JHEP06(2016)085 . , Springer June 6, 2016 May 10, 2016 June 15, 2016 : : : Accepted Received Published 10.1007/JHEP06(2016)085 arXiv:1509.00113 doi: [email protected] , Published for SISSA by [email protected] , . 3 1604.05308 The Authors. c AdS-CFT Correspondence, Gauge-gravity correspondence
For holographic CFT states near the vacuum, entanglement entropies for spa- , [email protected] 6224 Agricultural Road, Vancouver, BC, V6TE-mail: 1W9, Canada [email protected] Department of Physics and Astronomy, University of British Columbia, Open Access Article funded by SCOAP Keywords: ArXiv ePrint: first law. For two-dimensionalfor CFTs, the we second-order use contribution this toshow to the that entanglement derive this entropy a stress fromstates completely the tensor related explicit stress formula formula to tensor. can thebe We be vacuum reproduced reproduced by via by a theauxiliary a local de perturbative direct conformal Sitter solution CFT spacetime, transformation. to calculation extending the for a This first-order non-linear result result scalar in can wave also equation on an tial subsystems can be expressedof perturbatively local as an operators expansion dual ininformation to the light for one-point bulk functions CFT fields. statesgeneral Using and formula the canonical for connection energy between thisarbitrary quantum for expansion Fisher ball-shaped the up dual region, to spacetimes, extending second-order we the in describe first-order the a result one-point functions, given for by an the entanglement Abstract: Matthew J.S. Beach, Jaehoon Lee, Charles Rabideau and Mark Van Raamsdonk Entanglement entropy from one-pointholographic functions states in JHEP06(2016)085 i 2 25 15 and and Ψ | i 1 1 + Ψ i Ψ 1 | M Ψ | 2 , the superposition 2 9 Ψ 2 M 5 and 15 1 Ψ M 19 9 21 16 13 – 1 – and which properties of a CFT will guarantee that 1 23 ]. It is an interesting question to understand better 4 ], their entanglement entropies for arbitrary regions (at 3 17 – 1 7 ) are completely encoded in the extremal surface areas of an stress tensor contribution 2 N 1 5 are two such states, corresponding to different spacetimes . Thus, the set of “holographic states” is not a subspace, but some general subset. Even in holographic CFTs, it is clear that not all states will have this property. For example, if 4.4 Excited states around thermal background 4.1 Conformal transformations4.2 of the vacuum Entanglement state entropy4.3 of excited states Perturbative expansion 2.1 Relative entropy2.2 and quantum Fisher Canonical information energy 3.1 Example:3.2 CFT Example: scalar operator contribution 2 i 1 2 Ψ Ψ which CFT states have thisfamilies property, of low-energy states with geometric entanglement exist. | is not expected toM correspond to any single classical spacetime but rather to a superposition of asymptotically AdS spacetime.(functions In on general, a space the ofble space subsets asymptotically of of AdS the metrics possible AdS (functions entanglement boundary) ofof a is entropies geometrically-encodable few far entanglement spacetime larger entropy coordinates), than should soof the be this all space property present of quantum in possi- only field a theory tiny states fraction [ In holographic conformal fieldpretation theories, have a states remarkable withtanglement structure a entropy of simple formula entanglement: classical [ accordingleading gravity to dual order the inter- in holographic en- large B Rindler reconstruction for scalar operators in CFT 1 Introduction 5 Auxiliary de Sitter spaceA interpretation Direct proof of the positivity 4 Stress tensor contribution: direct calculation for CFT 3 Second-order contribution to entanglement entropy Contents 1 Introduction 2 Background JHEP06(2016)085 should A ) (1.1) A ( S 3 , 2 is a solution to the bulk equations i Ψ entanglement entropies | → ) . We will work perturbatively around the , z B µ x ( – 2 – α φ → ) and compare the results with a direct CFT calculation 1.1 dual to a holographic state asymptotics Ψ α φ M should be small enough so that the bulk extremal surface associated with ) could be applied to any state, but for states that are not holographic, i → A ) 1.1 µ x ]. ( 9 α 4 – 5 ) in cases where the gravitational equations are Einstein gravity with matter and i here indicates all light fields including the metric. α φ hO i → hO ( In this paper, our goal is to present some more explicit results for the gravity prediction The recipe ( According to the AdS/CFT dictionary, this boundary behavior is determined by the For a hint towards characterizing these holographic states, consider the gravity per- Ψ Results along these lines inAnother the interesting limit possibility is of that small the boundary one-point functions regions could or give constant boundary one-point data that functions is not Here, the region | 3 4 2 grav A be contained in thevalues; we part do of not the need spacetime this determined restriction if throughappeared we the are in equations working [ of perturbatively. motion by the boundaryconsistent with any solutionfor of the the radial classical evolution bulk problem obeys equations; certain this constraints. possibility exists since the “initial data” S the region is taken tovacuum state be to a obtain ball-shaped an region expression as a power series in the one-point functions of CFT the results will betest inconsistent for whether with a the CFT actualcarry state has CFT out a answers. the dual description procedure Thus,of well-described in we by the a have ( classical a entanglement spacetime: holographic. stringent entropies; if there is a mismatch for any region, the state is not where On the other hand, the bulk(and spacetime many itself other allows us non-local toallows quantities). calculated us entanglement (via Thus, entropies gravity calculations) the tolocal assumption determine properties that the entanglement a of entropies and state theby other is state the non- holographic (again, one-point at functions: least perturbatively) from the local data provided or even for thesay whole that spacetime. the Thus, bulk for spacetimebehavior geometries of (at dual the least various to in fields. holographic a states, perturbative sense) we can is encodedone-point functions in the of boundary low-dimension local operators associated with the light bulk fields. determined by evolution inat the the holographic timelike radial boundary direction,is of more with AdS. subtle, “initial In but data” this thea specified case, perturbative asymptotic the sense behavior (e.g. existence of perturbatively and thethe in uniqueness fields Fefferman-Graham deviations expansion). determines of from the a pure It metric AdS,is solution is at or