Positive gravitational subsystem energies from CFT cone relative entropies
Accepted: May
Positive gravitational subsystem energies from CFT cone relative entropies
Dominik Neuenfeld 0 1 2 3 4
Krishan Saraswat 0 1 2 3 4
Mark Van Raamsdonk 0 1 2 3 4
0 tum Gravity , Conformal Field Theory
1 6224 Agricultural Road, Vancouver , B.C. V6T 1W9 , Canada
2 Department of Physics and Astronomy, University of British Columbia
3 The surface @A is then described by
4 is constant. On the light cone , only the @
The positivity of relative entropy for spatial subsystems in a holographic CFT implies the positivity of certain quantities in the dual gravitational theory. In this note, we consider CFT subsystems whose boundaries lie on the lightcone of a point p. We show that the positive gravitational quantity which corresponds to the relative entropy for such a subsystem A is a novel notion of energy associated with a gravitational subsystem bounded by the minimal area extremal surface A~ associated with A and by the AdS boundary region A^ corresponding to the part of the lightcone from p bounded by @A. This generalizes the results of arXiv:1605.01075 for ballshaped regions by making use of the recent results in arXiv:1703.10656 for the vacuum modular Hamiltonian of regions bounded on lightcones. As part of our analysis, we give an analytic expression for the extremal surface in pure AdS associated with any such region A. We note that its form immediately implies the Markov property of the CFT vacuum (saturation of strong subadditivity) for regions bounded on the same lightcone. This gives a holographic proof of the result proven for general CFTs in arXiv:1703.10656. A similar holographic proof shows the Markov property for regions bounded on a lightsheet for nonconformal holographic theories de ned by relevant perturbations of a CFT.
AdSCFT Correspondence; Gaugegravity correspondence; Models of Quan

1 Introduction
2 Background
2.1
2.2
4.1
4.2
4.3
4.4
lightcone
Relative entropy in conformal eld theories
Gravity background
Light cone coordinates for AdS
HRRT surface in pure AdS
The bulk vector eld
Perturbative formulae for
H
4.4.1
4.4.2
Vanishing of the rst order expression
Relative entropy at second order
3 Bulk interpretation of relative entropy for general regions bounded on a
4 Perturbative expansion of the holographic dual to relative entropy
5 Holographic proof of the Markov property of the vacuum state
5.1
The Markov property for states on the nullplane
5.2 The Markov property for states on the lightcone
6 Discussion
A Equivalence of H on the boundary and the modular Hamiltonian
B The HRRT surface ending on the nullplane
C Calculation of the binormal
D HollandsWald gauge condition
1
Introduction
Via the AdS/CFT correspondence, it is believed that any consistent quantum theory of
gravity de ned for asymptotically AdS spacetimes with some xed boundary geometry B
corresponds to a dual conformal eld theory de ned on B
. Recently, it has been
understood that many natural quantum information theoretic quantities in the CFT correspond
to natural gravitational observables (see, for example [1], or [
2, 3
] for a review). Through
this correspondence, properties which hold true for the quantum information theoretic
{ 1 {
quantities can be translated to statements about gravitational physics. In this way, we
can obtain a alternative/deeper understanding of some known properties gravitational
systems, but also discover novel properties that must hold in consistent theories of gravity. A
particularly interesting quantum information theoretic quantity to consider is relative
entropy [4]. For a general state j i of the CFT, we can associate a reduced density matrix A
to a spatial region A by tracing out the degrees of freedom outside of A. Relative entropy
S( Ajj 0A), which we review in section 2, quanti es how di erent this state is from the
vacuum density matrix 0A reduced on the same region. Relative entropy is typically UV nite,
always positive, and has the property that it increases as we increase the size of the region
A (known as the monotonicity property). According to the AdS/CFT correspondence,
this should correspond to some quantity in the gravitational theory which also obeys these
positivity and monotonicity properties. This has previously been explored in [5{11].
As we review in section 2, by making use of the holographic formula relating CFT
entanglement entropies to bulk extremal surface areas (the \HRRT formula" [1, 12]), it
is possible to explicitly write down the gravitational quantity corresponding to relative
entropy as long as the vacuum modular Hamiltonian (HA0 =
log 0A) for the region A is
\local", that is, it can be written as a linear combination of local operators in the CFT.
Until recently, such a local form was only known for the modular Hamiltonian of
ballshaped regions [13]. For these regions, relative entropy has been shown to correspond to
an energy that can be associated with the bulk entanglement wedge corresponding to this
ball [8, 10].1 The positivity of relative entropy then implies an in nite family of positive
energy constraints (reviewed below) [11].
Ballshaped regions (of Minkowski space) have the property that their boundary lies
on the past lightcone of a point p and the future lightcone of some other point q. In the
recent work [14], it has been shown that the vacuum modular Hamiltonian for a region A
has a local expression so long as the boundary @A of A lies on the past lightcone of a point
p or the future lightcone of a point q.2 Thus, we have a much more general class of regions
for which the relative entropy and its properties can be interpreted gravitationally. The
main goal of the present paper is to explain this interpretation.
In the general case, we denote by A^ the region of the lightcone bounded by @A, as
shown in gure 1. The modular Hamiltonian can then be written as
HA0 =
Z
^ A(x)T (x) ;
A
(1.1)
where T
is the CFT stressenergy tensor,
is a volume form de ned in section 2, and
A(x) is a vector eld on A^ directed towards the tip of the cone and vanishing at the tip
To describe the gravitational interpretation of the relative entropy for region A, we
consider any codimension one spacelike surface
in the dual geometry such that
inter1The entanglement wedge is a region de ned by the union of spacelike surfaces with one boundary on
the HRRT surface from the ball and the other boundary on the domain of dependence of the ball at the
AdS boundary.
2The existence of such a region depends on the relativistic nature of the theory under consideration,
which guarantees the existence of a codimension0 domain of dependence.
