Holographic entanglement entropy inequalities beyond strong subadditivity

The vacuum entanglement entropy in quantum field theory provides nonperturbative information about renormalization group flows. Most studies so far have focused on the universal terms, related to the Weyl anomaly in even space-time dimensions, and the sphere free energy $F$ in odd dimensions. In this work we study the entanglement entropy on a sphere of radius $R$ in a large radius limit, for field theories with gravity duals. At large radius the entropy admits a geometric expansion in powers of $R$; the leading term is the well-known area term, and there are subleading contributions. These terms can be physical, they contain information about the full renormalization group flow, and they reproduce known monotonicity theorems in particular cases. We set up an efficient method for calculating them using the Hamilton-Jacobi equation for the holographic entanglement entropy. We first reproduce the known result for the area term, the coefficient multiplying $R^{d-2}$ in the entanglement entropy. We then obtain the holographic result for the $R^{d-4}$ term and establish its irreversibility. Finally, we derive the $R^{d-6}$ coefficient for holographic theories, and also establish its irreversibility. This result goes beyond what has been proved in quantum field theory based on strong subadditivity, and hints towards new methods for analyzing the monotonicity of the renormalization group in space-time dimensions bigger than four.


Introduction
Irreversibility theorems provide key insights into the nonperturbative structure of quantum field theories (QFTs). The first such theorem was proved by [1], and established the decrease of the central charge C in d = 1 + 1 dimensions along renormalization group flows. This theorem was rederived using quantum information tools in [2], and an extension of these methods allowed to establish the F-theorem in d = 2 + 1 dimensions [3]. In d = 3 + 1 dimensions, the A-theorem was proved by [4] using unitarity and the dilaton, and then in [5] based on strong subadditivity of the entanglement entropy.
This situation is somewhat puzzling. On the one hand, these proofs use quite different methods (euclidean, Lorentzian and information theoretic), and it is not clear how they are related. One would hope for some unifying understanding, but it has not emerged so far. Furthermore, none of the results extend to more than 3+1 space-time dimensions. A clue that these issues should have a positive resolution comes from QFTs with holographic duals. Indeed, for such theories, the null energy condition (NEC) allows to construct a running C-function and establish the irreversibility of holographic renormalization group (RG) flows [6][7][8]. The proposed C-function depends on the metric scale factor, and coincides with the universal A or F terms at fixed points. Further progress for understanding holographic RG flows in terms of the EE was made in [9,10]; we will revisit their method in the Appendix.
Motivated by these questions, in the present work we will analyze the entanglement entropy for holographic RG flows with the goal of finding new inequalities. We use holography because we want to access d > 4 dimensions, and because we hope that this may provide clues for future field theoretic approaches. We will go beyond previous irreversibility results by obtaining new inequalities for the geometric large radius terms of the holographic entanglement entropy. As we will discuss, these contain physical information about the renormalization group and about the effective gravitational action induced by integrating out the QFT degrees of freedom.
We consider Poincare invariant QFTs in d space-time dimensions. We assume that at short distances the theory is described by a conformal field theory; relevant perturbations trigger an RG flow, which we assume ends at an IR CFT (different from the UV one). We will analyze the EE associated to a spherical region of radius R. A and F are the so-called universal terms (related to the Weyl anomaly in even spacetime dimensions, and the sphere free energy in odd dimensions), and is a short distance cutoff. The R d−2k terms arise from geometric quantities of the boundary of the entangling region [11], and they will be the main focus in our work. The entropy and its µ coefficients diverge as the short distance cutoff → 0. However, we will be interested in comparing two different theories: the UV fixed point one, and the theory that undergoes the nontrivial RG flow. We will review below how this leads to a finite, cutoff-independent, entropy differece ∆S.
