#### Entanglement entropy from one-point functions in holographic states

Received: May
Entanglement entropy from one-point functions in
Matthew J.S. Beach 0 1
Jaehoon Lee 0 1
Charles Rabideau 0 1
Mark Van Raamsdonk 0 1
0 6224 Agricultural Road , Vancouver, BC, V6T 1W9 , Canada
1 Department of Physics and Astronomy, University of British Columbia
For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk elds. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the rst-order result given by the entanglement rst law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the rst-order result in arXiv:1509.00113.
AdS-CFT Correspondence; Gauge-gravity correspondence
1 Introduction Background 2 3
4
2.1
2.2
3.1
3.2
4.1
4.2
4.3
4.4
Conformal transformations of the vacuum state
Entanglement entropy of excited states
Perturbative expansion
Excited states around thermal background
Second-order contribution to entanglement entropy
Stress tensor contribution: direct calculation for CFT2
5
Auxiliary de Sitter space interpretation
A Direct proof of the positivity
B Rindler reconstruction for scalar operators in CFT2
1
Introduction
In holographic conformal eld theories, states with a simple classical gravity dual
interpretation have a remarkable structure of entanglement: according to the holographic
entanglement entropy formula [1{3], their entanglement entropies for arbitrary regions (at
leading order in large N ) are completely encoded in the extremal surface areas of an
asymptotically AdS spacetime. In general, the space of possible entanglement entropies
(functions on a space of subsets of the AdS boundary) is far larger than the space of
possible asymptotically AdS metrics (functions of a few spacetime coordinates), so this property
of geometrically-encodable entanglement entropy should be present in only a tiny fraction
of all quantum
eld theory states [4]. It is an interesting question to understand better
which CFT states have this property,1 and which properties of a CFT will guarantee that
families of low-energy states with geometric entanglement exist.
1Even in holographic CFTs, it is clear that not all states will have this property. For example, if j 1i and
j 2i are two such states, corresponding to di erent spacetimes M 1 and M 2 , the superposition j 1i+j 2i
is not expected to correspond to any single classical spacetime but rather to a superposition of M 1 and
M 2 . Thus, the set of \holographic states" is not a subspace, but some general subset.
{ 1 {
For a hint towards characterizing these holographic states, consider the gravity
perspective. A spacetime M
dual to a holographic state j i is a solution to the bulk equations
of motion. Such a solution can be characterized by a set of initial data on a bulk Cauchy
surface (and appropriate boundary conditions at the AdS boundary). The solution away
from the Cauchy surface is determined by evolving this initial data forwards (or backwards)
in time using the bulk equations. Alternatively, we can think of the bulk solution as being
determined by evolution in the holographic radial direction, with \initial data" speci ed
at the timelike boundary of AdS. In this case, the existence and uniqueness of a solution
is more subtle, but the asymptotic behavior of the elds determines the metric at least in
a perturbative sense (e.g. perturbatively in deviations from pure AdS, or order-by-order in
behavior of the various elds.
According to the AdS/CFT dictionary, this boundary behavior is determined by the
one-point functions of low-dimension local operators associated with the light bulk elds.
On the other hand, the bulk spacetime itself allows us to calculated entanglement entropies
(and many other non-local quantities). Thus, the assumption that a state is holographic
allows us (via gravity calculations) to determine the entanglement entropies and other
nonlocal properties of the state (again, at least perturbatively) from the local data provided
by the one-point functions:
j i ! hO (x )i !
asymptotics !
(x ; z) ! entanglement entropies S(A)
(1.1)
where
here indicates all light elds including the metric.2; 3
The recipe (1.1) could be applied to any state, but for states that are not holographic,
the results will be inconsistent with the actual CFT answers. Thus, we have a stringent
test for whether a CFT state has a dual description well-described by a classical spacetime:
carry out the procedure in (1.1) and compare the results with a direct CFT calculation
of the entanglement entropies; if there is a mismatch for any region, the state is not
holographic.4
In this paper, our goal is to present some more explicit results for the gravity prediction
Sgrav(hO i) in cases where the gravitational equations are Einstein gravity with matter and
A
the region is taken to be a ball-shaped region B. We will work perturbatively around the
vacuum state to obtain an expression as a power series in the one-point functions of CFT
2Here, the region A should be small enough so that the bulk extremal surface associated with A should
be contained in the part of the spacetime determined through the equations of motion by the boundary
values; we do not need this restriction if we are working perturbatively.
3Results along these lines in the limit of small boundary regions or constant one-point functions
appeared in [5{9].
4Another interesting possibility is that the one-point functions could give boundary data that is not
for the radial evolution problem obeys certain constraints.
entanglement (1)SB = hHBi, where
HB
log vBac = 2
Z
B
d
the second-order answer; in this case, it is less clear whether the gravity results from (1.1)
should hold for any CFT or whether they represent a constraint from holography. To
obtain explicit formulae at this order, we begin by writing
SB(j i) = SBvac +
hHBi
S( Bjj vBac)
which follows immediately from the de nition of relative entropy S( Bjj vBac) reviewed in
section 2 below. We then make use of a recent result in [11]: to second-order in
perturbations from the vacuum state, the relative entropy for a ball-shaped region in a holographic
state5 is equal to the \canonical energy" associated with a corresponding wedge of the
bulk spacetime. We provide a brief review of this in section 2 below. On shell, the latter
quantity can be expressed as a quadratic form on the space of rst-order perturbations to
pure AdS spacetime, so we have
hHBi
SB =
1
2
E (
;
) + O(
3) :
Rearranging this, we have a second-order version of (1.2):
) + O(
3
)
1
2
E (
Z
;
) + O(
3) :
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
SB(j i) = SBvac + (1)SB + (2)SB + O(
3
)
= SBvac +
= SBvac + 2
hHBi
Z
B
1
2
E (
d
d 1 R2
x
;
r
2
2R
hT00i
{ 3 {
As we review in section 2 below, the last term can be written more explicitly as
Z
E (
;
) =
!( g; $
g)
aTa(b2) b ;
where
is a bulk spatial region between B and the bulk extremal surface B~ with the
same boundary, ! is the \presymplectic form" whose integral de nes the symplectic form
5This second-order relative entropy is known as quantum Fisher information.
matter
elds, and
perturbations
on gravitational phase space, Ta(b2) is the matter stress tensor at second-order in the bulk
is a bulk Killing vector which vanishes on B~. The rst-order bulk
(including the metric perturbation) may be expressed in terms of the
boundary one-point functions via bulk-to-boundary propagators
Z
DB
(x; z) =
K (x; z; x0)hO (x0)i ;
(1.8)
where DB is the domain of dependence of the ball B. Given the one-point functions within
DB, we can use (1.8) to determine the linearized bulk perturbation in
and evaluate (1.7).