enough plausible order-by-order least to that in in determine in a many solution cases, nonperturbatively this to boundary some data finite distance into the bulk, spective. A spacetime of motion. Such asurface solution (and can appropriate be boundary characterizedfrom conditions by the a Cauchy at surface set the is of determined AdSin by initial boundary). time evolving data this using initial on The the data a solution bulk forwards bulk (or equations. away backwards) Cauchy Alternatively, we can think of the bulk solution as being JHEP06(2016)085 ) 1.1 (1.5) (1.6) (1.7) (1.2) (1.3) with the . ˜ B ) 3 ) reviewed in δφ B vac ( ρ . . O || ) ) 2 B 3 Our focus will be on i ρ ) + α ( δφ α ( S , hO b O ( 00 ) (1.4) , δφ T O below. On shell, the latter α (2) 2 ab ) + vac B r 2 + ) T α ρ δφ ]: to second-order in perturba- 3 a i ( || R − ξ E 2 00 11 B δφ 2 , δφ ): Σ 1 2 ( ρ T α ( R Z h O 2 S x 1.2 δφ ) − r 1 i − ( 3 ) − E R ) + 00 − d i − 2 δφ 1 2 α δg T d 2 ( B h ξ B R 2 = and the bulk extremal surface O H £ Z r , δφ x h 1 α B + π R B − − δg, S – 3 – 2 d B ( 2 δφ + ∆ ∆ ( d S ω = 2 R E B Σ x (2) 1 2 vac B Z 1 Z i − δ vac B S − π ρ B d + i − d H ) = h B B ) = log + 2 α B i S Z H Ψ , where h vac | B π (1) i , δφ ( ≡ − δ S α B ) = ∆ B B + + ∆ + 2 H S below, the last term can be written more explicitly as δφ h vac B ( H ) = δ 2 ρ i E vac vac vac B B B || Ψ = S S S | B ( ρ B is the “presymplectic form” whose integral defines the symplectic form = = ( B ) always gives the correct CFT result for ball-shaped regions, regardless S ) = S S ω i (1) 1.1 Ψ δ | ( B S ]: below. We then make use of a recent result in [ 2 is equal to the “canonical energy” associated with a corresponding wedge of the 10 5 This second-order relative entropy is known as quantum Fisher information. 5 where Σ is asame boundary, bulk spatial region between As we review in section Rearranging this, we have a second-order version of ( state bulk spacetime. We providequantity a can brief be review expressed of aspure this a AdS in quadratic spacetime, section form so on we the have space of first-order perturbations to which follows immediately fromsection the definition of relativetions entropy from the vacuum state, the relative entropy for a ball-shaped region in a holographic General second-order result forthe ball second-order entanglement answer; entropy. in thisshould case, hold it is for lessobtain any clear explicit CFT whether formulae the or at gravity this whether results order, from they we ( begin represent by a writing constraint from holography. To is the vacuum modular Hamiltonian fority a procedure ball-shaped ( region. Thus,of to first-order, whether the the grav- state is holographic. value [ This well-known expression isentanglement universal for all CFTs since it follows from the first law of operators. At first-order, the result depends only on the CFT stress tensor expectation JHEP06(2016)085 ). 1.7 (1.8) (1.9) (1.11) (1.10) ) start- , and the 1.7 ∂B ↔ −} + { , i ) + 0 2 2 i x . The first-order bulk ) ( x ˜ β B + 2 ≥ < x x ) is exactly reproduced. ( , 1 1 from the expectation values ihO with boundary i x x ) ) 0 ++ 0 B x 1.10 T x ( B D ( ) is always negative, as required α 2 2 ih α on the case of 1+1 dimensional ) ) ) 2 2 hO + 1 ) providing some explicit results for 3 hO x x ) 1.10 x 0 ) 0 ( In order to check the general formula 1.7 + − . Given the one-point functions within x ; B x, x ++ R R ( ( ( T 2 2 h x, z ) ) ) αβ ( 1 1 + 2 G α x x B , x K – 4 – ) via a direct CFT calculation by considering − + ) together provide a formal result for the ball D + 1 B ), our final expression involves integrals only over Z x R R D ( 1.8 ( ( B 1.10 Z 2 1.2 D ( K Z 2 2 is the matter stress tensor at second-order in the bulk + 2 ) = 1 2 π 6 cR − dx (2) ab 0 ), and ( x, z T ( = B α ) = Z 1.7 2 δφ + 1 , x ), ( 1 matter B dx x 0 S ( 1.6 is a bulk Killing vector which vanishes on B 2 Z (2) ) to determine the linearized bulk perturbation in Σ and evaluate ( ξ (including the metric perturbation) may be expressed in terms of the K δ 1 2 , we consider the matter terms in ( α 1.8 − δφ 3.2 = is the domain of dependence of the ball grav B is the central charge. In this special case, the conservation equations determine B S c D , so as in the first-order result ( (2) 0 δ In section We can also check the formula ( The expression ( , we can use ( B . This will not be the case for the terms involving matter fields, or in higher dimensions. B 0 case, the result takes the form with integrals over the entire domain of dependence region. states that are obtainedtwo from dimensions, states the with CFT anand vacuum arbitrary the by traceless conserved entanglement a stress-tensor entropy local can forout conformal be these this obtained, transformation. states calculation in can In section also 4, be and calculated show explicitly. thatthe We the quadratic carry result contributions ( of scalar operator expectation values. Here, as in the generic on B As a consistency check, weby show its that interpretation the as expression the ( second-order contribution to relative entropy. where the stress tensor expectation values throughout the region where the integrals cankernel be is taken given over by any spatial surface Explicit results for 1+1and dimensional provide CFTs. more explicitCFTs, results, carrying out we an focus expliciting calculation in from of section a the general gravitational boundary contributions to stress ( tensor. We find the result where D entanglement entropy of aone-point holographic functions. state, expanded to second-order in the boundary matter fields, and perturbations boundary one-point functions via bulk-to-boundary propagators on gravitational phase space, JHEP06(2016)085 (2.1) ) can be ]. 1.10 14 ]. We begin with ], it is shown that 11 13 ) with the CFT energy , In the latter case, and for µ ] it has been pointed out , we show that in the 1+1 ν 6 12 dx 5 , for which the the vacuum µ d µν T dx µ B ζ + 0 2 B Z dR = − of the CFT Recently, in [ ( B 2 ). More generally, we can write it as = 0. In section ) for the entanglement entropy at second- 2 dS B R ) are expected to hold only for holographic L R 1.3 1.1 1.10 ]. – 5 – = 2 15 , ds ) for the entanglement entropy of a ball with radius 13 ,H B ,R µ H x − ( e S = (1) ] that at least some of the contributions at this order can be reproduced by δ vac B 18 ρ – 16 acting as a source term at can be obtained as the solution to the equation of motion for a free scalar i ) While the explicit two-dimensional stress tensor contribution ( µ µ x x ( 00 , so reproducing these results via some local differential equation will require a T B h D Including the contributions from matter fields or moving to higher dimensions, the There is evidence in [ 6 and center CFT calculations in generalthe dimensions, results since there most they directly arise apply to from the CFT case where two the and perturbation three-point is to functions, the though theory rather than the state. a brief review of these results. 