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Figure 1: Conformal diagram of the maximally extended Schwarzschild spacetime and the past + + + domain of dependence D(u0,v0) of a partial Cauchy surface H (v0) i I (u0). ∪ ∪ is the Schwarzschild metric (in d > 3 dimensions) given by 2 d 3 2 d 3 1 2 2 2 ds = [1 (r0/r) − ]dt +[1 (r0/r) − ]− dr + r dΩ(rˆ) d 2 , (1.1) − − − S − d 2 where rˆ S − . We can consider, as usual, the “tortoise” coordinate defined by dr =[1 d 3∈ 1 ∗ +− (r0/r) − ]− dr. Then v = exp[κ(t + r )] is an affine parameter along the future horizon, H , and u = t r is a Bondi-type retarded∗ time-coordinate at future null infinity, I+. (Here κ = − ∗ + (d 3)/(2r0) is the surface gravity.) Consider the future H (v0) of a cut at v0 of the horizon − consisting of the points on the horizon having v-coordinate greater than the value v0. Similarly, + consider the future I (u0) of a cut at u0 of future null-infinity consisting of the points on the future null-infinity having u-coordinate greater than the value u0. Let A(u0,v0) be the algebra of observables for the linearized gravitational field associated with the “domain of dependence” + + + + of these regions, D(u0,v0) = D (H (v0) i I (u0)) with i future timelike infinity. See fig. − ∪ ∪ 1. Furthermore, let ω0 be a state which is defined to be the vacuum with respect to the “modes going through future null-infinity” and a thermal state at the Hawking temperature T = κ/2π with respect to “modes going through the future horizon”. Finally, consider a smooth, classical solution hab to the linearized Einstein equations whose initial data are compactly supported within D(u0,v0). Associated with such a solution, we can define a coherent state, ωh of the linearized gravitational field. Then, we can consider the relative entropy S(ω0/ωh) with respect to the partial observable algebra A(u0,v0). It is the amount of information which an observer having access to all observables in A(u0,v0) gains if she/he updates her/his belief about the system from state ω0 (black hole with no radiation) to ωh (black hole with radiation). The relative entropy between the vacuum state and a coherent state has also recently ap- peared in works by Casini et al. [7] and Longo [8] in the context of a free scalar field, and in [9] in the case of 1+1 conformal field theories. Here we apply their results (with some fairly trivial modifications) to the linearized gravitational field. Our main point is that the resulting formula can be rewritten in a suggestive form using Raychaudhuri’s equation. To frame our result, we imagine that we have a 1-parameter family of classical solutions to the full vacuum Einstein equations, gab(λ), such that gab is the Schwarzschild background |λ=0 (1.1), such that dgab/dλ = hab is the given linear perturbation, such that the second order |λ=0 3

perturbed expansion d2θ/dλ 2 , goes to zero when v ∞. By the Lemma 1 of [10], we can |λ=0 → – and should – assume that the linearized perturbation hab is in a gauge such that the ﬁrst order perturbed expansion, dθ/dλ = 0 on H+ vanishes2. |λ=0 Now we make a time translation by t > 0, thereby moving the region D(u0,v0) to D(u0 + κt t,v0e ) towards the future. Then the relative entropy, which depends on the algebra A(u0,v0) associated with this region, also changes – in fact it has to decrease by the well-known mono- tonicity of this quantity. To emphasize this, we write the relative entropy also as S(u0,v0) by abuse of notations. Actually, we shall show that3

2 δ A(v0) ′ 2 S(u0,v0) + = 2πδ F (u0) (1.2) 4 d 2 1 2 2 where a prime = . We also use the notation δgab = dgab/dλ ,δ gab = d gab/dλ ′ dt |λ=0 2 |λ=0 etc., and4 1 ab d 2 F (u0) = NabN dud − rˆ (1.3) + −32π ZI (u0) + where Nab is the news-tensor at I (see [11, 12, 13] for the precise deﬁnitions). The quantity S + A/4 on the left side is a version of Bekenstein’s generalized entropy, and the derivative of it with respect to a time parameter also appears in the quantum focussing condition [2]. The + quantity on the right side is the ﬂux of gravitational radiation through I (u0).