For an RG flow with a typical mass scale m, the UV CFT corresponds to m = 0. In this limit, the sphere EE has the form (1.1) with coefficients that we call µ U V CF T d−2k . For mR 1, the IR fixed point is approached, and the EE is also of the form (1.1) but with different coefficients µ IR d−2k . 1 We will be interested in the entropy difference ∆S(R) = S IR (R) − S U V CF T (R) = ∆µ d−2 R d−2 + ∆µ d−4 R d−4 + . . . , (1.2) with As discussed in Sec. 3.3, this becomes independent of the cutoff and hence is a property of the continuum theory. The entropy difference appears naturally in information-theoretic approaches to irreversibility [5,[12][13][14].
The ∆A or ∆F are intrinsic to the fixed points, namely they are independent of the RG flow that connected the UV and IR CFTs. This is what "universal" means in this context. The other quantities ∆µ d−2k in the large mR expansion do depend on the RG trajectory, but can still be physical (i.e. finite in the continuum limit). The simplest example is ∆µ d−2 , the coefficient of the area term. It corresponds to the renormalization of 1/G N when weakly coupling gravity to the QFT; equivalently, the flow of the area term in the EE coincides with the low energy QFT contribution to black hole entropy. Ref. [15] showed that where Θ(x) is the trace of the energy-momentum tensor. From here and reflection positivity or unitary, it follows that This result for ∆µ d−2 is also interesting because it contains the C-theorem as a particular case. Indeed, in the limit d → 2, ∆µ d−2 R d−2 → − 1 6 (c U V − c IR ) log(mR), and Eq. (1.6), established first in [16], is known as a sum-rule: the integrand in the right hand side depends on the RG trajectory, but its integral should only depend on the end-point central charges. Further progress on understanding the coefficients in the expansion of the EE was made in [5], which showed that The A-theorem is obtained for d → 4. Given these results, it is natural to conjecture that the higher order terms satisfy This would also imply the validity of A-theorems for all even d by setting 2k = d. So far there is no proof of this conjecture.
Our goal in this work is to analyze the ∆µ d−2k in theories with gravity duals. With this aim, we consider asymptotically AdS geometries that represent holographic RG flows. We will find that the Hamilton-Jacobi equation for the holographic EE as a function of the radius R and a radial cutoff provides an efficient method for obtaining the µ d−2k . The null energy condition will then allow us to establish new inequalities of the form (1.8). We will obtain holographic expressions for µ d−4 and µ d−6 , and will prove (1.8) for k = 1, 2, 3. The result for k = 3 is also conceptually important as it goes beyond current results based on strong subadditivity. It also gives further support for the conjecture (1.8) which, however, we will not be able to prove in general.
The work is structured as follows. In Sec. 2 we describe the holographic setup. In Sec. 3 we derive the Hamilton-Jacobi (HJ) equation for the holographic EE, we discuss properties of the large radius expansion, and we compare this with the linearized holographic RG. Secs. 4 and 5 contain the main results -the expressions for µ d−2 , µ d−4 , µ d−6 and the associated inequalities. Finally, in Sec. 6 we present our conclusions and future directions. Additional explicit calculations that supplement the main text are given in the Appendices, where in particular we explore an alternative method based on solving the minimal surface equation and matching UV and IR approximations.

Holographic setup
In this work we consider QFTs that are described by a UV CFT at very short distances, and a different IR CFT at long distances. The two fixed points are connected by an RG flow. For concreteness, we can think that this RG flow is triggered by perturbing the UV CFT by a single relevant deformation, where O is a primary operator of dimension ∆ < d. The relevant coupling g defines a mass scale typical for the RG, m ∼ g 1/(d−∆) . Multiple relevant operators can also be added, and this will not change our analysis. We will probe the properties of the RG flow using the entanglement entropy for a spherical region of radius R. We will focus on QFTs that admit gravity duals, where the EE is calculated by the Ryu-Takayanagi (RT) formula [17][18][19]. Let us briefly describe this next.

