Canonical energy is quantum Fisher information

Journal of High Energy Physics, Apr 2016

In quantum information theory, Fisher Information is a natural metric on the space of perturbations to a density matrix, defined by calculating the relative entropy with the unperturbed state at quadratic order in perturbations. In gravitational physics, Canonical Energy defines a natural metric on the space of perturbations to spacetimes with a Killing horizon. In this paper, we show that the Fisher information metric for perturbations to the vacuum density matrix of a ball-shaped region B in a holographic CFT is dual to the canonical energy metric for perturbations to a corresponding Rindler wedge R B of Anti-de-Sitter space. Positivity of relative entropy at second order implies that the Fisher information metric is positive definite. Thus, for physical perturbations to anti-de-Sitter spacetime, the canonical energy associated to any Rindler wedge must be positive. This second-order constraint on the metric extends the first order result from relative entropy positivity that physical perturbations must satisfy the linearized Einstein’s equations.

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Canonical energy is quantum Fisher information

HJE Canonical energy is quantum Fisher information Nima Lashkari 0 1 2 3 Mark Van Raamsdonk 0 1 2 0 6224 Agricultural Road, Vancouver , B.C., V6T 1W9 , Canada 1 77 Massachusetts Avenue , Cambridge, MA 02139 , U.S.A 2 Department of Physics and Astronomy, University of British Columbia 3 Center for Theoretical Physics, Massachusetts Institute of Technology In quantum information theory, Fisher Information is a natural metric on the space of perturbations to a density matrix, de ned by calculating the relative entropy with the unperturbed state at quadratic order in perturbations. In gravitational physics, Canonical Energy de nes a natural metric on the space of perturbations to spacetimes with a Killing horizon. In this paper, we show that the Fisher information metric for perturbations to the vacuum density matrix of a ball-shaped region B in a holographic CFT is dual to the canonical energy metric for perturbations to a corresponding Rindler wedge RB of Anti-de-Sitter space. Positivity of relative entropy at second order implies that the Fisher information metric is positive de nite. Thus, for physical perturbations to anti-de-Sitter spacetime, the canonical energy associated to any Rindler wedge must be positive. This second-order constraint on the metric extends the rst order result from relative entropy positivity that physical perturbations must satisfy the linearized Einstein's equations. AdS-CFT Correspondence; Gauge-gravity correspondence - 3 Constraints on spacetime geometry from relative entropy inequalities Relative entropy and Fisher information 2.1.1 Relative entropy in conformal eld theories Relative entropy in holographic conformal eld theories A fundamental identity Bulk integral for relative entropy First order results Second order results: the gravity dual of Fisher information Gravitational constraints from positivity of Fisher information Transformation to Hollands-Wald gauge Calculating canonical energy from h and V Example: perturbations to Poincare AdS3 1 Introduction 2 Background 2.1 2.2 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4 Examples 5 Discussion 1 Introduction In the AdS/CFT correspondence [ 1 ], the holographic entanglement entropy formula [2, 3] relates the entanglement structure of the CFT with the geometrical structure of the dual spacetime. On the CFT side, the entanglement structure obeys fundamental consistency constraints such as the strong subadditivity of entanglement entropy and the positivity and monotonicity of relative entropy.1 These translate to geometrical constraints that must be satis ed for geometries dual to consistent CFT states [5{9]. To leading order in perturbations away from the vacuum state, these constraints (speci cally the positivity of relative entropy) translate to the statement that the dual geometry must satisfy Einstein's equations to linear order in perturbations around AdS [10{12] (see also [13]). In this paper, we extend this work to give a complete characterization of the positivity of relative entropy constraints to second order in perturbations to the vacuum. We have a constraint for each ball-shaped region B in the CFT; these constraints imply the positivity of \canonical energy," a quantity quadratic in the metric perturbations to a Rindler wedge region RB associated with B. The results in this paper make use of an important identity in classical 1For a review, see for example [4]. { 1 { theories of gravity relating the gravity dual of relative entropy to the natural symplectic form on the space of perturbations to a metric [14]. We now present a concise summary of the background and results before giving an outline of the remainder of the paper. Fisher information in conformal eld theory. Consider a one-parameter family of states j ( )i of a CFT on Rd 1;1 with j (0)i the vacuum state. For any ball-shaped region B, de ne B( ) as the reduced density matrix for this region. We have that B(0) = e HB 1 Z where HB (the modular Hamiltonian for the subsystem B in the vacuum state) is the generator of a conformal Killing vector B acting in the causal diamond region DB associated with B, as shown in gure 1.2 For a ball of radius R, we have [15] where r is the distance to the center of the ball. For any state j ( )i we de ne as the di erence in entanglement entropy compared with the vacuum state. We also de ne EB = tr(HB B( )) tr(HB B(0)) as the di erence in the expectation value of the modular Hamiltonian. Both SB and EB are nite for well-behaved states. Positivity of relative entropy (reviewed below) gives the fundamental constraint that [6] Since = 0 represents a minimum for any family of perturbations, we must have (1.2) (1.3) (1.4) (1.5) (1.1) d d d 2 This can be promoted to a metric on perturbations Bj =0 to the unperturbed state. 