Relative entropy equals bulk relative entropy

Journal of High Energy Physics, Jun 2016

We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular flow is dual to the bulk modular flow in the entanglement wedge, with implications for entanglement wedge reconstruction.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP06%282016%29004.pdf

Relative entropy equals bulk relative entropy

Received: May Relative entropy equals bulk relative entropy Cambridge 0 2 U.S.A. 0 2 Princeton 0 2 U.S.A. 0 2 Princeton 0 2 U.S.A. 0 2 Santa Barbara 0 2 U.S.A. 0 2 Vancouver 0 2 Canada 0 2 Field Theories in Higher Dimensions 0 ow is dual to the 1 Kavli Institute for Theoretical Physics 2 Daniel L. Ja eris 3 Department of Physics and Astronomy, University of British Columbia 4 Jadwin Hall, Princeton University 5 Institute for Advanced Study 6 Center for Fundamental Laws of Nature, Harvard University We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular bulk modular ow in the entanglement wedge, with implications for entanglement wedge reconstruction. AdS-CFT Correspondence; 1/N Expansion; Models of Quantum Gravity 1 Introduction and summary of results Entanglement entropy, the modular Hamiltonian, and relative entropy 2.1 2.2 Modular Hamiltonian Relative entropy 2 3 4 5 6 4.1 4.2 4.3 6.1 6.2 6.3 6.4 6.5 Gravity dual of the modular Hamiltonian Regions with a local boundary modular Hamiltonian Linear order in the metric The graviton contribution Quadratic order for coherent states Smoothness of the full modular Hamiltonian in the bulk 5.2 Implications for entanglement wedge reconstruction Comments and discussion The relative entropy for coherent states Positivity of relative entropy and energy constraints Higher derivative gravity Beyond extremal surfaces Distillable entanglement A Subregions of gauge theories A.1 U(1) gauge theory A.2 Gravity Introduction and summary of results Recently there has been a great deal of e ort in elucidating patterns of entanglement for theories that have gravity duals. The simplest quantity that can characterize such patterns is the von Neumann entropy of subregions, sometimes called the \entanglement entropy". This quantity is divergent in local quantum eld theories, but the divergences are well understood and one can extract nite quantities. Moreover, one can construct strictly nite quantities that are well-de ned and have no ambiguities. A particularly interesting quantity is the so called \relative entropy" [ 1, 2 ]. This is a measure of distinguishability between two states, a reference \vacuum state" and an arbitrary state S( j ) = T r[ log log ] (1.1) { 1 { K If we de ne a modular Hamiltonian K = log , then this can be viewed as the free energy di erence between the state and the \vacuum" at temperature = 1, S( j ) = Relative entropy has nice positivity and monotonicity properties. It has also played an important role in formulating a precise version of the Bekenstein bound [ 3 ] and arguments for the second law of black hole thermodynamics [4, 5]. In some cases the modular hamiltonian has a simple local expression. The simplest case is the one associated to Rindler space, where the modular Hamiltonian is simply given by the boost generator. In this article we consider quantum eld theories that have a gravity dual. We consider an arbitrary subregion on the boundary theory R, and a reference state , described by a smooth gravity solution. can be the vacuum state, but is also allowed to be any state described by the bulk gravity theory. We then claim that the modular Hamiltonian corresponding to this state has a simple bulk expression. It is given by Kbdy = 4GN Areaext + Kbulk + + o(GN ) [Kbdy; ] = [Kbulk; ] 1The entanglement wedge is the domain of dependence of the region Rb. { 2 { backreaction. relative entropy The rst term is the area of the Ryu Takayangi surface S (see gure 1), viewed as an operator in the semiclassically quantized bulk theory, it includes the change in the extremal surface as we consider di erent states . This term was previously discussed in [6]. The o(G0N ) term Kbulk is the modular Hamiltonian of the bulk region enclosed by the RyuTakayanagi surface, Rb, when we view the bulk as an ordinary quantum eld theory, with suitable care exercised to treat the quadratic action for the gravitons. Finally, the dots represent local operators on S , which we will later specify. We see that the boundary modular Hamiltonian has a simple expression in the bulk. In particular, to leading order in the 1=GN expansion it is just the area term, which is a very simple local expression in the bulk. Furthermore, this simple expression is precisely what appears in the entropy. This modular Hamiltonian makes sense when we compute its action on bulk eld theory states which are related to by bulk perturbation theory. Roughly speaking, we consider a which is obtained from by adding or subtracting particles without generating a large Due to the form of the modular Hamiltonian (1.2), we obtain a simple result for the Kbulk in the interior of the entanglement wedge,1 (1.2) (1.3) (1.4) contribution to the entanglement entropy is computed by the area of an extremal surface S that ends at the boundary of region R. This surface divides the bulk into two, region Rb and its complement. Region Rb lives in the bulk and has one more dimension than region R. The leading correction to the boundary entanglement entropy is given by the bulk entanglement entropy between region Rb and the rest of the bulk. in Kbdy localized on S do not contribute to its action in the interior of the entanglement wedge, S being space-like to the interior. Note Kbulk is the bulk modular Hamiltonian associated to a very speci c subregion, that bounded by the extremal surface S. Implications of (1.4) for entanglement wedge reconstruction are described in section 5.2. The bulk dual of relative entropy for subregions with a Killing symmetry was considered before in [7{12]. In particular, in [12], the authors related