Gravity dual
The holographic dual is taken as Einstein-Hilbert gravity coupled to a scalar field φ dual to the relevant operator O: Additional scalars are straightforward to include. In order to represent an arbitrary RG flow between two fixed points, the potential should have two critical points at φ c such that , where c = UV, IR (see Figure 1). If φ = φ c , the gravity solution is AdS d+1 with radius c . We require that the matter sector satisfies the NEC. This energy condition  has been used previously in the holographic proof of the irreversibility of the universal term in arbitrary dimensions [6][7][8]. Moreover it is the weakest of the most common local conditions [20].
The solution to the scalar wave equation in AdS has two decay modes near the asymptotic boundary: a source-type term, and the VEV-type term [21]. The relevant deformation (2.1) is dual to turning on a source in the asymptotic boundary. This perturbs φ away from φ U V , which will then evolve radially towards φ IR . The background that describes this (preserving Poincare symmetry in the boundary) can be parametrized as To proceed, we will only need the following properties. First, note that when f (z) = 1 for all z, this geometry recovers the AdS d+1 spacetime with radius UV . For arbitrary f (z) we only require that f (z) → 1 near the asymptotic boundary z → 0. On the other hand, at large z, f → ( U V / IR ) 2 . Also, the NEC for the matter sector translates on the equations of motions to the monotonicity requirement f (z) > 0. So in summary, the main properties of the scale factor are with f (z) = 0 for AdS d+1 spacetimes. See Appendix A for more details.

Holographic Entanglement Entropy
For an entangling region Σ in a constant time slice, the holographic EE in asymptotically AdS spacetimes is given by the Ryu-Takayanagi prescription [17,18] where σ is a codimension-2 hypersurface in the bulk, anchored on ∂Σ and which minimizes the area; G (d+1) N is the (d + 1)-dimensional Newton constant. This result was established in [19]. To calculate the area of the minimal surface σ we exploit the fact that the bulk metric (2.3) is diagonal and that the boundary is time independent and spherically symmetric. The minimal surface is a (d − 1)-hypersurface parametrized by d − 2 angular variables Ω i and a radial function ρ(z). See Figure 2 for a schematic representation. Thus, S is given by where is a geometric cutoff that regulates the entropy, and z t is a bulk radial turning point. 2 The turning point z t arises at ρ = 0 as a consequence of spherical symmetry. The profile described by ρ(z) has the boundary condition The turning point obeys Note also that when the size of the entangling region R → 0, the minimal surface collapses to a point, and this gives zero area, We must evaluate S on the profile ρ(z) that minimizes the area. This is the solution for ρ(z) given by the Euler-Lagrange equation for (2.6), This is a non-linear second order ODE in z for ρ(z). It cannot be solved in closed form for a general metric f (z) except for d = 2.
At large R, (2.10) can be solved approximately in the UV and IR regions, and then both expressions can be matched in some overlapping regime [9,10]. We will discuss this in Appendix B . But we will find it more convenient to apply the Hamilton-Jacobi method at large R, and we turn to this next.

Hamilton-Jacobi approach
The Hamilton-Jacobi method for a mechanical degree of freedom q(t) gives an equation for its on-shell action S as a function of initial data q(t 0 ) = q 0 , with H the Hamiltonian. It appears naturally in holography with the radius playing the role of time, and the on-shell gravity action giving the large N partition function of the dual field theory. In this context, the HJ formulation is very convenient as it gives directly an equation for the on-shell action in terms of boundary data, without having to go through solving the bulk equations of motion. It has been applied extensively to the holographic RG, starting from [23], to Wilson loops [24] and to the entanglement entropy [25,26], among others.