2 Relative entropy of coherent states of the Weyl algebra

We first recall the definition of the Weyl algebra in an abstract setting adapted to the “null quantization” of the canonical commutation relations, for details see e.g. [14, 15, 16, 17]. This setting will be applied to both the future horizon and future null infinity below. To start, we ∞ d 2 define V0 = C (R S ,R), to be viewed as the space of null initial data for a wave equation. 0 × − On V , a symplectic form is defined by (rˆ Sd 2) ∈ − d 2 w(F1,F2) = (F1∂uF2 F2∂uF1)dud − rˆ. (2.1) R d 2 Z ZS − − The Weyl-algebra is defined by the relations W(F1)W(F2) = exp[ iw(F1,F2)/2]W(F1 +F2) and 1 − W(F)∗ = W ( F) = W(F)− . This algebra is next represented in a bosonic Fock space as fol- − 2 d 2 lows. The 1-particle space is defined to be H1 = L (R S ). An F V0 is mapped to H1 by + × − ∈ projecting onto the “positive frequency part” KF(k,rˆ) := √2kFˆ(k,rˆ), where a hat denotes the 5 ∞ Fourier transform of F(u,rˆ) in the variable u. On the bosonic Fock space H = C Hn, ⊕ ⊕n=1 with Hn the n-fold symmetrized tensor product of H1, we define bosonic creation operators by a(k,rˆ)∗ in the usual way (in the sense of distributions), with relation [a(k1,rˆ1),a(k2,rˆ2)∗] = d 2 δ(k1 k2)δ (rˆ1,rˆ2), and then we represent the Weyl operators by the unitary operators − − W(F) = exp(a(KF) a(KF)∗) (2.2) − 2This condition means that we are correctly identifying the coordinate location of the horizon to first order. 3We use units where G = c = h¯ = 1. 4For coupled Einstein-scalar field theory, there would be a contribution to the flux F from the matter fields. 5Our convention for the Fourier transform is Fˆ(k)= eikuF(u)du. R 4

H ∞ d 2 on with a(ψ) = 0 Sd 2 a(k)ψ(k)dkd − rˆ. For the vacuum state Ω0 in Fock space, one − 2 | i 2 then gets the formulaR ΩR 0 W(F)Ω0 = exp( KF /2), where we mean the L -norm on the right side. In fact, we canh write| this iL2-inner−k productk also as

1 F1(u1,rˆ)F2(u2,rˆ) d 2 (KF1,KF2) = − 2 du1du2d − rˆ. (2.3) π Z (u1 u2 i0) − − Since F1,F2 are real-valued, we have ℑ(KF1,KF2) = w(F1,F2)/2, which can be used to show that the formulas are consistent with the Weyl relations. For any u0, we define the algebra A(u0) 6 d 2 to be the weak closure of W(F) F V0,suppF (u0,∞) S . { | ∈ ⊂ × − } We next recall the definition of the relative entropy in terms of modular operators due to Araki. For details on operator algebras in general and references we refer to [18] and for a recent survey of operator algebraic methods in quantum information theory in QFT, we refer to [19]. A nice exposition directed towards theoretical physics audience is [20]. Let A be a v. Neumann algebra7 of operators on a Hilbert space8, H . We assume that H contains a “cyclic and separating” vector for A, that is, a unit vector Ω such that the set | i consisting of X Ω , X A is a dense subspace of H , and such that X Ω = 0 always implies X = 0 for any X| iA. We∈ say in this case that A is in “standard form” with| i respect to the given vector. A+ denotes∈ the set of positive, self-adjoint elements in A (which are always of the form X = Y Y for some Y A). ∗ ∈ In this situation, one can define the Tomita operator S on the domain dom(S) = X Ω X A by { | i| ∈ } SX Ω = X ∗ Ω (2.4) | i | i The definition is consistent due to the cyclic and separating property. It is known that S is a closable operator, and we denote its closure by the same symbol. This closure has a polar 1 decomposition denoted by S = J∆ 2 , with J anti-linear and unitary and ∆ self-adjoint and non- negative. Tomita-Takesaki theory concerns the properties of the operators ∆,J. The basic results of the theory are the following, see e.g. [18]:

1. JAJ = A′, where the prime denotes the commutant (the set of all bounded operators on 2 1 H commuting with all operators in A) and J = 1,J∆J = ∆− , it it it 2. If αt(X) = ∆ X∆ , then αtA = A and αtA = A for all t R, ∆ Ω = Ω for all t R. − ′ ′ ∈ | i | i ∈ 3. The positive, normalized (meaning ω(X) 0 X A+,ω(1) = 1) linear expectation functional ≥ ∀ ∈ ω(X) = Ω XΩ (2.5) h | i satisﬁes the KMS-condition relative to αt. This condition states that for all X,Y A, the bounded function ∈ it t FX Y (t) = ω(Xαt(Y )) Ω X∆ Y Ω (2.6) 7→ , ≡h | i 6In other words, the double commutant. 7An algebra of bounded operators that is closed in the topology induced by the size of matrix elements. The review of the expository material until eq. (2.12) follows [9]. 8We always assume that H is separable. 5

has an analytic continuation to the strip z C 1 < ℑz < 0 with the property that its boundary value for ℑz 1+ exists and{ is∈ equal| − to } → − FX Y (t i) = ω(αt(Y )X). (2.7) , −

4. Any normal (i.e. continuous in the weak∗-topology) positive linear functional ω′ on A has a unique vector representative Ω in the natural cone | ′i P♯ = ∆1/4X Ω X A+ = Xj(X) Ω X A , (2.8) { | i| ∈ } { | i| ∈ } where the overbar means closure and j(X) = JXJ. The state functional is thus ω′(X) = Ω XΩ for all X A. h ′| ′i ∈

A generalization of this construction is that of the relative modular operator, flow etc. For this purpose, let ω′ be a normal state on A, Ω′ its unique vector representative in the natural cone in H , which is assumed (for simplicity)| i to be cyclic and separating, too. Then we can consistently define S X Ω′ = X ∗ Ω (2.9) ω,ω′ | i | i 1 2 form the closure, and make the polar decomposition Sω,ω = J ∆ . The Araki relative ′ ω ω,ω′ entropy is defined by S(ω/ω′) = Ω (log∆ )Ω . (2.10) h | ω,ω′ i In the case of Type I factors (“quantum mechanics”), e.g. A = MN(C), the situation is this: State functionals are equivalent to density matrices ρ via ω(X) = Tr(Xρ), H is the algebra N N itself MN(C) = C C on which A acts by left multiplication. The state Ω corresponds to 1/2 ∼ ⊗ | i 1 ρ , the inner product is X Y = Tr(X ∗Y ), the modular operator is ∆ = ρ ρ− , and the |relativei modular operator is ρh | ρi 1 , where ρ is the density matrix associated⊗ with ω . Using ⊗ ′− ′ ′ this, one verifies S(ω/ω′) = Trρ(logρ logρ′). These formulae do not hold for type III factors which occur in quantum field theories.− The relative entropy has many beautiful properties. It is e.g. never negative, but can be infinite, is decreasing under completely positive maps, is jointly convex in both arguments, etc. The physical interpretation of S(ω/ω′) is the amount of information gained if we update our belief about the system from the state ω to ω′. In this paper, we are interested in the special case when ω′ = ωU , where

ωU (X) ω(U ∗XU) = UΩ XUΩ , (2.11) ≡ h | i and where U is some unitary operator from A. The corresponding vector representative in the natural cone is ΩU = Ujω(U) Ω , with jω (X) = Jω XJω . Going through the deﬁnitions, one | i 1/|2 i 1/2 ﬁnds immediately that jω (U)∆ω jω (U ∗) = ∆ , implying that ω,ω′