2Explicitly, we have for the ball of radius R centered at (t0; xi0). h ; i (0) + ; + i h ; i h ; i) : B = R h2(t t0)(xi B ~ RB is the intersection of the causal past and the causal future of the domain of dependence DB (boundary diamond). Solid blue paths indicate the boundary ow associated with HB and the conformal Killing vector . Dashed red paths indicate the action of the Killing vector . The second order statement of positivity of relative entropy is thus that Fisher Information metric is positive de nite. The Fisher information of perturbations near vacuum in conformal eld theory for ball-shaped regions is known to be related to 2-point function of the theory and universal [16]. Gravity interpretation. Now suppose that the CFT is holographic and that the oneparameter family of states j ( )i have gravity dual geometries M ( ) with M (0) equal to pure AdS. In this unperturbed geometry, the ball shaped-region B can be associated [17] with a Rindler wedge RB de ned as the intersection of the causal past and the causal future of DB, the boundary domain of dependence of B (see gure 1) (see also [18{20]). The boundary of this Rindler wedge is the extremal area surface B~ in the bulk with to the exterior of a hyperbolic Schwarzchild-AdS black hole for which B~ is the horizon.3 The wedge RB has a timelike Killing vector B vanishing on B~ that extends B into the bulk and de nes a \Rinder time" for the wedge. For the perturbed asymptotically AdS dual geometry M ( ) we can de ne B~( ) to be points in M~ that are spacelike separated from B~ towards the boundary [19, 21]. Thus, as we deform the CFT state, each wedge RB is deformed to RB( ) that can be viewed as a perturbed hyperbolic black hole. Using the holographic entanglement entropy formula, the CFT quantity SB corresponds to the change in area of B~ as the geometry is varied from M (0) to M ( ). As we review below, there is a natural gravitational energy EBgrav, 3This is related to the eld theory statement that a conformal transformation maps the region DB to hyperbolic space times time, mapping the vacuum density matrix on DB to the T = 1=(2 RH ) thermal state on hyperbolic space with curvature radius RH [15]. { 3 { calculated from the asymptotic metric near B, that can be associated with any RB( ) [22]. The eld theory quantity EB is related to the change in gravitational energy for RB as the geometry is varied from M (0) to M ( ). We can now translate the relative entropy constraint (1.2) to a gravitational statement. For any geometry M ( ) dual to a physical CFT state, we must have [6] Thus, for every ball B, the change in area of the extremal surface B~ is bounded by the change in gravitational energy for the region RB. At rst order, according to (1.3), these changes must be equal, so we have a gravitational rst law EBgrav Sgrav B 0 : EBgrav = Sgrav B governing perturbations of hyperbolic black holes. The recent work of [10] shows that the collection of these rst law statements for all B is equivalent to a single local bulk constraint, that the rst order perturbation satis es the linearized Einstein equation. A powerful method [11] to prove this rst order result makes use of a gravitational identity of Wald and Iyer [22] relating the di erence (1.6) to the integral of a bulk quantity over a surface B bounded by B and B~: d d ( EBgrav Sgrav)j =0 = B E^B( g) : Z B Here, E^B is a form that vanishes when the metric perturbation g satis es the linearized Einstein equations. Since the eld theory result (1.3) implies the vanishing of the left side here, we immediately have that all E^B integrals vanish. It is straightforward to show that this is impossible unless the metric perturbation satis es the linearized Einstein equations. The key technical tool in this paper is a result by Hollands and Wald [14] generalizing the gravitational identity (1.7) away from = 0. The full result takes the form d d ( EBgrav (1.8) Z B where again E^B is a quantity that vanishes when the metrics g( ) are on shell (i.e. satisfy the nonlinear gravitational equations), and W B is another integral over B de ned in terms of a natural symplectic form de ned on the space of perturbations to the metric g( ). The identity (1.8) allows us to rewrite the di erence of boundary integrals de ning the gravity dual of relative entropy (left side of (1.8)) as the integral over a bulk quantity. Specializing to the terms in (1.8) at order (i.e. the derivative of (1.8) at = 0), the result reduces to d 2 where EB( g; g) is a quadratic form on the metric perturbations known as \canonical energy." Essentially, it is the Rindler energy (associated with the Killing vector B) for the { 4 { (1.6) (1.7) Rindler wedge RB, including a gravitational piece (quadratic in the metric perturbation) and a matter contribution:4 E ( ; ) = Z a(Tagbrav + Tambatter)d b : (1.10) The left side of (1.9) is exactly the gravity dual of Fisher information (1.4). So we have that Fisher Information is dual to canonical energy. Consequently, the positivity of Fisher inHere, the \matter" contribution is actually Tambatter = 81 (Gab formation translates to the positivity of the canonical energy EB for each Rindler wedge B. gab), so this is a purely geometrical constraint, but we can rewrite this as the matter stress tensor assuming that Einstein's equations are satis ed. In this case the positivity of (1.10) can be interpreted as an energy condition restricting the behavior of the matter stress tensor in a consistent theory. It is quite natural that Fisher information and canonical energy are related to one another, since each de nes a natural metric on a space of perturbations, in one case to a density matrix, and in the other case to a metric satisfying the gravitational equations. This identi cation provides further evidence that the geometry of spacetime in quantum gravity is fundamentally related to the entanglement structure of the fundamental degrees of freedom. Outline. The remainder of this paper is organized as follows. In section 2, we provide in more detail the background material on relative entropy, quantum Fisher information, and