The expression (1.6), (1.7), and (1.8) together provide a formal result for the ball
entanglement entropy of a holographic state, expanded to second-order in the boundary
one-point functions.
Explicit results for 1+1 dimensional CFTs. In order to check the general formula
and provide more explicit results, we focus in section 3 on the case of 1+1 dimensional
CFTs, carrying out an explicit calculation of the gravitational contributions to (1.7)
starting from a general boundary stress tensor. We nd the result
with integrals over the entire domain of dependence region.
Recently, in [12] it has been pointed out
that the rst-order result (1)S(x ; R) for the entanglement entropy of a ball with radius
R and center x can be obtained as the solution to the equation of motion for a free scalar
eld on an auxiliary de Sitter space ds2 = LR2d2S ( dR2 + dx dx ) with the CFT energy
density hT00(x )i acting as a source term at R = 0. In section 5, we show that in the 1+1
dimensional case, the stress tensor term (1.10) for the entanglement entropy at
secondorder can also results from solving a scalar eld equation on the auxiliary de Sitter space
if we add a simple cubic interaction term. In an upcoming paper [13], it is shown that
this agreement extends to all orders for a suitable choice of the scalar eld potential. The
resulting nonlinear wave equation also reproduces the second-order entanglement entropy
HJEP06(21)85
near a thermal state in the auxiliary kinematic space recently described in [14].
Including the contributions from matter
elds or moving to higher dimensions, the
expression for entanglement entropy involves one-point functions on the entire causal
diamond DB, so reproducing these results via some local di erential equation will require a
more complicated auxiliary space that takes into account the time directions in the CFT.
This direction is pursued further in [13, 15].
Discussion.
While the explicit two-dimensional stress tensor contribution (1.10) can be
obtained by a direct CFT calculation for a special class of states, we emphasize that in
general the holographic predictions from (1.1) are expected to hold only for holographic
states in CFTs with gravity duals. It would be interesting to understand better whether all
of the second order contributions we considered here are universal for all CFTs or whether
they represent genuine constraints/predictions from holography.6 In the latter case, and for
the results at higher order in perturbation theory, it is an interesting question to understand
better which CFT states and/or which CFT properties are required to reproduce the results
through direct CFT calculations. This should help us understand better which theories
and which states in these theories are holographic.
2
Background
Our holographic calculation of entanglement entropy to second-order in the boundary
onepoint functions makes use of the direct connection between CFT quantum Fisher
information and canonical energy on the gravity side, pointed out recently in [11]. We begin with
a brief review of these results.
2.1
Relative entropy and quantum Fisher information
Our focus will be on ball-shaped subsystems B of the CFTd, for which the the vacuum
density matrix is known explicitly through (1.3). More generally, we can write it as
vac = e HB ;
B
HB =
Z
B0 B
T
;
(2.1)
6There is evidence in [16{18] that at least some of the contributions at this order can be reproduced by
CFT calculations in general dimensions, since they arise from CFT two and three-point functions, though
is the volume form on the surface perpendicular to a unit vector n , and B
is a conformal Killing vector de ned in the domain of dependence region DB, with B = 0
on @B. For the ball B with radius R and center x0 in the t = t0 slice, we have
R
[R2
(t
t0)2
(~x
By the conservation of the current B
T
associated with this conformal Killing vector, the
HJEP06(21)85
integral in (2.1) can be taken over any spatial surface B0 in DB with the same boundary
For excited states, the density matrix
measure of this di erence is the relative entropy
B will generally be di erent than vBac. One
S( Bjj vBac) = tr( B log B)
tr( B log vBac)
=
hHBi
SB ;
where HB is the vacuum modular Hamiltonian given in (2.1), SB =
tr( B log B) is the
entanglement entropy for the region B and
indicates the di erence with the vacuum
B( ) = vac +
B
hHBi) so the leading contribution to relative entropy appears at second-order
in . This quadratic in
1 with no contribution from
2
,
S( B( )jj vBac) =
2
h 1
;
1i vBac + O( 3) ;
h ;
i
1
2
tr
d
d
log( +
)
=0
:
This quadratic form, which is positive by virtue of the positivity of relative entropy, de nes
a positive-(semi)de nite metric on the space of perturbations to a general density matrix
. This is known as the quantum Fisher information metric.
Rearranging (2.4) and making use of (2.6), we have
SB = SBvac +
Z
B0 BhT i
2
h 1
;
1i vBac + O( 3) :
This general expression is valid for any CFT, but the O( 2) term involving the quantum
Fisher information metric generally has no simple expression in terms of local operator
expectation values. However, for holographic states we can convert this term into an
expression quadratic in the CFT one-point functions by using the connection between
quantum Fisher information and canonical energy.
For a one-parameter family of states near the vacuum, we can expand B as
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
Consider now a holographic state, which by de nition is associated with some dual
asymptotically AdS spacetime M . Near the boundary, we can describe M using a metric in
Fe erman-Graham coordinates as
z2
ds2 =
2
`AdS dz2 + dx dx + zd
(x; z)dx dx
(2.9)
where
(z; x) has a nite limit as z ! 0 and
= 0 for pure AdS.