Derivation
We begin by briefly reviewing how to obtain (3.1) for the holographic entropy (2.6), explaining some subtle points specific to our case. The solution to the equation of motion (2.10) gives a function ρ(z; , R) that depends explicitly on the boundary data and R. Similarly, the on-shell entropy is a function S = S( , R). In general, the derivatives ∂ z ρ and ∂ ρ are different. However, at z = , we have ρ(z = ; , R) = R. Since R is independent of , the total derivative dρ/d = 0 at z = , and hence ∂ρ ∂ This relation is needed for the following derivation.
To proceed, we compute the variation ∂ S( , R). This varies the endpoints as well as the integrand in (2.6), which on-shell depends on . There is no contribution from varying the endpoint z t , because the integrand vanishes at z = z t . On the other hand, varying the integrand, imposing the equation of motion and integrating by parts, gives a purely boundary term. Using here (3.2) and combining with the contribution from the endpoint at We recognize the right hand side as minus the Hamiltonian H = L − ∂L ∂ρ ρ at z = . To complete the derivation, we need the relation between the momentum conjugate to ρ(z) and ∂ R S. A calculation very similar to the one just described gives Using this equation to write ∂ z ρ in terms of ∂ R S and replacing into (3.3), we arrive at This is the desired HJ equation for the holographic entanglement entropy. It has to be solved in the domain ∈ (0, ∞), R ∈ (0, ∞) and, recalling (2.9), the boundary condition is We will use it to solve directly for S = S( , R) without recourse to the equations of motion. A conceptually important point is that, while originally was introduced as a very small and fixed geometric cutoff, in the HJ equation is varied in 0 < < ∞. This is very similar to what happens with the running Wilsonian cutoff, and is at the basis of the connection between the HJ equation and the holographic RG [23].
We will also use an equivalent form of (3.5), This is an eikonal equation, and it would be interesting to explore potential connections with the bit thread formulation [27], something that we leave for future work. 3

Exact solution for pure AdS
It is in general not possible to solve explicitly the HJ equation (3.5) for arbitrary f (z), but we will shortly see that it can be easily solved in a large R expansion. Before turning to that, however, we will focus on the case of pure AdS d+1 , f (z) = 1 for all z, where an exact solution can be obtained. We solve the equation by proposing that the entropy depends on the dimensionless combination y( , R) = R 2 + 1, S( , R) = S(y( , R)). 4 For R = 0 there is no entanglement entropy in this new variable so the boundary condition (3.6) is written as Now the new equation in one variable is integrable Using the condition (3.8) we get the final result The result is the incomplete beta function 5 (3.11) The expansion in powers of R gives We recognize the structure (1.1), with This particular combination is actually z t / and is motivated by the solution using the Euler-Lagrange equation in Appendix B, where it is clear that the surface that minimizes the area is a cut of a sphere of radius In particular, the logarithmic universal term is recovered for even d = 2k recalling that and It is worth noting that these coefficients are different from the ones obtained in the original calculation [18] -except for the universal terms that are identical. This is due to a different radial cutoff: in [18], the integral starts at z = , but the boundary condition is taken as ρ(z = 0) = R. In our case, we instead impose ρ(z = ) = R. One could view simply as a regulator, always taking → 0, and then we have just two different regularization schemes. However, in the HJ approach it is important that we vary over all scales in order to solve the HJ equation. So it is not necessarily the smallest scale in the problem. In fact, the EE with the cutoff procedure of [18] does not satisfy the HJ equation (3.7). We instead prefer to view the introduction of the Dirichlet wall at z = as a physical deformation of the boundary theory, akin to the T T deformation [22].