the tools to translate these to dual geometrical quantities in holographic theories. In section 3, we review the fundamental gravitational identity of Hollands and Wald that allows us to translate the gravitational expression dual to relative entropy (which can be expressed as a boundary integral over the surface B B~) to a bulk quantity. We review the de nition of canonical energy and show that this provides the gravity dual of quantum Fisher information. Finally, we express the positivity of Fisher information as an explicit constraint on the dual geometry, showing that it may be written in the form of an energy condition that must be obeyed by the matter stress tensor. In section 4 we provide some example calculations, discussing in general how to calculate canonical energy for an onshell metric perturbation given in a general gauge, and providing some explicit example calculations in AdS3. These calculation give explicit constraints on the second order metric for physical asymptotically AdS3 geometries. We check in particular that the constraints on the asymptotic metric exactly reproduce those calculated previously in [8]. We conclude in section 5 with a discussion. Note added: while this manuscript was in preparation, the paper [23] appeared, which discusses the gravitational interpretation of a di erent type of quantum information metric. The metric discussed there is de ned in terms of the inner product between states rather than the relative entropy between states, and the proposed gravity dual in [23] involves the volume of a spatial slice rather than the canonical energy. Thus, the two papers represent two independent elements in the quantum information / quantum gravity dictionary. 4Here, the gravitational contribution implicitly includes a term involving an integral over B~, as described in section 3. { 5 { In this section, we review in more detail the de nition of relative entropy and its positivity and monotonicity properties, starting from general quantum systems, and then specializing to the case of conformal eld theories. We then recall how the quantities entering into the formula for relative entropy are related to gravitational quantities in the case of holographic CFTs. 2.1 Relative entropy and Fisher information Relative entropy measures the distinguishability of a density matrix from some reference density matrix . It is de ned as S( jj ) = tr( log ) tr( log ) : The relative entropy is always nonnegative, equal to zero for identical states and increasing to in nity if has nonzero probability for a state orthogonal to the subspace of states in the ensemble described by . Further, relative entropy is monotonic: if A represents a subsystem of some quantum system B, and if A and A are the reduced density matrices for the subsystem obtained from B and B, then Detailed proofs of these results may be found in [4]. known explicitly. In this case, de ning the modular Hamiltonian These results are particularly useful when the density matrix for the reference state is S( Ajj A) S( Bjj B) : H = log( ) ; ( ) = 0 + with 0 = . To rst order in , it is straightforward to check that the relative entropy vanishes, a result known as the \ rst law of entanglement," [6] At the second order in , relative entropy is given by5 5Note that the terms involving 2 vanish by the entanglement rst law applied to the perturbation 2 2. { 6 { (2.1) (2.2) d d log( + ) + tr via (1.5). By the positivity of relative entropy, the quantum Fisher information is non-degenerate, non-negative and can be thought of as de ning a Riemannian metric on the space of states.6 Quantum Fisher information plays a central role in quantum state estimation which studies how to determine the density operator ( ) from measurements performed on n copies of the quantum system [24]. 2.1.1 Relative entropy in conformal eld theories In the rest of this paper, we focus on the case where our quantum system is a conformal eld theory on Rd 1;1, our reference state is the CFT vacuum, and our subsystems are the elds in ball-shaped regions. In this case, the modular Hamiltonian corresponding to the reduced density matrix for a ball is [6] This may be obtained most easily by noting that the domain of dependence region of the ball can be mapped by a conformal transformation to a Rindler wedge of Minkowski space. The modular Hamiltonian for this Rindler wedge in the CFT vacuum state is well-known to be the Rindler Hamiltonian (boost generator), and the modular Hamiltonian (2.6) is just the inverse conformal transformation applied to the Rindler Hamiltonian. For a ball B, the relative entropy between the reduced density matrix B in a general state and the vacuum density matrix B is then S( Bjj B) = 2 Z jxj<R d 2R hT0C0FTi SB : Note that while relative entropy is well-de ned for more general regions, it is only for ball-shaped regions that we can give an explicit form of the modular Hamiltonian as the integral of a local operator, and thus only in this case we will be able to translate relative (2.6) (2.7) entropy to a gravitational quantity. S( + k ) S( k + 6Using (2.4) it is straightforward to see that quantum Fisher information is symmetric in its arguments: We now consider the case of holographic conformal eld theories for which the RyuTakayanagi formula [2] and its covariant generalization by Hubeny, Rangamani, and Takayanagi (HRT) [3] holds. That is, we assume that there is a family of states j i and a related family of asymptotically AdS spacetimes M with boundary Rd 1;1 for which the entanglement entropy SA for any region A is proportional to the area of the minimal area extremal surface A~ in M M equivalent to the eld theory region A. The proportionality constant is related to (or can be used to de ne) the gravitational Newton constant GN as coordinates, which takes the form z2 ds2 = 2 `AdS dz2 + dx dx + zd 1 (z; x) where (z; x) has a nite limit as z ! 