The relative entropy S( Bjj vBac) can be computed at leading order in large N by
making use of the holographic entanglement entropy formula, which relates the entanglement
entropy for a region A to the area of the minimal-area extremal surface A~ in M with
SA
Area(A~)
hHBi term in relative entropy to a gravitational quantity, since
it implies that the expectation value of the CFT stress tensor is related to the asymptotic
behaviour of the metric through [19]
(2.10)
(2.11)
(2.12)
(2.13)
h
T i =
d`dAd1S
Thus, for holographic states, we can write
For a one-parameter family of holographic states j ( )i near the CFT vacuum, the
dual spacetimes M ( ) can be described via a metric and matter elds
= (g; matter)
with some perturbative expansion
g = gAdS +
g1 + 2 g2 + O( 3) ;
matter =
matter + 2
1
matter + O( 3) :
2
By the result (2.8) from the previous section, the second-order contribution to entanglement
entropy is equal to the leading order contribution to relative entropy. This is related to a
gravitational quantity via (2.12). The main result in [11] is that this second-order quantity
can be expressed directly as a bulk integral over the spatial region
the integrand is a quadratic form on the linearized bulk perturbations g1 and
To describe the general result, consider the region
between B and B~ in pure AdS
spacetime, and de ne RB as the domain of dependence of this region, as shown in gure 1.
Alternatively, RB is the intersection of the causal past and the causal future of DB; it
can be thought of as a Rindler wedge of AdS associated with B. On RB, there exists a
Killing vector which vanishes at B~ and approaches the conformal Killing vector B at the
matter.
1
between B and B~ where
boundary. In Fe erman-Graham coordinates, this is
[R2
z
2
(t
t0)2
(~x
(2.14)
R
{ 7 {
t
B
HJEP06(21)85
extremal surface B~.
blue lines indicate the ow of B, and the red lines B. The surface
lies between B and the
The vector B is timelike hence de nes a notion of time evolution within the region RB;
the \Rindler time" associated with this Rindler wedge.
The \canonical energy", dual to relative entropy at second-order, can be understood
as the perturbative energy associated with this time, as explained in [20]. This is quadratic
in the perturbative bulk elds including the graviton, and given explicitly by
E ( g1;
1) = W ( 1; $ B
1)
=
=
=
Z
Z
Z
!full ( 1; $ B
! ( g1; $ B g1) +
!( g1; $ B g1)
1)
Z
Z
!matter ( 1; $ B
1)
In the rst line, W is the symplectic form associated with the phase space of gravitational
solutions on
, and $ B
1 is the Lie derivative with respect to
on
1, the rst-order
perturbation in metric and matter elds. The symplectic form is equal to the integral over
of a \presymplectic" form !full which splits into a gravitational part and a matter part as in
the third line. The matter part can be written explicitly in terms of Ta(b2), the matter stress
tensor at quadratic order in the elds, while the gravitational part ! is given explicitly by
!( 1; 2) =
1
In deriving (2.15) it has been assumed that the metric perturbation has been expressed
in a gauge for which the coordinate location of the extremal surface B~ does not change
(2.15)
(2.16)
{ 8 {
(so that B continues to vanish there), and the vector B continues to satisfy the Killing
equation at B~. Thus, we require that
BjB~( ) = 0;
$ B g( )jB~( ) = 0:
(2.17)
(2.18)
As shown in [20], it is always possible to satisfy these conditions; we will see an explicit
example below.
3
Second-order contribution to entanglement entropy
Using the result (1.7), we can now write down a general expression for the ball entanglement
entropy of a general holographic state up to second-order in perturbations to the vacuum
state, in terms of the CFT one-point functions. According to (2.8) and (2.15), the
secondorder term in the entanglement entropy for a ball B can be expressed as an integral over
the bulk spatial region
between B and the corresponding extremal surface B~, where the
integrand is quadratic in rst-order bulk perturbations.
These linearized perturbations are determined by the boundary behavior of the elds
via the linearized bulk equations. In general, to determine the linearized perturbations
in the region
(or more generally in the Rindler wedge RB), we only need to know the
boundary behavior in the domain of dependence region DB, as discussed in detail in [21].
The relevant boundary behaviour of each bulk eld is captured by the one-point function
of the corresponding operator. We can express the results as
Z
DB
( 1) (x; z)j =
ddx0K (x; z; x0)hO (x0)iCF T
(3.1)
where K (x; z; x0) is the relevant bulk-to-boundary propagator. As discussed in [21{23],
K
should generally be understood as a distribution to be integrated against consistent
CFT one-point functions, rather than a function. Since the expression (3.1) is linear in
the CFT expectation values, the result (1.7) is quadratic in these one-point functions and
represents our desired second-order result.
To summarize, for a holographic state, the second-order contribution to entanglement
entropy in the expansion (2.8) is the leading order contribution to the relative entropy
S( Bjj vBac). This is dual to canonical energy, given explicitly by:
(2)SB =
h 1
;
1i vBac =
E ( 1
;
1) =
!( g1; $ B g1)+
BaTa(b2) b : (3.2)
1
2
and these can be expressed in terms of the CFT one-point functions on DB as (3.1).
Example: CFT2 stress tensor contribution
In this section, as a sample application of the general formula, we provide an explicit
calculation of the quadratic stress tensor contribution to the entanglement entropy for
{ 9 {
In two dimensions, these constraints can be expressed most simply using light-cone
coordinates x
= x
t, where we have
h
T
i = 0 :
Thus, a general CFT stress tensor can be described by the two functions, hT++(x+)i and
h
T
(x ) .
i
Assuming that the state is holographic, there will be some dual geometry of the
form (2.9). According to (2.11), the stress tensor expectation values determine the
asymptotic form of the metric as
++(x; 0) = 8
GN
`AdS
hT++(x+)
i
(x; 0) = 8
GN
`AdS
h
T
(x )
i
Now, suppose that our state represents a small perturbation to the CFT vacuum, so that
the stress tensor expectation values and the asymptotic metric perturbations are governed
by a small parameter :
++(x; 0)
h+(x+)
holographic states in two-dimensional conformal eld theories. This arises from the rst
term in (1.7).
For a general CFT state, the stress tensor is traceless and conserved,
ab = hab + ($V g)ab = hab + raVb + rbVa :
(3.10)
Then the metric perturbation throughout the spacetime is determined by this asymptotic
behavior by the Einstein equations linearized about AdS. Here, we need only the
components in the eld theory directions, which give
1
The solution in our Fe erman-Graham coordinates with boundary behaviour (3.6) is
(+1+)(x; z) =
h+(x+)
(1) (x; z) =
h (x )
with the linearized perturbation
(1) independent of z.