Large R expansion
We will now use the HJ equation to solve for the entropy in the large R expansion. Let us first analyze some properties of this expansion.
The RG flow introduces some characteristic mass scale m ∼ g 1/(d−∆) (see (2.1)), which in the gravity side translates into the value of z for which f (z) starts to differ appreciably from the UV value f (0) = 1. At long distances, the theory flows to an IR fixed point; in gravity language, f (z → ∞) ∼ 2 U V / 2 IR . This regime is probed by the EE with radius R 1/m. As reviewed in Sec. 1, near a fixed point the EE admits a geometric expansion (3. 16) In the semiclassical bulk description, the leading contribution to the EE is proportional to γ d , as in (2.6). 6 To simplify the following formulas, we have made γ d explicit here in front of the large R expansion. The coefficients µ d−2k here then differ from their QFT counterparts (1.1) by this factor of γ d . The expansion is valid for positive integer powers of R. The terms that decay with R (the '. . .' in the last line) generically have non-integer powers; these are related to the specific dimensions of the leading irrelevant operators that control the approach to the IR fixed point. We will not consider such terms in this work; Refs. [9,10] studied them in holographic theories.
The entropy is dimensionless, so µ d−2k has dimensions of 1/(length) d−2k to compensate. On dimensional grounds then, for a dimensionless functionμ d−2k These expressions are not known in general; however, their parametric dependence may be understood near a fixed point using conformal perturbation theory. In particular, near the UV fixed point g 1/(d−∆) we expect, up to order one constants, where we used that the leading perturbation arises at order g 2 OO [29]. The first term is an ultraviolet divergence coming from the UV CFT. The simplest example is the divergent area term, but such divergences also afflict other terms in the geometric large R expansion. The second term, on the other hand, is a consequence of the relevant perturbation. Recalling that ∆ < d, this term will diverge for → 0 in the window These divergences are familiar from renormalizable interactions in QFT. They may also be understood from the gravitational action induced by the QFT. We will instead restrict to so that there are no UV divergences associated to the relevant perturbation. While the µ's are cutoff dependent, the difference ∆µ d−2k is finite as → 0, Therefore, by comparing the large radius expansion of the UV fixed point and of the theory with nontrivial RG flow we obtain physical (cut-off independent) entropy coefficients ∆µ d−2k .
They are a property of the continuum theory. 7 The inequalities below will refer to these coefficients.
Let us now obtain the differential equations for the coefficients µ d−2k . For this, we replace (3.16) into (3.5) and set to zero the coefficient multiplying each independent power R d−2k .
Denoting → z here, this gives linear differential equations µ d−2k (z) in terms of lower µ's. In particular, the first three are Therefore, the HJ equation in the large R expansion gives first order equations that can be solved iteratively. The area coefficient µ d−2 is sourced by the inhomogeneous area term in the HJ equation, and the higher order coefficients are in turn sourced by a nontrivial µ d−2 . In Secs. 4 and 5 we will solve these equations and establish inequalities for the resulting solutions.

Linearized approximation
The Wilsonian RG can often be seen as an infinitesimal step in the exact RG [30][31][32] and holographic RG equations [23,28,33]. It is then interesting to consider a linearized version of the EE HJ equation in order to derive a flow interpretation. The connection with the RG was stressed in [25].
With this aim, let us consider the first step in the large radius expansion, writing where S area cancels the inhomogenous right hand term in (3.7), (3.24) The HJ equation forŜ then reads This is still the complete HJ equation now forŜ; it contains both linear and quadratic terms, and shows that S area acts like a source term forŜ. The idea now is to neglect the terms that are quadratic in derivatives ofŜ, We will check the validity of this approximation shortly. This equation may be interpreted as a geometric flow equation for the EE, relating the change in cutoff to the geometric deformation ∂ RŜ and the area term.
The linear differential equation can be solved by the method of characteristics. The general solution iŝ where G(η) is some arbitrary function we should fix with boundary conditions, and 0 comes from a choice of integration constant in the method. In order to assess the validity of the linearized approximation, we expandŜ at large R in terms of the µ d−2p , and plug into (3.26). This gives ODEs term by term in the 1/R expansion; the first are (3.28) As opposed to the full nonlinear ODEs for the µ d−2p , these keep only terms that are linear in (µ d−4 , µ d−6 , . . .). Relating (3.22) and (3.28), we find that µ d−4 is captured completely by the linearized approximation, while from the order µ d−6 the linearized approximation fails. The quadratic terms that are neglected (3.26) turn out to be comparable to (3.28) beginning at O(R d−6 ), and for this reason the linearized approximation is not useful for such non-universal terms. The equations (3.22) are still linear differential equations in the unknowns, and hence resemble RG beta functions for the entropy coefficients. But getting these right requires taking into account more nonlinear terms beyond the leading area contribution.