d it S(ω/ωU ) = U ∗Ω (log∆)U ∗Ω = i U ∗Ω ∆ U ∗Ω , (2.12) −h | i dt h | i t=0

where ∆ is the modular operator of the original state ω. 6

Now we specialize to the following situation: A = A(u0) is the v. Neumann closure d 2 of the partial Weyl algebra W(F) suppF (u0,∞) S − . Ω = Ω0 is the vacuum. This state is cyclic and separating{ by| the Reeh-Schlieder⊂ × theorem,} | i and| thereforei the modular operator exists. U = W (F) = W ( F) is a speciﬁc Weyl-operator for some F V0 sup- ∗ − ∈ ported in the interval (u0,∞). We call the corresponding state functionals ω0 (vacuum) and ωF (X) = ω(W(F)XW(F)∗) (coherent state). We wish to compute S(ω0/ωF ) relative to the algebra A(u0). As one may expect, the general relation (2.12) can be expressed in the case at hand in terms of the 1-particle quantities only. This is done as follows. Let K H1 be the image under ⊂ K of all possible G V0 such that the support of G is contained in (u0,∞). Since V0 is a real ∈ vector space, K is only a real linear subspace of H1. However, the 1-particle version of the Reeh-Schlieder theorem states that K + iK is dense in H1 and K iK = 0. Such a real subspace is called “standard”. For a standard real subspace of a Hilbert∩ space one can deﬁne 1-particle versions of the Tomita-operators on the dense domain K + iK by [21]

1/2 SK (ψ + iφ) = ψ iφ, φ,ψ K , SK = JK ∆ , (2.13) − ∈ K and these objects have properties analogous to those of the Tomita-operators for a v. Neumann algebra in standard form. In the case at hand, the 1-particle modular operator acts by dilations of the coordinate u around the point u0, more precisely

it 2πt ∆K KG = K(G Λt), Λt (u,rˆ)=(u0 + e− (u u0),rˆ), G V0, (2.14) ◦ − ∀ ∈ by the 1-particle version of the Bisognano-Wichmann theorem [22]. Furthermore, the full mod- it it ular operator ∆ = Γ(∆K ) is the second-quantized version of the 1-particle modular operator. In fact, one can show in general [8] (Proposition 3.1)

d it d it Ω0 W(F)∗∆ W(F)Ω0 = (KF,∆K KF) , (2.15) dt h | i dt t=0 t=0

which when evaluating the right side gives in combination with eqs. (2.3), (2.14) and (2.12) the

relation ∞ 2 d 2 S(ω0/ωF ) = 2π (u u0)∂uF(u,rˆ) dud − rˆ. (2.16) d 2 Zu0 ZS − −

3 Relative entropy for scalar and gravitational perturbations

We now apply the abstract setting of the previous section to scalar (first) and then gravitational a perturbations of Schwarzschild. Consider first a classical solution ∇ ∇aφ = 0 to the massless Klein-Gordon (KG) equation with smooth, compactly supported initial data on some Cauchy surface Σ, i.e. a surface stretching from the bifurcation surface to spatial infinity such as the t = 0 surface in the coordinates (r ,t,rˆ). The symplectic form w is given by ∗ a w(φ1,φ2) = (φ1∇aφ2 φ2∇aφ1)dΣ (3.1) ZΣ − 7