{ 2 {
A
^
A
sects the AdS boundary at A^ and is bounded in the bulk by the HRRT surface A~ (the
minimal area extremal surface homologous to A). This is illustrated in
gure 2. Next,
we de ne a timelike vector eld
in a neighborhood of
with the properties that
approaches A at the AdS boundary and behaves near the extremal surface A~ like a Killing
vector associated with the local Rindler horizon at A~. The timelike vector eld
a particular choice of time on the surface
and we can de ne an energy H
represents
associated
with this. While generally there are many choices for the surface
and the vector eld ,
we can show that all of them lead to the same value for the energy H . It is this quantity
that corresponds to the CFT relative entropy S( Ajj 0A).3
The independence of H
on the surface
used to de ne it can be understood as a
bulk conservation law for this notion of energy. In the case of a ballshaped region [11],
3In this paper, we focus on the leading contribution to the CFT relative entropy at large N and make
can be de ned on the full entanglement wedge
for A, i.e. the union of spacelike surfaces ending on A~ and on any A0 in DA, so we can
think of the energy H
as being associated with the entire entanglement wedge. In the
more general case considered in this paper, the collection of allowed surfaces
generally
still de ne a codimension zero region WA of the bulk spacetime (equivalent to the bulk
domain of dependence of any particular
), but this region intersects the boundary only
on the lightlike surface A^ rather than the whole domain of dependence region DA.
In section 4, we consider the limit where the geometry is a small deformation away
from pure AdS. For pure AdS, we show that the extremal surface A~ associated with a
region A whose boundary lies on the lightcone of p always lies on the bulk lightcone of p.
Thus, in a limit where perturbations to AdS become small, the wedge WA collapses to the
portion A^bulk of this lightcone between p and A~. We present an analytic expression for the
extremal surface A~ and a canonical choice for the vector eld
on A^bulk. In terms of these,
we can write an explicit expression for the leading perturbative contribution to the energy
H , which takes the form of an integral over A^bulk quadratic in the bulk eld perturbations.
In section 5, we point out that the explicit form of the extremal surface A~ in the pure
AdS case (in particular, the fact that it lies on the bulk lightcone) leads immediately to
a holographic proof of the Markov property for subregions of a CFT in its vacuum state,
namely that for two regions A and B the strong subadditivity inequality
S(A) + S(B)
S(A \ B)
S(A [ B)
0;
(1.2)
is saturated if their boundaries lie on the past or future lightcone of the same point p.
This was shown for general CFTs in [14], so it had to hold in this holographic case. The
holographic proof extends easily to cases where the eld theory is Lorentzinvariant but
nonconformal, for example a CFT deformed by a relevant perturbation. In this case, the
statement holds for subregions A, B whose boundaries lie on a nullplane.
We conclude in section 6 with a discussion of some possible future directions.
2
2.1
Background
Relative entropy in conformal eld theories
For a general quantum system or subsystem described by a density matrix , the relative
entropy quanti es the di erence between
and a reference state . It is de ned as
S( jj ) = tr( log )
tr( log ) ;
which can be shown to be nonnegative as well as vanishing if and only if
= .
Relative entropy also obeys a monotonicity property: when A is a subsystem of the
original system, the relative entropy satis es
S( Ajj A)
S( jj ) ;
{ 4 {
(2.1)
(2.2)
where A and A are the reduced density matrices for the subsystem de ned from
and
respectively.
entropy as [4]
Using the de nition S( ) =
tr( log ) of entanglement entropy and H
=
log( )
of the modular Hamiltonian associated with , we can rewrite the expression for relative
S( k ) =
hH i
S
where
indicates a quantity calculated in the state
minus the same quantity calculated
in the reference state .
For a conformal eld theory in the vacuum state, the modular Hamiltonian of a
ballshaped region takes a simple form [13]. For a ball B of radius R centered at x0 in the
spatial slice perpendicular to the unit timelike vector u , the modular Hamiltonian is
^ dx d 1 is a volume form and B is the conformal
where
Killing vector
= (d 11)!
B =
R
[R2
(x
x0)2]u + [2u (x
x0) ](x
x0)
;
with some fourvelocity u . The modular Hamiltonian is the same for any surface B0 with
the same domain of dependence as B.
Using the expression (2.4) in (2.3), the relative entropy for a state
compared with the
vacuum state may be expressed entirely in terms of the entanglement entropy and the stress
tensor expectation value. For a holographic theory in a state with a classical gravity dual,
these quantities can be translated into gravitational language using the HRRT formula
(which also implies the usual holographic relation between the CFT stressenergy tensor
expectation value and the asymptotic bulk metric [15]). Thus, the CFT relative entropy for
a ballshaped region corresponds to some geometrical quantity in the gravitational theory
with positivity and monotonicity properties. In [10] and [11], this quantity was shown to
have the interpretation of an energy associated with the gravitational subsystem associated
with the interior of the entanglement wedge associated with the ball.
Recently, Casini, Teste, and Torroba have provided an explicit expression for the
vacuum modular Hamiltonian of any spatial region A whose boundary lies on the lightcone
of a point [14]. To describe this, consider the case where @A lies on the past lightcone of
a point p and let A^ be the region on the lightcone that forms the future boundary of the
domain of dependence of A. For x 2 A^, de ne a function f (x) that represents what fraction
of the way x is along the lightlike geodesic from p through x to @A (so that f (p) = 0 and
f (x) = 1 for x 2 @A). Now, de ne a lightlike vector eld on A^ by
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
In general, we cannot extend the vector eld away from the surface A^ such that the
expression (2.7) remains valid when integrated over an arbitrary surface A0 with @A0 = @A.
In equation (4.3) we give an explicit expression for HA in a convenient coordinate frame.
Using this expression in (2.3), we can express the relative entropy for the region A in a
form that can be translated to a geometrical quantity using the HRRT formula. We would
again like to understand the gravitational interpretation for this positive quantity.
2.2
Gravity background
We now focus on states in a holographic CFT dual to some asymptotically AdS spacetime
with a good classical description. For any spatial subsystem A of the CFT, there is a
corresponding region on the boundary of the dual spacetime (which we will also call A).
The HRRT formula asserts that the CFT entanglement entropy for the spatial subsystem
A in a state j i, at leading order in the 1=N expansion, is equal to 1=(4GN ) times the area
of the minimal area extremal surface A~ in the dual spacetime which is homologous to the
region A on the boundary.
