Entanglement entropy from one-point functions in holographic states

For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv/1509.00113.


Introduction
In holographic conformal field theories, states with a simple classical gravity dual interpretation have a remarkable structure of entanglement: according to the holographic entanglement entropy formula [1][2][3], their entanglement entropies for arbitrary regions (at leading order in large N ) are completely encoded in the extremal surface areas of an asymptotically AdS spacetime. In general, the space of possible entanglement entropies (functions on a space of subsets of the AdS boundary) is far larger than the space of possible asymptotically AdS metrics (functions of a few spacetime coordinates), so this property of geometrically-encodable entanglement entropy should be present in only a tiny fraction of all quantum field theory states [4]. It is an interesting question to understand better which CFT states have this property 1 , and which properties of a CFT will guarantee that families of low-energy states with geometric entanglement exist.
For a hint towards characterizing these holographic states, consider the gravity perspective. A spacetime M Ψ dual to a holographic state |Ψ is a solution to the bulk equations of motion. Such a solution can be characterized by a set of initial data on a bulk Cauchy surface (and appropriate boundary conditions at the AdS boundary). The solution away from the Cauchy surface is determined by evolving this initial data forwards (or backwards) in time using the bulk equations. Alternatively, we can think of the bulk solution as being determined by evolution in the holographic radial direction, with "initial data" specified at the timelike boundary of AdS. In this case, the existence and uniqueness of a solution is more subtle, but the asymptotic behavior of the fields determines the metric at least in a perturbative sense (e.g. perturbatively in deviations from pure AdS, or order-by-order in the Fefferman-Graham expansion). It is plausible that in many cases, this boundary data is enough to determine a solution nonperturbatively to some finite distance into the bulk, or even for the whole spacetime. Thus, for geometries dual to holographic states, we can say that the bulk spacetime (at least in a perturbative sense) is encoded in the boundary behavior of the various fields.
According to the AdS/CFT dictionary, this boundary behavior is determined by the one-point functions of low-dimension local operators associated with the light bulk fields. On the other hand, the bulk spacetime itself allows us to calculated entanglement entropies (and many other non-local quantities). Thus, the assumption that a state is holographic allows us (via gravity calculations) to determine the entanglement entropies and other non-local properties of the state (again, at least perturbatively) from the local data provided by the one-point functions: (1.1) where φ here indicates all light fields including the metric. 2,3 The recipe (1.1) could be applied to any state, but for states that are not holo-graphic, the results will be inconsistent with the actual CFT answers. Thus, we have a stringent test for whether a CFT state has a dual description well-described by a classical spacetime: carry out the procedure in (1.1) and compare the results with a direct CFT calculation of the entanglement entropies; if there is a mismatch for any region, the state is not holographic. 4 In this paper, our goal is to present some more explicit results for the gravity prediction S grav A ( O α ) in cases where the gravitational equations are Einstein gravity with matter and the region is taken to be a ball-shaped region B. We will work perturbatively around the vacuum state to obtain an expression as a power series in the one-point functions of CFT operators. At first-order, the result depends only on the CFT stress tensor expectation value [10]: This well-known expression is universal for all CFTs since it follows from the first law of entanglement δ (1) is the vacuum modular Hamiltonian for a ball-shaped region. Thus, to first-order, the gravity procedure (1.1) always gives the correct CFT result for ball-shaped regions, regardless of whether the state is holographic.

General second-order result for ball entanglement entropy
Our focus will be on the second-order answer; in this case, it is less clear whether the gravity results from (1.1) should hold for any CFT or whether they represent a constraint from holography. To obtain explicit formulae at this order, we begin by writing which follows immediately from the definition of relative entropy S(ρ B ||ρ vac B ) reviewed in Section 2 below. We then make use of a recent result in [11]: to second-order in perturbations from the vacuum state, the relative entropy for a ball-shaped region in a holographic state 5 is equal to the "canonical energy" associated with a corresponding wedge of the bulk spacetime. We provide a brief review of this in Section 2 below. On shell, the latter quantity can be expressed as a quadratic form on the space of first-order perturbations to pure AdS spacetime, so we have Rearranging this, we have a second-order version of (1.2): As we review in Section 2 below, the last term can be written more explicitly as where Σ is a bulk spatial region between B and the bulk extremal surfaceB with the same boundary, ω is the "presymplectic form" whose integral defines the symplectic form on gravitational phase space, T ab is the matter stress tensor at second-order in the bulk matter fields, and ξ is a bulk Killing vector which vanishes onB. The first-order bulk perturbations δφ α (including the metric perturbation) may be expressed in terms of the boundary one-point functions via bulk-to-boundary propagators where D B is the domain of dependence of the ball B. Given the one-point functions within D B , we can use (1.8) to determine the linearized bulk perturbation in Σ and evaluate (1.7). The expression (1.6), (1.7), and (1.8) together provide a formal result for the ball entanglement entropy of a holographic state, expanded to second-order in the boundary one-point functions.