0. With this assumption, we can compute the relative entropy of a holographic state j i using the dual geometry M . The term S is exactly the di erence in area of the extremal surface A~ in the geometry M compared with the geometry Mjvaci = AdSd+1. To calculate the term hHBi, we can use the fact that the HRT formula implies [11] that the CFT stress tensor expectation value is related to the asymptotic behavior of the metric (2.9) as A useful explicit description of the spacetimes M is the metric in Fe erman-Graham Using this, we have Thus, for holographic states, we have h T i = T grav d`d 3 j j 2R 00(x; z = 0) Egrav S( Bjj vBac) = EB SB = EBgrav Sgrav B where and Mvac. EBgrav is de ned by the boundary integral (2.11) and B Sgrav is de ned via (2.8) as the area di erence for the extremal surface with boundary @B between the geometries M 3 Constraints on spacetime geometry from relative entropy inequalities For a holographic CFT state j i with a gravity dual geometry M , equation (2.12) provides a geometrical interpretation for the relative entropy with the vacuum state for a ball-shaped region B. The positivity of relative entropy thus implies the positivity of EBgrav Sgrav B (2.8) (2.9) (2.10) (2.11) (2.12) for every ball-shaped region B in every Lorentz frame, while monotonicity implies that it must increase if the size of the ball is increased. If one of these constraints fails to hold for some spacetime M this spacetime cannot be related to any consistent state of a holographic CFT. In other words, it is unphysical. Though (2.12) already allows us to write down these constraints explicitly and check them for any geometry, understanding the nature of these constraints in general is di cult in the present form, with relative entropy expressed as a di erence of boundary terms on B and B~. In [10, 11], it was shown that to leading order in perturbations away from pure AdS, the set of nonlocal constraints can be recast as local constraints on the metric, and that these local constraints are precisely Einstein's equations linearized about AdS. We will now see that very similar technology can be used to rewrite the relative entropy constraints more generally, allowing a more straightforward interpretation of their implications. 3.1 A fundamental identity To proceed, we will make use of a fundamental gravitational identity described recently by Hollands and Wald [14]. Consider a one-parameter family of metrics gab( ), and an arbitrary vector eld Xa. Consider also a general gravitational Lagrangian L (not necessarily the actual Lagrangian for our physical system). Then the identity takes the form !L(g; dg=d ; LX g) + E^ L(g; dg=d ) = d L(g; dg=d ) (3.1) where !L(g; h1; h2) is a d-form whose integral over a Cauchy surface de nes a natural symplectic form on the space of perturbations to a metric for the theory with Lagrangian L, E^ L is a d-form that vanishes if the equations of motion associated with L are satis ed for g( ), and L is a (d 1) form whose integral over the boundary regions B and the associated bulk extremal surface B~ can be related respectively to a gravitational energy E and entropy S associated with L. This identity will allow us to rewrite the (d 1)dimensional integrals on B and B~ de ning Egrav Sgrav in terms of a d-dimensional bulk integral on a bulk spacelike surface bounded by B B~. To de ne the quantities appearing in the fundamental identity, consider the Lagrangian L expressed as a (d + 1)-form, where is the volume form For later use, we also de ne the lower-dimensional forms Under a variation of elds this Lagrangian form varies as = 1 (d + 1)! g a1 ad+1 dxa1 ^ ^ dxad+1 : c1:::ck = 1 (d k + 1)! g c1:::ckak+1 ad+1 dxak+1 ^ ^ dxad+1 : p p L = L : { 9 { Current conservation implies that this form can be expressed as a total derivative plus a term that vanishes when the equations of motion are satis ed, In terms of these quantities, we have g; g = QX (g) iX g; g : (3.5) Finally, the term in (3.1) involving the equations of motion is de ned to be ^ EL g; g = iX E(g) All of these quantities depend on which gravitational Lagrangian we choose. For the case of pure Einstein gravity with a cosmological constant, we have [11, 14] where Eg = 0 give the equations of motion for the elds, and is a boundary term typically called the symplectic potential current form. In this expression, g is taken to represent both the metric and any other elds appearing in the Lagrangian L. The term involving is the total derivative term that is produced by integration by parts when deriving the action. The form ! in (3.1) is de ned in terms of by This \symplectic current form" plays an important role in the covariant phase space formulation of the theory. If restricted to on-shell perturbations, it is closed and non-degenerate, and is used to de ne a natural symplectic form on the space of perturbations around a classical solution g,7 To de ne the form appearing in (3.1), we consider the Noether current associated to di eomorphisms generated by the vector eld X. Expressed as a di erential form, this is (3.3) (3.4) 7As described in [14], it is possible to introduce canonically conjugate variables so that the symplectic form becomes simply W (g; 1g; 2g) = 16 1 Z ph[ 1hab 2pab 1hab 2pab] : ! = 16 raXb ab a(gacgbd P abcdef = gaegfbgcd gadgbc)rd d d gbc aP abcdef ( b2crd e1f 2 1 gadgbegfc 2 Using (3.5) and the equations above, we nd that ( ; X) = 1 16 ab ac rcXb 1 2 ccraXb + r 2 1 gbcgaegfd + 1 gbcgadgef 2 (3.6) b acXc rc acXb + r a ccXb : (3.7) Using the fundamental identity (3.1) we now show that the rst derivative @ S(g( )jjg0) can be written as an integral over a spacelike surface bounded by B and B~. This was argued in general in [11]. Next, we have that This follows by the vanishing of on B~, which gives Z Z B Bulk integral for relative entropy We now consider a one-parameter family of asymptotically AdS spacetimes M ( ), and the family of extremal surfaces B~( ) associated with some xed ball-shaped boundary region B. In [14], it was shown that it is always possible to choose metrics g( ) such that the extremal surface B~( ) has a xed coordinate location, and such that the Killing vector Ba de ned in section 1 continues to satisfy ( B)jB~ = (ra( B)b + rb( B)a)jB~ = 0 : That is, continues to behave as a Killing vector near the extremal surface B~. Consider the gravitational expression for relative entropy evaluated for this family of and the result This holds in the unperturbed spacetime since Q is the Noether charge associated with the Killing vector B, which de nes the Wald entropy of the bifurcate Killing horizon B~. As shown in [14], this continues to hold in the perturbed spacetime because of the gauge condition (3.8). jB~ = Q jB~ ; Z B Q = A : 1 4 (3.11) HJEP04(216)53 Combining these results, we have that d d S(g( )jjg0) = (Egrav(g( )) Sgrav(g( ))) = = = d d Z Z Z d + Z Z B Finally, using the identity (3.1), we obtain d d S(g( )jjg0) = W g; d d g; L g d d g d d iX E(g) where the last line makes use of the identity (3.1). This is the fundamental relation that we will make use of below when translating constraints on relative entropy to constraints on geometry. Primarily, we will make use of this identity for the case where the Lagrangian is chosen to be that for pure Einstein gravity with cosmological constant, so that all quantities in the expression above are purely gravitational quantities. However, we can alternatively choose to consider the case where the various quantities are de ned with respect to the Lagrangian for Einstein gravity coupled to matter. In this case, assuming that curvature tensors do not appear in the matter part of the Lagrangian, the results (3.9) and (3.10) remain valid, so the expression (3.12) is also correct when W , E, and C are constructed starting from the full Lagrangian including matter. In this case, the terms involving E(g) and CX (g) vanish on shell, since these are built from the tensors appearing in the full equations of motion. Thus, we have that W full g; d d g; L g = W g; g; L g d d + Z iX E(g) d d g d d where the expressions on the right are purely gravitational. First order results We rst consider the result (3.12) evaluated at = 0. Since is a Killing vector of the unperturbed metric, we have L g = 0 so the term W (g; dd g; L g) vanishes. Also, the unperturbed AdS metric satis es the vacuum Einstein equations, so the term iX (E(g) dd g) also vanishes. Thus, using (3.6), we have d d S(g( )jjg0)j =0 = CX (g) = Z d d 2 Z a dEagb b d (3.14) where Eagb are the gravitational equations. Positivity of relative entropy in the CFT implies that the relative entropy is minimized for the vacuum state, so the rst order variation must vanish. Gravitationally, this implies that the left side of (3.14) must vanish, so we have that Z d a dEagb b = 0 : As shown in [11], if this holds for all regions associated with any ball B in any Lorentz frame, we must have that dEagb=d = 0, that is, the metric g( ) must satisfy the Einstein equation to rst order in . Thus, for spacetimes M ( ) which geometrically encode the entanglement entropies of CFT states via the HRT formula, the constraints of relative entropy positivity at rst order in are precisely the linearized gravitational equations. Second order results: the gravity dual of Fisher information Next, consider the derivative of the result (3.12) evaluated at = 0. De ning = dg=d j =0 L g = E(g) = d=d (E(g)) = 0) as the rst order metric perturbation we nd (using Consider rst the case where we have a holographic CFT dual to some known theory of Einstein gravity coupled to matter, and where the quantities Eg and W are de ned with respect to the full Lagrangian. Then Eg represent the full equations of motion for the theory, which should vanish for the one-parameter family of eld con gurations g( ) dual to holographic CFT states j ( )i. Thus, we have simply: d 2 d 2 S(g( )jjg0)j =0 = W (g; ; L ) The left side is precisely the gravitational dual of the Fisher Information h ; i, while the right side was de ned in [14] as the canonical energy E ( g; g) W (g; ; L ) : h B; Bi = EB( g; g) : Thus, for holographic CFTs in the classical limit, we have that Fisher Information is dual to canonical energy, More generally, we can promote EB to a bilinear form on perturbations, EB( g1; g2) W (g; g1; L g2) ; which can be shown to be symmetric. This quantity is dual to the Fisher Information metric de ned above, h ( B)1; ( B)2i = EB( g1; g2) : Since the Fisher information and the Fisher information metric must be non-negative, it must be that the corresponding gravitational quantities are also non-negative. Thus, the positivity of relative entropy at second order implies the positivity of canonical energy. Speci cally, for any one parameter family g( ) of physical asymptotically AdS spacetimes, and for any ball-shaped region B on the boundary, we must have EB( g; g) > 0. It should be possible to demonstrate this directly in speci c consistent classical theories of gravity. (3.15) (3.16) (3.17) In general, the expression for canonical energy de ned in the previous section depends on both the metric and the matter elds for the theory. However, using the result (3.13), it is always possible to rewrite it (using the equations of motion) as a purely gravitational expression. De ning the gravitational part of canonical energy E grav( ; ) = W grav(g; ; L ) ; we nd from (3.13) that E ( ; ) = EB grav( ; ) 2 Z a 0 (3.18) HJEP04(216)53 section 4 below. Eagb in (3.6)) This gives This gives a purely geometrical constraint on asymptotically AdS spacetimes that can arise in consistent theories for which the HRT formula holds (expected to be theories with Einstein gravity coupled to matter in the classical limit). Note that EB is calculated using only the rst order perturbation = dg=d j =0, which must solve the Einstein equations linearized about AdS. The second term involves also the metric at second order. Thus, we can think of the relation (3.18) as constraining the O( 2) terms in the metric in terms of the O( ) terms. We provide some explicit examples in Another useful form of the constraint is obtained from the expression (3.18) by making use of the general expression for the gravitational equations (recalling the normalization of where Ta(b2) are the terms in the matter stress