2.1 Relative entropy andOur quantum focus Fisher will information bedensity matrix on is ball-shaped known subsystems explicitly through ( 2 Background Our holographic calculation of entanglementpoint entropy to functions second-order makes in use the oftion boundary the and one- direct canonical connection energy between on CFT quantum the Fisher gravity informa- side, pointed out recently in [ they represent genuine constraints/predictions from holography. the results at higher order inbetter perturbation which theory, CFT it states is and/or an which interestingthrough CFT question properties to direct understand are required CFT to calculations. reproduceand the which results This states should in these help theories us are understand holographic. better which theories Discussion. obtained by a directgeneral CFT the calculation holographic for predictionsstates a from in special CFTs ( with class gravity duals. ofof the It states, second would we be order interesting emphasize contributions to we that understand considered better in here whether are all universal for all CFTs or whether expression for entanglement entropy involvesmond one-point functions on themore entire complicated causal auxiliary dia- spaceThis that direction takes into is account pursued the further time in directions [ in the CFT. order can also resultsif from solving we a add scalar athis field simple agreement equation extends cubic on to interaction the allresulting auxiliary term. orders nonlinear de for wave Sitter In a equation space an suitable alsonear choice upcoming reproduces a of thermal the paper the state second-order [ scalar in entanglement field the entropy potential. auxiliary The kinematic space recently described in [ that the first-order result R field on an auxiliarydensity de Sitter space dimensional case, the stress tensor term ( Auxiliary de Sitter space interpretation. JHEP06(2016)085 B ζ = 0 (2.5) (2.6) (2.7) (2.8) (2.2) (2.3) (2.4) B . One ζ ) is the , and B vac B µ ρ ρ n as log , with . B ρ B t B ∂ ρ D ] 2 . ) tr( ) 0 − ~x 3 λ ( = − ) . , slice, we have with the same boundary , O ~x B ) . 1 ( 0 vac B 3 S B t + − ) ρ λ d 3 − ( D ν =0 ), = λ ) term involving the quantum 2 vac B λ ( O ) 2
t log ρ dx in ) , 0 i 2.1 λ O t 2 B 1 + 0 ( ρ + B − O δρ λδρ vac B 2 , δρ t tr( ρ , 1 ( in the i + δρ is defined as 1 ∧ · · · ∧ B − 2 µ 0 δρ − σ 1 h S ) λ x 2 ν 2 , δρ B ∆ will generally be different than λ R 1 + ρ [ log( dx ), we have 1 1 B − δρ π R − d i − ρ h log d ν dλ 2.6 – 6 – ν 2 B + B i associated with this conformal Killing vector, the λ λ δρ ··· and ∆ indicates the difference with the vacuum δρ i H ρ 1 ν h ∂ µν + ) µ νν T B and center i 0 ) = T h tr x µ µ vac B B B 1 2 R = ∆ ζ ζ vac B ρ 1)! − 0 ρ ) = tr( ≡ i 1 − B || x ) σ Z vac B ) = d i λ )( ( ρ λ ( 0 + ( || with no contribution from t B = B B ρ 1 − ρ ρ vac ( B ν ( δρ, δρ t δρ S S h ( with radius S π = ) and making use of ( R B 2 B − 2.4 S ) so the leading contribution to relative entropy appears at second-order = i ) can be taken over any spatial surface B B is the volume form on the surface perpendicular to a unit vector ζ 2.1 H µ h is the vacuum modular Hamiltonian given in ( is the CFT stress tensor operator and δ µ B µν n = H T . For the ball B . Rearranging ( For a one-parameter family of states near the vacuum, we can expand For excited states, the density matrix . This quadratic in S ∂B B λ . This is known as the quantum Fisher information metric. (1) This general expression isFisher valid for information any metric CFT, generallyexpectation but has the values. no simpleexpression However, expression for quadratic in holographic in termsquantum states the of Fisher we CFT information local and can one-point operator canonical convert functions energy. this by term using into the an connection between a positive-(semi)definite metric onσ the space of perturbations to a general density matrix where This quadratic form, which is positive by virtue of the positivity of relative entropy, defines The first-order contribution to relative entropy vanishesδ (this is the first law ofin entanglement where entanglement entropy for thestate. region measure of this difference is the relative entropy By the conservation of the current integral in ( as so that is a conformal Killing vectoron defined in the domain of dependence region where JHEP06(2016)085 . ) ) 1 . ; it with (2.9) B 2.10 (2.12) (2.10) (2.14) (2.13) (2.11) where at the matter D ˜ M matter 1 by mak- B B ζ g, φ δφ N in t ∂ in pure AdS and ˜ ] = ( A 2 . ˜ B and B ) , there exists a α 0 B 1 φ ~x using a metric in R AdS δg ) ). The result ( and − ˜ A N M ~x B ( ν G . On . − (4 dx / ) 2 B µ , Area( N ) ) 3 near the CFT vacuum, the 0 ) λ G t dx − 3 i ( 4 ) ) AdS λ . − O ) ( M λ ˜ ) t A O x, z ( ˜ + A ( Ψ( ] is that this second-order quantity | = 0) + − µν 11 2 2 . Γ z Area( δg d ) matter 2 x, z Area( 2 z ˜ ( − − A λ N δφ 2 − µν + 2 M G + ν R Γ λ ) µ [ 4 1 ˜ N A 1 π + R Area( dx 0) – 7 – − µ d AdS 0 and Γ = 0 for pure AdS. λδg πG ≡ x, ] + ( dx d` i + A 16 → ∂ matter 1 µν S + ) z ] i 0 Γ = 2 and approaches the conformal Killing vector ) can be computed at leading order in large = (Area( AdS x µ B i term in relative entropy to a gravitational quantity, since g λδφ 19 ζ ˜ dz A vac B B i − µν ρ S B = = B ). The main result in [ i T || Z h x g H 2 B h z 2 AdS . Near the boundary, we can describe N ρ 1 2.12 ` ( + ( − as the domain of dependence of this region, as shown in figure matter M d AdS S z πG to the area of the minimal-area extremal surface = φ B d` z∂ 2 16 R A ) can be described via a metric and matter fields )[ ds λ 0 ( t is the intersection of the causal past and the causal future of ) = ) from the previous section, the second-order contribution to entanglement − M B t vac B ( R ) has a finite limit as ρ 2.8 , π || R 2 B z, x ρ ∂A ( − ( µν S = B ξ To describe the general result, consider the region Σ between For a one-parameter family of holographic states The relative entropy Alternatively, can be thought ofKilling as vector a which vanishes Rindler at boundary. wedge In of Fefferman-Graham AdS coordinates, associated this with is gravitational quantity via ( can be expressed directly as athe bulk integral integrand over the is spatial a region quadratic Σ between form on the linearizedspacetime, bulk and perturbations define By the result ( entropy is equal to the leading order contribution to relative entropy. This is related to a dual