Explicit results for 1+1 dimensional CFTs
In order to check the general formula and provide more explicit results, we focus in Section 3 on the case of 1+1 dimensional CFTs, carrying out an explicit calculation of the gravitational contributions to (1.7) starting from a general boundary stress tensor. We find the result where the integrals can be taken over any spatial surface B with boundary ∂B, and the kernel is given by where c is the central charge. In this special case, the conservation equations determine the stress tensor expectation values throughout the region D B from the expectation values on B , so as in the first-order result (1.2), our final expression involves integrals only over B . This will not be the case for the terms involving matter fields, or in higher dimensions. As a consistency check, we show that the expression (1.10) is always negative, as required by its interpretation as the second-order contribution to relative entropy. We can also check the formula (1.10) via a direct CFT calculation by considering states that are obtained from the CFT vacuum by a local conformal transformation. In two dimensions, states with an arbitrary traceless conserved stress-tensor can be obtained, and the entanglement entropy for these states can also be calculated explicitly. We carry out this calculation in section 4, and show that the result (1.10) is exactly reproduced.
In Section 3.2, we consider the matter terms in (1.7) providing some explicit results for the quadratic contributions of scalar operator expectation values. Here, as in the generic case, the result takes the form with integrals over the entire domain of dependence region.

Auxiliary de Sitter Space Interpretation
Recently, in [12] it has been pointed out that the first-order result δ (1) S(x µ , R) for the entanglement entropy of a ball with radius R and center x µ can be obtained as the solution to the equation of motion for a free scalar field on an auxiliary de Sitter space with the CFT energy density T 00 (x µ ) acting as a source term at R = 0. In Section 5, we show that in the 1+1 dimensional case, the stress tensor term (1.10) for the entanglement entropy at second-order can also results from solving a scalar field equation on the auxiliary de Sitter space if we add a simple cubic interaction term. In an upcoming paper [13], it is shown that this agreement extends to all orders for a suitable choice of the scalar field potential. The resulting nonlinear wave equation also reproduces the second-order entanglement entropy near a thermal state in the auxiliary kinematic space recently described in [14].
Including the contributions from matter fields or moving to higher dimensions, the expression for entanglement entropy involves one-point functions on the entire causal diamond D B , so reproducing these results via some local differential equation will require a more complicated auxiliary space that takes into account the time directions in the CFT. This direction is pursued further in [13,15].

Discussion
While the explicit two-dimensional stress tensor contribution (1.10) can be obtained by a direct CFT calculation for a special class of states, we emphasize that in general the holographic predictions from (1.1) are expected to hold only for holographic states in CFTs with gravity duals. It would be interesting to understand better whether all of the second order contributions we considered here are universal for all CFTs or whether they represent genuine constraints/predictions from holography. 6 In the latter case, and for the results at higher order in perturbation theory, it is an interesting question to understand better which CFT states and/or which CFT properties are required to reproduce the results through direct CFT calculations. This should help us understand better which theories and which states in these theories are holographic.

Background
Our holographic calculation of entanglement entropy to second-order in the boundary one-point functions makes use of the direct connection between CFT quantum Fisher information and canonical energy on the gravity side, pointed out recently in [11]. We begin with a brief review of these results.

