#### Relative entropy equals bulk relative entropy

Received: May
Relative entropy equals bulk relative entropy
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Field Theories in Higher Dimensions
0 ow is dual to the
1 Kavli Institute for Theoretical Physics
2 Daniel L. Ja eris
3 Department of Physics and Astronomy, University of British Columbia
4 Jadwin Hall, Princeton University
5 Institute for Advanced Study
6 Center for Fundamental Laws of Nature, Harvard University
We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular bulk modular ow in the entanglement wedge, with implications for entanglement wedge reconstruction.
AdS-CFT Correspondence; 1/N Expansion; Models of Quantum Gravity
1 Introduction and summary of results
Entanglement entropy, the modular Hamiltonian, and relative entropy
2.1
2.2
Modular Hamiltonian
Relative entropy
2
3
4
5
6
4.1
4.2
4.3
6.1
6.2
6.3
6.4
6.5
Gravity dual of the modular Hamiltonian
Regions with a local boundary modular Hamiltonian
Linear order in the metric
The graviton contribution
Quadratic order for coherent states
Smoothness of the full modular Hamiltonian in the bulk
5.2 Implications for entanglement wedge reconstruction
Comments and discussion
The relative entropy for coherent states
Positivity of relative entropy and energy constraints
Higher derivative gravity
Beyond extremal surfaces
Distillable entanglement
A Subregions of gauge theories
A.1 U(1) gauge theory
A.2 Gravity
Introduction and summary of results
Recently there has been a great deal of e ort in elucidating patterns of entanglement for
theories that have gravity duals. The simplest quantity that can characterize such patterns
is the von Neumann entropy of subregions, sometimes called the \entanglement entropy".
This quantity is divergent in local quantum eld theories, but the divergences are well understood and one can extract nite quantities.
Moreover, one can construct strictly
nite quantities that are well-de ned and have no ambiguities. A particularly interesting
quantity is the so called \relative entropy" [
1, 2
]. This is a measure of distinguishability
between two states, a reference \vacuum state"
and an arbitrary state
S( j ) = T r[ log
log ]
(1.1)
{ 1 {
K
If we de ne a modular Hamiltonian K =
log , then this can be viewed as the free
energy di erence between the state
and the \vacuum"
at temperature
= 1, S( j ) =
Relative entropy has nice positivity and monotonicity properties. It has also played an
important role in formulating a precise version of the Bekenstein bound [
3
] and arguments
for the second law of black hole thermodynamics [4, 5].
In some cases the modular hamiltonian has a simple local expression. The simplest
case is the one associated to Rindler space, where the modular Hamiltonian is simply given
by the boost generator.
In this article we consider quantum eld theories that have a gravity dual. We consider
an arbitrary subregion on the boundary theory R, and a reference state , described by
a smooth gravity solution.
can be the vacuum state, but is also allowed to be any
state described by the bulk gravity theory. We then claim that the modular Hamiltonian
corresponding to this state has a simple bulk expression. It is given by
Kbdy =
4GN
Areaext + Kbulk +
+ o(GN )
[Kbdy; ] = [Kbulk; ]
1The entanglement wedge is the domain of dependence of the region Rb.
{ 2 {
backreaction.
relative entropy
The rst term is the area of the Ryu Takayangi surface S (see
gure 1), viewed as an
operator in the semiclassically quantized bulk theory, it includes the change in the extremal
surface as we consider di erent states . This term was previously discussed in [6]. The
o(G0N ) term Kbulk is the modular Hamiltonian of the bulk region enclosed by the
RyuTakayanagi surface, Rb, when we view the bulk as an ordinary quantum
eld theory, with
suitable care exercised to treat the quadratic action for the gravitons. Finally, the dots
represent local operators on S
, which we will later specify.
We see that the boundary
modular Hamiltonian has a simple expression in the bulk. In particular, to leading order
in the 1=GN expansion it is just the area term, which is a very simple local expression in
the bulk. Furthermore, this simple expression is precisely what appears in the entropy.
This modular Hamiltonian makes sense when we compute its action on bulk
eld theory
states
which are related to
by bulk perturbation theory. Roughly speaking, we consider
a
which is obtained from
by adding or subtracting particles without generating a large
Due to the form of the modular Hamiltonian (1.2), we obtain a simple result for the
Kbulk in the interior of the entanglement wedge,1
(1.2)
(1.3)
(1.4)
contribution to the entanglement entropy is computed by the area of an extremal surface S that ends
at the boundary of region R. This surface divides the bulk into two, region Rb and its complement.