Satisfying the gauge conditions.
We would now like to evaluate the metric contribution to (3.2)
This formula assumes the gauge conditions (2.17) which di er from the Fe erman-Graham
gauge conditions we have been using so far. Thus, we must nd a gauge transformation to
bring our metric perturbation to the appropriate form. In general, we can write
(3.3)
(3.4)
is the desired metric perturbation satisfying the gauge condition, and h is the
perturbation in Fe erman-Graham coordinates (equivalent to
for d = 2).
The procedure for nding an appropriate V and evaluating (3.9) is described in detail
in [11], but we review the main points here. De ning coordinates (XA; Xi) so that the
extremal surface lies at some xed value of XA with Xi describing coordinates along the
surface, the gauge condition (2.17) (equivalent to requiring that the coordinate location of
the extremal surface remains xed) gives
ririVA + [ri; rA]V i + rihiA
1
2 rAhii
while the condition (2.18) that B continues to satisfy the Killing equation at B~ gives
(hiA + riVA + rAVi) jB~ = 0 ;
A
h D
12 ADhCC + rAVD + rDV A
D
DrC V CC
= 0 :
To solve these, we rst expand our general metric perturbation in a Fourier basis.
h (t; x; z) =
Z h + +h^+(k)eikx+ +
h^ (k)eikx i dk ;
with a gauge choice hza(t; x; z) = 0.
For each of the basis elements, we use the equations (3.11), (3.12) and (3.13) to
determine V and its rst derivatives at the surface V . For these calculations, it is useful to
de ne polar coordinates (z; x) = (r cos ; r sin ). Since the gauge conditions are linear in
V , the conditions on V for a general perturbation are obtained from these by taking linear
combinations as in (3.14),
Va(t; x; z) =
Z hV^a+(k)eikx+ + V^a (k)eikx i dk :
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
After requiring Va remain nite at
=
2 , we nd
e ikt
e ikt
k3r2 cos2
k3r2 cos2
e ikt
2 k2 r cos
e ikt
k3r2 cos
i cos(kr) + sin sin(kr)
i
sin(kr)
i sin cos(kr)
(k2r2 cos2
1)eikr sin
2
(k2r2 cos2 sin
+ ikr cos2
+ 2i sin )eikr sin !
2
(2 + k2r2 cos2
2 ikr sin )eikr sin
2 sin(kr)
k3r2
2i cos(kr)
+ 2kr sin +r3k3 sin cos2 +i r2k2 cos2
kr2 + 2 eikr sin
(3.16)
using
where
identity
where
Thus, we have where
[ ; V ]a = b
!(g; ; $ g) = d ( ; X)
!(g; ; $
) = !(g; h; $ h) + d
= (h + $V g; [ ; V ])
($ h; V ) :
E =
Z
!(g; h; $ h) +
(h; V ) :
Z
and we have used that $ g = 0. We can simplify this expression using the gravitational
( ; X) =
1
V^t+(k; t; r; ). The results here give the behavior of V and its derivatives
only at the surface B~ (r = R in polar coordinates). Elsewhere, V can be chosen arbitrarily,
but we will see that our calculation only requires the behavior at B~.
Evaluating the canonical energy.
Given the appropriate V , we can evaluate (3.9)
!(g; ; $
) = !(h + $V g; $ B (h + $V g))
= !(g; h; $ h) + !(g; h + $V g; $[ ;V ]g)
!(g; $ h; $V g)
Finally, choosing V so that it vanishes at B, we can rewrite (3.9) as
In this nal expression, we only need V and its derivatives at the surface B~. Thus, we can
now calculate the result explicitly for a general perturbation. In the Fourier basis, the nal
result in terms of the boundary stress tensor is
where the kernel is
K^2(k1; k2) =
`AdSK3(K
k2). We note in particular that the result splits into a
left-moving part and a right-moving part with no cross term.
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
Transforming back to position space
g ;
(3.26)
where the kernel K2 is symmetric under exchange of x1 and x2 , and has support only on
x
i 2 [ R; R]. Focusing only on the domain of support, we have
Using the relation c = 3`AdS=(2GN ) between the CFT central charge and the gravity
parameters, we recover the result (1.10) from the introduction.
Like the leading order result in (2.8), the integrals can be taken over any surface B0
with boundary @B. The fact that we only need the stress tensor on a Cauchy surface for
DB is special to the stress tensor in two dimensions, since the conservation relations allow
us to nd the stress tensor expectation value everywhere in DB from its value on a Cauchy
surface. For other operators, or in higher dimensions, the result will involve integrals over
the full domain of dependence. We will see an explicit example in the next subsection.
Positivity of relative entropy requires E to be positive which requires the kernel to be
positive semi-de nite. As we show in appendix A, one can demonstrate that the positivity
explicitly, providing a check of our results. An alternative proof of positivity is given in
section 5. As a more complete check, we will show in section 4 that this result can be
reproduced by a direct CFT calculation for the special class of states that can be obtained
from the vacuum state by a local conformal transformation.
3.2
Example: scalar operator contribution
We now consider an explicit example making use of the bulk matter eld term in (1.7) in
order to calculate the terms in the entanglement entropy formula quadratic in the scalar
operator expectation values. The discussion for other matter elds would be entirely parallel.
This example is more representative, since the formula will involve scalar eld expectation
values in the entire domain of dependence DB, i.e. a boundary spacetime region rather
than just a spatial slice. The results here are similar to the recent work in [16{18], but we
present them here to show that they follow directly from the canonical energy formula.