Sphere free energy
The large R expansion (3.16) must be modified for odd space-time dimensions d and has to include a term F ( ), therefore F is a constant. In a CFT this is the constant term of the free energy of a d dimensional euclidean sphere [34,35].
The term F can be fully determined at fixed points in our holographic context from the expression (3.11) for the entropy when f (z) is constant. More precisely, are the constant terms in the UV and IR expansions. This gives the inequality 8 because of UV ≥ IR as a consequence of the NEC (2.4) and the positivity of the prefactor for d odd. This F-theorem was obtained by [7,8] in an holographic analysis for all d odd and by [3,5] using QFT methods involving the SSA in d = 3. At the moment there is no QFT proof for F-theorems in d > 3 spacetime dimensions.

Analysis of the R d−2 and R d−terms
In this section we calculate the µ d−2 (z) and µ d−4 (z) terms in the large R expansion of the EE. The area term has been calculated for holographic theories in [36] using a different method based on stress-tensor correlators; we will reproduce their result. The holographic result for µ d−4 (z) has not appeared in the literature, as far as we are aware. For each coefficient we give expressions in terms of a single integral involving f (z) that allow to prove the inequalities ∆µ d−2 ≤ 0 and ∆µ d− using the NEC (2.4). We will also rewrite these coefficients in a way that makes manifest the anomaly result when d → 2 and d → 4 respectively. This "anomaly oriented" procedure will also be useful for our analysis of ∆µ d−6 ≤ 0 in Sec.5.

∆µ d−2 (z)
The equation for µ d−2 (z) is The integration constant should be independent of R, and z; moreover for z = → 0, this must reproduce the UV CFT expansion 13). Therefore, this integrates to The holographic value for F matches with the universal term a * d = π defined in [7,8] via and recalling that due to the Euler's reflection formula.
Subtracting the contribution of pure AdS, f (v) = 1, gives the inequality for the area term because f (z) ≥ 1 for all z. The decrease of the area term coefficient along RG flows has been proved in QFT in [13,15]. The holographic calculation using stress-tensor correlators was carried out in [36]. Our HJ method is extremely simple and gives the same result. 9 In the limit d → 2, (4.3) should reproduce the holographic C-theorem, and so it should only depend on UV and IR CFT quantities and not on the full RG flow. To exhibit this, we integrate by parts the 1/v d−1 factor, where we define For future use, note that since µ d−2 (z) > 0 for all z, Finally, taking the limit z = → 0 and recalling that f ( ) → 1 and g d−2 = 0 for AdS, 10 When d → 2, g d−2 becomes a total derivative and the logarithmic term in ∆S is recovered, giving raise to the weak version of the holographic C-theorem. The central charge is identified via c = 3

2G
(3) N in d = 2 [37]. 9 To match their result, the appropriate change of variables is UV . 10 In general ∆µ d−2k has potentially the most divergent contributions coming from terms that goe like