it to the classical canonical energy. In fact, we argue below that the bulk modular hamiltonian is equal to the canonical energy in this case. This result extends that discussion to the quantum case. Note (1.2) and (1.3) are valid for arbitrary regions, with or without a Killing symmetry. In addition, we are not restricting to be the vacuum state. Recently a di erent extension of [12] has been explored in [13], which extends it to situations where one has a very large deformation relative to the vacuum state. That discussion does not obviously overlap with ours. This paper is organized as follows. In section two, we recall de nitions and properties of entanglement entropy, the modular Hamiltonian, and relative entropy. In section three, we present an argument for the gravity dual of the modular hamiltonian and the bulk expression for relative entropy. In section four, we discuss the case with a U(1) symmetry, relating to previous work. In section ve, we discuss the ow generated by the boundary modular hamiltonian in the bulk. We close in section six with some discussion and open questions. 2 Entanglement entropy, the modular Hamiltonian, and relative entropy We consider a system that is speci ed by a density matrix . This can arise in quantum eld theory by taking a global state and reducing it to a subregion R. We can compute the von Neumann entropy S = T r[ log ]. Due to UV divergences this is in nite in quantum eld theory. However, these divergences are typically independent of the particular state we consider, and when they depend on the state, they do so via the expectation value of an operator. See [14, 15]. { 3 { It is often useful to de ne the modular hamiltonian K log . From its de nition, it is not particularly clear why this is useful | it is in general a very non-local complicated operator. However, for certain symmetric situations it is nice and simple. The simplest case is a thermal state where K = H=T , with H the Hamiltonian of the system. Another case is when the subregion is the Rindler wedge and the state is the vacuum of Minkowski space, when K is the boost generator. This is a simple integral of a local operator, the stress tensor. For a spherical region in a conformal eld theory, we have a similarly simple expression, which is obtained from the previous case by a conformal transformation [16]. In free eld theory one can also obtain a relatively simple expression that is bilocal in the elds [17] for a general subregion of the vacuum state. In this paper we consider another case in which simpli cation occurs. We consider a quantum system with a gravity dual and a state that can be described by a gravity solution. We will argue that the modular Hamitonian is given by the area of the Ryu-Takayanagi minimal surface plus the bulk modular Hamiltonian of the bulk region enclosed by the Ryu-Takayanagi surface. 2.2 Relative entropy Modular Hamiltonians also appear in the relative entropy Srel( j ) = tr (log log ) = hK i S (2.1) was (2.2) where K = log is the modular Hamiltonian associated to the state . If a thermal state, the relative entropy would be the free energy di erence relative to the thermal state. As such it should always be positive. Relative entropies have a number of interesting properties such as positivity and monotonicity [ 1 ]. Moreover, while the entanglement entropy is not well de ned for QFT's, relative entropies have a precise mathematical de nition [ 2 ]. If = + , then, because of positivity, the relative entropy is zero to rst order in . This is called the rst law of entanglement: S = hK i The rst law can be used to determine the modular Hamiltonian. By considering an arbitrary rst order deformation of the density matrix, we can determine all matrix elements of the modular Hamiltonian. Then we can use this modular Hamiltonian on states which are not small deformations of the original state. When we consider a gauge theory, the de nition of entanglement entropy is ambiguous. If we use the lattice de nition, there are di erent operator algebras that can be naturally associated with a region R [14]. Di erent choices give di erent entropies. These algebras di er in the elements that are kept when splitting space into two, so that ambiguities are localized on the boundary of the region, @R. One natural way of de ning the entanglement entropy is by xing a set of boundary conditions and summing over all possibilities, since there is no physical boundary. This was carried out for gauge elds in [18, 19, 51] and gives { 4 { the same result as the euclidean prescription of [ 20 ]. However, the details involved in the de nition of the subalgebra are localized on the boundary. Because of the monotonicity of relative entropy, these do not contribute to the relative entropy (see section 6 of [14] for more details). In the case of gravitons we expect that similar results should hold. We expect that we similarly need to x some boundary conditions and then sum over these choices. For example, we could choose to x the metric uctuations on the Ryu-Takayanagi surface, viewing it as a classical variable, and then integrate over it. As argued in [14], we expect that the detailed choice should not matter when we compute the relative entropy. See appendix A for more details. As we mentioned above, it often occurs that two di erent possible de nitions of the entropy give results that di er by the expectation value of a local operator, S( ) =tr( O) + S~( ). A trivial example is the divergent area term which is just a number. In these cases the two possible modular Hamiltonians are related by S( ) = tr ( O) + S~( ) ! K = O + K~ (2.3) necessary that O is the same operator for the states and . 