tensor at second order in . Thus, the positivity of Fisher information constrains the behavior of the matter stress-energy tensor that should hold in any consistent theory. As a more explicit example, if we use Fe erman-Graham coordinates ds2 = (dz2 + dx dx )=z2, and consider the ball B = ft = 0; j~xj Rg, the Killing vector B is B = 2 R (t so the constraint (3.20) is [R2 z 2 (t t0)2 (~x (3.21) Z Since the gravitational contribution to canonical energy must be positive on its own, we see that positivity of the matter stress-tensor (more generally, the weak energy condition) will guarantee that the relative entropy constraint is satis ed. However, it is also possible R 4 below. Here to satisfy (3.22) with a certain amount of negative energy. Thus, the positivity of relative entropy implies a somewhat weaker integrated energy condition, as pointed out for special cases in [7, 8]. We give some more explicit examples derived from this constraint in section Finally, we note that equation (84) in [14] gives an illuminating expression for the gravitational part of canonical energy, EB grav( ; ) = Z aTagbrav(2) b Z d 2 B d 2 Q (g + ): Tagbrav(2) = d 2 d 2 Eagb(g + )j =0 HJEP04(216)53 is the expression quadratic in the rst order metric perturbation that provides the source term in the equation determining the second order perturbation when solving Einstein's equations perturbatively. Thus, we have Up to the boundary term, this is exactly the \Rindler energy" associated with the Killing vector B in the wedge RB, including perturbative contributions from both the metric perturbation and the matter elds. Thus, it is indeed the \canonical" expression for energy computed with respect to the timelike Killing vector in the background geometry. 4 Examples In this section, we provide some examples to illustrate the calculation of canonical energy for perturbations to asymptotically AdS spacetimes. Such calculations are necessary to provide a more explicit form of the energy condition (3.18), to check that the condition is satis ed for particular cases, or to prove that this condition is satis ed in general for a speci c theory (e.g. pure gravity). 4.1 Transformation to Hollands-Wald gauge The main challenge for calculations is that the results of section 3 (and of [14]) make use of the assumed gauge choice that the extremal surfaces B~ for the family of spacetimes g( ) all have the same coordinate description and that the Killing vector B of the unperturbed spacetime continues to satisfy (3.8). It will be useful to have a procedure that allows us to calculate the canonical energy for a perturbation given in some more general gauge. Thus, suppose that g is some background satisfying the equations of motion, h is some perturbation satisfying the linearized equations about the background g (but not necessarily the gauge condition), and K is the Killing vector in the unperturbed space. Then there is some metric perturbation satisfying the gauge condition that is related to h by a gauge transformation, = h + LV g To determine the required gauge transformation V , we begin with the condition that the original extremal surface remains extremal under the perturbation . To derive an explicit condition on V , it is convenient to choose coordinates for the unperturbed spacetime such that the extremal surface is described by Xi = i XA = X0A where i are the coordinates that we use to parametrize the surface, and X0A are constants. Then our condition is that for the area functional A(X + X; g + ), the term at order X vanishes both for = 0 and at linear order in . In calculating this term, we can use the simpli cation that all derivatives of XA( ) vanish. The nal result is HJEP04(216)53 LK hjB~ = 0 ; ( cbraKc + c arbKc)B~ = 0 : ( cb ac + c a bc)B~ = 0 Explicitly, this gives ab = (h + LV g)ab = hab + raVb + rbVa : ririVA + [ri; rA]V i + rihiA 1 2 rAhii ~ B = 0 : The condition that K continues to satisfy LK gB~ = r(aKb)jB~ = 0 in the perturbed i ri A 1 2 rA ii ~ B where i runs over the directions along the surface B~ and A runs over the transverse directions. We obtain a condition on V by the substitution geometry gives or explicitly, Since r(aKb) = 0 and r[aKb] / ab, this is equivalent to where ab = nanb 1 2 n2anb1 is the binormal to the surface B~. Taking the various components of this expression in the normal and tangential directions, we nd ( iA)B~ = 0 A D Finally, using (4.2), we nd that the conditions on V are 1 ADhC C + rAVD + rDV A A DrC V C (hiA + riVA + rAVi)B~ = 0 To summarize, given a metric perturbation h, the equations (4.3) and (4.6) determine the conditions on V so that the gauge transformation gives the metric perturbation equivalent to h but satisfying the gauge conditions. (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) The canonical energy is calculated using the de nition (3.17) together with (3.4) and (3.6), where the metric perturbation is assumed to obey the gauge constraint. Using the results of the previous section, we can write for some arbitrary metric perturbation using (4.2), where V is required to satisfy the conditions equations (4.3) and (4.6) at the surface B~. We will now see that the canonical energy can be evaluated using the same expression as in (3.17), applied to h, plus a boundary integral that depends on h and V . To begin, we note that !(g; ; LK ) = !(g; h + LV g; LK (h + LV g)) = !(g; h; LK h) + !(g; h + LV g; L[K;V ]g) !(g; LK h; LV g) (4.7) HJEP04(216)53 where we have used that LK g = 0 and In the nal expression, the commutator of vector elds is de ned as Using the fundamental identity (3.1), we have for any g and satisfying the equations of motion, where is given in (3.7). The second and third terms in (4.7) take the form of the left side of (4.8), so all can be written as derivatives of a form. Thus, we can write LK LV g = [LK ; LV ]g = L[K;V ]g : !(g; ; L g) = d ( ; X) !(g; ; LK ) = !