spacetimes with some perturbative expansion behaviour of the metric through [ Thus, for holographic states, we can write This yields immediately that ∆ also allows us to relate theit ∆ implies that the expectation value of the CFT stress tensor is related to the asymptotic where Γ ing use of theentropy holographic for entanglement entropy a formula, region boundary which relates the entanglement Consider now a holographic state,totically which by AdS definition is spacetime associatedFefferman-Graham with coordinates some as dual asymp- 2.2 Canonical energy JHEP06(2016)085 ; B R (2.15) and the . B ef g ad g , the first-order bc ) does not change 1 g 1 , the matter stress 2 1 ˜ δφ ]. This is quadratic B δφ (2) is given explicitly by on the boundary. The ab + B 20 ξ T on ω B fd £ ξ g , 1 ae g . δφ ( b bc g 2 1 (2.16) (2) ) ab ˜ T − B 2 ef matter a B γ ω ξ ef d g Σ . The surface Σ lies between Σ ∇ ) Z B Z cd Σ 1 1 bc ξ g γ − δφ ab ) + ) ) − g 1 B 1 1 ξ 2 1 δg 1 ef δg £ δφ – 8 – γ B , − BB B B ξ d 1 ξ x ξ z £ fc ∇ £ £ δφ , g , , ( 2 bc 1 1 1 B be γ ( g δg D δg ( δφ full ( ( ad t associated to the ball-shaped region ω ω ω g Σ which splits into a gravitational part and a matter part as in , and the red lines 1 2 Σ Σ Σ B abcdef Z Z Z W B R is the Lie derivative with respect to ζ P − a full = = = 1 cd ω ) = g N δφ 1 fb B ξ 1 g πG , δφ £ is the symplectic form associated with the phase space of gravitational ae 1 16 g Σ δg . ) it has been assumed that the metric perturbation has been expressed ( = ˜ W B E is timelike hence defines a notion of time evolution within the region ) = 2 2.15 B ξ , γ 1 abcdef γ . The Rindler wedge P ( ω The “canonical energy”, dual to relative entropy at second-order, can be understood In deriving ( in a gauge for which the coordinate location of the extremal surface perturbation in metric and matter fields.of The a symplectic “presymplectic” form form is equalthe to the third integral line. over Σ The mattertensor part at can quadratic be order written in explicitly the in fields, terms while of the gravitational part In the first line, solutions on Σ, and in the perturbative bulk fields including the graviton, and given explicitly by extremal surface The vector the “Rindler time” associated with this Rindler wedge. as the perturbative energy associated with this time, as explained in [ Figure 1 blue lines indicate the flow of JHEP06(2016)085 ]. ], 21 23 (3.1) (3.2) – (2.17) (2.18) . ). 21 b 3.1 , where the (2) ab ˜ ) is linear in T B as ( ), the second- a B ξ 3.1 B Σ 2.15 D Z 1 2 )+ CFT ) and ( 1 i ) 0 δg 2.8 x B ), we only need to know the ( ξ α , as discussed in detail in [ B £ continues to satisfy the Killing B , R 1 hO B D ) 0 ξ δg (including the metric perturbation, x ( can be expressed as an integral over ; , . ω α B Σ δφ x, z Z = 0 = 0 ( ) ) 1 2 α λ λ ( ( − K 0 ˜ ˜ B B | | x ) is quadratic in these one-point functions and ) d – 9 – B λ ) = d ξ ( 1 1.7 B g D B ξ , δφ Z and the corresponding extremal surface 1 £ = B δφ ( Σ | E ) ) is the leading order contribution to the relative entropy 2 1 − x, z 2.8 ( = α stress tensor contribution ) 1 2 vac B ρ δφ i ( ), we can now write down a general expression for the ball entanglement 1 ) is the relevant bulk-to-boundary propagator. As discussed in [ 1.7 0 , δρ ], it is always possible to satisfy these conditions; we will see an explicit 1 x ; 20 . Thus, we require that δρ ˜ continues to vanish there), and the vector B x, z −h ). This is dual to canonical energy, given explicitly by: ( B ξ α = vac B K ρ B || S To summarize, for a holographic state, the second-order contribution to entanglement These linearized perturbations are determined by the boundary behavior of the fields should generally be understood as a distribution to be integrated against consistent B ρ (2) α ( δ and these can be expressed in terms of3.1 the CFT one-point Example: functions on CFT In this section,calculation as of a the sample quadratic application stress of tensor the contribution general to formula, the we entanglement provide entropy an for explicit S This is quadratic in the linearized perturbations K CFT one-point functions, ratherthe than CFT expectation a values, function. therepresents result our Since ( desired the second-order expression result. ( entropy in the expansion ( of the corresponding operator. We can express the results as where via the linearized bulkin equations. the region In Σboundary general, (or behavior more to in generally determine the inThe domain the the relevant of linearized boundary dependence Rindler perturbations behaviour region wedge of each bulk field is captured by the one-point function entropy of a general holographicstate, state in terms up of to the second-orderorder CFT in term one-point perturbations in functions. to the Accordingthe the entanglement to bulk vacuum entropy ( spatial for region a Σintegrand ball between is quadratic in first-order bulk perturbations. example below. 3 Second-order contribution toUsing entanglement the entropy result ( equation at As shown in [ (so that JHEP06(2016)085 and (3.6) (3.7) (3.9) (3.3) (3.4) (3.5) i (3.10) ) + ) is x ( 3.6 ++ i ) T h − x ( −− . T ) (3.8) h ) . − − N x a x ( AdS ( . G V ` − b − . ) π 1 ∇ . λh λh δg + = 0 B ≡ b ξ i = 0 ) = . V 0) = 8 £ . a 0) , ++ z µν 1 x, ∇ x, z T x, ( Γ = 0 ( h ( ρ δg + i − ( ∂ −− (1) −− ∂ ) which differ from the Fefferman-Graham ρ −− ab µν Γ ∂ h grav T = µ ω i 2.17 = ∂ ) + Σ h – 10 – i ab )Γ −− Z )Γ µν ) ) = T + + 1 2 Γ g + h We would now like to evaluate the metric contribu- i x z x x V ( + − µ ( independent of ∂ ( ∂ µ 3 + £ + z = T ++ (1) µν ( = h λh λh z T + ( i h ∂ − grav B ≡ 3 ab 1 ), the stress tensor expectation values determine the asymp- N + ) = z S h AdS T 0) G ` h (2) = x, z 2.11 δ x, π ( ( ab : γ (1) ++ ++ λ Γ Γ 0) = 8 , where we have t x, ( x ++ Γ ). = ) . i ). According to ( 1.7 ) 3.2 x − 2.9 x ( Assuming that the state is holographic, there will be some dual geometry of the For a general CFT state, the stress tensor is traceless and conserved, −− T This formula assumes the gaugegauge conditions conditions ( we have beenbring using our so metric far. perturbation Thus, to we the must find appropriate a form. gauge In transformation general, to we can write with the linearized perturbation Γ Satisfying the gauge conditions. tion to ( The solution in our Fefferman-Graham coordinates with boundary behaviour ( Then the metric perturbationbehavior throughout the by spacetime the is Einstein determinednents equations in by linearized this the about asymptotic field AdS. theory Here, directions, we which need give only the compo- Now, suppose that ourthe state stress represents tensor a expectation small values perturbationby and a to the small the asymptotic parameter CFT metric vacuum, perturbations so are that governed form ( totic form of the metric as dinates Thus, a general CFTh stress tensor can be described by the two functions, term in ( In two dimensions, these constraints can be expressed most simply using light-cone coor- holographic states in two-dimensional conformal field theories. This arises from the first JHEP06(2016)085 is the (3.16) (3.13) (3.14) (3.15) (3.12) θ ) to de- h gives sin θ ˜ ! ) B 3.13 ikr sin θ ) so that the e 2 kr i r sin ikr 1) 3 e k , . ,X − ikr 2 A e dk , 2 sin( ) and ( θ ) = 2). i X 2 θ = 0 = 0 − − d + 2 ) is described in detail ˜ ˜ θ B B = 0 (3.11) 3.12 2 | cos