For pure AdS, when the CFT region is a ball B, the spatial region
between B
and B~ forms a natural \subsystem" of the gravitational system, in that there exists a
timelike Killing vector B de ned on the domain of dependence D
of
and vanishing
on B~. At the boundary of AdS, this reduces to the vector B appearing in the modular
Hamiltonian (2.4) for B. The vector B gives a notion of time evolution which is con ned
to D . From the CFT point of view, this time evolution corresponds to evolution by the
modular Hamiltonian (2.4) within the domain of dependence of B, which by a conformal
transformation can be mapped to hyperbolic space times time.
For states which are small perturbations to the CFT vacuum state, it was shown in [10]
that the relative entropy for a ball B at second order in perturbations to the vacuum state
corresponds to the perturbative bulk energy associated with the timelike Killing vector B
in D
(known as the \canonical energy" associated with this vector).
This result was extended to general states in [11]. While there are no Killing vectors
for general asymptotically AdS geometries, it is always possible to de ne a vector eld B
that behaves near the AdS boundary and near the extremal surface in a similar way to the
behavior of the Killing vector B in pure AdS. Speci cally, we impose conditions
a
jB =
a
B
;
r
[a b]jB~ = 2 nab;
jB~ = 0 ;
g ! g + L g :
{ 6 {
where nab is the binormal to the codimension two extremal surface B~. Given any such
vector eld, we can de ne a di eomorphism
This represents a symmetry of the gravitational theory, so we can de ne a corresponding
conserved current and Noether charge. The resulting charge H turns out to be the same
for any vector eld satisfying the conditions (2.8){(2.10). It can be interpreted as an
(2.8)
(2.9)
(2.10)
(2.11)
the ow (2.11) in the phase space formulation of gravity. The main result of [11] is that
the CFT relative entropy for a state j i comparing the reduced density matrix
vacuum counterpart (vac) is equal to the di erence of this gravitational energy between
B to its
S( Bjj B
(vac)) = H (M )
H (AdS) :
where L is the Lagrangian density and is de ned by
J = (L g)
L ;
L(g) = d ( g) + E(g) g :
We will review the derivation of this identity in the next section when we generalize it to
our case.
To write H explicitly, we start with the Noether current (expressed as a dform)
HJEP06(218)5
Thus, for a background satisfying the gravitational equations, we have
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
Here, is any spacelike surface bounded by the HRRT surface B~ and by a spacelike surface
@M on the AdS boundary with the same domain of dependence as B. For a ballshaped
region B, the quantity H is independent of both the bulk surface
(as a consequence of
di eomorphism invariance) and also the spacelike surface @M at the boundary of AdS (as
a consequence of the fact that B de nes an asymptotic symmetry).
The quantity K in the boundary term is de ned so that
(
K) =
As explained in [11], this ensures that the di erence (2.12) does not depend on the
regularization procedure used to calculated the energies and perform the subtraction.
We can rewrite H completely as a boundary term using the fact that onshell, J can
be expressed as an exact form [11]
Here, E(g) are the equations of motion obtained in the usual way by varying the action.
The Noether current is conserved o shell for Killing vector elds and onshell for any
vector eld ,
Then, up to a boundary term, the energy H is de ned in the usual way as the integral of
the Noether charge over a spatial surface:
dJ = E(g) L g:
Z
K:
J = dQ :
{ 7 {
This shows that the de nition of H is independent of the details of the vector eld
in
the interior of . In our derivations below, it will be useful to have a di erential version
of this expression that gives the change in H under onshell variation of the metric. By
combining (2.19) with (2.17), we obtain
The interpretation of H as a Hamiltonian for the phase space transformation
associated with (2.11) can be understood by recalling that the symplectic form on this phase
space is de ned by
HJEP06(218)5
Z
H =
( Q
)
where the dform ! is de ned in terms of
In terms of ! we have that for an arbitrary onshell metric perturbation
!(g; 1g; 2g) = 1 (g; 2g)
2 (g; 1g) :
To begin, we choose a bulk vector eld
satisfying
a
jA^ =
a
A
;
r
[a b]jA~ = 2 nab;
jA~ = 0:
{ 8 {
(2.20)
(2.21)
(2.22)
(2.23)
(3.1)
(3.2)
(3.3)
(3.4)
This amounts to the usual relation dH = vH
between a Hamiltonian (in this case H )
and its corresponding vector eld (in this case L g) via the symplectic form
.
3
Bulk interpretation of relative entropy for general regions bounded on
a lightcone
Consider now a more general spacelike CFT subsystem A whose boundary lies on some
lightcone. In this case  unless the boundary is a sphere  there is no longer a conformal
Killing vector de ned on the domain of dependence region DA and we cannot write the
boundary modular Hamiltonian as in (2.4) where the result is independent of the surface
B^. Nevertheless, we have a similar expression (2.7) for the modular Hamiltonian as a
weighted integral of the CFT stress tensor over the lightcone region A^ (shown in gure 1).
Thus, making use of the formula (2.3) for relative entropy, together with the holographic
entanglement entropy formula and the holographic dictionary for the stressenergy tensor,
we can translate the CFT relative entropy to a gravitational quantity. In this section, we
show that this can again be interpreted as an energy di erence,
S( Ajj vAac) = H (M )
H (AdS)
for an energy H
extremal surface A~.
associated with a bulk spatial region
bounded by A^ and the bulk
We will now evaluate
H for this vector eld starting from (2.20) and
nd that it
matches with a holographic expression for the change in relative entropy. First, we evaluate
the part at the AdS boundary. Explicit calculations in the FG gauge, which are done in
appendix A, show that
where
was de ned in the previous section and
Q
jz!0 =
d
Using the standard holographic relation between the asymptotic metric and the CFT stress
tensor expectation value, we obtain
Z
^
A
Here, HA^ is the boundary modular Hamiltonian for the region A, so this term represents
the variation in the modular Hamiltonian term in the expression (2.3) for relative entropy.