Relative entropy and quantum Fisher information
Our focus will be on ball-shaped subsystems B of the CFT d , for which the the vacuum density matrix is known explicitly through (1.3). More generally, we can write it as where T µν is the CFT stress tensor operator and is defined as so that n µ µ is the volume form on the surface perpendicular to a unit vector n µ , and ζ B is a conformal Killing vector defined in the domain of dependence region D B , with ζ B = 0 on ∂B. For the ball B with radius R and center x µ 0 in the t = t 0 slice, we have By the conservation of the current ζ µ B T µ ν associated with this conformal Killing vector, the integral in (2.1) can be taken over any spatial surface B in D B with the same boundary as B.
For excited states, the density matrix ρ B will generally be different than ρ vac B . One measure of this difference is the relative entropy where H B is the vacuum modular Hamiltonian given in (2.1), is the entanglement entropy for the region B and ∆ indicates the difference with the vacuum state. For a one-parameter family of states near the vacuum, we can expand ρ B as The first-order contribution to relative entropy vanishes (this is the first law of entanglement δ (1) S B = δ H B ) so the leading contribution to relative entropy appears at second-order in λ. This quadratic in δρ 1 with no contribution from δρ 2 , This quadratic form, which is positive by virtue of the positivity of relative entropy, defines a positive-(semi)definite metric on the space of perturbations to a general density matrix σ. This is known as the quantum Fisher information metric. Rearranging (2.4) and making use of (2.6), we have This general expression is valid for any CFT, but the O(λ 2 ) term generally has no simple expression in terms of local operator expectation values. However, for holographic states we can convert this term into an expression quadratic in the CFT one-point functions by using the connection between quantum Fisher information and canonical energy.

Canonical energy
Consider now a holographic state, which by definition is associated with some dual asymptotically AdS spacetime M . Near the boundary, we can describe M using a metric in Fefferman-Graham coordinates as where Γ µν (z, x) has a finite limit as z → 0 and Γ = 0 for pure AdS. The relative entropy S(ρ B ||ρ vac B ) can be computed at leading order in large N by making use of the holographic entanglement entropy formula, which relates the entanglement entropy for a region A to the area of the minimal-area extremal surfacẽ A in M with boundary ∂A, (2.10)

This yields immediately that ∆S
The result (2.10) also allows us to relate the ∆ H B term in relative entropy to a gravitational quantity, since it implies that the expectation value of the CFT stress tensor is related to the asymptotic behaviour of the metric through [19] T Thus, for holographic states, we can write For a one-parameter family of holographic states |Ψ(λ) near the CFT vacuum, the dual spacetimes M (λ) can be described via a metric and matter fields φ α = (g, φ matter ) with some perturbative expansion By the result (2.8) from the previous section, the second-order contribution to entanglement entropy is equal to the leading order contribution to relative entropy. This is related to a gravitational quantity via (2.12). The main result in [11] is that this second-order quantity can be expressed directly as a bulk integral over the spatial region Σ between B andB where the integrand is a quadratic form on the linearized bulk perturbations δg 1 and δφ matter To describe the general result, consider the region Σ between B andB in pure AdS spacetime, and define R B as the domain of dependence of this region, as shown in figure 1. Alternatively, R B is the intersection of the causal past and the causal future of D B ; it can be thought of as a Rindler wedge of AdS associated with B. On R B , there exists a Killing vector which vanishes atB and approaches the conformal Killing vector ζ B at the boundary. In Fefferman-Graham coordinates, this is The vector ξ B is timelike hence defines a notion of time evolution within the region R B ; the "Rindler time" associated with this Rindler wedge. The "canonical energy", dual to relative entropy at second-order, can be understood as the perturbative energy associated with this time, as explained in [20]. This is quadratic in the perturbative bulk fields including the graviton, and given explicitly by (2. 15) In the first line, W Σ is the symplectic form associated with the phase space of gravitational solutions on Σ, and £ ξ B δφ 1 is the Lie derivative with respect to ξ on δφ 1 , the first-order perturbation in metric and matter fields. The symplectic form is equal to the integral over Σ of a "presymplectic" form ω f ull which splits into a gravitational part and a matter part as in the third line. The matter part can be written explicitly in terms of T (2) ab , the matter stress tensor at quadratic order in the fields, while the gravitational part ω is given explicitly by In deriving (2.15) it has been assumed that the metric perturbation has been expressed in a gauge for which the coordinate location of the extremal surfaceB does not change (so that ξ B continues to vanish there), and the vector ξ B continues to satisfy the Killing equation atB. Thus, we require that As shown in [20], it is always possible to satisfy these conditions; we will see an explicit example below.
3 Second-order contribution to entanglement entropy Using the result (1.7), we can now write down a general expression for the ball entanglement entropy of a general holographic state up to second-order in perturbations to the vacuum state, in terms of the CFT one-point functions. According to (2.8) and (2.15), the second-order term in the entanglement entropy for a ball B can be expressed as an integral over the bulk spatial region Σ between B and the corresponding extremal surfaceB, where the integrand is quadratic in first-order bulk perturbations. These linearized perturbations are determined by the boundary behavior of the fields via the linearized bulk equations. In general, to determine the linearized perturbations in the region Σ (or more generally in the Rindler wedge R B ), we only need to know the boundary behavior in the domain of dependence region D B , as discussed in detail in [21]. The relevant boundary behaviour of each bulk field is captured by the one-point function of the corresponding operator. We can express the results as where K α (x, z; x ) is the relevant bulk-to-boundary propagator. As discussed in [21][22][23], K α should generally be understood as a distribution to be integrated against consistent CFT one-point functions, rather than a function. Since the expression (3.1) is linear in the CFT expectation values, the result (1.7) is quadratic in these one-point functions and represents our desired second-order result.
To summarize, for a holographic state, the second-order contribution to entanglement entropy in the expansion (2.8) is the leading order contribution to the relative entropy S(ρ B ||ρ vac B ). This is dual to canonical energy, given explicitly by: (3.2) This is quadratic in the linearized perturbations δφ α (including the metric perturbation, and these can be expressed in terms of the CFT one-point functions on D B as (3.1).