and is independent of the choice of Cauchy surface by the usual argument based on Gauss’ theorem. Now consider deforming Σ to the degenerate “surface” consisting of the union of H+, I+ and (formally) future timelike infinity i+, see fig. 1. By taking a suitable limit, one may hope that d 2 d 2 w(φ1,φ2) = (F1∂vF2 F2∂vF1)dvd − rˆ + (F1∂uF2 F2∂uF1)dud − rˆ. (3.2) ZH+ − ZI+ − d 2 + d 2 Here d − rˆ is the integration element of the unit radius round sphere in the case of I and r0− times that in the case of H+. F denotes the restriction of φ to the future horizon in the case of + d/2 1 + H , and the restriction of r − φ to future null infinity in the case of I . The latter restriction is immediately seen to exist by a standard conformal transformation of the Schwarzschild metric to an unphysical metric smooth at I+. However, it is not obvious that the above integrals are absolutely convergent for u,v ∞ and that the expression (3.2) really agrees with the original definition (3.1) of the symplectic→ form over Σ, i.e. that there is no “leakage of symplectic flux” through i+. This is shown in Thm. 2.2 of [23] [which is using non-trivial pointwise decay results on φ due to [24, 25] in d = 4, and is extendable to d > 4 in view of [26]]. More precisely, these authors establish that if one enlarges V0 to suitable subspaces VH respectively + + VI in each case so that it contains the restrictions of φ to I respectively H , then the map d/2 1 φ (φ + ,r φ + ) VH VI, taking values in this enlarged space, is a symplectic and 7→ |H − |I ∈ × injective map. In view of the mentioned decay results, it is possible e.g. to choose VH for some ∞ d 2 d 2 v0 > 0 to consist of those F C (R S − ;R) such that supp(F) (v0,∞] S − with a decay ∈ 3/2+ε × ⊂ 9 × of the type F(v) C(logv)− as v ∞, for arbitrarily small ε > 0 and with a decay of | | ≤ 1 → ∂vF(v) faster by a factor of v− . It is possible to choose VI for some u0 to consist of those | ∞| d 2 d 2 F C (R S − ;R) such that supp(F) (u0,∞] S − for some v0 > 0 with a decay of the ∈ × 1/2 ⊂ × 1/2 type F(u) Cu for u ∞, with a decay of ∂uF(u) faster by a factor of u . | | ≤ − → | | − The second formula (3.2) for the symplectic form shows by comparison with (2.1) that one may quantize the theory by two copies of the Weyl algebra (for I+ and H+) as described in the previous section, now based on the symplectic space VH VI. Furthermore, one may define a “vacuum” state by making the above Fock space construction× for both copies. See [23] for a more detaild description of this procedure and especially Prop. 3.3(b) of [23] for the precise definition of the map K in this case. Thereby, we get algebras of the form A(u0,v0) := A + (v0) H ⊗ AI+ (u0), with v0 > 0,u0 R. Such an algebra is associated with the region D(u0,v0) = + + + ∈ D (H (v0) i I (u0)) as described in the introduction, see fig. 1. This von Neumann al- − ∪ ∪ gebra is by construction large enough to contain the Weyl unitaries W(F) W(FH+ ) W(FI+ ) ≡ d/2 1 ⊗ associated with the null data coming from the restriction map φ (φ +,r φ + ) = F 7→ |H − |I ∈ VH VI of a smooth solution φ with compact support on Σ, provided the causal future of the × support of the initial data is within D(u0,v0). Note that, owing to the relationship between Killing time t and affine time v on H+, a time translation by t corresponds to changing v to eκt v. By the usual arguments (see e.g. [14, 16]), the “vacuum” state defined on AH+ corresponds to a thermal state with respect to Killing time translations at the Hawking temperature T = κ/2π.10 Furthermore, under a translation by κt Killing time t, our region D(u0,v0) is moved into D(u0 +t,e v0). 9C depends on this ε. 10 By the arguments of [27, 23, 15], the state ω0 is a Hadamard state in the exterior region of the Schwarzschild spacetime, which in general becomes singular on H+, by analogy with the Unruh-vacuum. 8

a A classical solution to ∇ ∇aφ = 0 such that the causal future of the support of its initial data is contained in D(u0,v0) gives rise to a coherent state on A(u0,v0) by the construction of d/2 1 + the previous section, taking F =(φ H+ ,r − φ I+ ) VH VI to be the restriction of φ to I + | | ∈ × respectively H (characteristic data), by taking U = W(F) = W (F + ) W(F + ) = UH UI. ∗ H ∗ ⊗ I ∗ ⊗ We call the coherent state ωφ in the present situation to emphasize its origin from the classical solution, φ. To justify the name “coherent state” for that state, we recall (see e.g. [32]) that the quantum KG ﬁeld Φ is related to the Weyl operators formally by W(F) = exp(iw(F,Φ)), where + + F VH VI corresponds to the characteristic initial data on H ,I of some classical solution, and∈ w is× as in eq. (3.2). By the Weyl relations, one can then show, with F the characteristic initial data of the classical solution φ:

W(F)Φ(x)W(F)∗ = Φ(x) + φ(x)1, (3.3)

so in particular ωF (Φ(x)) = ω0(W(F)Φ(x)W(F)∗) = φ(x), as expected from a coherent state. By eqs. (2.12), (2.15), (2.16) applied to the two copies A(u0,v0) := AH+ (v0) AI+ (u0) we get using the additivity of the relative entropy under the tensor product: ⊗