Region Rb lives in the bulk and has one more dimension than region R. The leading correction to
the boundary entanglement entropy is given by the bulk entanglement entropy between region Rb
and the rest of the bulk.
in Kbdy localized on S do not contribute to its action in the interior of the entanglement
wedge, S being space-like to the interior. Note Kbulk is the bulk modular Hamiltonian
associated to a very speci c subregion, that bounded by the extremal surface S. Implications
of (1.4) for entanglement wedge reconstruction are described in section 5.2.
The bulk dual of relative entropy for subregions with a Killing symmetry was considered
before in [7{12]. In particular, in [12], the authors related it to the classical canonical
energy. In fact, we argue below that the bulk modular hamiltonian is equal to the canonical
energy in this case. This result extends that discussion to the quantum case. Note (1.2)
and (1.3) are valid for arbitrary regions, with or without a Killing symmetry. In addition,
we are not restricting
to be the vacuum state. Recently a di erent extension of [12] has
been explored in [13], which extends it to situations where one has a very large deformation
relative to the vacuum state. That discussion does not obviously overlap with ours.
This paper is organized as follows. In section two, we recall de nitions and properties
of entanglement entropy, the modular Hamiltonian, and relative entropy. In section three,
we present an argument for the gravity dual of the modular hamiltonian and the bulk
expression for relative entropy. In section four, we discuss the case with a U(1) symmetry,
relating to previous work. In section
ve, we discuss the
ow generated by the
boundary modular hamiltonian in the bulk. We close in section six with some discussion and
open questions.
2
Entanglement entropy, the modular Hamiltonian, and relative entropy
We consider a system that is speci ed by a density matrix . This can arise in quantum
eld theory by taking a global state and reducing it to a subregion R. We can compute the
von Neumann entropy S =
T r[ log ]. Due to UV divergences this is in nite in quantum
eld theory. However, these divergences are typically independent of the particular state
we consider, and when they depend on the state, they do so via the expectation value of
an operator. See [14, 15].
{ 3 {
It is often useful to de ne the modular hamiltonian K
log . From its de nition, it
is not particularly clear why this is useful | it is in general a very non-local complicated
operator. However, for certain symmetric situations it is nice and simple.
The simplest case is a thermal state where K = H=T , with H the Hamiltonian of
the system. Another case is when the subregion is the Rindler wedge and the state is the
vacuum of Minkowski space, when K is the boost generator. This is a simple integral of
a local operator, the stress tensor. For a spherical region in a conformal eld theory, we
have a similarly simple expression, which is obtained from the previous case by a conformal
transformation [16]. In free eld theory one can also obtain a relatively simple expression
that is bilocal in the elds [17] for a general subregion of the vacuum state.
In this paper we consider another case in which simpli cation occurs. We consider a
quantum system with a gravity dual and a state that can be described by a gravity solution.
We will argue that the modular Hamitonian is given by the area of the Ryu-Takayanagi
minimal surface plus the bulk modular Hamiltonian of the bulk region enclosed by the
Ryu-Takayanagi surface.
2.2
Relative entropy
Modular Hamiltonians also appear in the relative entropy
Srel( j ) = tr (log
log ) =
hK i
S
(2.1)
was
(2.2)
where K
=
log
is the modular Hamiltonian associated to the state . If
a thermal state, the relative entropy would be the free energy di erence relative to the
thermal state. As such it should always be positive.
Relative entropies have a number of interesting properties such as positivity and
monotonicity [
1
].
Moreover, while the entanglement entropy is not well de ned for QFT's,
relative entropies have a precise mathematical de nition [
2
].
If
=
+
, then, because of positivity, the relative entropy is zero to rst order in
. This is called the rst law of entanglement:
S = hK i
The rst law can be used to determine the modular Hamiltonian. By considering an
arbitrary
rst order deformation
of the density matrix, we can determine all matrix
elements of the modular Hamiltonian. Then we can use this modular Hamiltonian on
states which are not small deformations of the original state.
When we consider a gauge theory, the de nition of entanglement entropy is ambiguous.