We suppose that the CFT has a scalar operator of dimension
with expectation value
with mass m2 =
(
d) and asymptotic behavior
hO(x)i. According to the usual AdS/CFT dictionary, this corresponds to a bulk scalar eld
where
is a constant depending on the normalization of the operator O. The leading
e ects of the bulk scalar eld on the entanglement entropy (3.2) come from the matter
term in the canonical energy
(x; z) !
z hO(x)i ;
(3.28)
(3.29)
Using the explicit form of B from (2.14) and from (2.2), this gives (for a ball centered
at the origin)
This expression is valid for a general bulk matter eld. For a scalar eld, we have
1
where gab is the background AdS metric and 1 represents the solution of the linearized
scalar eld equation on AdS,
n d 1
o
with boundary behavior as in (3.28). This solution is given most simply in Fourier space,
where we have
1(k; z) =
2
( + 1) Z
(2 )d
k02>~k2
ddk
d
=2 z 2 J
qk2
0
~k2z
hO(k)i ;
(3.33)
(3.30)
(3.31)
(3.32)
(3.34)
(3.35)
(3.36)
(3.37)
k2
0
Z
eik x
~k2
This reproduces previous results in the literature [5, 17].
where
=
d=2, but we can formally write a position-space expression using a
bulk-toboundary propagator K(x; z; x0) as [24, 25]
1(x; z) =
dx0K(x; z; x0)hO(x0)i :
The integral here is over the boundary spacetime, however it has been argued (see, for
example [21, 22]) that to reconstruct the bulk eld throughout the Rindler wedge RB (and
speci cally on
), we need only the boundary values on the domain of dependence region.
We recall some explicit formulae for this \Rindler bulk reconstruction" in appendix B.
Combining these results, we have a general expression for the scalar eld contribution to
entanglement entropy at second-order in the scalar one-point functions,
(2)SBscalar =
`dAd1S Z R dz Z
2
0 zd 1
x2<R2 z2
d
d 1
x (R2
R
z
2
x2)
m2
z2 1
2
where 1 is given in (3.33) or (3.34) .
As a simple example, consider the case where the scalar eld expectation value is
constant. In this case it is simple to solve (3.32) everywhere to nd that
Inserting this into the general formula (3.35), and performing the integrals, we obtain
1(x; z) = hOiz :
(2)SBscalar =
`dAd1S 2
4
hOi2R2
d 2
(
d
2
1
2
(
) (
+ 32 )
d2 + 1)
:
In section 3.1, we used the equivalence between quantum Fisher information and canonical
energy to obtain an explicit expression for the second-order stress tensor contribution to
the entanglement entropy for holographic states in two-dimensional CFTs. This is
applicable for general holographic states, whether or not other matter elds are present in the
dual spacetime (in which case there are additional terms in the expression for entanglement
entropy). In special cases where there are no matter elds, the spacetime is locally AdS and
we can understand the dual CFT state as being related to the vacuum state by a local
conformal transformation. We show in this section that in this special case, we can reproduce
the holographic result (3.27) through a direct CFT calculation, providing a strong
consistency check. We note that the result does not rely on taking the large N limit or on any
special properties of the CFT, so the formula holds universally for this simple class of states.
Our approach will be to develop an iterative procedure to express the entanglement
entropy as an expansion in the stress tensor expectation value for this special class of states.
We evaluate the entanglement entropy for these states from a correlation function of twist
operators obtained by transforming the result for the vacuum state.7 Similarly, the stress
tensor expectation values follow directly from the form of the conformal transformation.
Inverting the relationship between the required conformal transformation and the stress
tensor expectation value allows us to express the entanglement entropy as a perturbative
expansion in the expectation value of the stress tensor. Similar CFT calculations have also
been used recently in [13].
4.1
Conformal transformations of the vacuum state
In two-dimensional CFT, under a conformal transformation w = f (z), the stress tensor
(4.1)
(4.2)
(4.3)
(4.4)
2
000(z) 0(z) +
3 00(z)2 + 3 0(z)2 000(z) + 3 0(z) 00(z)2 +
The CFT vacuum is invariant under the SL(2; C) subgroup of global conformal
transformations. However, for transformations which are not part of this subgroup, the vacuum
state transforms into excited states. The action of the full conformal group includes the
full Virasoro algebra which involves arbitrary products and derivatives of the stress tensor
Id
transforms as
derivative
expanded as
fz +
(z); zg =
000(z)
2
2
Here c is the central charge of the CFT and the inhomogeneous part is the Schwarzian
T 0(w) =
dw
dz
T (z) +
c
12 ff (z); zg :
ff (z); zg
f 000(z)
f 0(z)
3f 00(z)2
2f 0(z)2
:
For an in nitesimal transformation f (z) = z +
(z), the Schwarzian derivative can be
These states capture the gravitational sector of the gravity dual. Other excited states can
be obtained by the action of other primary operators and their descendants. However we
restrict ourselves to the class states that are related to `pure gravity' excitations, which are
the states obtained by conformal transformation of the vacuum state.
We denote the excited state as jf i = Uf j0i where Uf is the action of a conformal
transformation on the vacuum j0i. The expectation value of the stress tensor for the state
perturbed state jf i is
hf jT (z)jf i = h0jUfy T (z) Uf j0i = h0jT 0(w)j0i =
df
dz
where we used that h0jT (z)j0i = 0. The anti-holomorphic component of the stress tensor
T (z) is similarly related to the anti-holomophic part of the conformal transformation f .
To leading order in a conformal transformation near the identity, this equation relates
the conformal transformation to hT (z)i by a third-order ordinary di erential equation.
The three integration constants correspond to the invariance of hT (z)i under the global
conformal transformations. Thus we have an invertible relationship between the conformal
transformations modulo their global part and hT (z)i, at least near the identity.
4.2
Entanglement entropy of excited states
In a two-dimensional CFT, the entanglement entropy can be explicitly computed using the
replica method [27, 28]. The computation can be reduced to a correlation function of twist
operators
, which are conformal primaries with weight (hn; hn) = 2c4 (n
1=n; n
1=n).
The Renyi entropy is
exp (1
n)S(n)
= h +(z1)
(z2)i = (z2
z1) 2hn (z2
z1) 2hn :
(4.6)
The entanglement entropy is obtained by taking the n ! 1 limit of S(n).