2 This implies that relative entropies are unambiguous, S( j ) = S~( j ). For the equality of relative entropies, it is not necessary for O to be a state independent operator. It is only 3 Gravity dual of the modular Hamiltonian A leading order holographic prescription for computing entanglement entropy was proposed in [ 21, 22 ] and it was extended to the next order in GN in [ 23 ] (see also [ 24 ]). The entanglement entropy of a region R is the area of the extremal codimension-two surface S that asymptotes to the boundary of the region @R, plus the bulk von Neumann entropy of the region enclosed by S, denoted by Rb. See gure 1. Aext(S) 4GN Sbdy(R) = + Sbulk(Rb) + SWald like (3.1) SWald like indicates terms which can be written as expectation values of local operators on . They arise when we compute quantum corrections [ 23 ], we discuss examples below. We can extract a modular Hamiltonian from this expression. We consider states that can be described by quantum eld theory in the bulk. We consider a reference state , which could be the vacuum or any other state that has a semiclassical bulk description. We consider other states which likewise can be viewed as semiclassical states built around the bulk state for . To be concrete we consider the situation where the classical or quantum elds of are a small perturbation on so that the area is only changed by a small amount. Now the basic and simple observation is that both the area term and the SWald like are 2In other words, if we consider a family of states, with and in that family, then O should be a state independent operator within that family. { 5 { HJEP06(21)4 expectation values of operators in the bulk e ective theory. Therefore, for states that have a bulk e ective theory, we can use (2.2), (2.3) to conclude that contains both the classical area as well as any changes in the area that result from the condition, this area is a gauge invariant observable in the gravity theory.3 Note that the area changes as we change the state, but we can choose a gauge where the position of the extremal surface is xed. Finally S^Wald like are the operators whose expectation values give us SWald like. Interestingly, all terms that can be written as local operators drop out when we consider (3.2) (3.3) (4.1) the relative entropy. The relative entropy has a very simple expression Sbdy( j ) = Sbulk( j ) Note that the term going like 1=GN cancels out and we are only left with terms of order G0N . There could be further corrections proportional to GN which we do not discuss in this article. It is tempting to speculate that perhaps (3.3) might be true to all orders in the GN expansion (i.e. to all orders within bulk perturbation theory). Of course, using the equation for the entropy (3.1) and (3.2) we can check that the rst law (2.2) is obeyed. In the next section we discuss this in more detail for a spherical subregion in the vacuum. 4 Regions with a local boundary modular Hamiltonian For thermal states, Rindler space, or spherical regions of conformal eld theories we have an explicit expression for the boundary modular Hamiltonian. In all these cases there is a continuation to Euclidean space with a compact euclidean time and a U(1) translation symmetry along Euclidean time. We also have a corresponding symmetry in Lorentzian signature generated by a Killing or (conformal Killing) vector . The modular Hamiltonian is then given in terms of the stress tensor as Kbdy = ER is over a boundary space-like slice. When the theory has a gravity dual, the bulk state is also invariant under a bulk Killing vector . In this subsection we will discuss (3.2) for R ( :Tbdy), where the integral states constructed around . For this discussion it is useful to recall Wald's treatment of the rst law [25{27] 3If we merely de ne a surface by its coordinate location in the background solution, then a pure gauge contribution. Here Alin is the rst order variation in the area due to a metric uctuation g. And is any Cauchy slice in the entanglement wedge Rb. Equation (4.1) is a tautology, it arises by integrating by parts the linearized Einstein tensor. It is linear in g and we can write it as an operator equation by sending g ! small uctuations in the metric in the semiclassically quantized theory. g^, where g^ is the operator describing Linear order in the metric Eg( g) that [25{27] For clarity we will rst ignore dynamical gravitons, and include them later (we would have HJEP06(21)4 nothing extra to include if we were in three bulk dimensions). We consider matter elds with an o(G0N ) stress tensor in the bulk, assuming the matter stress tensor was zero on the background.4 Such matter elds produce a small change in the metric that can be obtained by linearizing the Einstein equations around the vacuum. These equations say = T mat, where T mat is the stress tensor of matter. Inserting this in (4.1) we nd ER = terms of the stress tensor due to the presence of a Killing vector with the right properties at the entangling surface S . Notice that we can disregard additive constants in both the area and E , which are the values for the state . We only care about deviations from these values. This is basically the inverse of the argument in [28]. This shows how (3.2) works in this symmetric case. The term S^Wald like in (3.2) arises in some cases as we discuss below. Let us now discuss the S^Wald like term. There can be di erent sources for this term. A simple source is the following. The bulk entanglement entropy has a series of divergences which include an area term, but also terms with higher powers of the curvature. Depending on how we extract the divergences we can get certain terms with nite coefcients. Such terms are included in SWald like. A di erent case is that of a scalar eld with a coupling 2(R R0) where R is the Ricci scalar in the bulk, and R0 the Ricci scalar on the unperturbed background, the one associated to the state . Then there exists an additional term in the entropy of the form S^Wald like = 2 2. If we compute the entropy as the continuum limit of the one on the lattice, then it will be independent of . Under these conditions the bulk modular Hamiltonian is also independent of and is given by the canonical stress tensor, involving only rst derivatives of the