Next, we look at the part of (2.20) coming from the other boundary of , at the
extremal surface. By condition (3.4) we have that
1
integral over Q . Q can be brought into the form 16 r
a b
we obtain the entanglement entropy using the HRRT conjecture,
vanishes on A~ and we are left with the
^ab [16] and by virtue of (3.3)
The argument that the latter two conditions can be satis ed is the same as in [11], making
use of the fact that we can de ne Gaussian null coordinates near the surface A~. To enforce
the rst condition, we will make use of Fe ermanGraham (FG) coordinates for which the
nearboundary metric takes the form
1
ds2 =
z2 (dz2 + dx dx + zd (d)dx dx + O(zd+1))
and choose a vector eld expressed in these coordinates as
=
A + z 1 + z2 2 + : : :
z = z 1z + z2 2z + : : : :
(3.5)
(3.6)
(3.7)
(3.8)
(3.10)
(3.11)
(3.12)
Combining both contributions to (2.20), we have that
Z
~
A
1
4GN
Z
~
A
Q =
= S:
where the variation corresponds to an in nitesimal variation of the CFT state. Integrating
this from the CFT vacuum state up to the state j i, we have that
Thus, we have established that for a boundary region A with @A on a lightcone, the CFT
relative entropy is interpreted in the dual gravity theory as an energy associated with the
timelike vector eld .
The energy H is naturally associated with a certain spacetime region of the bulk,
foliated by spatial surfaces bounded by the boundary lightcone region A^ and the bulk
extremal surface A~. That such spatial surfaces exist is a consequence of the fact that the
extremal surface A~ always lies outside the causal wedge of the region A (the intersection
of the causal past and the causal future of the domain of dependence of A) [17].
4
Perturbative expansion of the holographic dual to relative entropy
(4.2)
(4.3)
In this section, we consider the expression for H in the case where the CFT state is a small
perturbation of the vacuum state so that the density matrix can be written perturbatively
as A = vac +
A
with metric g ( ) = g(0) + g(1) + 2g(2) + : : : .
1 + 2 2 + : : : . In this case, the CFT state will be dual to a spacetime
We recall that relative entropy vanishes up to second order in perturbations; making
use of the expression (2.23), we will check that the gravitational expression for relative
entropy also vanishes up to second order for general regions A bounded on a light cone.
We then further make use of (2.23) to derive a gravitational expression dual to the rst
nonvanishing contribution to relative entropy, expressing it as a quadratic form in the rst
order metric perturbation.
4.1
Light cone coordinates for AdS
It will be convenient to introduce coordinates for AdSd+1 tailored to the light cone on
which the boundary of A lies. Starting from standard Poincare coordinates with metric
ds2 =
1
z2 dz2
dt2 + d~x2 ;
(t; ; ) = (t; ; 1; : : : ; d 2) centered at ~x = 0 and de ne
= t
we assume that the point p whose light cone contains @A is at ~x = z = 0 and t =
where 0+ is an arbitrary constant. On the AdS boundary, we introduce polar coordinates
+ =
0+ and some function
=
( i). With
these coordinates, the vector eld (2.6) de ning the boundary modular ow takes the form
and the modular Hamiltonian (2.7) may be written explicitly as
HA = 4
d
d
Z Z 0
+
( i)
d 1
+
0
( i)
( i)
T
:
For the choice ( i) =
0+ the region A is a ball of radius 0+ centered at the origin on the
t = 0 slice and the expression reduces to the usual expression for a modular Hamiltonian
of such a ballshaped region.
jA^ =
0
+
0
)(
2
In the bulk, we similarly de ne polar coordinates (t; r; ; 1; : : : ; d 2) where ( ; z) =
r(cos ; sin ) and de ne r
t
r so that the bulk light cone of the point p is r+ =
We will see below that for pure AdS, the extremal surface A~ lies on this bulk light cone
+
0
on a surface that we will parameterize as r
The AdSd+1 line element in these coordinates reads
( ; i), where ( = 0; i) is the function
ds2 =
1
sin2
4dr+dr
(r+
r )2 + d 2 + cos2 gij d id j ;
where gij is the metric on the unit d
In this section, we derive an analytic expression for the extremal surface A~ in pure AdS
whose boundary is the region @A on the lightcone of p. This will be useful in giving more
explicit expressions for the relative entropy at leading order in perturbations.
We choose static gauge, parameterizing the surface using the spacetime coordinates
and i and describing its pro le in the other directions by
( ; i). The equations which
determine its location are
8p
ab
sin2 (r+
!
:
(4.4)
HJEP06(218)5
(4.5)
(4.6)
+
0
(4.7)
The solution which corresponds to ballshaped entangling surfaces is well known to be
located at 2 + z2 = const. In order to obtain the solution for entangling surfaces of
arbitrary shape (but still on a lightcone) we substitute the expression for the induced
metric and separate the equation for r into
1
1
f ( ; i)
=
1
pg
: (4.8)
the lightcone r+ =
surface is
0+ and r
Let us make the ansatz that even away from the boundary the extremal surface lives on
=
( ; i). The induced metric of this codimension two
ab =
1
sin2
a b + cos2 gij ai bi ;
where a; b 2 f ; 1; : : : ; d 2g and i; j 2 f
g
1; : : : ; d 2 . This metric is independent of r ;
we will see in section 5 that this is related to the Markov property of CFT subregions with
boundary on a lightcone.
Since the induced metric is independent of r , the left hand side of the equations of
motion (4.5) vanishes and we can see from the right hand side that the ansatz r+ =
solves the equations. The remaining equation for f ( ; i)
+
0
( ; i) reads
p
sin2
ab
1
!
:
To x the constants in (4.10) it helps to use intuition from the solutions in the case where
the boundary of a subregion is located on a nullplane instead of a lightcone (see appendix
B). In that case it is clear that e ects from perturbations away from a constant entangling
surface on the extremal surface die o as z ! 1. Under a transformation which maps the
Rindler result to a ballshaped region, the distant part of the extremal surface corresponds
to
=
=2. Consequently, we require that hn( =2) ! 0 for n
1 and hn( =2) = 1 for
n = 0. At the same time, for
! 0 we need that hn( ) is constant and di erent from zero.