∆µ d−4 (z)
After the previous warm-up, we now move to the R d−4 coefficient, for which there was no known holographic expression. Starting from the equation, the solution integrates to (4.10) The expression (4.10) satisfies µ d−4 < 0 for all z. Therefore, the NEC gives the following inequality, Indeed, the AdS contribution is explicitly larger than the one corresponding to the holographic RG flow, since f (z) > 1 is a monotonically increasing function.
Ref. [5] proved ∆µ d−4 ≤ 0 in QFT using the strong subadditivity for multiple boosted spheres. Here we have obtained the same result for theories with gravity duals and the NEC, in the large R expansion. It would be interesting to relate (4.10) to euclidean boundary correlators. This would help towards an euclidean proof of the A-theorem, with a corresponding sum rule.
Following what we did before for µ d−2 , let us rewrite (4.11) in order to make the holographic A-theorem manifest when d → 4. From (4.4), we have (4.12) Replacing into (4.9) and integrating, Finally, integrating by parts with respect to 1/v d−3 in the first term, and defining we arrive at Note that from the integration by parts, we obtain the bound (4.17) Finally setting z = → 0 and subtracting the pure AdS result, we obtain the desired inequality The last term is positive semidefinite because of (4.6). It contributes by a finite amount to the entropy and depends on the full RG flow. In the window (3.20) for k = 2, this is finite for → 0 and hence we have a physical cut-off independent coefficient.
When d → 4, g d−4 becomes a total derivative and the logarithmic term in ∆S is recovered, In this section we give an integral expression of µ d−6 (z) in terms of f (z) and provide a proof of the inequality ∆µ d−6 ≤ 0 following an anomaly oriented procedure. This is the first term in the large R expansion whose RG flow is not constrained by the boosted strong subadditivity. Surpisingly, we find that the NEC in the large R expansion is sufficient to establish the inequality.
For the present case the equation is Unlike what happens with the coefficients in Sec. 4, this is the first time there is a competition between two terms of opposite sign (because of (4.2) and (4.10)). Using (4.4) and (4.15), expanding the two terms in (5.1), and integrating, .
Adding up the first two terms arising from both integrals gives and because of (5.3) Finally, we arrive at .
Upon setting z = → 0, and subtracting the pure AdS contribution, we find .
Given (3.20), this is finite when → 0. From to the inequalities (4.6) and (4.17), it satisfies the inequality ∆µ d−6 ≤ 0 . (5.8) Therefore, we have found that the NEC implies the decrease of the R d−6 term along holographic RG flows. Finally, when d → 6 the function g d−6 becomes a total derivative and the logarithmic term is recovered It would be interesting to translate (5.7) into stress tensor correlators. This expression already has the desired features of satisfying the inequality (5.8) and reproducing a sum rule for d → 6. So it may provide hints towards an A-theorem for d = 6.

Conclusions and future directions
In this work we analyzed the coefficients ∆µ d−2k in the large radius expansion (1.1) of the EE for field theories with gravity duals. These coefficients are finite for RG flows triggered by operators with dimension ∆ < d+2 2 . Unlike the universal A and F terms, the ∆µ d−2k contain information about the full RG flow connecting the UV and IR fixed points. Starting from the Ryu-Takayanagi formula, we derived a Hamilton-Jacobi equation (3.7) for the holographic EE S( , R). The cut off modifies the EE not just by cutting the integral, but also modifying the minimal surfaces, allowing to take any value. At large radius, the HJ equation reduces to first order differential equations for the µ d−2k , which are straightforward to solve. We derived explicit holographic results for µ d−2 , µ d−4 , µ d−6 and established the inequalities The first two inequalities have been proved before for general QFTs using the boosted strong subadditivity [5] (SSA). On the other hand, the inequality ∆µ d−6 ≤ 0 is new and goes beyond the SSA.
Let us end by discussing future directions opened by these results. We have obtained integral expressions for ∆µ d−4 and ∆µ d−6 as a function of the metric scale factor. These expressions are a starting point for deriving expressions in terms of euclidean stress-tensor correlators. This would be interesting for different reasons. It could suggest a sum rule for the 4d A-theorem, generalizing the sum rule of d = 2 (which uses T µ µ (x)T ν ν (0) ) to d = 4. The d = 6 result in terms of stress-tensor correlators, on the other hand, may shed light on a possible A-theorem for that dimensionality.
Another important lesson is that the NEC together with a large radius expansion give rise to the inequality ∆µ d−6 ≤ 0 that is stronger than current results using SSA. It would be very useful to translate this into quantum information conditions. It suggests thinking in terms of a large radius expansion in field theory.
Finally, it would be important to push the holographic analysis to the higher coefficients ∆µ d−2k . Our results support the conjecture and it would be nice to prove this for theories with gravity duals. The analysis for ∆µ d−8 and higher coefficients appears to be significantly more involved, possibly requiring new tools so that it can be made systematic.