2 d 2 2 d 2 S(ω0/ωφ ) = 2π (u u0)∂uφ˜(u,rˆ) dud − rˆ + 2π (v v0)∂vφ(v,rˆ) dvd − rˆ, (3.4) ZI+ − ZH+ −

d/2 1 where φ˜ = limI+ r − φ. At this stage, S(ω0/ωφ ) could still be infinite, but we now argue that the right side is actually finite. Finiteness of the second integral immediately follows from the characterization of the space VH in which FH+ = φ H+ lives. Finiteness of the first inte- | d/2 1 gral does not follow from the characterization of the space VI where F + = r φ + lives, I − |I but we can argue as follows, restricting for simplicity attention to d = 4. Let F (u1,u2) = u2 (∂ φ˜)2dud2rˆ be the flux through I+ between u ,u . By a result of [24], Thm. 7.1 (18), u1 u 1 2 2 FR (u1,u2) Cmax u1,1 − for all u2 u1. Let α > 1. The first integral in (3.4) is bounded ≤ { } ≥ ∞ α F α α above by (without loss of generality u0 1) ∑N=0(N + 1) (u0 + N ,u0 +(N + 1) ) 2 α ∞ α ≥ ≤ ≤ Cmax u− ,4 ∑ (N + 1) < ∞. { 0 } N=0 − A completely analogous analysis can be made in principle for the case of a gravitational perturbation hab, i.e. a smooth solution of the same type to the linearized vacuum Einstein equations. However, to make our analysis completely rigorous, we would need to justify the decay of gravitational perturbations at I+ and H+, for instance when proving the analog of (3.2). Here one can use recent results by [28]. Since these results concern the Teukolski equation rather than linearized Einstein equation, one would additionally have to represent a gravitational perturbation in terms of Hertz-type potential solutions to Teukolsky’s equation [29, 30]. Such an analysis would go beyond the scope of this short note, and one could, at any rate, always consider solutions which, in some gauge, are smooth and of compact support at I+ and H+, obtained by a characteristic initial value problem as considered e.g. in [31].11 We shall therefore proceed more formally, assuming that the integrals in questions converge, as they did in the case of the scalar field. c c c c The linearized Einstein equations are ∇ ∇chab + ∇a∇bh c ∇c∇ah b ∇c∇bh a = 0. The symplectic forms on the pair of null surfaces I+ respectively−H+ are given− in eqs. (97) respectively (102) of [10], which uses results of [11], where we note that the boundary terms are

11Such solutions would not arise from compactly supported initial data in the interior, though. 9

absent in our setting. Going through the analogous steps as for a scalar ﬁeld and using formulas such as (96) and (101) of [10], one ﬁnds

1 1 ab d 2 S(ω0/ωh) = (u u0)δNabδN dud − rˆ 2π 32π ZI+(u ) − 0 (3.5) 1 ab d 2 + (v v0)δσabδσ dvd − rˆ, + 8π ZH (v0) − + where δNab is the perturbed news tensor of the perturbation hab = δgab on I [11], and where + δσab is the perturbed shear on H . On H+, we now consider the Raychaudhuri equation (see e.g. [32]) for the 1-parameter family gab(λ) described in the introduction,

d 1 2 ab θ(λ) = θ(λ) σab(λ)σ (λ) 8π Tvv(λ). (3.6) dv −d 2 − − − Taking two derivatives with respect to the family parameter λ, using that θ = δθ = 0 on H+ + as well as σab = 0 on H in the background, and using that the stress tensor vanishes in the d 2 ab + present situation, we see dv δ θ = δσabδσ on H . This formula is now used in the second term in eq. (3.5), giving −

1 1 ab d 2 S(ω0/ωh) = (u u0)δNabδN dud − rˆ 2π 32π ZI+(u ) − 0 (3.7) 1 d 2 d 2 (v v0) δ θ dvd − rˆ. + −8π ZH (v0) − dv κt Finally, we time translate the region D(u0,v0) by t to D(u0 +t,e v0), take a derivative of (3.7) d 2 d 2 d 2 with respect to t, and use the relation δ A(v0) = + δ θ dvd − rˆ, which uses our − dv0 H (v0) dv assumption that δ 2θ(v) 0 as v ∞. This then immediatelyR gives the claim (1.2) made in the introduction12. → → We may use the same argument if we have a linear gravitational field and a massless scalar Klein-Gordon field. For classical solutions hab,φ of the type described for both theories, we now get a coherent state ωh,φ . The relative entropy with the reference state ω0 is given by the sum d 2 ab of (3.5) and (3.4). The second order Raychaudhuri equation now gives dv δ θ = δσabδσ 2 + − − 8π(∂vφ) on H , and the flux F now also contains a contribution from the scalar field. With this contribution included, we get the same equation as stated in the introduction (1.2). Note that since the horizon area does not change to first and zeroth order in the classical perturbation (hab,φ) defining the coherent state, and since the background news tensor is zero, we may also write (1.2) in the more suggestive form