eld. However, the combination of Kbulk + S^Wald like = R ( :T grav( )), where T grav( ) is the standard stress tensor that would appear in the right hand side of Einstein's equations. T grav( ) does depend on . The dependent contribution is a total derivative which evaluates to 2 2 at the extremal surface. A related discussion in the eld theory context appeared R S 4This discussion can be simply extended when there is a non-zero but U(1)-symmetric background stress tensor to obtain the bulk modular Hamiltonian. We expect that we can view the propagating gravitons as one more eld that lives on the original background, given by the metric g . In fact, we can expand Einstein's equations in terms of g = g + g2 + h. Here h, which is of order pGN , represents the dynamical graviton eld and obeys linearized eld equations. g2 takes into account the e ects of back-reaction and obeys the equation E( g2) = T grav(h) + T matter where T grav(h) comes simply from expanding the Einstein tensor (plus the cosmological constant) to second order and moving the quadratic term in h to the right hand side. h obeys the homogeneous linearized equation of motion, so the term linear in h in the equation above vanishes. We can now use equations (44-46) in [30], which imply that (4.3) (4.4) (4.5) (4.6) Kbdy;1+2 = E1+2 = A^lin(h + g2) + A^quad(h) 4GN + Ecan where Kbdy;1+2 is the boundary modular Hamitonian (or energy conjugate to translations) expanded to quadratic order in uctuations. Similarly, the area is expanded to linear and quadratic order. Finally, Ecan is the bulk canonical energy5 de ned by Ecan = R !(h; L h)+matter contribution, where ! is the symplectic form de ned in [30]. From this expression we conclude that the modular Hamiltonian is the canonical energy Kbulk = Ecan We can make contact with the previous expression (4.2) as follows. If we include the gravitons by replacing T mat ! T mat + T grav(h) in (4.2), then we notice that we get Alin( g2), without the term Aquad(h). However, one can argue that (see eq. (84) of [30]) Z ( :T grav(h)) = Ecan(h) + Aquad(h) 4GN thus recovering (4.4). for quantizing the graviton eld. In appendix A we discuss in more detail the boundary conditions that are necessary 4.3 Quadratic order for coherent states The problem of the gravity dual of relative entropy was considered in [12] in the classical regime for quadratic uctuations around a background with a local modular Hamiltonian. They argued that the gravity dual is equal to the canonical energy. Here we rederive their result from (3.3). ei R ^+ ^ We simply view a classical background as a coherent state in the quantum theory. j i, where j i is the state associated to . 6 We see that in free eld theory 6Here could be O(1=p GN ) as long as the backreaction is small. 5This di ers from the integral of the gravitational stress tensor by boundary terms. { 8 { we can view coherent states as arising from the action of a product of unitary operators, one acting inside the region and one ouside. For this reason nite coherent excitations do not change the bulk von Neumann entropy of subregions, or Sbulk = 0. Thus, the contribution to the bulk relative entropy comes purely from the bulk Hamiltonian, which we have argued is equal to canonical energy (4.5) . Therefore, in this situation we recover the result in [12] Sbdy( j ) = Sbulk( j ) = Kbulk Sbulk = Kbulk = Ecanonical (4.7) 5 The modular hamiltonian generates an automorphism on the operator algebra, the modular ow. Consider the unitary transformation U(s) = eiKs. Even if the modular hamiltonian is not technically an operator in the algebra, the modular ow of an operator, O(s) U(s)OU( s), stays within the algebra. For a generic region, the modular ow might be complicated, see [31] for some discussion about modular ows for fermions in 1 + 1 dimensions. However, in our holographic context it can help us understand subregionsubregion duality. In particular, it can help answer the question of whether the boundary region R describes the entanglement wedge or only the causal wedge [32{35]. The entanglement wedge is the causal domain of the spatial region bounded by the interior of S. From (1.2), we have that [Kbdy; ] = [Kbulk; ] (5.1) where is any operator with support only in the interior of the entanglement wedge, and where on the right-hand side we have suppressed terms subleading in GN . On the lefthand side terms in Kbdy localized on S have dropped out, similarly as in (3.3). Thus the boundary modular ow is equal to the bulk modular ow of the entanglement wedge, the causal wedge does not play any role. 5.1 Smoothness of the full modular Hamiltonian in the bulk If the global state is pure, one may also consider the ow generated by the total modular operator, Kbdy;Total = Kbdy;R KbdyR, which should be a smooth operator without any ambiguities. It annihilates the global state. From our full formula for the bulk dual of the modular Hamiltonian we see that Kbdy;Total = Kbulk;Total + o(GN ). For problems that have a U(1) symmetry, such as thermal states and Rindler or spherical subregions of CFTs, we know the full boundary modular Hamiltonian E . We can de ne a time coordinate which is translated by the action of E in the boundary theory. In these situations the bulk state also has an associated symmetry generated by the Killing vector . We can choose coordinates so that we extend in the bulk and simply translates in the bulk. Then the bulk modular Hamiltonian is the bulk operator that performs a translation of the bulk elds along the bulk direction. Let us now consider an eternal black hole and the thermo eld double state [36]. This state is invariant under the action of HR HL. Let us now consider the action of only { 9 { HJEP06(21)4 HJEP06(21)4 (a) (b) modular Hamiltonian e itKbulk;R we get a new state on the horizontal line that has a singularity at the horizon. (b) The area term introduces a kink, or a relative boost between the left and right sides. Then the state produced by the full