Example: CFT 2 stress tensor contribution
In this section, as a sample application of the general formula, we provide an explicit calculation of the quadratic stress tensor contribution to the entanglement entropy for holographic states in two-dimensional conformal field theories. This arises from the first term in (1.7).
For a general CFT state, the stress tensor is traceless and conserved, In two dimensions, these constraints can be expressed most simply using light-cone coordinates x ± = x ± t, where we have Thus, a general CFT stress tensor can be described by the two functions, T ++ (x + ) and Assuming that the state is holographic, there will be some dual geometry of the form (2.9). According to (2.11), the stress tensor expectation values determine the asymptotic form of the metric as Now, suppose that our state represents a small perturbation to the CFT vacuum, so that the stress tensor expectation values and the asymptotic metric perturbations are governed by a small parameter λ: Then the metric perturbation throughout the spacetime is determined by this asymptotic behavior by the Einstein equations linearized about AdS. Here, we need only the components in the field theory directions, which give The solution in our Fefferman-Graham coordinates with boundary behaviour (3.6) is with the linearized perturbation Γ µν independent of z.

Satisfying the gauge conditions
We would now like to evaluate the metric contribution to (3.2) This formula assumes the gauge conditions (2.17) which differ from the Fefferman-Graham gauge conditions we have been using so far. Thus, we must find a gauge transformation to bring our metric perturbation to the appropriate form. In general, we can write where γ is the desired metric perturbation satisfying the gauge condition, and h is the perturbation in Fefferman-Graham coordinates (equivalent to Γ for d = 2). The procedure for finding an appropriate V and evaluating (3.9) is described in detail in [11], but we review the main points here. Defining coordinates (X A , X i ) so that the extremal surface lies at some fixed value of X A with X i describing coordinates along the surface, the gauge condition (2.17) (equivalent to requiring that the coordinate location of the extremal surface remains fixed) gives while the condition (2.18) that ξ B continues to satisfy the Killing equation atB gives To solve these, we first expand our general metric perturbation in a Fourier basis.
with a gauge choice h za (t, x, z) = 0. For each of the basis elements, we use the equations (3.11), (3.12) and (3.13) to determine V and its first derivatives at the surface V . For these calculations, it is useful to define polar coordinates (z, x) = (r cos θ, r sin θ). Since the gauge conditions are linear in V , the conditions on V for a general perturbation are obtained from these by taking linear combinations as in (3.14), After requiring V a remain finite at θ = ± π 2 , we find where the V ± solutions are related throughV + r (k; t, r, θ) =V − r (k; −t, r, θ) andV − t (k; t, r, θ) = −V + t (k; −t, r, θ). The results here give the behavior of V and its derivatives only at the surfaceB (r = R in polar coordinates). Elsewhere, V can be chosen arbitrarily, but we will see that our calculation only requires the behavior atB.