(g; h; LK h) + d = (h + LV g; [K; V ]) (LK h; V ) : = B~ Z (4.8) (4.9) (4.10) (4.11) where R B In the integral (3.4) de ning canonical energy, the integral over d can be converted B) using Stokes' theorem. Since the conditions on V are localized to B~, we can always choose V to vanish at the other boundary so that = 0. In this case, we have EB = !(g; h; LK h) + (h; V ) : Z Thus, given a metric perturbation h in some general gauge, we can compute the canonical energy for the region associated with a ball B by nding V satisfying the conditions (4.3) and (4.6) and vanishing near B and evaluating (4.11). Note that we don't need the explicit form of V everywhere; rather, we need only determine V (and some of its derivatives) at the surface B~. We now consider the speci c example of perturbations to AdS3. It will be convenient to use polar coordinates for the unperturbed metric 1 ds2 = r2 cos2 ( dt2 + dr2 + r2d 2) ; B~ = ft = 0; r = Rg so that the extremal surface for a region B = x 2 [ R; R]; t = 0 is given as with chosen as the embedding coordinate. In these coordinates, the Killing vector K = B in the unperturbed geometry is B = R 2 R For perturbations to the background, the condition (4.12) for the surface to remain extremal become while the condition for to satisfy the Killing vector condition on B~ are Translating these to the explicit conditions (4.6) and (4.3) on V give cos cos 1 1 sin sin r jB~ = 0 t ~ B = 0 : trjB~ = t jB~ = r jB~ = ( tt + rr)jB~ = 0 2 2 r Vr = htr 1 2 ht hr (htt + hrr) All of these equations are required to hold on the surface r = R. Given a perturbation hab we must then use these equations to determine V and its derivatives on this surface, which allows us to calculate the canonical energy for this perturbation using (4.11). Homogeneous perturbations. As an example, we consider a perturbation to the planar black hole geometry. In Fe erman-Graham coordinates, this geometry is described by ds2 = 1 z2 (dz2 + (1 + z2=2)2dx2 (1 z2=2)2dt2) : (4.16) (4.12) HJEP04(216)53 (4.13) (4.14) (4.15) In the polar coordinates that we are using, the perturbation to rst order in hrr = sin2 htt = hr = r sin cos h = r2 cos2 : (4.17) To solve (4.15), we can choose V of the form In this case, we nd that the equations (4.15) are satis ed if and only if the following conditions are satis ed at B~: We choose C1 = C2 = 0 in order that V is well-behaved at the boundary (where cos( ) ! 0). Fortunately, these are the only properties of V that will be required for our calculation. We are now ready to calculate the canonical energy using (4.11). Making use of the de nition (3.6), we nd 2 tan Vr + r sin cos 1 6 1 6 (cos2 (cos2 4 r Vr + 2 2) cos sin 2) + 2 C2 cos2 cos2 C2 cos2 2 sin 3 cos = 0 = 0 = 0 + C1 sin cos2 2 C1 sin cos2 C1 cos These require that: Vr(R; )jr=R = R (4.18) (4.19) (4.21) (4.22) HJEP04(216)53 1 3 1 R dr Z = Combining (4.20) and (4.21) as in (4.11) to calculate the (gravitational part of) canonical energy we nd In the case of pure gravity, or where no other elds are turned on in the bulk, this is the complete result for the canonical energy associated with the wedge RB for a ball B of radius R. The positivity of Fisher information required that this be positive, so we see that the constraints are satis ed. !(g; h; L h) = Z !(g; h; L h) = 2 1 r4 cos3 dr ^ d ; 2 1 r4 cos3 = 2 15 R4 : (4.20) so that gives Using (4.10) and (3.6), we nd that jB~ = R4 12 cos3( )(2 cos2( ) 3)d + rdr where r depends on the speci c form of V but is not needed for our calculation. This Comparison with relative entropy. As a check we now compare the result with the second derivative Egrav Sgrav about pure AdS. Using the metric (4.16), we can compute the extremal surface B for arbitrary and compare its area with the unperturbed result. Using calculations in [8] we have that S( ) Svac = 2 1 2G 4 Z z0 dz < 8 0 z : q 1 1 z2f(z0) z02f(z) 1 9 = ; ln 2R z0 3 5 where f (z) = (1 + =2z2)2, and z0 is related to R by HJEP04(216)53 Working perturbatively in , we nd To nd E, we use that and Combining these and using that ht(t0) = , we get R = Z z0 0 1 q f2(z)z02 f(z0)z2 f (z) : S( ) Svac = 2 + O( 3) : h T i = R2 6G R4 90G 1 E = 2 Z R R2 x 2 R 2R hTtti : E = R2 6G : E S = d 2 d 2 ( E S) = Thus, to second order in , we nd that the relative entropy is so (setting G to 1), This agrees precisely with our expression above. Constraints for theories with matter. The result (4.22) gives the canonical energy EB associated with the homogeneous rst order perturbation (4.17) in the case where the metric is the only eld turned on in the bulk. Since we expect that the geometry (4.16) corresponds (for positive ) to a physically consistent state (the thermal state of a holographic CFT), the positivity of canonical energy was fully expected; our calculation serves as a consistency check for the HRT formula. More generally, consider a theory with Einstein gravity coupled to matter. First order perturbations to pure AdS are still governed by the linearized Einstein equations, since the matter stress tensor typically has only quadratic and higher order terms in the elds. Thus, the perturbation (4.17) still represents a consistent deformation in this case. However at second and higher order, the metric can di er from (4.16) in the case when matter elds are present. In this case, the full expression (3.18) for canonical energy includes contributions from the second order terms in the metric, or equivalently, via (3.20), from the matter stress-energy tensor. In the latter form, equation (3.22) together with (4.22) give that the positivity constraint is: Z dzdx (R2 z 2 Rz x2) T0(02) R4 45 : To express this directly as a constraint on the geometry in the case of a static, translation-invariant spacetime, we write the metric g( ) as ds2 = ds2AdS + (dx2 + dt2) + 2(ht(t2)(z)dt2 + h(x2x)(z)dx2) + O( 3) : Then after integrating over x and integrating by parts to eliminate z derivatives on h (assuming h vanishes at the z = 0), (3.18) gives 0 pR2 z2 8R5 45 : This constraint must hold for all possible R. As a special case, we can consider this constraint in the limit of small R to place constraints on the coe cients of h(x2x) expanded as a power series in z. We have checked that this precisely reproduces the constraints