right side Hamiltonian is non-singular, and locally equal to the vacuum state. S the right side boundary Hamiltonian HR. 7 It was argued in [37] that this corresponds to the same gravity solution but where the origin of the time direction on the right side is changed. This implies that the Wheeler de Witt patch associated to tL = tR = 0 looks as in gure 2(b), after the action of e itHR On the other hand, if we consider the bulk quantum eld theory and we act with only the right side bulk modular Hamiltonian Kbulk;R we would produce a state that is singular at the horizon. By the way, it is precisely for this reason that algebraic quantum eld theorists like to consider the total modular Hamiltonian instead. It turns out that the change in the bulk state is the same as the one would obtain if we were quantizing the bulk eld theory along a slice which had a kink as shown in gure 2(b). Interestingly the area term in the full modular Hamiltonian (3.2) has the e ect of producing such a kink. In other words, the area term produces a shift in the coordinate, or a relative boost between the left and right sides [38]. The action of only the area term or only KBulk;R would lead to a state that is singular at the horizon, but the combined action of the two produces a smooth state, which is simply the same bulk geometry but with a relative shift in the identi cation of the boundary time coordinates.8 Let us go back to a general non-U(1) invariant case. Since the bulk modular Hamiltonian reduces to the one in the U(1)-symmetric case very near the bulk entangling surface , we expect that the action of the full boundary modular Hamiltonian, including the area term, will not be locally singular in the bulk | though it can be singular from the boundary point of view due to boundary UV divergences. 5.2 Implications for entanglement wedge reconstruction One is often interested in de ning local bulk operators as smeared operators in the boundary. This operator should be de ned order by order in GN over a xed background and should be local to the extent allowed by gauge constraints. If we consider a t = 0 slice in 7Here left and right denote the two copies in the thermo eld double state. 8We thank D. Marolf for discussions about this point. the vacuum state, then we can think of a local bulk operator (X) as a smeared integral of boundary operators [39] Z bdy (X) = dxd 1dt G(Xjx; t)O(x; t) + o(GN ) (5.2) One would like to understand to what extent this operator can be localized to a subregion in the boundary. Given a region in the boundary R, we have been associating a corresponding region in the bulk, the so-called entanglement wedge which is the domain of dependence of Rb, D[Rb]. There is another bulk region one can associate to R, the causal wedge (with spacelike slice RC ) which is the set of all bulk points in causal contact with D[R], [40]. RC is generically smaller than Rb [35, 41]. In situations with a U(1) symmetry, such as a thermal state or a Rindler or spherical subregion of a CFT, we have time-translation symmetry and a local modular Hamiltonian that generates translations in the time . We can express bulk local operators in the entanglement wedge (which coincides with the causal wedge) in terms of boundary operators localized in D[R] [39, 42]9 (X) = Z R dyd 1 Z d G0(Xjy; )O(y; ) + o(GN ) ; X 2 Rb (5.3) A natural proposal for describing operators in that case is that we can replace in (5.3) by the modular parameter s. In other words, we consider modular ows of local operators on the boundary, de ned as OR(x; s) U(s)OR(x; 0)U 1(s) A simple case in which Rb is larger than RC is the case of two intervals in a 1+1 CFT such that their total size is larger than half the size of the whole system, see gure 3. Here, it is less clear how to think about the operators in the entanglement wedge. We would like to use the previous fact that the modular ow is bulk modular ow to try to get some insight into this issue. The modular ow in the entanglement wedge will be non-local, but highly constrained: the bulk modular hamiltonian is bilocal in the elds [17]. If we have an operator near the boundary of the causal wedge and modular evolve it, it will quickly develop a non zero commutator with a nearby operator which does not lie in the causal wedge. Alternatively, an operator close to the boundary of the entanglement wedge will have an approximately local modular ow. It will follow the light rays emanating from the extremal surface and it can be on causal contact with the operators in the causal wedge. See gure 3. So we see that to reconstruct the operator in the interior of the entanglement wedge, one necessarily needs to understand better the modular ow. It seems natural to conjecture that one can generalize (5.3) to two intervals (or general regions) by considering the modular parameter instead of Rindler time, ie the simplest generalization of the AdS/Rindler formula which accounts for the non-locality of the modular hamiltonian would be (X) = Z R dx Z dsG00(Xjx; s)O(x; s) ; X 2 Rb (5.4) 9It is sometimes necessary to go to Fourier space to make this formula precise [42, 43]. surface is the dotted black line, while the boundary of RC is the blue dashed line (color online). In a), the shaded region denotes the de ning spatial slice Rb of the entanglement wedge. In b), the shaded region is the de ning spatial slice RC of the causal wedge. The modular ow of an operator close to the Ryu-Takayanagi surface will be approximately local, so that 1(s) will be almost local and, after some s, it will be in causal contact with C1. This ow takes the operator out of this slice to its past or to its future. Alternatively, if we consider an operator near the boundary of the causal wedge C2, it is clear that, under modular ow, [ C2(s); 2] 6= 0. Here G00 is a function that should be worked out. It will depend on the bilocal kernel that describes the modular Hamiltonian for free elds [17]. So we see that to reconstruct the operator in the