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 ballshaped region B in a holographic CFT is dual to the canonical energy metric for perturbations to a corresponding Rindler wedge RB of AntideSitter space. Positivity of relative entropy at second order implies that the Fisher information metric is positive de nite. Thus, for physical perturbations to antideSitter spacetime, the canonical energy associated to any Rindler wedge must be positive. This secondorder constraint on the metric extends the rst order result from relative entropy positivity that physical perturbations must satisfy the linearized Einstein's equations.
AdSCFT Correspondence; Gaugegravity 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 HollandsWald 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
ballshaped 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 oneparameter family of
states j ( )i of a CFT on Rd 1;1 with j (0)i the vacuum state. For any ballshaped 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 wellbehaved 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 ballshaped regions is known to be related to 2point 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 shapedregion 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 SchwarzchildAdS 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 nondegenerate, nonnegative 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 ballshaped 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 wellknown
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 wellde ned for more general regions, it is only for
ballshaped 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 ermanGraham
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 ballshaped
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 ballshaped 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 oneparameter 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 dform 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 dform 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 ddimensional
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 lowerdimensional 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 onshell perturbations, it is closed and nondegenerate,
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 oneparameter family of asymptotically AdS spacetimes M ( ), and the
family of extremal surfaces B~( ) associated with some xed ballshaped 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 oneparameter 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 nonnegative, it
must be that the corresponding gravitational quantities are also nonnegative. 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 ballshaped 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 stressenergy tensor
that should hold in any consistent theory.
As a more explicit example, if we use Fe ermanGraham 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 stresstensor (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 HollandsWald 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 ermanGraham 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 wellbehaved 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 stressenergy 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,
translationinvariant 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 stressenergy tensor.
By construction, the stressenergy 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 stressenergy 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 stressenergy 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
stressenergy 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 rstorder 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 wellde ned in the classical (or semiclassical) limit of the gravity theory to
quantities which are completely wellde 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 quantummechanical regime, beyond the semiclassical approximation.
Acknowledgments
We thank Stefan Hollands, Hirosi Ooguri and Bob Wald for helpful conversations. We
acknowledge the support of the KITP during the programs \Entanglement in
StronglyCorrelated Quantum Matter" and \Quantum Gravity: from UV to IR" where some of
this work was done. The research of MVR and NL is supported in part by the Natural
Sciences and Engineering Research Council of Canada and by FQXi. The work of MVR
was supported by grant 376206 from the Simons Foundation.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Theor. Phys. 38 (1999) 1113 [hepth/9711200] [INSPIRE].
[2] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,
Phys. Rev. Lett. 96 (2006) 181602 [hepth/0603001] [INSPIRE].
[3] V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement
entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
[4] M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge
University Press, Cambridge U.K. (2010).
[5] R. Callan, J.Y. He and M. Headrick, Strong subadditivity and the covariant holographic
entanglement entropy formula, JHEP 06 (2012) 081 [arXiv:1204.2309] [INSPIRE].
9For other recent interesting examples of speci c connections between natural concepts and quantities
in quantum information and natural quantities in gravitational theories, see for example [9, 23, 30{32].
HJEP04(216)53
08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
[7] J. Lin, M. Marcolli, H. Ooguri and B. Stoica, Locality of gravitational systems from
entanglement of conformal eld theories, Phys. Rev. Lett. 114 (2015) 221601
[arXiv:1412.1879] [INSPIRE].
conditions from entanglement inequalities, JHEP 06 (2015) 067 [arXiv:1412.3514]
[10] N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from
entanglement `thermodynamics', JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
[11] T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from
entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
[12] B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement,
arXiv:1405.2933 [INSPIRE].
[arXiv:1412.5648] [INSPIRE].
[13] T. Faulkner, Bulk emergence and the RG ow of entanglement entropy, JHEP 05 (2015) 033
[14] S. Hollands and R.M. Wald, Stability of black holes and black branes, Commun. Math. Phys.
321 (2013) 629 [arXiv:1201.0463] [INSPIRE].
entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
[15] H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement
[16] N. Lashkari, Modular Hamiltonian of excited states in conformal eld theory,
arXiv:1508.03506 [INSPIRE].
[17] B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler quantum gravity,
Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].
[18] R. Bousso, S. Leichenauer and V. Rosenhaus, Lightsheets and AdS/CFT, Phys. Rev. D 86
(2012) 046009 [arXiv:1203.6619] [INSPIRE].
[19] B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a
density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
[20] V.E. Hubeny and M. Rangamani, Causal holographic information, JHEP 06 (2012) 114
[arXiv:1204.1698] [INSPIRE].
[21] A.C. Wall, Maximin surfaces and the strong subadditivity of the covariant holographic
entanglement entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
[22] V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical
black hole entropy, Phys. Rev. D 50 (1994) 846 [grqc/9403028] [INSPIRE].
[23] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between
quantum states and gaugegravity duality, Phys. Rev. Lett. 115 (2015) 261602
[arXiv:1507.07555] [INSPIRE].
[24] D. Petz and C. Ghinea, Introduction to quantum Fisher information, in Quantum probability
and related topics, vol. 1, World Scienti c, Singapore (2011), pg. 261.
entanglement in AdS/CFT, JHEP 05 (2014) 029 [arXiv:1401.5089] [INSPIRE].
[arXiv:1310.6659] [INSPIRE].
(2014) 044 [arXiv:1310.5713] [INSPIRE].
HJEP04(216)53
entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
of the length of a curve, JHEP 06 (2015) 157 [arXiv:1410.1540] [INSPIRE].
AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
(2014) 126007 [arXiv:1406.2678] [INSPIRE].
[1] J.M. Maldacena , The largeN limit of superconformal eld theories and supergravity , Int. J. [8] N. Lashkari , C. Rabideau , P. SabellaGarnier and M. Van Raamsdonk , Inviolable energy [9] J. Bhattacharya , V.E. Hubeny , M. Rangamani and T. Takayanagi , Entanglement density and gravitational thermodynamics , Phys. Rev. D 91 ( 2015 ) 106009 [arXiv: 1412 .5472] [INSPIRE]. [25] S. Banerjee , A. Bhattacharyya , A. Kaviraj , K. Sen and A. Sinha , Constraining gravity using [26] S. Banerjee , A. Kaviraj and A. Sinha , Nonlinear constraints on gravity from entanglement , Class. Quant. Grav . 32 ( 2015 ) 065006 [arXiv: 1405 .3743] [INSPIRE]. [27] J. Camps , Generalized entropy and higher derivative gravity , JHEP 03 ( 2014 ) 070 [30] B. Czech , P. Hayden , N. Lashkari and B. Swingle , The information theoretic interpretation [32] D. Stanford and L. Susskind , Complexity and shock wave geometries , Phys. Rev. D 90