A Gravity dual equations of motion
The action for an arbitrary minimally coupled scalar field φ(z) in the bulk is (A.1) Here we consider the Einstein-Hilbert action with the standard Gibbons-Hawking boundary term and some arbitrary potential V (φ) for the scalar field. The equations of motion are If the potential has a critical point dV It satisfies the boundary conditions if The minimal surface is a hemisphere centered at z = ρ = 0, with radius z U V t . This is the radius at z = 0, which is different than the size R of the entangling region at z = .
Similarly, in the IR theory, f (z) = 2 U V / 2 IR and the solution reads In this case the boundary conditions imply that These solutions are valid for all d.

B.1.2 Two dimensional case and c-theorem
When d = 2 the ρ(z) dependence disappears in the equation (B.1) leaving an equation involving just ρ (z) and ρ (z) . Therefore the variational problem contains a conserved quantity and the solution can be written as The turning point is fixed by As a consequence of the NEC (2.4), f (z) ≥ 1, so for an arbitrary z t R ≤ zt dz 1 we have z t ≤ z IR t . The combined inequalities give The first derivative of S with respect to R can be written as an expression in terms ρ(z) evaluated at the cut-off , see (3.4). In particular for d = 2, This defines a running c-function by comparing (B.11) between the UV fixed point and the holographic flow, (B.12) In the large R limit, z t ≈ U V IR R and the expression extracts the universal coefficient in d

B.2 Larger dimensions
For dimensions d > 2 the variational problem associated to (B.1) does not have a conserved quantity, and the equation of motion cannot be solved analytically for arbitrary f (z). Instead, we will follow a matching procedure similar to what was originally done by [9,10]. The idea is to write two different expansions for ρ(z). The first expansion is written in powers of R in the large R limit, where each coefficient depends on z (Section B.2.1). This expansion is valid near z = 0 but it is not suitable near the turning point z t , so all coefficients in the expansion have an unknown parameter. The second expansion is written in powers of the IR mass scaleμ. We refer to this as the large z expansion (Section B.2.2). There we can impose the turning point boundary condition but not ρ( ) = R. In the limit where R and z are both large, both expansions overlap and this allows to fix the unknown parameters (Section B.2.3). In Fig. 3, we depict a schematic profile for ρ(z) including all the scales present, and the overlap range. 0 . The UV is defined for < z < µ −1 , where is an UV cut-off and µ an UV energy scale; the IR is defined forμ < z where,μ is an IR energy scale. The matching procedure is done when z is betweenμ −1 < z < R, where R is the sphere's radius of the spatial region at the boundary.
for some coefficients d 2k . Then (B.33) suggests that the b n coefficients must be identified by looking at the terms ∼ z d /R n in the large z expansion of ρ(z) (B.21). Since σ 1 (z) is of order (μz) −2α only the square root term might contribute to b n . Its expansion for large z is To make computations shorter, we work with non-integer dimensions d, so we can effectively consider b n = 0 for all n. Taking the limit to the integer dimensions recovers the correct result.

B.3 Entropy expansion
Having obtained the approximate solution ρ(z) at large R, we are ready to evaluate the entropy. To make more explicit the R dependence of the entropy using the Euler-Lagrange approach, we rewrite (2.6) as . (B.36) Both terms ρ(z) R and ρ (z) contain only inverse odd powers of R, thus the large R limit of the integrand reproduces the structure (1.1). The upper limit of integration z t is also Rdependent, and this only contributes to finite terms because at first order z t ∼ R. So in order to obtain the µ d−2k it is sufficient to take z t → ∞.
The first term from this expansion reproduces the area law and coincides with the result of the HJ formalism (4.2) when z t → ∞: The second term depends on ρ 1 (z), so it needs to be evaluated using the solution (B.16) with b 1 = 0. In particular, we have