These constraints are easily solved with c1 = 0; c2 = 1. Introducing a normalization factor
to ensure that hn(0) = 1, we are left with
hn( ) = cosn
( d+2n ) (
d 1+n )
2
( 2
d 1 + n) ( )
2
d 2F1
n
2
1 n d
; ;
2
1 + 2n
2
; cos2
:
r
=
( ; i) with
In conclusion this shows that extremal surfaces in the bulk are located at r+ = 0+ and
+
( ; i) = 0
1
C0 + P1
n=1
Pl Cn;lhn( ) ln( i) :
Here, n runs over spherical harmonics in d
2 dimensions and l over their respective
degeneracy. They intersect the boundary at
Here, we followed our conventions and used indices i; j for the angular coordinates i. If
we write f1 = h( ) ( i) we nd that the left hand side can be solved if ( i) is a spherical
harmonic. In d
2 dimensions, the eigenvalues of the Laplacian on Sd 2 are given by
n(3
d
n) for the nth harmonic. Every level n has a corresponding set of degenerate
eigenfunctions ln with l = 1; : : : ; 2n+d 3 n+d 4 [18]. The left hand side reads
n n 1
This di erential equation can be solved in terms of hypergeometric functions,
h( ) = c1 cos3 d n
2F1
+ c2 cosn
2F1
n
2
2
d
2
1 n d
; ;
2
Thus, the constants Cn;l are determined in terms of the function parameterizing the
boundary surface by performing the spherical harmonic expansion
As a simple example, one choice of surface involving only the n = 1 harmonics for the
AdS4 case takes the form
+
( ) = 0
+
;
( ) = 0
+
2 0+p1
1 +
cos
2
;
and correspond to ballshaped regions in a reference frame boosted relative to the original
one by velocity
in the xdirection.
+
( i) = 0
C0 + P1
n=1
1
Pl Cn;l ln( i) :
+
0
1
( i)
1
n=1 l
= C0 + X
X Cn;l ln( i) :
4.3
Our next step is to provide an explicit expression for the vector eld on the extremal surface
which obeys equations (3.2){(3.4), such that the quantity H is dual to relative entropy.
Using (4.2), the explicit form of equation (3.2) is
jA^ =
= n n
Equation (3.3) requires knowledge of the unit binormal
but thanks to the knowledge about the expression for the extremal surface which we found
in the preceding section it is possible to calculate it explicitly. Here, n1;2 denote two
orthogonal normal vectors to the RT surface. The calculation is delegated to appendix C.
The nonzero components of the unit binormal read
n
+
= g
+ ;
n
a
=
where a again runs over coordinates ( ; i). One possible choice of a vector eld satisfying
the boundary conditions given by equations (3.3) is:4
(4.16)
(4.17)
(4.18)
(4.19)
=
2 ( 0+
vector
in the case when
component (along
the lightcone) is nonzero, and this has the same qualitative behavior as the vector
on the
boundary lightcone. It is immediately clear that conditions (3.2) and (3.4) are satis ed.
It is also straightforward to verify the condition involving the unit binormal using the fact
that for a torsion free connection we have r
r
Calculating the Lie derivative of the metric with respect to this vector eld gives zero
on the light cone r+ = 0+ but not away from the light cone. This is in contrast to the case of
a ballshaped region, where the Lie derivative vanished everywhere inside the entanglement
wedge.
4Upon expanding the sums in equation (4.19) it looks like the i components of the vector eld diverge
as
! 2 and for d > 3 as i
! 0;
due to the metric on the S
d 2 sphere. However, these divergences
can be shown to be mere coordinate singularities: from equation (4.11) we see that @i
cos . Hence
the i components of the vector eld go only as cos 1 . This happens as a consequence of the coordinate
singularity at
=
=2 in polar coordinates which can be removed by going into Poincare coordinates (t; z; ~x).
Similar arguments also hold for singularities due to the S
d 2 metric. Coordinate independent quantities
like the norm of the spatial part of the vector eld remain nite as can be seen from inspecting the metric.
4.4
To write an explicit perturbative expression for
H , we begin with the onshell result
1
2
1
2
1
2
= g g g
g g g
g g
g
g g g
+
g g g : (4.22)
Z
H =
Here, the symplectic dform ! is explicitly given by
d
d
At second order, we have
d
2
d 2 S(g( )jjg0)j =0 =
Z
d
d
d
d
and ^ depend on the metric they will have a series expansions in
when we express the metric as a series. Also in this case we will use sub or superscripts
in parenthesis to indicate the order of the term in . Here and in the following we will use
to denote covariant derivatives with respect to g (0) = g(0).
It will be convenient for us to choose a gauge for the metric perturbations such that
the extremal surface stays at the same coordinate location for any variation of the metric.
It was shown in [16] that this is always possible. In this case, we have at rst order
H(1) =
Z
d
d
We will see in the next section that this vanishes, in accord with the general vanishing of
relative entropy at rst order (also known as the rst law of entanglement).
1
2
=0
:
(4.20)
(4.23)
(4.24)
(4.25)
(4.26)
We will calculate this more explicitly in section (4.4.2).
4.4.1
Vanishing of the rst order expression
In this section, we demonstrate that our gravitational expression for the relative
entropy vanishes for rst order perturbations as required. Expanding the rst order
expression (4.23) for ! yields
Z
d
d
where repeated lower case letters a; b imply summation over angular coordinates ( ; i).
Using the de nition of the Lie derivative
and the fact that since gab is independent of r
all Christo el symbols of the form
vanish at leading order, the problem reduces to a problem of only the angular coordinates.
Substituting the general form of a from equation (4.19) and using that ga(0b) =
a(0b) we end
1
0
!
g(0) and g(1) are the bulk metric and its perturbation and a(0b), a(1b) are the induced metric
and the induced metric perturbation, respectively. This expression is proportional to the
equation for an extremal surface, equation (4.7), and therefore vanishes.
If we drop the assumption that the Einstein equations are satis ed, one can show
that the rst law of entanglement entropy implies that the Einstein equations hold at rst
order around pure AdS. This was done in [15] where only ballshaped CFT subregions were
considered. Utilizing more general subregions bounded by a lightcone does not yield new
(in)equalities at rst order.
4.4.2
Relative entropy at second order
We will now provide a more explicit expression for the leading perturbative contribution to
relative entropy, which appears at second order in the perturbations. Starting from (4.24)
and using our explicit expression for !, we obtain four potentially contributing terms,
We obtain
up with
d
2
The rst and third terms vanish because of our rst order results of section 4.4.1. The
last term is reminiscent of the standard canonical energy associated with the interior of the
entanglement wedge, except that is no longer a Killing vector. The non zero contributions
take the form
(2)H =
Z
Here, a; b; c; d run over angular coordinates, ; run over all coordinates. Note that
although we are calculating relative entropy at second order, the expression only depends on
rst order metric perturbations. Due to the fact that
is no longer a Killing vector eld,
=0
ab
(4.27)
(4.28)
(4.29)
(4.30)
HJEP06(218)5
we appear to have a contribution in addition to the rst term which appears for the case
of ballshaped regions.