Svac = lim S(n) = lim (1
n) 1 log(z2
z1) 2hn (z2
z1) 2hn
n!1
n!1
c
12
=
log
(z2
z1)2(z2
z1)2
For the excited states obtained by conformal transformations z ! w = f (z) the Renyi
entropy is
exp (1
n)Se(xn)
= hf j +(z1)
(z2)jf i
(4.7)
(4.8)
=
df
dz z1
hn df
hn df
hn df
hn
dz z2
dz z1
dz z2
h0j +(z1)
(z2)j0i : (4.9)
Here z1; z2 are the points f (z1) = f (z1) =
R, f (z2) = f (z2) = R. The entanglement
entropy of the excited state is
Sex = lim Se(xn) =
n!1
c
12
log
f 0(z1)f 0(z2)f 0(z1)f 0(z2)(z2
z1)2(z2
z1)2
2 2
:
(4.10)
Therefore the change in entanglement entropy respect to the vacuum state is
S
Sex
Svac =
c
12
+
log
c
12
log
f 0(f 1(R))f 0(f 1
( R))(f 1(R)
By inverting (4.5), the conformal transformation required to reach the state jf i can
be expressed as a function of the expectation value of the stress tensor. Plugging this f
into (4.11), allows us to express the entanglement entropy as a function of the expectation
value of the stress tensor alone, as we set out to do.
In practice, we will invert (4.5) order by order in a small conformal transformation
and express the entanglement entropy as an expansion in the resulting small stress tensor.
The second-order term in this expansion will be the Fisher information metric.
In the following, we will focus on the holomorphic term in (4.5), noting that the
antiHJEP06(21)85
holomorphic part follows identically.8
Perturbative expansion
Consider a conformal transformation perturbation near the identity transformation
w = f (z) = z + f1(z) + 2f2(z) + 3f3(z) +
;
(4.12)
where
is a small expansion parameter.
In this expansion,
1c2 hT (w)i = f1000(w)+ 2
32 f100(w)2
and the entanglement entropy is
12
c
Sex = log
f 0(z1)f 0(z2)(z2
z1)2
2
3f10 (w)f1000(w) + f2000(w) f1(w)f10000(w) +O( 3) ; (4.13)
= log (2R)2
2
+
+ 2
+ O( 3) :
R (f10 ( R) + f10 (R)) + f1( R) f1(R)
R
(f1(R) f1( R))2
4R2
+
f1( R)f10 ( R) + f1(R)f10 (R) + f2( R) f2(R)
R
Linear order. To rst-order in , the stress tensor is given by
8Note that the potential cross-term between left and right moving contributions vanished in the
gravitational computation of (2)S.
(4.15)
so that change in the expectation value of the modular Hamiltonian becomes
hHBi =
dz (R2
z2)f1000(z)
z2)f100(z) + 2 zf10 (z)
f1(z)
R
R
R(f10 (R) + f10 ( R))
(f1(R)
f1( R)) :
From (4.11) we also have that the rst-order change in entanglement entropy is
Comparing with (4.16) we see that the rst law of entanglement holds
(1)S = hHBi :
Second-order. The second-order change in entanglement entropy gives the second-order
relative entropy as the modular Hamiltonian is linear in the expectation value of the stress
tensor. This is the quantum Fisher metric in the state space, which is dual to the canonical
energy in gravity [11]. In this section, we obtain the expression for canonical energy from
the CFT side and
nd an exact match to the results of section 3.1.
Our procedure so far yields the entanglement entropy of a subregion in terms of a
perturbative expansion in small stress tensor expectation value
S =
K1(z)hT (z)i
Z
dz
B 2
+ fz $ zg :
To obtain K2(z1; z2), we need to invert the relationship in (4.13) order by order, the
lower order solutions fi 1; fi 2;
f1 becoming sources for the i-th order solution.
Taking the explicit expression for hT (z)i to simplify solving the di erential equations,
is su cient to extract the Fourier transformed kernel.
The rst-order solution is
f1(z) = F1 + F2z + F3z2 +
12i
c
eik1z
;
where Fi are constants that corresponds to the global part of the conformal transformation
and do not e ect the
nal result. We take these constants to be zero for simplicity. The
second-order solution is
f2(z) =
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
With these solutions, we obtain
K~1(k) =
k2
2 sin (kR) kR cos (kR)
kR
;
as well as
K~2(k1; k2) =
c
where
is a window function with support x 2 [ R; R].
The second-order position space kernel is
Z
K1(z) =
dk K~1(k)e ikz =
R2
z
2
R
W (R; z)
W (R; x)
(sgn (R + x) + sgn (R
x))
2
K2(z1; z2) =
6 2 ((R
cR2
z1)2(R + z2)2
(R + z1)2(R
z2)2
R
R
z2
Taking the inverse Fourier transformation of K~1(k)
This can be obtained by a conformal transformation from the vacuum with
h jT j i =
2
c
this reproduces the kernel for canonical energy in (3.27).
t = 0 slice, z = z = x and our result becomes
This result holds for regions de ned on any spatial slice of the CFT. If we choose the
Changing variables using x1 = x
r, x2 = x + r, the kernel is simply
SE(2E) =
Z
B
dx1
dx2 K2(x1; x2) [hT++(x1)ihT++(x2)i + hT
(x1)ihT
(x2)i] :
The anti-holomorphic part is the same with z ! z, and the cross term vanishes. With the
relation
c =
3`AdS
2GN
K2(x; r) = K2(x; r) =
jrj)2
Excited states around thermal background
A similar analysis can be applied to perturbations around a thermal state with temperature
T =
1. If we denote homogeneous thermal state j i, the stress tensor one-point function is
On top of this transformation, one could also apply an in nitesimal conformal
transformation to obtain non-homogeneous perturbation around thermal state.
A similar computation as the previous section leads to the rst-order kernel
2
K1 (z) =
sinh( 2 R ) sinh
(R
z)
sinh
(R + z)
;
(4.32)
which is the modular hamiltonian of thermal state in 2d CFT.
Furthermore, the second-order kernel is
K2 (z1; z2) =
Consistency check: homogeneous BTZ perturbation
As a check, consider the homogeneous perturbation example, where hT i = hT i = 8GN .9 In
AdS3 this is a perturbation towards the planar BTZ geometry
ds2 =
1
z2 dz2 + (1 + z2=2)2dx2
(1
z2=2)2dt2
in Fe erman-Graham coordinates. Holographic renormalization (2.11) tells us the stress
tensor expectation value of the dual CFT is
1
2
hTtti =
hT i + hT i =
As the black hole corresponds to the thermal state in CFT, the dual state be obtained by
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
the conformal transformation (4.31).
entropy with respect to the vacuum is
First, applying (4.11) for this conformal transformation, the change in entanglement
S =
R2
6G
which matches the previous known results [5, 11].