Evaluating the canonical energy
Given the appropriate V , we can evaluate (3.9) using and we have used that £ ξ g = 0. We can simplify this expression using the gravitational identity ω(g, γ, £ ξ g) = dχ(γ, X) (3.19) where Finally, choosing V so that it vanishes at B, we can rewrite (3.9) as In this final expression, we only need V and its derivatives at the surfaceB. Thus, we can now calculate the result explicitly for a general perturbation. In the Fourier basis, the final result in terms of the boundary stress tensor is where the kernel iŝ . We note in particular that the result splits into a left-moving part and a right-moving part with no cross term. Transforming back to position space where the kernel K 2 is symmetric under exchange of x ± 1 and x ± 2 , and has support only on x ± i ∈ [−R, R]. Focusing only on the domain of support, we have Using the relation c = 3 AdS /(2G N ) between the CFT central charge and the gravity parameters, we recover the result (1.10) from the introduction. Like the leading order result in (2.8), the integrals can be taken over any surface B with boundary ∂B. The fact that we only need the stress tensor on a Cauchy surface for D B is special to the stress tensor in two dimensions, since the conservation relations allow us to find the stress tensor expectation value everywhere in D B from its value on a Cauchy surface. For other operators, or in higher dimensions, the result will involve integrals over the full domain of dependence. We will see an explicit example in the next subsection.
Positivity of relative entropy requires E to be positive which requires the kernel to be positive semi-definite. As we show in Appendix A, one can demonstrate that the positivity explicitly, providing a check of our results. An alternative proof of positivity is given in Section 5. As a more complete check, we will show in Section 4 that this result can be reproduced by a direct CFT calculation for the special class of states that can be obtained from the vacuum state by a local conformal transformation.

Example: Scalar operator contribution
We now consider an explicit example making use of the bulk matter field term in (1.7) in order to calculate the terms in the entanglement entropy formula quadratic in the scalar operator expectation values. The discussion for other matter fields would be entirely parallel. This example is more representative, since the formula will involve scalar field expectation values in the entire domain of dependence D B , i.e. a boundary spacetime region rather than just a spatial slice. The results here are similar to the recent work in [16][17][18], but we present them here to show that they follow directly from the canonical energy formula.
We suppose that the CFT has a scalar operator of dimension ∆ with expectation value O(x) . According to the usual AdS/CFT dictionary, this corresponds to a bulk scalar field with mass m 2 = ∆(∆ − d) and asymptotic behavior where γ is a constant depending on the normalization of the operator O. The leading effects of the bulk scalar field on the entanglement entropy (3.2) come from the matter term in the canonical energy Using the explicit form of ξ B from (2.14) and from (2.2), this gives (for a ball centered at the origin) 00 (x, z) .

(3.30)
This expression is valid for a general bulk matter field. For a scalar field, we have where g ab is the background AdS metric and φ 1 represents the solution of the linearized scalar field equation on AdS, with boundary behavior as in (3.28). This solution is given most simply in Fourier space, where we have where ν = ∆ − d/2, but we can formally write a position-space expression using a bulk-to-boundary propagator K(x, z; x ) as [24,25] The integral here is over the boundary spacetime, however it has been argued (see, for example [21,22]) that to reconstruct the bulk field throughout the Rindler wedge R B (and specifically on Σ), we need only the boundary values on the domain of dependence region. We recall some explicit formulae for this "Rindler bulk reconstruction" in Appendix B. Combining these results, we have a general expression for the scalar field contribution to entanglement entropy at second-order in the scalar one-point functions, where φ 1 is given in (3.33) or (3.34) . As a simple example, consider the case where the scalar field expectation value is constant. In this case it is simple to solve (3.32) everywhere to find that (3.36) Inserting this into the general formula (3.35), and performing the integrals, we obtain .