interior of the entanglement wedge, it is necessary to understand better modular ows in the quantum eld theory of the bulk. To make these comments more precise, a more detailed analysis would be required, which should include a discussion about gravitational dressing and the constraints. We leave this to future work. Here we have discussed how the operators in the entanglement wedge can be though of from the boundary perspective. However, note that from (3.2) (and consequently the formula for the relative entropy), it is clear that one should think of the entanglement wedge as the only meaningful candidate for the \dual of R", see also [32]. If we add some particles to the vacuum in the entanglement wedge Rb (which do not need to be entangled with Rb), the bulk relative entropy will change. According to (1.3), the boundary relative entropy also changes and, therefore, state is distinguishable from the vacuum, even if we have only access to R. 6 6.1 Comments and discussion The relative entropy for coherent states If we consider coherent states, since their bulk entanglement entropy is not changed, the relative entropy will just come from the di erence in the bulk modular hamiltonian. Since our formulation is completely general, one could in principle compute it for any reference region or state and small perturbations over it. A particularly simple case would be the relative entropy for an arbitrary subregion between the vacuum and a coherent state of matter. To second order in the perturbation, one only needs to work out how the modular hamiltonian for the free elds [17] looks like for that subregion of AdS, and then evaluate it in the coherent state background. Positivity of relative entropy and energy constraints Our formula (3.3) implies that the energy constraints obtained from the positivity of the relative entropy can be understood as arising from the fact that the relative entropy has to be positive in the bulk. Higher derivative gravity Even though we focused on Einstein gravity, our discussion is likely to apply to other theories of gravity. The modular hamiltonian will likely be that of an operator localized on the entangling surface plus the bulk modular Hamiltonian in the corresponding entanglement wedge. Thus the relative entropy will be that of the bulk. There could be subtleties that we have not thought about. 6.4 Beyond extremal surfaces A. Wall proved the second law by using the monotonicity of relative entropy [4, 5]. If we consider two Cauchy slices 0 ; t>0 outside a black hole, then Srel;t < Srel;0 is enough to prove the generalized second law. Interestingly, section 3 of [30] shows the \decrease of canonical energy": Ecan(t) < Ecan(0). The setup (Cauchy slices) that they both consider is the same. Due to the connection between relative entropy and canonical energy, [12], we expect a relation between these two statements. This does not obviously follow from what we said due to the following reason. Here we limited our discussion to the entanglement wedge. In other words, we are always considering the surface S to be extremal. We expect that the discussion should generalize to situations where the surface S is along a causal horizon. The question is: what is the precise boundary dual of the region exterior to such a horizon? Even though we can think about the bulk computation, we are not sure what boundary computation it corresponds to. A proposal was made in [44], and perhaps one can understand it in that context. entanglement. Being able to de ne relative entropies for regions which are not bounded by minimal surfaces is also crucial to the interesting proposal in [45] to derive Einstein's equations from (a suitable extension to non-extremal surfaces of) the Ryu-Takayanagi formula for 6.5 Distillable entanglement In the recent papers [46, 47] it was argued that for gauge elds, only the purely quantum part of the entanglement entropy corresponds to distillable entanglement. The \classical" piece that cannot be used as a resource corresponds to the shannon entropy of the center variables of [14]. Our terms local in S are the gravitational analog of this classical piece and one might expect that a bulk observer with access only to the low-energy e ective eld theory can only extract bell pairs from the bulk entanglement. This seems relevant for the AMPS paradox [48{50]. Acknowledgments We thank H. Casini, M. Guica, D. Harlow, T. Jacobson, N. Lashkari, J. Lin, D. Marolf, H. Ooguri, M. Rangamani, and A. Wall for discussions. The work of D.L.J. is supported in part by NSFCAREER grant PHY-1352084 and a Sloan Fellowship. A.L. was supported in part by the US NSF under Grant No. PHY-1314198. J.M. was supported in part by U.S. Department of Energy grant de-sc0009988 and by the It from Qubit grant from the Simons Foundation. J.S. was supported at KITP in part by the National Science Foundation under Grant No. NSF PHY11-25915, and is now supported in part by the Natural Sciences and Engineering Research Council of Canada and by grant 376206 from the Simons Foundation. A A.1 Subregions of gauge theories U(1) gauge theory The problem of de ning the operator algebra of a subregion of a gauge theory was considered in [14]. It was shown that for a lattice gauge theory there are several possible de nitions of the subalgebra. It was further found that the subalgebra can have a center, namely some operators that commute with all the other elements of the subalgebra. In this case we can view the center as classical variables. Calling the classical variables xi, then for each value of xi we have a classical probability pi and a density matrix i for each irreducible block. The relative entropy between two states is then S( j ) = H(pjq) + X piS( ij i) i (A.1) where pi; qi are the probabilities of variables xi in the state and respectively. H is the classical (Shanon) relative entropies of two probability distributions, H = Pi pi log(pi=qi). In the continuum we expect that the relative entropy is nite and independent of the microscopic details regarding the precise de nition of the algebra [ 2 ]. These microscopic details have a continuum counterpart. When we consider a region R we would like to be able to de ne a consistent quantum theory within the subregion. In particular, imagine that we consider all classical solutions restricted to the subregion. Then we de ne a presymplectic product between two such solutions, which we will use to quantize the gauge orbits. This presymplectic product should be gauge invariant so that it does not depend on the particular representative. Let us consider a free Maxwell eld. The presymplectic product is given by integrating Z (A1; A2) = !