The linear order equals hHBi as expected from the entanglement rst law.
The second-order term gives the quantum Fisher information or the canonical energy
E =
d
2
d 2
( E
S)
=0
=
R4
45GN
:
Using the formula using the second-order kernel (4.19) and (4.27), we obtain the same
canonical energy
E = 2
d
2
1 Z dz1 Z dz2 K2(x1; x2)hT ihT i
=0
=
R4
45GN
:
9 = 2 2 sets the temperature.
S+ since S follows identically.
iary de Sitter space with metric
In [
12
], it was pointed out that the leading order perturbative expression (1.2) for
entanglement entropy, expressed as a function of the center point x and radius R of the ball B,
is a solution to the wave equation for a free scalar eld on an auxiliary de Sitter space,
with hT00(x)i acting as a source.
It was conjectured that higher order contributions might be accounted for by local
propagation in this auxiliary space with the addition of self-interactions for scalar eld.
In this section, we show that for two-dimensional CFTs, the second-order result (1.10)
can indeed be reproduced by moving to a non-linear wave equation with a simple cubic
interaction to this scalar eld. A slight complication is that we actually require two-scalar
elds; one sourced by the holomorphic stress tensor T++, and the other sourced by the
anti-holomorphic part T
; the perturbation to the entanglement entropy is then the sum
of these two scalars, S = S+ + S , reproducing both terms in (1.10). We will focus on
To reproduce the second-order results for entanglement entropy, we consider an
auxiland consider a scalar eld S+ with mass m2L2dS =
2 and action
ds2dS =
L2dS
R2
The equation of motion is
As shown in [
12
], the rst-order perturbation (1.2) obeys the linearized wave equation
2
rdS
m2
S+(R; x) =
12
cL2dS ( S+(R; x))2 :
2
rdS
m2 (1)S+(R; x) = 0 :
We can immediately check that the second-order perturbation (1.10) is consistent with the
nonlinear equation by acting with the dS wave equation on the second-order kernel (4.27)
2
rdS
m2 K2(x1
x; x2
x) =
24
cL2dS K1(x1
x)K1(x2
x) :
Integration against the CFT stress tensor then gives (5.3).
Alternatively, introducing the retarded10 bulk-to-bulk propagator [29]
GdS( ; x; 0; x 0) =
2 + 0
2
(x
x 0)2
4 0
10These propagators are de ned to be non-zero only within the future directed light-cone. This is
important in reproducing both the support and the exact form of K2(x1; x2).
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
and bulk-to-boundary propagator
!0
KdS( ; x; x 0) = lim
4
lim GdS( ; x; 0; x 0) =
we can show directly that the solution with boundary behavior
2
(x
x0)2
;
δS+
x1
0!
4
Z
(R, x)
δS+
g3
δS+
x2
(5.7)
(5.8)
(5.9)
#2
:
(5.11)
x − R
x + R
cubic interaction given by (5.2). The bold (red) line is the conformal boundary of de Sitter which is
identi ed with a time slice of the CFT. S+ is sourced by the CFT stress tensor on this boundary.
where pjgdSj and the squared expression are manifestly positive and the bulk-to-bulk
propagator (5.6) is positive over the range of integration where (y
x0)2
(R
)2. That this
expression is negative is required by the positivity of relative entropy, since we showed above
that
(2)S represents the leading order perturbative expression for the relative entropy.
Recently, it has been realized that the modular Hamiltonian in certain non-vacuum
states in two dimensional CFTs can be described by propagation in a dual geometry [14]
For R ! 0 gives
at rst-order and
(2)S+(R; x0) =
12 Z
cL2dS dS
S+ =
3 hT++iR2 + O(R3) :
(1)S+(R; x0) =
dx KdS(R; x0; x)hT++(x)i
d 0dx0pjgdSj GdS(R; x0; 0; x0)
dx KdS( 0; x0; x)hT++(x)i
Z
2
; (5.10)
at second-order, where the latter term comes from the diagram shown in gure 2.
The integrals can be performed directly to show that these results match with the
expressions (1.2) and (1.10) respectively.
manifestly negative. More explicitly, we have
A useful advantage of writing the second-order result in the form (5.10) is that it is
(2)S+(R; x0) =
cL2dS
d dypjgdSj
R2 + 2 (x0 y)2 "Z
R
By
dx KdS ( ; y; x)hT++(x)i
matching the kinematic space found previously in [30{33]. We nd that the results of
section 4.4 can be explained by the same interacting theory (5.2) on this kinematic space.
The kinematic space dual to the thermal state is11
ds2 =
4 2L2dS
2 sinh2 2 R
(5.12)
The second-order perturbation to the entanglement entropy from (4.33) obeys the wave
equation (5.3) with the same interactions in this kinematic space.
We could imagine adding additional elds propagating in de Sitter to capture the
contributions to the entanglement entropy from scalar operators discussed in section 3.2.
However, unlike the contribution from the stress tensor, this contribution involves integration of
the one-point functions over the full domain of dependence DB. In higher-dimensions, this
will also be true for the stress tensor contribution. The R = 0 boundary of the auxiliary
de Sitter space does not include the time direction of the CFT, so any extension of these
results to contributions of other operators or higher dimensional cases will require a more
sophisticated auxiliary space. Promising work in this direction is discussed in [13, 15].
Acknowledgments
We thank Michal Heller, Ali Izadi Rad, Nima Lashkari, Don Marolf, Robert Myers, and
Philippe Sabella-Garnier for helpful discussions. This research is supported in part by the
Natural Sciences and Engineering Research Council of Canada, and by grant 376206 from
the Simons Foundation.