(S + A/4)′ = 2πF , (3.8) 12As mentioned before, the arguments given here would be straightforwardly generalized to Kerr spacetimes with no serious obstraction. For extremal Kerr spacetimes, the surface gravity vanishes, κ = 0, and the horizon affine parameter is given by v0 = t +r with r appropriately defined. In this case the time translation by t should be ∗ ∗ taken simply as D(u0,v0) D(u0 +t,v0 +t), and one would obtain the same claim (1.2). The formula (3.7) should also hold for stationary, asymptotically→ AdS black holes with the elimination of the first integral of right-hand side as the news δNab vanishes due to the reflecting nature of the AdS boundary conditions. Then the formula (1.2) represents simply a conservation of the generalized entropy, see [36] for similar looking formulas. 10

which holds up to and including second order in the perturbation, where S is the relative entropy between the vacuum state and the coherent state. This equation relates an information theoretic quantity on the left side to the Bondi news tensor giving the ﬂux on the right side. It is tempting to speculate that this formula continues to hold in perturbative quantum gravity to all orders, and perhaps even for full quantum gravity. It would also be interesting to relate our formula to, e.g. [33, 34, 35].13 Acknowledgements: Part of this work was carried out while S.H. was visiting IHES, Paris. It is a pleasure to thank IHES for hospitality and ﬁnancial assistance. S.H. is grateful to the Max-Planck Society for supporting the collaboration between MPI-MiS and Leipzig U., grant Proj. Bez. M.FE.A.MATN0003. The work of A.I. was supported in part by JSPS KAKENHI Grants No. 15K05092.

References

[1] R. Longo and F. Xu, “Comment on the Bekenstein bound,” J. Geom. Phys. 130, 113 (2018)

[2] R. Bousso, Z. Fisher, S. Leichenauer and A. C. Wall, “Quantum focusing conjecture,” Phys. Rev. D 93, no. 6, 064044 (2016)

[3] R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer and A. C. Wall, “Proof of the Quantum Null Energy Condition,” Phys. Rev. D 93, no. 2, 024017 (2016)

[4] H. Casini and M. Huerta, “ A c-theorem for the entanglement entropy,” J. Phys. A 40, 7031 (2007)

[5] A. C. Wall, “A Survey of Black Hole Thermodynamics,” arXiv:1804.10610 [gr-qc].

[6] M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, Springer Lecture Notes in Physics (2017)

[7] H. Casini, S. Grillo and D. Pontello, “Relative entropy for coherent states from Araki formula,” arXiv:1903.00109 [hep-th].

[8] R. Longo, “Entropy of Coherent Excitations,” arXiv:1901.02366 [math-ph].

13Some other interesting connections between the news and information theoretic quantities may also be worth investigating. For instance, in [37], we found a relation between the supertranslation, T associated with a burst or radiation emitted by the collision of particles. Interestingly, T can be written as

T (rˆ)= 2E ∑ η(i)ρ(i) logρ(i), ρ(i) = (E(i) rˆ p(i))/E (3.9) (i) in, out − · where E is the total energy of all particles, η = for in/outgoing particles, and (E ,p ) the four-momentum (i) ± (i) (i) of particle i. Note that ρ(i) for the in- and outgoing momenta are two probability distributions (this follows from four-momentum conservation{ } and the fact that E p ). Furthermore, the right side has the form of 2E times (i) ≥ | (i)| the v. Neumann entropy difference of the in- and out probability distributions ρ(i) . Perhaps this is more than an accident. { } 11

[9] S. Hollands, “Relative entropy for coherent states in chiral CFT,” arXiv:1903.07508 [hep- th].