(A1; A2) = Z (A1 ^ F 2 A ^ F 1) 2 (A.2) where A1 = A1 dx is a gauge eld con guration. Here we imagine that both A1 and A2 are solutions to the equations of motion. is any spacelike surface. Demanding gauge invariance amounts to the statement 0 = (A; d ) = Z ^ F (A.3) where @ is the boundary of the spacelike surface. We have used the equations of motion for F and integrated by parts. In order to make this vanish we need some boundary conditions. In particular, let us concentrate on the boundary conditions required at the boundary of corresponding to the boundary of a region S = @ . One possible boundary condition is to set Ai = Aicl for components along the surface, where Aicl is a classical gauge eld on the surface. In this case, it is natural to set = 0 on the surface. We can quantize the problem for each xed Aicl and then integrate over all Aicl. These values of Aicl are the \center" variables xi in the above discussion. This is called the \magnetic" center, since the gauge eld Aicl de nes a magnetic eld F = dAcl on the surface. There are other possibilities, such as xing the electric eld, or \electric center", where the perpendicular electric eld is xed. These would correspond to speci c choices on the lattice. Since we expect that relative entropy is a nite and smooth function of the shape of the region, [14] has shown that the detailed boundary condition does not matter, as long as we choose something that makes physical sense. Recently, [18, 19] carried out explicitly the eld theory calculation, being careful with the center variables. A.2 Gravity Here we consider the problem of de ning a subregion in a theory of Einstein gravity. We consider only the problem at the quadratic level where we need to consider free gravitons moving around a xed background (which obeys Einstein's equations). These gravitons can be viewed as a particular example of a gauge theory. We can also compute the symplectic form, as given in [27], and then impose that the symplectic inner product between a pure gauge mode and another solution to the linearized equations vanishes. Here the gauge transformations are reparametrizations, generated by a vector eld . Note that is not a killing vector, it is a general vector eld and it should not be confused with discussed in section 4. Writing the metric as g + g, where g is the background metric and g is a small uctuation. Then the gauge transformation acts as g ! g + L g, where L is the Lie derivative. Then, as shown in [30], there is a simple expression for the sympectic product with a such a pure gauge mode Z !( g; L g) = Q : (g; g) (A.4) with Q and (g; g) given in eqns (32) and (17) of [30]. We would like to choose boundary conditions on the surface which make the right hand side zero. We choose boundary conditions similar to the \magnetic" ones above. Namely, we x the metric along the entangling surface S to gij = ij . We treat ij as classical and then integrate over it. This is enough to make all terms in (A.4) vanish. Let us be more explicit. By a change of coordinates we can always set the metric to have the following form near the entangling surface. For simplicity we write it in Euclidean space, but the same is true in Lorentzian signature ds2 = d 2 + [ 2 + o( 4)](d + aidyi)2 + hij dyidyj (A.5) here ai and hij can be functions of and , with a regular expansion around = 0. In these coordinates the extremal surface S is always at = 0, both for the original metric and the perturbed metric. Extremality implies that the trace of the extrinsic curvature is zero, or KA = hij @XA hij = 0, where XA = (X1; X2) = ( cos ; sin ). This is true for the background and the uctuations KA = 0; KA = 0 (A.6) HJEP06(21)4 which ensures that even on the perturbed solution we are considering the minimal surface. These conditions ensure that the splitting between the two regions is de ned in a gauge invariant way. We demand that all uctuations are given in the gauge (A.5). Thus, near = 0, g leads to ai and hij . We now further set a boundary condition that hij = ij where ij is a classical function which we will later integrate over. For de ning the quantum problem we will view it as being classical. We will quantize the elds in the subregion for xed values of ij and then integrate over the classical values of ij . With these boundary conditions we see that all terms in (A.4) vanish. In fact, (A.4), has three terms10 Z !( h; L g) = hQ( ) i (g; h) ai i + hii + B AB (A.7) 1 2 Since the uctuation of the metric is zero at the entangling surface, hij = 0, we see that many terms vanish. In addition, since we are setting hij = 0, it is also natural to restrict the vector elds so that i = 0 on the surface. This ensures that the rst term in (A.7) vanishes. Note that the middle term is related to the fact that the area generates a shift in the coordinate . After all the area is the Noether charge associated to such shifts [26, 27]. The extremality condition makes sure that we are choosing a (generically) unique surface for each geometry. We then treat the induced geometry on the surface as a classical variable, quantize the metric in the subregion, and then sum over this classical variable. In this region, we seem to have a gauge invariant symplectic product. We have not explicitly computed the entanglement entropy for gravitons with these choices, but we expect that it should lead to a well de ned problem and that relative entropies will be nite. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 10We did not keep track of the numerical coe cients in front of each of the three terms. = Z Z Kyoto 1976 (1976) 809 [INSPIRE]. Rev. D 82 (2010) 124019 [arXiv:1007.1493] [INSPIRE]. HJEP06(21)4 [5] A.C. Wall, A proof of the generalized second law for rapidly changing elds and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [arXiv:1105.3445] [INSPIRE]. [6] D.L. Ja eris and S.J. Suh, The gravity duals of modular hamiltonians, arXiv:1412.8465 [7] D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP [arXiv:1412.1879] [INSPIRE]. [INSPIRE]. [8] N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement `thermodynamics', JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE]. [9] T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE]. [10] J. Lin, M. Marcolli, H. Ooguri and B. Stoica, Locality of gravitational systems from entanglement of conformal eld theories, Phys. Rev. Lett. 114 (2015) 221601 [11] N. Lashkari, C. Rabideau, P. Sabella-Garnier and M. Van Raamsdonk, Inviolable energy conditions from entanglement inequalities, JHEP 06 (2015) 067 [arXiv:1412.3514] [12] N. Lashkari and M. Van Raamsdonk, Canonical energy is quantum Fisher information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE]. [13] N. Lashkari, J. Lin, H. Ooguri, M. van Raamsdonk, B. Stoica, to appear. [14] H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge elds, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE]. [15] H. Casini, F.D. Mazzitelli and E. Teste, Area terms in entanglement entropy, Phys. Rev. D 91 (2015) 104035 [arXiv:1412.6522] [INSPIRE]. entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE]. [16] H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement [17] H. Casini and M. Huerta, Entanglement entropy in free quantum eld theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE]. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE]. [18] W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. [19] W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic eld, arXiv:1506.05792 [INSPIRE]. HJEP06(21)4 classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE]. [25] J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE]. [gr-qc/9307038] [INSPIRE]. [27] V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE]. [28] B. Swingle and M. Van Raamsdonk, Universality of Gravity from Entanglement, arXiv:1405.2933 [INSPIRE]. [29] J. Lee, A. Lewkowycz, E. Perlmutter and B.R. Safdi, Renyi entropy, stationarity and entanglement of the conformal scalar, JHEP 03 (2015) 075 [arXiv:1407.7816] [INSPIRE]. [30] S. Hollands and R.M. Wald, Stability of black holes and black branes, Commun. Math. Phys. 321 (2013) 629 [arXiv:1201.0463] [INSPIRE]. [31] H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. 26 (2009) 185005 [arXiv:0903.5284] [INSPIRE]. [32] A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE]. [33] B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE]. [34] R. Bousso, S. Leichenauer and V. Rosenhaus, Light-sheets and AdS/CFT, Phys. Rev. D 86 (2012) 046009 [arXiv:1203.6619] [INSPIRE]. [35] M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE]. [36] J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE]. (2013) 781 [arXiv:1306.0533] [INSPIRE]. [gr-qc/9312002] [INSPIRE]. [37] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 [38] S. Carlip and C. Teitelboim, The o -shell black hole, Class. Quant. Grav. 12 (1995) 1699 [39] A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE]. [arXiv:1204.1698] [INSPIRE]. entanglement entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE]. in holography, JHEP 05 (2014) 053 [arXiv:1403.3426] [INSPIRE]. [arXiv:1211.6767] [INSPIRE]. AdS/CFT, JHEP 03 (2014) 118 [arXiv:1309.3610] [INSPIRE]. (2016) 136 [arXiv:1510.07455] [INSPIRE]. entanglement of distillation for gauge theories, arXiv:1511.04369 [INSPIRE]. JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE]. [arXiv:1412.2730] [INSPIRE]. [1] M. Ohya and D. Petz , Quantum entropy and its use, corrected 2nd printing, Text and Monographs in Physics, Springer Study Edition, Springer ( 2004 ). [2] H. Araki , Relative entropy of states of von Neumann algebras , Publ. Res. Inst. Math. Sci. [3] H. Casini , Relative entropy and the Bekenstein bound , Class. Quant. Grav . 25 ( 2008 ) 205021 [4] A.C. Wall , A Proof of the generalized second law for rapidly-evolving Rindler horizons , Phys. [20] D.N. Kabat , Black hole entropy and entropy of entanglement, Nucl . Phys. B 453 ( 1995 ) 281 [21] S. Ryu and T. Takayanagi , Holographic derivation of entanglement entropy from AdS/CFT , Phys. Rev. Lett . 96 ( 2006 ) 181602 [ hep -th/0603001] [INSPIRE]. [22] V.E. Hubeny , M. Rangamani and T. Takayanagi , A covariant holographic entanglement entropy proposal , JHEP 07 ( 2007 ) 062 [arXiv: 0705 .0016] [INSPIRE]. [23] T. Faulkner , A. Lewkowycz and J. Maldacena , Quantum corrections to holographic entanglement entropy , JHEP 11 ( 2013 ) 074 [arXiv: 1307 .2892] [INSPIRE]. [24] T. Barrella , X. Dong , S.A. Hartnoll and V.L. Martin , Holographic entanglement beyond [26] R.M. Wald , Black hole entropy is the Noether charge , Phys. Rev. D 48 ( 1993 ) 3427 [40] V.E. Hubeny and M. Rangamani , Causal holographic information, JHEP 06 ( 2012 ) 114 [42] I.A. Morrison , Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property [43] K. Papadodimas and S. Raju , An infalling observer in AdS/CFT , JHEP 10 ( 2013 ) 212 [45] T. Jacobson , Entanglement equilibrium and the Einstein equation , Phys. Rev. Lett . 116 [48] A. Almheiri , D. Marolf , J. Polchinski and J. Sully , Black holes: complementarity or


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP06%282016%29004.pdf

Daniel L. Jafferis, Aitor Lewkowycz, Juan Maldacena. Relative entropy equals bulk relative entropy, Journal of High Energy Physics, 2016, 4, DOI: 10.1007/JHEP06(2016)004