A
Direct proof of the positivity
Consider the left moving part of perturbation h+(x+) / T++(x+). The real space h+(x)
must be real valued functions for a perturbation of AdS3. We can expand h+(x) in a Taylor
series h+(x) = P1
n=0 anxn so that the canonical energy is given by
E
X
n
X anam
m
Z Z
B B
dx1dx2 x1nx2mK2(x1; x2)
X
n
m
X anamRn+m+4
An;m :
(A.1)
where the proportionality factor is up to a positive constant and
An;m =
1
(n + m + 3)(n + m + 1) >
8
>
>
<
0
1
(n+1)(m+1)
>
: nmnm(n++n2+)(mm++32)
if n + m odd
if n; m even
if n; m odd
(A.2)
which is clearly non-negative and symmetric in n; m.
11The kinematic space dual to the BTZ black hole was rst described in [30, 31]. The explicit form of
the metric in the coordinates we are using can be found in [14].
To show that the canonical energy is positive, we need to show the matrix M with
entries given by An;m = An 1;m 1
12 is positive de nite. To do so, we will use proof by
induction and Sylvester's criterion which states that a square matrix M is positive de nite
if and only it has a positive determinant and all the upper-left submatrices also have a
positive determinant.
Proof by induction
Suppose that the N
N matrix MN whose components are given by An;m is positive
de nite. Then consider the block matrix constructed as
HJEP06(21)85
MN+1 = AN+1;N+1
MN B !
BT 1
(A.3)
(A.4)
(A.5)
where B is a N -column vector with entries given by Ai;N+1. Since MN is positive-de nite,
it has a positive determinant and all the upper-left submatrices of MN also have a positive
determinant by Sylvester's criterion. To show that MN+1 is positive-de nite, we need only
show it has a positive determinant since all the upper-left submatrices are already known .
The determinant of MN+1 can be evaluated using the formula
det(MN+1) = AN+1;N+1 2 det(MN )
det(MN + BT B)
so it is su cient to show
det(MN + BT B) < 2 det(MN ) :
We denote the eigenvalues of MN + BT B by i
M+Bwhere they are ordered from largest
to smallest
1
M+B
2
M+B
: : :
non-zero eigenvalue is given by
N
M+B: Since BT B is a rank-one matrix, the sole
= Tr(BT B) = PiN=1 Ai;N+1
0. Since BT B is positive
semi-de nite, there exists an upper bound on det(MN +BT B) given by the Weyl inequality
M+B
i
1
2
iM + iwhere i
M are the eigenvalues of MN in order from largest to smallest
: : :
N . We then expand the determinant as
det(MN + BT B) = Y
1
M+B N
M
Y
i=1
iM =
1 +
M
1
det(MN ) :
(A.6)
So it remains to show that 1
0 to complete the proof. The maximum eigenvalue
1M is bound from below by the minimum sum of a column of MN through the
PerronFrobenius theorem (equivalently Gershgorin circle theorem). For the matrix MN , the
minimum sum of a column vector is simply the sum of the N -th column PiN=1 Ai;N since
Ai;j decreases with i and j. Therefore it remains to show
N
i=1
M
i
M+B
B
N
X
i=1
Ai;N
2
Ai;N+1
0 :
(A.7)
12The inelegant notation change is due to conventional matrix notation starting at n = 1, while the
Taylor series starts at n = 0.
X A2i;N
(N+1)=2
X
i=1
(N+1)=2
X
i=1
N=2
i=1
We split this sum up into two cases. The rst case is if N is even. Then we have
since the nal term in Pi(=N1 1)=2 A22i 1;N+1 is zero. Explicitly analyzing the coe cients, we
see that A2i;N
A22i 1;N+1 is always positive for all i 2 f1 : : : N=2g, so clearly the entire
sum is positive. In the case of odd N , the sum becomes
Each term in this sum is also positive, so we have shown
M
1
B
0. The expressions
in (A.8) and (A.9) are not obviously positive, but they reduce to some polynomial equations
which can be shown to be positive. Therefore we've shown det(MN + BT B) < 2 det(MN ),
thus MN+1 is positive-de nite given that MN is. Since M1 is positive-de nite we completed
the proof by induction. The kernel for canonical energy is explicitly positive-semide nite
as required by the positivity of relative entropy.
B
Rindler reconstruction for scalar operators in CFT2
In this appendix we nd an expression for the matter contribution to the second-order
perturbation to the entanglement entropy of a ball B using Rindler reconstruction so as to
only use the one-point functions of the scalar operator in the domain of dependence DB.
We specialize to two dimensional CFTs in order to obtain a more explicit expression which
can be compared to the gravitational contribution (1.10). Further discussions of Rindler
reconstruction can be found in the literature [21, 22, 24, 25, 34].
Coordinates on the Rindler wedge RB of radius R can be given by (r; ; ) which map
back into Poincare coordinates by
where 1 < r < 1.
using [21]
(r; ; ) =
Z
The scalar eld dual to an operator O can be reconstructed in this Rindler wedge
f!;k(r) = r
1
d!dk e i! ik f!;k(r)O!;k ;
1
r2
i !
2
2F1
2
i(! + k)
2
2
i(! + k)
2
; ; r 2
z =
t =
x =
r cosh
r cosh
r cosh
Rpr2
1 sinh
R
+ pr2
+ pr2
Rr sinh
+ pr2
1 cosh
1 cosh
1 cosh
;
;
;
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
where O!;k is the Fourier transform of the CFT expectation value of the operator
This can be expressed in terms of the operator in the original coordinates
O!;k =
Z
d d ei! +ik
hO( ; )i :
dtdx h(R + x + t)i k+2! (R
x
t) i k+2!
(R
x + t)i !2 k (R + x
t)i k 2! i
hO(t; x)i ;
O!;k =
Z
DB
1 Z 1
(B.6)
(B.7)
2)
(B.8)
HJEP06(21)85
where the region of integration is only over the domain of dependence DB.
for (2)Sscalar which only depends on the expectation value of O in DB,
This form of the scalar eld can be combined with (3.29) to obtain an an expression
(2)Sscalar =
drdkd!1d!2 rpr2 1 f!1;k(r)f!2; k(r)
r2
!1!2 +
1
k
2
r2 + (
+ r
2
1 f !01;k(r)f !02; k(r) O!1;kO!2; k :
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