However, we have not yet imposed the HollandsWald gauge condition on the rst
order metric perturbations, for which the coordinate location of the extremal surface is the
same as in the case of pure AdS. We have additional gauge freedom on top of this, and
it may be that for a suitable gauge choice, the nal term in the expression above can be
eliminated. We have checked that this is the case for a planar black hole in AdS4. We
discuss this, as well as the procedure of choosing the HollandsWald gauge condition in
more detail in appendix D.
5
Holographic proof of the Markov property of the vacuum state
In [14] it was pointed out that the vacuum states of subregions of a CFT bounded by curves
=
A and
=
B on the lightcone + = 0+ saturate strong subadditivity, i.e.
HJEP06(218)5
SA + SB
SA\B
SA[B = 0:
This is also known as the Markov property. Moreover, even for CFTs deformed by relevant
perturbations, the reduced density matrices for regions A and B describe Markov states
if A and B have their boundary on a nullplane. In its most general form the proof used
that the modular Hamiltonians for such regions obey
HA + HB
HA\B
HA[B = 0;
which can be proven using methods of algebraic QFT. In this section we will give a
holographic proof of the Markov property which uses the RyuTakayanagi proposal for
entanglement entropy. We will start with the proof for a subregion of a deformed CFT with
boundary on a nullplane and after that also show the property for subregions of CFTs
with boundary on a lightcone.
5.1
The Markov property for states on the nullplane
The vacuum state of a deformed CFT is dual to a geometry of the form
ds2 = f (z)dz2 + g(z)( 2dx+dx + dx?dx? ):
and x+ = x+(~x?; z) simpli es the equation to
An undeformed CFT corresponds to the special case f (z) = g(z) = z12 . The entanglement
entropy of a subregion A can then be calculated using the RT prescription, following the
same steps as in section B. We assume that the boundary @A is described by x
= const
and x+ = x+(~x?). To describe the corresponding extremal surface we go to static gauge,
where z and x? are our coordinates and x (z; x?) is the embedding. The ansatz x
= const
(5.1)
(5.2)
(5.3)
(5.4)
The relevant solution to this equation in the case of pure AdS is discussed in appendix B
and is given by
Here, Kd=2 is the modi ed Bessel function of the second kind and the coe cients aki are
given in terms of the entangling surface x+(0; xi ) as
More generally, the induced metric on the extremal surface in the bulk is
?
ak =
Z d
d 2x
ds2 = f (z)dz2 + g(z)(dx?dx? )
(5.5)
(5.6)
(5.7)
and independent of the embedding x+(~x?; z). Thus, it is clear that the areas of all extremal
surfaces ending on x
= const are the same, potentially up to terms which depend on
how the area of the extremal surface is regularized as we approach the boundary. The
standard prescription given by cutting o
z at some distance
away from the boundary
gives a universal cuto term for all such extremal surfaces and therefore the entanglement
entropies for all regions with boundary on x
are identical and strong subadditivity is
saturated. Our argument is an explicit version of very similar arguments which have been
used to show the saturation of the Quantum Null Energy condition [19].5
5.2
The Markov property for states on the lightcone
If we consider an arbitrary region on the lightcone we expect the Markov property to hold
for undeformed CFTs, since the lightcone is conformally equivalent to the nullplane. The
solution for an extremal surface in pure AdS ending on a lightcone at the boundary was
already discussed in section 4.2. Consider the case where we have two di erent entangling
surfaces given by
=
A( i) and
=
B( i). We have seen before that the metric
on the extremal surface is in fact r independent. However, again the dependence on the
entangling surface can enter through regularization of the integral and would show up in
the cuto dependent term.
In the coordinates of our choice ; i the divergent term in the area comes from the
integral over . Following the standard way of regulating the surface integral we introduce
a cuto
z = , which translates into cutting o the integral at
= r
. From this is
follows that if we choose the canonical way of regulating the entropy, the
integral runs
from ( 0+2
)
to =2.
The entropy which is proportional to the area term can now be calculated using the
explicit form of the induced metric, equation (4.6), and is given by
Z p
=
Z
d
Z =2
d
cosd 2
sind 1 :
(5.8)
5We thank Adam Levine for pointing this out to us.
(5.9)
A
(5.10)
HJEP06(218)5
The only way the shape of the entangling surface appears is through the cuto , i.e. the
surface area can be expanded as
A =
0
X
where the coe cients cn are the same for all entangling surfaces. In the light of
equation (5.8) saturation of strong subadditivity for two regions on a lightcone de ned by
and
B is guaranteed if
Z
d
( 0+
A( i)) + ( 0+
max( 0+
A( i); 0+
B( i))
B( i))
min( 0+
A( i); 0+
B( i))
= 0;
which is trivially pointwise true. This again shows that strong subadditivity is saturated,
or in other words, reduced density matrices for regions on the lightcone describe Markovian
states. For more details on the form of the coe cients cn in the expansion, see [20].
The authors of [14] also speculated about the possibility of introducing a cuto to
regulate the area of extremal surfaces such that the area of the extremal surfaces of subregions
on the lightcone are all exactly equal. The previous discussion explicitly shows that
choosing to introduce a cuto
=
instead of z =
realizes such a regularization procedure in
which all entanglement entropies for regions on the lightcone are in fact the same.
6
Discussion
The results of this paper imply that for any classical asymptotically AdS spacetime arising
in a consistent theory of quantum gravity, the energy
H
must be positive and must not
decrease as we increase the size of region A. It would be interesting to understand if it
is possible to prove this result directly in general relativity, by requiring that the matter
stressenergy tensor satisfy some standard energy condition.
It may be useful to point out that there is a di erential quantity whose positivity
implies all the other positivity and monotonicity results considered here. If we consider a
deformation of the region A by an in nitesimal amount v( ), where v is some vector eld
on @A pointing along the lightcone away from p, the change in relative entropy to rst
order must take the form
S( Ajj vAac) =
Z
v( )SA( )
(6.1)
The monotonicity property implies that the quantity SA( ) must be positive for all A and
all .6 It would be interesting to make use of our results to come up with a more explicit
expression for the gravitational analogue of the quantity SA( ). One approach to providing
a GR proof of the subsystem energy theorems would be to prove positivity of this.