sin dk . ikx ˜ ) 2 3.9 i B e sin i i kr
r ) ), ( V 2 − C C k − ikr A k ( + 2 i θ V e ( i ikx − ) θ ∇ 2 i 3.11 C h e ˆ θ h 2 ) A describing coordinates along the + ∇ − − ν k cos ( δ i ) ∇ D sin A cos D 2 2 − − µ 1 2 ) V δ a k X kr δ i ˆ 2 V − kr r − ikr . For these calculations, it is useful to ikr ∇ + + 2 i A A sin( V i + + + h with and evaluating ( + θ V cos( i θ − + ). Since the gauge conditions are linear in , we find ikx iA θ θ D A θ θ V ∇ ikx 2 π e h 2 e ) 2 sin ∇ ( X ) + k sin θ sin k ( i + i cos – 11 – ( 2 cos ) + sin = + V θ + 2 D ] a ˆ h − θ θ, r r ) kr ˆ V cos A V ) + ν 2 h A 2 sin δ k kr ∇ r kr cos 3 + , µ ∇ 2 cos( i continues to satisfy the Killing equation at Z k δ r i k ) (equivalent to requiring that the coordinate location of 3 h + λ ∇ ( cos( B r sin( − (2 + i ξ C Z + C − 2 + [
) = 0. 2.17 θ ) = ( h λ ) = θ θ θ A θ D 2 2 A V sin i δ z, x ) = cos ) that ikt ikt ikt t, x, z 2 1 ), cos cos cos ikt ∇ t, x, z r ( for a general perturbation are obtained from these by taking linear i kr − − − remain finite at ( 2 2 2 − 2 e e 2 e − a e r r r za ∇ a V k 2.18 3 3 3 V h 3.14 V D t, x, z 2 k k k A + ( h µν h ) = ) = ) = ) = and its first derivatives at the surface t, r, θ t, r, θ t, r, θ t, r, θ ; ; ; ; is the desired metric perturbation satisfying the gauge condition, and V k k k k ( ( ( ( γ ], but we review the main points here. Defining coordinates ( − − − − θ θ r t After requiring For each of the basis elements, we use the equations ( To solve these, we first expand our general metric perturbation in a Fourier basis. The procedure for finding an appropriate ˆ ˆ ˆ ˆ V V V V 11 t r ∂ , the conditions on ∂ termine define polar coordinates ( V combinations as in ( with a gauge choice the extremal surface remains fixed) gives while the condition ( perturbation in Fefferman-Graham coordinates (equivalent to Γ for in [ extremal surface lies atsurface, some the fixed gauge value condition of ( where JHEP06(2016)085 ) 3.9 . (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (3.18) ) and b , X c c ) κ t, r, θ γ g − a V ; cos . Thus, we can ∇ , k £ 3 K ˜ ( B + − h, and its derivatives K r b ξ ˆ V V £ ) cos X ↔ −} +8 . ac g, K ˜ ( , we can evaluate ( K 4 γ + B c can be chosen arbitrarily, { ω κ ) as V ) = . + − + V ) sin 3.9 . i ) ) 3 − ∇ 4 ) g ) c κ ] 2 K dρ k t, r, θ X + ( ; h, V a c ξ,V 2 [ ξ ) (3.19) k a ξ h, V κ ) + ( b ( + 4) ++ £ γ £ 2 h ∂ ρ b + ( 2 T r (3.17) ξ b g, ˜ K ˆ χ κ B V ∇ γ, X ih )) £ 6 V V ( ) Z g − 2 ( 1 + − £ − V k dχ b 4 ]) ( − a ) + + £ g, h, X K 5 ( h V a ++ ξ b + ) = ω , we can rewrite ( (5 ξ, V K T ∂ [ g ∇ £ ( g, h h h b B – 12 – c ξ − ( ( c ξ ) g, Given the appropriate £ ω and its derivatives at the surface B ) = 3 2 γ ξ V ) γ = 1 2 g, h, κ ξ , k V £ £ ( ). We note in particular that the result splits into a a 1 ) + ] 2 £ − ω g, γ, k + g, + h k ( ( b ξ Σ V 2 h ω N K − Z X ( £ ξ, V ˆ £ ( [ K c g, γ, G 1 χ 3 ( ). The results here give the behavior of 2 = ) k 4 + ∇ ω ( in polar coordinates). Elsewhere, κ = = 0. We can simplify this expression using the gravitational E R dk ac h g, h, R ( ( ρ 2 γ R g − ξ t, r, θ ω ω π Z ≡ 1 = K £ − ( = ; ab r 3 dk 256 , κ ( k ) = ) ( so that it vanishes at K γ N 2 ˜ Z + B ξ t k solutions are related through ˆ V 1 V £ = πG AdS + ` − E 1 16 k V g, γ, ( ( ) = ) = R 2 ω ) = ≡ , k 1 t, r, θ k γ, X ; K ( ( k 2 χ ( ˆ K − t ˆ In this final expression, wenow only calculate need the result explicitlyresult for a in general terms perturbation. of In the the boundary Fourier stress basis, tensor the final is with left-moving part and a right-moving part with no cross term. where the kernel is where Finally, choosing Thus, we have and we have usedidentity that where where V only at the surface but we will see that our calculationEvaluating only requires the the canonical behaviorusing at energy. where the JHEP06(2016)085 0 B ) in (3.29) (3.28) (3.26) (3.27) 1.7 ], but we 18 – , . The leading 16 O . ↔ −} 2 2 + x { that this result can be ≥ < x ) come from the matter + , and has support only on i 1 1 4 from its value on a Cauchy ) x x 2 3.2 + 2 x B x ( 2 2 D ) ) . 2 2 b and ++ x x , T 1 i + − (2) ih x ab ) ) T x R R + 1 ( ( ( a B 2 2 x ξ , one can demonstrate that the positivity ) ) ( hO 1 1 , i.e. a boundary spacetime region rather Σ A x x ∆ Z B ++ T 1 2 to be positive which requires the kernel to be D − + γz h ) E R R – 13 – = ( ( → + 2 ), the integrals can be taken over any surface ) ( ) from the introduction. , x 2.8 + 1 ) between the CFT central charge and the gravity N x, z matter x B N AdS ( G ( 1.10 S ` 2 2 φ G 2 π K (2) 4 (2 R δ / + 2 dx ) = AdS 0 2 ` ) and asymptotic behavior B d , x Z 1 + 1 − = 3 x is symmetric under exchange of ( dx 2 c 2 0 . The fact that we only need the stress tensor on a Cauchy surface for K B K Z ∂B = ∆(∆ = 2 ]. Focusing only on the domain of support, we have E m is a constant depending on the normalization of the operator . As a more complete check, we will show in section R,R 5 γ . According to the usual AdS/CFT dictionary, this corresponds to a bulk scalar field − i [ ) We suppose that the CFT has a scalar operator of dimension ∆ with expectation value Positivity of relative entropy requires Like the leading order result in ( Transforming back to position space is special to the stress tensor in two dimensions, since the conservation relations allow x ∈ ( B i where effects of the bulkterm scalar in the field canonical on energy the entanglement entropy ( hO with mass erator expectation values. The discussionThis for example other is matter more fields representative, wouldvalues since be in the entirely formula parallel. the will entire involvethan scalar domain just field a of expectation spatial dependence slice.present them The here results to here show are that similar they to follow the directly recent work from in the [ canonical energy formula. from the vacuum state by a local conformal3.2 transformation. Example: scalarWe operator now consider contribution an explicitorder example to calculate making the use terms in of the the entanglement entropy bulk formula matter quadratic in field the term scalar in op- ( the full domain of dependence. We will see anpositive semi-definite. explicit example As in we the showexplicitly, in next providing appendix subsection. a checksection of our results.reproduced An by alternative a proof direct CFT of calculation positivity for is the special given class in of states that can be obtained with boundary D us to find the stresssurface. tensor expectation For other value everywhere operators, in or in higher dimensions, the result will involve integrals over Using the relation parameters, we recover the result ( where the kernel x JHEP06(2016)085 . B (and (3.37) (3.36) (3.33) (3.34) (3.30) (3.31) (3.32) B R , 2 1 . i φ ) ) 2 2 k . ( z m x, z ( hO + (3.35) γ + 1) ) 2 (2) 00 ) 2 2 d T 1 x , ) z φ ) 2 − ) 2 z − 3 2 2 1 x k , ~ ∂ 2 φ z . 