Stress tensor contribution: direct calculation for CFT 2
In Section 3.1, we used the equivalence between quantum Fisher information and canonical energy to obtain an explicit expression for the second-order stress tensor contribution to the entanglement entropy for holographic states in two-dimensional CFTs. This is applicable for general holographic states, whether or not other matter fields are present in the dual spacetime (in which case there are additional terms in the expression for entanglement entropy). In special cases where there are no matter fields, the spacetime is locally AdS and we can understand the dual CFT state as being related to the vacuum state by a local conformal transformation. We show in this section that in this special case, we can reproduce the holographic result (3.27) through a direct CFT calculation, providing a strong consistency check. We note that the result does not rely on taking the large N limit or on any special properties of the CFT, so the formula holds universally for this simple class of states. Our approach will be to develop an iterative procedure to express the entanglement entropy as an expansion in the stress tensor expectation value for this special class of states. We evaluate the entanglement entropy for these states from a correlation function of twist operators obtained by transforming the result for the vacuum state. 7 Similarly, the stress tensor expectation values follow directly from the form of the conformal transformation. Inverting the relationship between the required conformal transformation and the stress tensor expectation value allows us to express the entanglement entropy as a perturbative expansion in the expectation value of the stress tensor. Similar CFT calculations have also been used recently in [13].

Conformal transformations of the vacuum state
In two-dimensional CFT, under a conformal transformation w = f (z), the stress tensor transforms as Here c is the central charge of the CFT and the inhomogeneous part is the Schwarzian derivative For an infinitesimal transformation f (z) = z + λ (z), the Schwarzian derivative can be expanded as The CFT vacuum is invariant under the SL(2, C) subgroup of global conformal transformations. However, for transformations which are not part of this subgroup, the vacuum state transforms into excited states. The action of the full conformal group includes the full Virasoro algebra which involves arbitrary products and derivatives of the stress tensor Id ∼ 1, T, ∂ m T, T 2 , T ∂ n T, · · · . (4.4) These states capture the gravitational sector of the gravity dual. Other excited states can be obtained by the action of other primary operators and their descendants. However we restrict ourselves to the class states that are related to 'pure gravity' excitations, which are the states obtained by conformal transformation of the vacuum state. We denote the excited state as |f = U f |0 where U f is the action of a conformal transformation on the vacuum |0 . The expectation value of the stress tensor for the state perturbed state |f is where we used that 0|T (z)|0 = 0. The anti-holomorphic component of the stress tensorT (z) is similarly related to the anti-holomophic part of the conformal transformationf .
To leading order in a conformal transformation near the identity, this equation relates the conformal transformation to T (z) by a third-order ordinary differential equation. The three integration constants correspond to the invariance of T (z) under the global conformal transformations. Thus we have an invertible relationship between the conformal transformations modulo their global part and T (z) , at least near the identity.

Entanglement entropy of excited states
In a two-dimensional CFT, the entanglement entropy can be explicitly computed using the replica method [27,28]. The computation can be reduced to a correlation function of twist operators Φ ± , which are conformal primaries with weight (h n ,h n ) = c 24 (n − 1/n, n − 1/n).
The Rényi entropy is The entanglement entropy is obtained by taking the n → 1 limit of S (n) .
For the excited states obtained by conformal transformations z → w = f (z) the Rényi entropy is (4.9) Here z 1 , z 2 are the points f (z 1 ) =f (z 1 ) = −R, f (z 2 ) =f (z 2 ) = R. The entanglement entropy of the excited state is (4.10) Therefore the change in entanglement entropy respect to the vacuum state is By inverting (4.5), the conformal transformation required to reach the state |f can be expressed as a function of the expectation value of the stress tensor. Plugging this f into (4.11), allows us to express the entanglement entropy as a function of the expectation value of the stress tensor alone, as we set out to do.
In practice, we will invert (4.5) order by order in a small conformal transformation and express the entanglement entropy as an expansion in the resulting small stress tensor. The second-order term in this expansion will be the Fisher information metric.
In the following, we will focus on the holomorphic term in (4.5), noting that the anti-holomorphic part follows identically. 8

Perturbative expansion
Consider a conformal transformation perturbation near the identity transformation where λ is a small expansion parameter. In this expansion, 13) and the entanglement entropy is (4.14)