6A special case of this positivity result was utilized in the proof of the averaged null energy condition
in [21].
The Markov property discussed in section 5 suggests that it should be interesting to
consider (for general states) the gravitational dual of the combination S(A) + S(B)
B)
S(A \ B) of entanglement entropies for regions A and B on a lightcone. Since strong
subadditivity is saturated for the vacuum state, this gravitational quantity will vanish for
pure AdS, but must be positive for any nearby physical asymptotically AdS spacetime
according to strong subadditivity. Thus, strong subadditivity for these regions on a light
cone will lead to a constraint on gravitational physics that appears even when considering
small perturbations away from AdS. For twodimensional CFTs, this quantity was already
considered previously in [5, 8]; the analysis there suggests that this gravitational constraint
takes the form of a spatially integrated nullenergy condition. See [2] for some additional
discussion of gravitational constraints from strong subadditivity.
Acknowledgments
We would like to thank Alex May for helpful discussions. This work was supported in part
by the Natural Sciences and Engineering Research Council of Canada and by the Simons
Foundation. DN is supported by a UBC Four Year Doctoral Fellowship.
A
Equivalence of H on the boundary and the modular Hamiltonian
In this appendix we will show that H reduces to the modular Hamiltonian on the boundary,
even in the case of a deformed entangling surface. We take the in nitesimal di erence
between pure AdS and another spacetime that satis es the linearized Einstein's equations
around pure AdS, i.e. we want to calculate
Q
boundary. We can nd in the appendix of [11] that
on a constant z slice near the
=
1
16 GN
^ab
1
2
gac
rc
b
gcr
c a b + c b ga
c
r
b
rc gca + b a gcc : (A.1)
r
The next step is to expand the sum over a and b. As we approach the boundary we consider
volume elements on constant z slices and thus the term involving the volume element ^
vanishes. In Fe ermanGraham gauge ( gzc = 0) we nd
given in equation (3.6).
tion (A.2) are
Now all we need to do is nd the leading order behaviour near z = 0. To this e ect we
assume that the vector elds have a asymptotic expansion near the conformal boundary
We also take gab = z
d 2 (adb) + zd 1 (d+1) + : : :. The leading order terms of
equa
ab
Q
Q
(A.2)
g
:
(A.3)
=
1
16 GN
+
1
16 GN
^ z
1
2
g r
z
^ z
g
r
z
1
2
r
z g
g r
c
r
z +
gcz +
r
z g
rc gcz
z
r
g
+ z
r
d
16 GN
^ z
(d)
zd+1 + : : : = O(1);
The HRRT surface ending on the nullplane
In order to derive the HRRT surface which ends on a curve located on a boundary
nullplane, we split the coordinates into x
= t x (here x is the spatial direction parallel to the
nullplane), boundary directions xi orthogonal to the nullplane, and the bulk coordinate
z. The metric on the Poincare patch in these coordinates is
(A.4)
(A.5)
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
1
;
We choose static gauge for the coordinates on our extremal surface, such that x
=
x (z; xi?). The entangling surface on the boundary is then given by x
The equations which determine the embeddings x (z; xi?) are given by
= x (0; xi?).
where the induced metric is denoted by ab. Having the extremal surface ending on a
boundary nullplane means that either x+ or x
are constant. Without loss of generality,
we choose x
= x
0 = const. This reduces the two equations (B.2) to a single equation for
x+(z; xi?). Making the ansatz x+(z; xi?) = hk(z)gk(xi?) we can separate the equation into
where we use the fact that for a CFT traceless stressenergy tensor implies that
and ^ z = O z (d+1) . Finally, employing the relation between the metric perturbation in
FG coordinates and the stressenergy tensor,
and the de nition of given in section 2 we arrive at
h
T i =
d
16 GN
(d)
z=0
Q
=
h
T i
+ O(z):
?gk(xi?):
The general solutions for the functions hk(z) and gk(xi?) are given by
gk(x?) = akieikixi?;
hk(z) = ckzd=2Id=2(zk) + dkzd=2Kd=2(zk);
xi. I and K
where k = jkij and xi?ki denotes the Euclidean inner product between the vectors ki and
denote the modi ed Bessel functions of rst and second kind, respectively.
We also de ne h0 = limz!0 hk(z). It turns out that we do not want the full solution for
hk. Intuitively, it is clear that the e ect of deformations of the entangling surface on the
boundary should die o as z ! 1. At the same time we also require that the shape of the
extremal surface is uniquely determined by boundary conditions. The asymptotic behavior
of hk as z ! 1 and z ! 0 is
z!1
lim hk(z) = ck
zli!m0 hk(z) = dk2 2
1
2 k
r
We can only ful ll above requirements if we set ck = 0. Hence any extremal surface ending
The normalization is chosen such that
determines ak in terms of the entangling surface x+(0; x?).
C
Calculation of the binormal
The binormal n
is de ned as
( ; i)) = 0. A convenient set of tangent vectors is given by
where n1 and n2 are orthogonal
1 normalized normal vectors to the extremal surface.
To calculate them start by calculating the d
1 tangent vectors to the surface which will
g
+
0 ) = 0 and
and so on for all i. It is easy to see that these vectors form an orthonormal basis on the
RyuTakayanagi surface. Requiring that n1 and n2 are orthogonal to all tangent vectors,
g n1;2ta = 0. This requirement is ful lled by choosing
n1+;2 = g
+ ;
a
n1;2 =
where a stands again for all angular components. The condition that n1 and n2 be
orthogonal and normalized to +1 and
1, respectively, is obeyed provided we choose
1
2
n
1 =
(1
n
2 =
1
2
One can check that the only nonzero components of the binormal are given by:
n
+
= g
+ ;
n
a
=
(B.6)
(B.7)
(B.8)
(B.9)
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
2
~
A
(0) ab
~
A
:
As a warmup consider a ballshaped entangling surface with a corresponding extremal
surface at r+ = 0+; r
=
0+ placed in a planar black hole background,
HollandsWald gauge condition
as a perturbation of pure AdS, we can choose a gauge where g(1a)j
= 0 = g(1)
j
In this appendix, we argue that for the example of a planar black hole in AdS4, considered
which at
the same time is compatible with HollandsWald gauge. In this case, the nal term in our
second order expression (4.30) for the relative entropy vanishes.