2 − − + ( i 2 2 0 m )Γ(∆ − = 0 ) z 2 0 k 2 1 ), this gives (for a ball centered 2 ) φ x + Γ(∆ + 1 − ( q R 2 − 1 2 φ 2.2 ( 2 i z φ 2 d m ]. hO ∂ π d R R ν . ) ( ∂ 0 J x − 1 17 x ∆ 1 2 d π R , + ( ; φ φ ∆Γ( z − 5 from ( z c 2 x represents the solution of the linearized µ 2 d ) 1 ∂ 2 d ∂ − 1 1 − x, z ) everywhere to find that d µ cd hOi 2 ν/ φ d ( φ z g ∂ 0 γ Ω d ( µ − ∂ K 2 2 x 0 ), and performing the integrals, we obtain 2 ( + z 3.32 k ab 2∆ ~ ) and µ ] g − – 14 – ) = o ik dx R − T 2 λ ) β x h sinh sinh ( R 11 S 2) 2 , + ( + / 5 ∆ ) ) π K 2 i 1 1 4 G – 20 – z z 2 T − π R λz h − + β β 90 2 dz R R E 2 ( ( π λ π π B 1 sinh 2 (∆ Z as expected from the entanglement first law. ) ). − 2 2 2 2 i 1 = + (1 + 2 d π β πR G dλ B i 2 2 2 dz R 6 β 4.31 tt H sinh sinh 2 λ = dz h B T h δ Z E = 2 ) sinh( 1 2 1 z ) for this conformal transformation, the change in entanglement β δS πR 2 2 = 2 2 ) = ( 2 d β z 4.11 2 dλ ( β ds 1 24 K = 2 sinh c E )= sets the temperature. 2 2 , z β π 1 2 z this is a perturbation towards the planar BTZ geometry ( = β The linear order equals The second-order term gives the quantum Fisher information or the canonical energy First, applying ( Furthermore, the second-order kernel is A similar computation as the previous section leads to the first-order kernel On top of this transformation, one could also apply an infinitesimal conformal trans- 3 2 λ 9 K canonical energy Using the formula using the second-order kernel ( entropy with respect to the vacuum is which matches the previous known results [ As the black hole correspondsthe to conformal the transformation thermal ( state in CFT, the dual state be obtained by in Fefferman-Graham coordinates.tensor Holographic expectation renormalization value ( of the dual CFT is Consistency check: homogeneous BTZ perturbation As a check, consider theAdS homogeneous perturbation example, where which is the modular hamiltonian of thermal state in 2d CFT. formation to obtain non-homogeneous perturbation around thermal state. JHEP06(2016)085 , ) ) B (5.2) (5.3) (5.4) (5.5) (5.6) (5.1) 4.27 1.10 ) for entan- . of the ball 1.2 ) x R ] . − 29 ). We will focus on 3 2 ) x + ( ) is consistent with the 1 . 1.10 δS 2 K ( ) 2 )) ) 1.10 x 0 and radius 2 dS 4 x − x cL . R, x ). 1 − ( 2 , and the other sourced by the . x ). + ( + x 0 , x 1 ( 1 2 ++ 2 ) δS x ηη 5.3 T K ) = 0 2 and action − ( ( + 4 dx 2 2 − 0 2 dS K 2 dS + δS 24 η R, x ) obeys the linearized wave equation = ( 12 2 cL ( 2 cL + + − 1.2 2 dS 2 m dR S bulk-to-bulk propagator [ L η 1 2 2 − ) = (1) 10 ) = − δ m – 21 – x 2 ) + 2 dS 2 R, x − + R ) = ( L 0 m 2 + , reproducing both terms in ( δS = ( − − , x 0 δS a x, x η 2 dS δS 2 dS ; ∇ 2 − with mass ) ∇ ds + m 1 + η, x + x + ( − ( δS 2 ; the perturbation to the entanglement entropy is then the sum δS δS ( dS 2 dS K a G = −− ∇ ∇ 2 T 1 2 m δS = − L 2 dS ], the first-order perturbation ( follows identically. ∇ acting as a source. 12 − i ) δS x ( 00 T ], it was pointed out that the leading order perturbative expression ( h since Alternatively, introducing the retarded To reproduce the second-order results for entanglement entropy, we consider an auxil- It was conjectured that higher order contributions might be accounted for by local 12 These propagators are defined to be non-zero only within the future directed light-cone. This is impor- + 10 tant in reproducing both the support and the exact form of nonlinear equation by acting with the dS wave equation on the second-order kernel ( Integration against the CFT stress tensor then gives ( As shown in [ We can immediately check that the second-order perturbation ( The equation of motion is and consider a scalar field anti-holomorphic part of these two scalars, δS iary de Sitter space with metric propagation in this auxiliaryIn space this with section, thecan we addition indeed show of be that self-interactions reproduced forinteraction for to by two-dimensional scalar this moving CFTs, field. scalar to the field.fields; a A second-order one non-linear slight result sourced complication wave ( is equation by that with the we a holomorphic actually simple require stress two-scalar cubic tensor In [ glement entropy, expressed as ais function a of the solution center towith point the wave equation for a free scalar field on an auxiliary de Sitter space, 5 Auxiliary de Sitter space interpretation JHEP06(2016)085 ] . 14 2 (5.8) (5.9) (5.7) # (5.10) (5.11) i ) , x . ( 2 2 . That this i ++ , 2 ) ) T ) is that it is 2 x h η ( ) 0 ) x x − ++ ; 5.10 T R − h ( R ) η x η, y x ( ( + ≤ ; 0 x 2 − i dS ) ) , x 2 0 0 x η η x ( ( π dx K − dS ++ y = y T . B h field propagates in de Sitter with a ) Z ) ) 3 0 dx K " x + R ; 2 + 2 ( Z 0 , x δS ) x 0 y O δS η ) ; 0 − + R, x 0 3 ( ) , x 2 g x 0 . The η, x ( η dS R ( S ; i − R η R, x 0 + dS ( 2 (2) δ η ++ G δS – 22 – dx K T R, x + is sourced by the CFT stress tensor on this boundary. h ( + 2 → 1 Z 0 π + 3 lim dS η R x δS 4 | G δS | ) = = π dS 0 4 dS g + | g | − p δS R, x p 0 0 ( . The bold (red) line is the conformal boundary of de Sitter which is + → ) dx ) respectively. 0 S dηdy R 5.2 dη (1) ( Z − ) = lim δ 1.10 0 dS 2 dS x 3 x Z ; cL 2 dS − 12 η, x ) and ( cL ( )= and the squared expression are manifestly positive and the bulk-to-bulk prop- dS 0 1.2 | represents the leading order perturbative expression for the relative entropy. )= 0 K ) is positive over the range of integration where ( dS S 0 gives g R, x . Feynman diagram which computes | ( 5.6 (2) R, x ( δ + → p + S − S Recently, it has been realized that the modular Hamiltonian in certain non-vacuum The integrals can be performed directly toA useful show advantage that of these writing results the match second-order with result the in the form ( R (2) (2) δ δ agator ( expression is negative is required by thethat positivity of relative entropy, since we showed above states in two dimensional CFTs can be described by propagation in a dual geometry [ manifestly negative. More explicitly, we have where at second-order, where the latter term comes from theexpressions diagram ( shown in figure For at first-order and we can show directly that the solution with boundary behavior cubic interaction given by identified with a time slice of the CFT. and bulk-to-boundary propagator Figure 2 JHEP06(2016)085 ) x ( + ]. (A.1) (A.2) (5.12) h . How- 15 , . 