Linear order
To first-order in λ, the stress tensor is given by so that change in the expectation value of the modular Hamiltonian becomes From (4.11) we also have that the first-order change in entanglement entropy is Comparing with (4.16) we see that the first law of entanglement holds

Second-order
The second-order change in entanglement entropy gives the second-order relative entropy as the modular Hamiltonian is linear in the expectation value of the stress tensor. This is the quantum Fisher metric in the state space, which is dual to the canonical energy in gravity [11]. In this section, we obtain the expression for canonical energy from the CFT side and find an exact match to the results of Section 3.1.
Our procedure so far yields the entanglement entropy of a subregion in terms of a perturbative expansion in small stress tensor expectation value To obtain K 2 (z 1 , z 2 ), we need to invert the relationship in (4.13) order by order, the lower order solutions f i−1 , f i−2 , · · · f 1 becoming sources for the i-th order solution.
Taking the explicit expression for T (z) to simplify solving the differential equations, is sufficient to extract the Fourier transformed kernel. The first-order solution is where F i are constants that corresponds to the global part of the conformal transformation and do not effect the final result. We take these constants to be zero for simplicity. The second-order solution is (4.22) With these solutions, we obtaiñ as well as 24) with K ≡ R(k 1 + k 2 ) and κ ≡ R(k 1 − k 2 ). Taking the inverse Fourier transformation ofK 1 (k) is a window function with support x ∈ [−R, R].
The second-order position space kernel is The anti-holomorphic part is the same with z →z, and the cross term vanishes. With the relation c = 3 AdS 2G N (4.28) this reproduces the kernel for canonical energy in (3.27). This result holds for regions defined on any spatial slice of the CFT. If we choose the t = 0 slice, z =z = x and our result becomes Changing variables using x 1 = x − r, x 2 = x + r, the kernel is simply

Excited states around thermal background
A similar analysis can be applied to perturbations around a thermal state with temperature T = β −1 . If we denote homogeneous thermal state |β , the stress tensor one-point function is This can be obtained by a conformal transformation from the vacuum with On top of this transformation, one could also apply an infinitesimal conformal transformation to obtain non-homogeneous perturbation around thermal state.
A similar computation as the previous section leads to the first-order kernel which is the modular hamiltonian of thermal state in 2d CFT. Furthermore, the second-order kernel is (4.33)

Consistency check : homogeneous BTZ perturbation
As a check, consider the homogeneous perturbation example, where T = T = λ 8G N . 9 In AdS 3 this is a perturbation towards the planar BTZ geometry in Fefferman-Graham coordinates. Holographic renormalization (2.11) tells us the stress tensor expectation value of the dual CFT is As the black hole corresponds to the thermal state in CFT, the dual state be obtained by the conformal transformation (4.31).
First, applying (4.11) for this conformal transformation, the change in entanglement entropy with respect to the vacuum is which matches the previous known results [5,11]. The linear order equals δ H B as expected from the entanglement first law. The second-order term gives the quantum Fisher information or the canonical energy (4.37) Using the formula using the second-order kernel (4.19) and (4.27), we obtain the same canonical energy (4.38)