[10] S. Hollands and R. M. Wald, “Stability of Black Holes and Black Branes,” Commun. Math. Phys. 321, 629 (2013)

[11] S. Hollands and A. Ishibashi, “Asymptotic ﬂatness and Bondi energy in higher dimensional gravity,” J. Math. Phys. 46, 022503 (2005)

[12] S. Hollands, A. Ishibashi and R. M. Wald, “BMS Supertranslations and Memory in Four and Higher Dimensions,” Class. Quant. Grav. 34, no. 15, 155005 (2017)

[13] S. Hollands and A. Thorne, “Bondi mass cannot become negative in higher dimensions,” Commun. Math. Phys. 333, no. 2, 1037 (2015)

[14] B. S. Kay and R. M. Wald, “Theorems on the Uniqueness and Thermal Properties of Sta- tionary, Nonsingular, Quasifree States on Space-Times with a Bifurcate Killing Horizon,” Phys. Rept. 207, 49 (1991).

[15] C. Dappiaggi, V. Moretti and N. Pinamonti, “Hadamard States From Light-like Hypersur- faces,” SpringerBriefs Math. Phys. 25 (2017)

[16] S. Hollands and R. M. Wald, “Quantum ﬁelds in curved spacetime,” Phys. Rept. 574, 1 (2015)

[17] R. M. Wald, Quantum ﬁeld theory in curved spacetime and black hole thermodynamics, U. of Chicago Press (1994)

[18] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I. Springer (1987) O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics II. Springer (1997)

[19] S. Hollands and K. Sanders, Entanglement measures and their properties in quantum ﬁeld theory, SpringerBriefs in Mathematical Physics (2018) arXiv:1702.04924 [quant-ph].

[20] E. Witten, “Notes on Some Entanglement Properties of Quantum Field Theory,” arXiv:1803.04993 [hep-th].

[21] R. Brunetti, D. Guido and R. Longo, “Modular localization and Wigner particles,” Rev. Math. Phys. 14, 759 (2002)

[22] J. J. Bisognano and E. H. Wichmann, “On the Duality Condition for Quantum Fields,” J. Math. Phys. 17, 303 (1976)

[23] C. Dappiaggi, V. Moretti and N. Pinamonti, “Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime,” Adv. Theor. Math. Phys. 15, no. 2, 355 (2011) 12

[24] M. Dafermos and I. Rodnianski, “The black hole stability problem for linear scalar perturbations,” XVIth International Congress on Mathematical Physics, P. Exner (ed.), World Scientiﬁc, London, 2009, pp. 421-433

[25] J. Luk, “Improved decay for solutions to the linear wave equation on a Schwarzschild black hole,” Annales Henri Poincare 11, 805 (2010)

[26] V. Schlue, “Decay of linear waves on higher dimensional Schwarzschild black holes,” Anal. PDE 6 (2013) 515-600

[27] S. Hollands, “Aspects of quantum ﬁeld theory in curved spacetime,” PhD Thesis, U. of York (2000)

[28] M. Dafermos, G. Holzegel and I. Rodnianski, “The linear stability of the Schwarzschild solution to gravitational perturbations,” arXiv:1601.06467 [gr-qc].

[29] P. L. Chrzanowski, “Vector Potential and Metric Perturbations of a Rotating Black Hole,” Phys. Rev. D 11, 2042 (1975).

[30] L. S. Kegeles and J. M. Cohen, “Constructive Procedure For Perturbations Of Space- times,” Phys. Rev. D 19, 1641 (1979).

[31] P. T. Chrusciel and T. T. Paetz, “Characteristic initial data and smoothness of Scri. I. Framework and results,” Annales Henri Poincare 16, no. 9, 2131 (2015)

[32] R. M. Wald, General Relativity, U. of Chicago Press (1984)

[33] D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. Josephine Suh. “Relative entropy equals bulk relative entropy.” JHEP, 06:004 (2016)

[34] X. Dong, D. Harlow, and A. C. Wall, “Reconstruction of Bulk Operators within the En- tanglement Wedge in Gauge-Gravity Duality.” Phys. Rev. Lett., 117(2):021601 (2016)

[35] T. Faulkner and A. Lewkowycz. “Bulk locality from modular ﬂow.” JHEP, 07:151 (2017)

[36] N. Lashkari and M. Van Raamsdonk, “Canonical Energy is Quantum Fisher Information,” JHEP 1604, 153 (2016)

[37] D. Garﬁnkle, S. Hollands, A. Ishibashi, A. Tolish and R. M. Wald, “The Memory Effect for Particle Scattering in Even Spacetime Dimensions,” Class. Quant. Grav. 34, no. 14, 145015 (2017)