3.2 13 ) in a Taylor n,m x ( A + h odd ) obeys the wave +4 even odd m ]. The explicit form of m + 4.33 n 31 + , R ). The real space n n, m n, m 30 m + ) on this kinematic space. a if if . x . In higher-dimensions, this n ( a 5.2 B 2 = 0 boundary of the auxiliary ++ D ]. We find that the results of m +2) T dx +3 X R 33 m m . We can expand + – +1) ∝ n + 3 2 X m ) n 30 1 +2)( ]. . + + n ∼ ( dR x 14 +1)( ) ( nm − n 2 11 + ( nm 0 if n, m h , x 1 x β πR ( 2 2 – 23 – 2 dS + 1) K L 2 2 m 2 m π x 4 + n 1 sinh x n 2 2 1 β dx 1 = + 3)( 2 dx so that the canonical energy is given by m ds B n Z + x B n n ( a Z m = =0 a ∞ n n a n,m P A ) with the same interactions in this kinematic space. m X ) = can be explained by the same interacting theory ( 5.3 x n ( X + 4.4 h We could imagine adding additional fields propagating in de Sitter to capture the con- E ∼ The kinematic space dual to the BTZ black hole was first described in [ 11 which is clearly non-negative and symmetric in the metric in the coordinates we are using can be found in [ where the proportionality factor is up to a positive constant and Consider the left movingmust part be of real valued perturbation functions forseries a perturbation of AdS Natural Sciences and Engineering Researchthe Council Simons of Foundation. Canada, and by grant 376206 from A Direct proof of the positivity Acknowledgments We thank Michal Heller,Philippe Ali Sabella-Garnier Izadi for Rad, helpful Nima discussions. Lashkari, This Don research Marolf, is supported Robert in Myers, part and by the the one-point functions over thewill full domain also of be dependence truede for Sitter the space stress does tensorresults not contribution. to include contributions The the of time othersophisticated direction operators auxiliary of or space. the higher Promising CFT, dimensional work so cases in any will this extension require direction of a is these more discussed in [ The second-order perturbation toequation the ( entanglement entropy from ( tributions to the entanglement entropy fromever, scalar unlike operators the discussed contribution in from section the stress tensor, this contribution involves integration of section The kinematic space dual to the thermal state is matching the kinematic space found previously in [ JHEP06(2016)085 with since , the (A.5) (A.6) (A.7) (A.3) (A.4) N M i,N M is positive A is positive- B = 1, while the . =1 N i ) T n n,m N P B A is positive definite M ) is positive-definite, also have a positive B through the Perron- N M T N det( N B M M 0. Since M + M 1 β ≥ λ -th column N ) given by the Weyl inequality . N M +1 is a rank-one matrix, the sole . Since ) ! B 1 + is positive-definite, we need only N T . in order from largest to smallest +1 i,N 1 B B det( 0 B M A T +1 where they are ordered from largest N i,N = N T − + ≥ N B B A ) =1 M B + N i M N M i M N λ M i 2 det( M P +1 λ M N =1 < Since i Y +1 2 i,N ) by . ) = A B ,N B B B + B M − 2 det( T +1 + – 24 – T T λ M 1 N B M N B λ B whose components are given by 0 to complete the proof. The maximum eigenvalue i,N λ A +1 + is positive definite. To do so, we will use proof by + A ≤ ≥ N ,N = ≥ N N 12 B B M +1 1 = Tr( + M N =1 +1 β M i − N X ... β N M i can be evaluated using the formula − A ,m λ 1 M det( ≥ are the eigenvalues of M 1 − +1 N =1 n λ i Y ) = matrix B . Therefore it remains to show N M i j A + λ +1 M N 2 M ) = N = λ × B and M = 0. T i . We then expand the determinant as ≥ n n,m N B where N i A B det( λ β + + -column vector with entries given by ≥ + N 1 M N λ M M i ... λ is a ≥ det( ≤ 2 B λ B decreases with We denote the eigenvalues of The determinant of To show that the canonical energy is positive, we need to show the matrix is bound from below by the minimum sum of a column of The inelegant notation change is due to conventional matrix notation starting at + ≥ 12 i,j M 1 M i 1 Taylor series starts at So it remains to showλ that Frobenius theorem (equivalently Gershgorinminimum sum circle of theorem). aA column vector For is the simply matrix the sum of the λ λ to smallest non-zero eigenvalue is given by semi-definite, there exists an upper bound on det( so it is sufficient to show where it has a positive determinantdeterminant and by all Sylvester’s the criterion. upper-left To submatricesshow show of it that has a positive determinant since all the upper-left submatrices are already known . Suppose that the definite. Then consider the block matrix constructed as entries given by induction and Sylvester’s criterion whichif states that and a only square matrix itpositive has determinant. a positive determinant andProof all by the induction upper-left submatrices also have a JHEP06(2016)085 . ), B N D (B.5) (B.1) (B.2) (B.3) (B.4) (A.8) (A.9) M , 2 det( ) which map . < 2 ) − +1 r B r, τ, φ T 0. The expressions i,N B +1 2 2 ; ∆; , so clearly the entire ≥ ) } A ,N 2 + k 1 2 B − N − i β + 2 2 2 M ]. is even. Then we have ,N ω − A 1 ( i ...N/ − N i 34 − M 1 , , , , 2 1 λ + is positive-definite we completed τ τ τ A ). Further discussions of Rindler i,N 1 25 2 ∈ { 2 , ∆ 2 can be given by ( i , M A =1 using Rindler reconstruction so as to τ ) i N/ X 1.10 24 R , k 2 1 cosh 1 cosh 1 cosh B + =1 φ + n/ i 2 X − − − 22 , ω 2 2 2 1 sinh = ( r r r N,N 21 i , R can be reconstructed in this Rindler wedge sinh is. Since − √ √ √ A +1 2 − N of radius is zero. Explicitly analyzing the coefficients, we ω,k O r + + + = Rr – 25 – ,N 2 O 1 ∆ M B φ φ φ √ ) +1 − +1 r i R R ( 2 2 ,N 1 , the sum becomes 1 i,N A F cosh cosh cosh 2 2 − ω,k N 2 2 i r r r f / 2 2 A 2 ω 2 i A = = = =1 ikφ +1) − i =1 2 X i t N/ X − is always positive for all / z x N ( 1) 2 − 1 iωτ − r − − +1 N =1 ,N ( i − 1 ,N i,N − 1 1 2 P i 2 − A i 2 2 A 2 dωdk e ∆ 2 A =1 − / i . N/ X ) are not obviously positive, but they reduce to some polynomial equations Z r − ∞ =1 +1) i X A.9 N ) = ) = i,N ( r is positive-definite given that 2 ( A ] +1 < r < ω,k r, τ, φ N f ) and ( 21 ( φ M The scalar field dual to an operator Coordinates on the Rindler wedge We split this sum up into two cases. The first case is if A.8 back into Poincar´ecoordinates by only use the one-pointWe functions specialize of to two the dimensional scalar CFTscan operator in be order in compared to the obtain to domainreconstruction a the of more can gravitational explicit dependence be expression contribution found which ( in the literature [ as required by the