from positivity of relative entropy obtained in [8]. As discussed in [8], for the case of homogeneous perturbations to AdS3, it is possible to come up with stronger constraints by demanding positivity of relative entropy with the reference state chosen to be the thermal state T with the same energy expectation value as the state j i. For such a thermal state, the modular Hamiltonian for an interval is an integral over the region of an expression linear in components of the stress-energy tensor. By construction, the stress-energy tensor expectation values match for j i and T , so EB in (1.2) vanishes, and the second order constraint of relative entropy positivity becomes 0. Now, let g( ) and gT ( ) be metrics describing the spacetimes SA(j i)) we nd that dual to j i and T . Taking the di erence of the equation (3.15) applied to the two states, SA(j i)) = 2 Z a d2Eagb b ; since the rst order perturbations and the Egrav depend only on the boundary stress tensor and are thus the same for both solutions. Therefore, rewriting the Einstein tensor here in terms of the matter stress tensor using (3.19), we have that the positivity constraint is precisely Z that is, the Rindler energy computed from the second order matter stress-energy tensor must be positive for each Rindler wedge. For the example of a spatial interval, the explicit (4.23) HJEP04(216)53 (4.24) (4.25) HJEP04(216)53 constraint (4.24) on the second order metric is strengthened to 0 pR2 z2 2R5 15 : 5 In this paper, we have shown the canonical energy for perturbations to Rindler wedges of pure AdS spacetime may be identi ed with the quantum Fisher information which compares the density matrix for the corresponding boundary region with the vacuum density matrix for the same region. Conversely, for any CFT states j ( )i whose entanglement entropies are encoded holographically in dual spacetimes M ( ) via the covariant holographic entanglement entropy formula, the Fisher information of a ball B must equal the canonical energy associated with the region RB in the spacetime M ( ) . This statement does not make any additional assumptions beyond the HRT formula; in particular, it does not assume a full AdS/CFT correspondence. In the context of a consistent theory of quantum gravity for asymptotically AdS spacetimes, the positivity of quantum Fisher information in the dual CFT implies that canonical energy for each region RB must be positive for physically consistent spacetimes. In the case of pure gravity, or speci c examples of gravity coupled to matter, it should be possible to check this positivity explicitly for all allowed perturbations to AdS; partial results along these lines were given in [6, 25, 26]. More generally, we can view these constraints as conditions on the stress-energy tensor that must be satis ed for any spacetime in any consistent theory. Speci cally, equation (3.20) generalizes partial results for the energy condition arising from positivity of relative entropy at second order given in [7, 8]. This condition is implied by the weak energy condition but is a weaker integrated version. The condition may be interpreted as requiring the positivity of Rindler energy for all possible wedges RB. The present work focuses on constraints on asymptotically AdS spacetimes at second order in perturbations around pure AdS arising from positivity of relative entropy. These can be viewed as a special case of a general set of constraints on arbitrary asymptotically AdS spacetimes from the monotonicity of relative entropy.8 In a forthcoming paper, we will describe how the technology of Hollands and Wald can be used to describe these most general relative entropy constraints as inequalities on bulk integrals involving the matter stress-energy tensor. While the explicit examples in this paper have focused on Einstein gravity coupled to matter, the Wald formalism applies to general covariant theories of gravity. For these more general theories, the entanglement entropy formula must be generalized [27, 28], but we expect that all the main results carry over as they did in the case of the rst order analysis [11]. It would also be interesting to extend the analysis in this work to the semiclassical level (as for the rst-order analysis in [12]), where the holographic entanglement entropy formula includes a contribution from entanglement entropy of bulk quantum elds [29]. 8In this context, all positivity constraints follow from the monotonicity constraints. It would be interesting to understand the gravity interpretation of quantum Fisher information more generally, e.g. for perturbations around other solutions to Einstein equations. On the other hand, there are many other contexts where canonical energy is wellde ned, e.g. for perturbations to black holes in AdS or in more general spacetimes. It would be interesting to understand whether in these cases also canonical energy may be identi ed with Fisher information in some underlying quantum system. Assuming this to be the case might provide hints on the Hilbert space structure of the underlying quantum theory for cases where we currently do not have a nonperturbative description. The identi cation of canonical energy with the Fisher information provides another link between quantum information theory and gravitational physics in the context of the AdS/CFT correspondence.9 Such identi cations allow us to promote geometrical quantities which are well-de ned in the classical (or semiclassical) limit of the gravity theory to quantities which are completely well-de ned in the full quantum theory provided by the CFT dual. Making use of these identi cations should help us to ask physical questions about gravity in a fully quantum-mechanical regime, beyond the semiclassical approximation. Acknowledgments We thank Stefan Hollands, Hirosi Ooguri and Bob Wald for helpful conversations. 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Nima Lashkari, Mark Van Raamsdonk. Canonical energy is quantum Fisher information, Journal of High Energy Physics, 2016, 153, DOI: 10.1007/JHEP04(2016)153