HollandsWald gauge is determined by requiring that the extremal surface in the
deformed spacetime sits at the same coordinate location than the extremal surface in the
undeformed spacetime. In particular this means that
( ; );
r
:
The requirement that also after a perturbation of the metric the extremal surface A~ sits at
its old coordinate location translates into
(D.1)
(D.2)
(D.3)
(D.4)
(D.5)
(D.6)
(D.7)
(D.8)
(D.9)
(D.10)
(1
zd)dt2 +
dz2
(1
zd)
+ dx2 ;
ab (1)
(0) ab
~
A
at leading order in . The equations for the extremal surface now become at rst order
We can use the symmetry of the perturbation under time translations and regularity at the
boundary to
nd a vector eld v that generates a di eomorphism g ! Lvg which locates
the extremal surface in the perturbed geometry at the same coordinate location as the
extremal surface in the unperturbed geometry.
This di eomorphism brings the metric perturbation into the form
ds2 =
8
r )3 cos3 cot d 2:
v+ =
v
=
v =
v = 0:
64
64
64
r )
1 + sin2
sin
sin (1 + sin2 )(r+
r )2;
sin (1 + sin2 )(r+
r )2;
The only nonvanishing components of the metric in the new coordinates are g+ ; g
g . In particular, we have that g(1a) = 0 = g(1) . The main bene t of these coordinates
and
is that equation (D.5) holds automatically. Hence at least for a ballshaped entangling
surface we are in HollandsWald gauge and the extremal surface is located at r
=
It can be see from the metric that lines of constant r
are lightlike and therefore we know
that the new entangling surface still is on the bulk lightcone of a point p at the boundary.
+
0
.
From this we can conclude that the entanglement wedge associated to any region
bounded by a lightcone does not contain any point outside the causal wedge. As we have
seen this is true for ballshaped regions. A deformation of the entangling surface cannot
change this, since the boundary domain of dependence is smaller than that of some
ballshaped region. At the same time, the extremal surface cannot lie within the causal domain
of dependence and therefore we must conclude that the extremal surface also lies on the
lightcone.
This means that the transformations (D.6){(D.9) bring the RT surface to its correction
r+ location. The only additional adjustment we need to make to the coordinate system is
to reparameterize r
around the extremal surface, e.g. by rescaling the r
coordinate in
an angledependent way.
To
nd a solution to the general HollandsWald gauge condition, equation (D.3), we
alter the pluscomponent of the vector
eld, v+ ! v+ + v~+( ; ), around the extremal
surface such that it shifts the extremal surface into its new correct location on the lightcone.
This vector
eld can be chosen such that at the extremal surface A~ it remains constant
along r
and r+ and thus depends only on
and . It should be clear that such a solution
exists, since at the boundary the correction v~+( ; ) vanishes and is smooth everywhere
else. More formally, in this case equation (D.3) reduces to
16
16
( ; ))2) +
( ; ))2
1
cos
~
A
~
A
(D.11)
For small deformations of the ball shaped entangling surface we can write ( ; ) as a
series expansion in the deformations. At rst order, this gives us a linear PDE which can
be solved. Higher orders become inherently nonlinear and thus this equation is in general
very hard to solve. An interesting observation one can make for small n = 1 deformations
of the entangling surface is that the linear order correction is zero.
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[1] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,
Phys. Rev. Lett. 96 (2006) 181602 [hepth/0603001] [INSPIRE].
[2] M. Van Raamsdonk, Lectures on gravity and entanglement, in Proceedings, Theoretical
Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and
Strings (TASI 2015), Boulder CO U.S.A., 1{26 June 2015, World Scienti c, Singapore,
(2017), pg. 297 [arXiv:1609.00026] [INSPIRE].
(2017) 1 [arXiv:1609.01287] [INSPIRE].
08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
entanglement in AdS/CFT, JHEP 05 (2014) 029 [arXiv:1401.5089] [INSPIRE].
HJEP06(218)5
[6] S. Banerjee, A. Kaviraj and A. Sinha, Nonlinear constraints on gravity from entanglement,
Class. Quant. Grav. 32 (2015) 065006 [arXiv:1405.3743] [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].
[INSPIRE].
[8] N. Lashkari, C. Rabideau, P. SabellaGarnier and M. Van Raamsdonk, Inviolable energy
conditions from entanglement inequalities, JHEP 06 (2015) 067 [arXiv:1412.3514]
[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].
[10] N. Lashkari and M. Van Raamsdonk, Canonical energy is quantum Fisher information,
JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].
[11] N. Lashkari, J. Lin, H. Ooguri, B. Stoica and M. Van Raamsdonk, Gravitational positive
energy theorems from information inequalities, PTEP 2016 (2016) 12C109
[arXiv:1605.01075] [INSPIRE].
[12] V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement
entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
[13] H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement
entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
[14] H. Casini, E. Teste and G. Torroba, Modular Hamiltonians on the null plane and the Markov
property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
[15] 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].
[16] S. Hollands and R.M. Wald, Stability of black holes and black branes, Commun. Math. Phys.
321 (2013) 629 [arXiv:1201.0463] [INSPIRE].
[17] 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].
[18] C.R. Frye and C.J. Efthimiou, Spherical harmonics in p dimensions, arXiv:1205.3548
[INSPIRE].
[19] J. Koeller, S. Leichenauer, A. Levine and A. ShahbaziMoghaddam, Local modular
hamiltonians from the quantum null energy condition, Phys. Rev. D 97 (2018) 065011
[arXiv:1702.00412] [INSPIRE].
halfspaces and the averaged null energy condition, JHEP 09 (2016) 038 [arXiv:1605.08072]
[3] M. Rangamani and T. Takayanagi , Holographic entanglement entropy, Lect. Notes Phys . 931 [4] D.D. Blanco , H. Casini , L. Y. Hung and R.C. Myers , Relative entropy and holography , JHEP [5] S. Banerjee , A. Bhattacharyya , A. Kaviraj , K. Sen and A. Sinha , Constraining gravity using [20] H. Casini , E. Teste and G. Torroba, All the entropies on the lightcone , arXiv: 1802 . 04278 [21] T. Faulkner , R.G. Leigh , O. Parrikar and H. Wang , Modular hamiltonians for deformed