positivity of relative entropy. B Rindler reconstruction forIn scalar this operators appendix in weperturbation CFT to find the an entanglement entropy expression of a for ball the matter contribution to the second-order Each term in thisin sum ( is also positive,which so can we be have shown shown tothus be positive. Therefore we’vethe shown proof det( by induction. The kernel for canonical energy is explicitly positive-semidefinite sum is positive. In the case of odd using [ where 1 see that since the final term in JHEP06(2016)085 , , (B.6) (B.7) (B.8) JHEP , 2) , − (2006) ] D 90 08 ]. +∆(∆ . , 2 2 B i r k ) JHEP , SPIRE D , Phys. Rev. + Inviolable energy BPS geometries B IN , 1 t, x 2 8 ( D ][ ]. / ω ]. − 1 1 Constraining gravity using hO 2 in ω ω arXiv:1412.3514 2 r + i [ k . O ω and i ]. − ) to obtain an an expression SPIRE 2 . − SPIRE k − 4 k i ) ) IN geometries − i IN / ) , t r ) 1 2 ][ ( 5 t ][ 3.29 ω k − SPIRE D τ, φ − − O - x , ( IN 1 Relative entropy and holography 2 ,k x (2015) 067 ω 1 ][ − D ]. hO f ω + arXiv:1401.5089 ) A covariant holographic entanglement 06 R O r [ ( ikφ R ( ( ) + ω ,k 2 r k + 1 SPIRE ( k 2 ω − iωτ i k ]. ]. IN holography for ω f JHEP ) i − t 3 , ][ , ) – 26 – 2 1 t arXiv:0705.0016 0 arXiv:1405.3743 ω + Nonlinear constraints on gravity from entanglement [ (2014) 029 − f [ + AdS x SPIRE ) 2 SPIRE dτdφ e r r x hep-th/0603001 IN IN 05 ( + [ Z p ][ − ,k ][ 1 R r ]. 0 ( = ω R 2 h f ( ), which permits any use, distribution and reproduction in JHEP (2007) 062 Aspects of holographic entanglement entropy Holographic derivation of entanglement entropy from AdS/CFT ω,k dω 1 × , 1 O SPIRE Entanglement entropy and dtdx 07 − (2015) 065006 IN 2 B arXiv:1507.00945 r ][ D [ 32 (2006) 181602 Z drdkdω + CC-BY 4.0 JHEP = 96 ∞ , 1 arXiv:1405.6185 arXiv:1305.3182 This article is distributed under the terms of the Creative Commons Z [ [ ω,k 4 1 O − (2015) 004 which only depends on the expectation value of ]. = is the Fourier transform of the CFT expectation value of the operator 11 hep-th/0605073 scalar ω,k [ (2013) 060 scalar S SPIRE O S IN Class. Quant. Grav. (2014) 066004 JHEP 08 entanglement in AdS/CFT entropy proposal conditions from entanglement inequalities [ Phys. Rev. Lett. 045 (2) This form of the scalar field can be combined with ( S. Giusto and R. Russo, S. Giusto, E. Moscato and R. Russo, D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, S. Banerjee, A. Bhattacharyya, A. Kaviraj, K. Sen and A.S. Sinha, Banerjee, A. Kaviraj and A. Sinha, V.E. Hubeny, M. Rangamani and T. Takayanagi, N. Lashkari, C. Rabideau, P. Sabella-Garnier and M. Van Raamsdonk, S. Ryu and T. Takayanagi, S. Ryu and T. Takayanagi, (2) δ δ [8] [9] [5] [6] [7] [3] [4] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. for where the region of integration is only over the domain of dependence This can be expressed in terms of the operator in the original coordinates where JHEP06(2016)085 , ]. Erratum (2015) 033 [ SPIRE 05 J. Stat. Mech. IN , ]. , ][ ]. Gravitation from ]. Null geodesics, local JHEP ]. , SPIRE Commun. Math. Phys. SPIRE , IN SPIRE (2007) 106001 IN ][ IN ]. ][ SPIRE ][ (2013) 064057 IN ]. A stereoscopic look into the (2016) 061602 D 75 ][ ]. SPIRE arXiv:1312.7856 D 88 IN [ 116 Holographic representation of local Local bulk operators in AdS/CFT: a SPIRE ][ ]. Entanglement, holography and causal IN Holographic de Sitter geometry from SPIRE ]. ]. ][ IN Phys. Rev. ][ , hep-th/0606141 SPIRE arXiv:1511.05179 [ Equivalence of emergent de Sitter spaces from IN (2014) 051 ]. Phys. Rev. [ SPIRE SPIRE arXiv:1602.01380 ]. , ][ hep-th/9403108 IN IN [ [ 03 [ Shape dependence of entanglement entropy in ][ – 27 – Phys. Rev. Lett. SPIRE ]. Towards a derivation of holographic entanglement Geometric and renormalized entropy in conformal SPIRE , Bulk locality and quantum error correction in Canonical energy is quantum Fisher information IN arXiv:1403.3426 IN ][ JHEP [ (2016) 088 ][ , ]. arXiv:1411.7041 SPIRE (2016) 105 (2006) 066009 [ (1994) 443 04 IN arXiv:1102.0440 [ ]. [ Entanglement entropy and quantum field theory 04 Stability of black holes and black branes ]. ]. ]. SPIRE hep-th/0612053 D 74 IN [ (2014) 053 B 424 JHEP SPIRE , arXiv:1604.02687 JHEP SPIRE SPIRE SPIRE arXiv:1508.00897 IN (2015) 163 05 , , [ IN IN IN [ ][ (2011) 036 hep-th/0405152 ][ ][ 04 [ arXiv:1201.0463 05 Entanglement entropy of excited states in conformal perturbation theory and [ Phys. Rev. Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property JHEP , , Bulk emergence and the RG flow of entanglement entropy Modular Hamiltonian of excited states in conformal field theory Nucl. Phys. (2007) 129902] [ JHEP , arXiv:1606.03307 (2016) 153 , , JHEP , 04 arXiv:1604.03110 D 75 (2013) 629 , (2004) P06002 arXiv:1209.4641 arXiv:1412.5648 arXiv:1509.00113 arXiv:1508.03506 field theory 06 bulk operators holographic description of theibid. black hole interior in holography CFT operators and AdS/CFT for[ subregions 321 AdS/CFT entanglement in holographic CFTs the Einstein equation conformal field theories bulk [ [ diamonds conformal field theory entropy JHEP entanglement in conformal field theory C. Holzhey, F. Larsen and F. Wilczek, P. Calabrese and J.L. Cardy, A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, N. Lashkari, R. Bousso, B. Freivogel, S. Leichenauer, V. Rosenhaus and C. Zukowski, A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, S. Hollands and R.M. Wald, A. Almheiri, X. Dong and D. Harlow, I.A. Morrison, T. Faulkner, R.G. Leigh and O. Parrikar, T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J.T. Sully, Faulkner, A.J. Speranza, J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, C.T. Asplund, N. Callebaut and C. Zukowski, N. Lashkari and M. Van Raamsdonk, J. de Boer, M.P. Heller, R.C. Myers and Y. Neiman, H. Casini, M. Huerta and R.C. Myers, [27] [28] [25] [26] [23] [24] [20] [21] [22] [18] [19] [15] [16] [17] [13] [14] [11] [12] [10] JHEP06(2016)085 , JHEP D 90 , D 90 , (2013) 212 Phys. Rev. 10 Phys. Rev. , , JHEP , Integral geometry and holography Tensor networks from kinematic space ]. ]. ]. – 28 – SPIRE SPIRE SPIRE IN IN IN ][ ][ ][ An infalling observer in AdS/CFT ]. ]. ]. Holographic definition of points and distances SPIRE SPIRE SPIRE IN IN IN [ [ ][ A tensor network quotient takes the vacuum to the thermal state arXiv:1402.7080 arXiv:1409.4473 arXiv:1505.05515 [ [ [ Holographic representation of local operators in de Sitter space (2015) 175 arXiv:1211.6767 [ arXiv:1510.07637 arXiv:1512.01548 (2014) 024061 (2014) 106005 10 B. Czech et al., B. Czech, L. Lamprou, S. McCandlish and J. Sully, K. Papadodimas and S. Raju, B. Czech and L. Lamprou, B. Czech, L. Lamprou, S. McCandlish and J. Sully, X. Xiao, [32] [33] [34] [30] [31] [29]