Auxiliary de Sitter space interpretation
In [12], it was pointed out that the leading order perturbative expression (1.2) for entanglement entropy, expressed as a function of the center point x and radius R of the ball B, is a solution to the wave equation for a free scalar field on an auxiliary de Sitter space, with T 00 (x) acting as a source. It was conjectured that higher order contributions might be accounted for by local propagation in this auxiliary space with the addition of self-interactions for scalar field. In this section, we show that for two-dimensional CFTs, the second-order result (1.10) can indeed be reproduced by moving to a non-linear wave equation with a simple cubic interaction to this scalar field. A slight complication is that we actually require twoscalar fields; one sourced by the holomorphic stress tensor T ++ , and the other sourced by the anti-holomorphic part T −− ; the perturbation to the entanglement entropy is then the sum of these two scalars, δS = δS + + δS − , reproducing both terms in (1.10). We will focus on δS + since δS − follows identically.
To reproduce the second-order results for entanglement entropy, we consider an auxiliary de Sitter space with metric and consider a scalar field δS + with mass m 2 L 2 dS = −2 and action The equation of motion is As shown in [12], the first-order perturbation (1.2) obeys the linearized wave equation We can immediately check that the second-order perturbation (1.10) is consistent with the nonlinear equation by acting with the dS wave equation on the second-order kernel (4.27) Integration against the CFT stress tensor then gives (5.3). Alternatively, introducing the retarded 10 bulk-to-bulk propagator [29] G dS (η, x; η , and bulk-to-boundary propagator we can show directly that the solution with boundary behavior at first-order and 10) at second-order, where the latter term comes from the diagram shown in Figure 2. The integrals can be performed directly to show that these results match with the expressions (1.2) and (1.10) respectively.
A useful advantage of writing the second-order result in the form (5.10) is that it is manifestly negative. More explicitly, we have where |g dS | and the squared expression are manifestly positive and the bulk-to-bulk propagator (5.6) is positive over the range of integration where (y − x 0 ) 2 ≤ (R − η) 2 . That this expression is negative is required by the positivity of relative entropy, since we showed above that −δ (2) S represents the leading order perturbative expression for the relative entropy.
Recently, it has been realized that the modular Hamiltonian in certain non-vacuum states in two dimensional CFTs can be described by propagation in a dual geometry [14] matching the kinematic space found previously in [30][31][32][33]. We find that the results of Section 4.4 can be explained by the same interacting theory (5.2) on this kinematic space. The kinematic space dual to the thermal state is 11 The second-order perturbation to the entanglement entropy from (4.33) obeys the wave equation (5.3) with the same interactions in this kinematic space. We could imagine adding additional fields propagating in de Sitter to capture the contributions to the entanglement entropy from scalar operators discussed in Section 3.2. However, unlike the contribution from the stress tensor, this contribution involves integration of the one-point functions over the full domain of dependence D B . In higher-dimensions, this will also be true for the stress tensor contribution. The R = 0 boundary of the auxiliary de Sitter space does not include the time direction of the CFT, so any extension of these results to contributions of other operators or higher dimensional cases will require a more sophisticated auxiliary space. Promising work in this direction is discussed in [13,15].
which is clearly non-negative and symmetric in n, m.
To show that the canonical energy is positive, we need to show the matrix M with entries given by A n,m = A n−1,m−1 12 is positive definite. To do so, we will use proof by induction and Sylvester's criterion which states that a square matrix M is positive definite if and only it has a positive determinant and all the upper-left submatrices also have a positive determinant.

Proof by Induction
Suppose that the N × N matrix M N whose components are given by A n,m is positivedefinite. Then consider the block matrix constructed as where B is a N -column vector with entries given by So it remains to show that λ M 1 −β B ≥ 0 to complete the proof. The maximum eigenvalue λ M 1 is bound from below by the minimum sum of a column of M N through the Perron-Frobenius theorem (equivalently Gershgorin circle theorem). For the matrix M N , the minimum sum of a column vector is simply the sum of the N -th column N i=1 A i,N since A i,j decreases with i and j. Therefore it remains to show We split this sum up into two cases. The first case is if N is even. Then we have since the final term in (N −1)/2 i=1 A 2 2i−1,N +1 is zero. Explicitly analyzing the coefficients, we see that A 2i,N − A 2 2i−1,N +1 is always positive for all i ∈ {1..N/2}, so clearly the entire sum is positive. In the case of odd N , the sum becomes Each term in this sum is also positive, so we have shown λ M 1 − β B ≥ 0. The expressions in (A.8) and (A.9) are not obviously positive, but they reduce to some polynomial equations which can be shown to be positive. Therefore we've shown det(M N + B T B) < 2 det(M N ), thus M N +1 is positive-definite given that M N is. Since M 1 is positive-definite we completed the proof by induction. The kernel for canonical energy is explicitly positive-semidefinite as required by the positivity of relative entropy.

B Rindler reconstruction for scalar operators in CFT 2
In this appendix we find an expression for the matter contribution to the second-order perturbation to the entanglement entropy of a ball B using Rindler reconstruction so as to only use the one-point functions of the scalar operator in the domain of dependence D B . We specialize to two dimensional CFTs in order to obtain a more explicit expression which can be compared to the gravitational contribution (1.10). Further discussions of Rindler reconstruction can be found in the literature [21,22,24,25,34].