ in Eq.(25)is a positivedefinite operator when acting on normalizable functions on

the hyperbolic space, though this does not quite imply the

positivity of the energy density.Comparison with information theoretic analysis.We

now discuss to what extent we can recover the monoto-nicity and convexity(12)of the relative entropy from the

following general property of the relative entropy. Consider

a density matrix [with, 0, tr 1], and twoincrements h; l, given by matrices with hh, l l,trh trl 0. If the matrices ; h; l satisfy; h ; l 0, the relative entropy satisfies

Shj l hh l; 121h li; 26

where the right-hand-side is the Fisher metric, with theHilbert-Schmidt inner productha; bi trab. Thus, thesecond-order term is non-negative definite, and the quad-

ratic form only vanishes forhl.The entanglement density matrices R and 0R

discussed in this Letter have additional properties for smallR. Since Hmod is given by the integral (8), the Taylorexpansion ofTtt around~x~x0 givesHmodh0Rd .Therefore, the density matrix for the vacuum state can be

expanded as


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0R 1

N h0

0Rd ; 27

where tr1N and h00h0 1=Ntrh0, so that

trh000. ForR, we postulate

R 1N

X

i

liRi hRd ; 28

so that the small R expansion of the relative entropy SPiR2i si expected from the holographic computation

above is reproduced. Here, trli0 and is are scalingdimensions of relevant operators, i< d.

The right-hand-side of Eq. (26)becomes

Xij

N

2hli; ljiRij : 29

Thus, the leading-order term ofS1j0 is

S1j0 N

2jl1j2R21; 30

where1minifig. Its first and second derivatives in Rhave leading term

RS1j0 N1jl1j2R211;

2RR1R R2S1j0 N

2jl1j2421 1R212:

31

The first is manifestly positive, and the second is non-negative provided 1 1=2, which is satisfied by ourassumption1> d=2 ford 2.

Our holographic analysis shows that the positivity and

convexity of the relative entropy hold for subleading termsup to OR2d. On the other hand, corrections to Eq. (30)may involve not only quadratic terms with ij < 2d,but also cubic terms withijk < 2d, etc. It appearsthat additional assumptions on the density matrices arerequired to explain the convexity from this point of view.

In this Letter, we gained new insight into how spacetime

emerges in the holographic setup. We found Eq. (25) to

express the local energy density in the gravitational theoryusing quantum entanglement data in the boundary theory,and that universal properties of entanglement such as the

positivity and monotonicity of the relative entropy can beinterpreted as positivity conditions on the bulk energy

density.

We thank N. Bao, D. Gaiotto, D. Harlow, T. Hartman,

P. Hayden, N. Hunter-Jones, C. Keller, D. Kutasov, H.Liu, Y. Nakayama, S. Pufu, P. Sulkowski, T. Takayanagi,

M. Van Raamsdonk, and E. Witten for useful discussion.J. L. acknowledges support from the Sidney Bloomenthalfellowship at the University of Chicago. M. M. is cur-

rently supported by NSF Grants No. PHY-1205440,

No. DMS-1201512, and No. DMS-1007207. H. O. and

B. S. are supported in part by the Walter Burke Institutefor Theoretical Physics at Caltech, by U.S. DOE GrantNo. DE-SC0011632, and by a Simons Investigator award.

The work of H. O. is also supported in part by the

WPI Initiative of MEXT of Japan and JSPS Grant-in-Aid for Scientific Research C-26400240. He also thanksthe hospitality of the Aspen Center for Physics and the

National Science Foundation, which supports the Centerunder Grant No. PHY-1066293. B. S. is supported in part

by a Dominic Orr Graduate Fellowship. J. L., H. O., andB. S. would like to thank the Institute for Advanced Study,Princeton University, and the Simons Center for Geometry

and Physics for hospitality. J. L. also thanks Caltech forhospitality.

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