If we use the lattice de nition, there are di erent operator algebras that can be naturally
associated with a region R [14]. Di erent choices give di erent entropies. These algebras
di er in the elements that are kept when splitting space into two, so that ambiguities are
localized on the boundary of the region, @R. One natural way of de ning the entanglement
entropy is by
xing a set of boundary conditions and summing over all possibilities, since
there is no physical boundary. This was carried out for gauge elds in [18, 19, 51] and gives
{ 4 {
the same result as the euclidean prescription of [
20
]. However, the details involved in the
de nition of the subalgebra are localized on the boundary. Because of the monotonicity of
relative entropy, these do not contribute to the relative entropy (see section 6 of [14] for
more details).
In the case of gravitons we expect that similar results should hold. We expect that we
similarly need to x some boundary conditions and then sum over these choices.
For example, we could choose to
x the metric
uctuations on the Ryu-Takayanagi
surface, viewing it as a classical variable, and then integrate over it. As argued in [14], we
expect that the detailed choice should not matter when we compute the relative entropy.
See appendix A for more details.
As we mentioned above, it often occurs that two di erent possible de nitions of the
entropy give results that di er by the expectation value of a local operator, S( ) =tr( O) +
S~( ). A trivial example is the divergent area term which is just a number. In these cases
the two possible modular Hamiltonians are related by
S( ) = tr ( O) + S~( )
! K = O + K~
(2.3)
necessary that O is the same operator for the states
and .
2
This implies that relative entropies are unambiguous, S( j ) = S~( j ). For the equality of
relative entropies, it is not necessary for O to be a state independent operator. It is only
3
Gravity dual of the modular Hamiltonian
A leading order holographic prescription for computing entanglement entropy was proposed
in [
21, 22
] and it was extended to the next order in GN in [
23
] (see also [
24
]).
The
entanglement entropy of a region R is the area of the extremal codimension-two surface S
that asymptotes to the boundary of the region @R, plus the bulk von Neumann entropy of
the region enclosed by S, denoted by Rb. See gure 1.
Aext(S)
4GN
Sbdy(R) =
+ Sbulk(Rb) + SWald like
(3.1)
SWald like indicates terms which can be written as expectation values of local operators on
. They arise when we compute quantum corrections [
23
], we discuss examples below.
We can extract a modular Hamiltonian from this expression. We consider states that
can be described by quantum
eld theory in the bulk. We consider a reference state ,
which could be the vacuum or any other state that has a semiclassical bulk description. We
consider other states
which likewise can be viewed as semiclassical states built around the
bulk state for . To be concrete we consider the situation where the classical or quantum
elds of
are a small perturbation on
so that the area is only changed by a small amount.
Now the basic and simple observation is that both the area term and the SWald like are
2In other words, if we consider a family of states, with and in that family, then O should be a state
independent operator within that family.
{ 5 {
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expectation values of operators in the bulk e ective theory. Therefore, for states that have
a bulk e ective theory, we can use (2.2), (2.3) to conclude that
contains both the classical area as well as any changes in the area that result from the
condition, this area is a gauge invariant observable in the gravity theory.3 Note that the
area changes as we change the state, but we can choose a gauge where the position of the
extremal surface is xed. Finally S^Wald like are the operators whose expectation values
give us SWald like.
Interestingly, all terms that can be written as local operators drop out when we consider
(3.2)
(3.3)
(4.1)
the relative entropy. The relative entropy has a very simple expression
Sbdy( j ) = Sbulk( j )
Note that the term going like 1=GN cancels out and we are only left with terms of order
G0N . There could be further corrections proportional to GN which we do not discuss in
this article. It is tempting to speculate that perhaps (3.3) might be true to all orders in
the GN expansion (i.e. to all orders within bulk perturbation theory).
Of course, using the equation for the entropy (3.1) and (3.2) we can check that the
rst law (2.2) is obeyed. In the next section we discuss this in more detail for a spherical
subregion in the vacuum.
4
Regions with a local boundary modular Hamiltonian
For thermal states, Rindler space, or spherical regions of conformal eld theories we have
an explicit expression for the boundary modular Hamiltonian. In all these cases there is
a continuation to Euclidean space with a compact euclidean time and a U(1) translation
symmetry along Euclidean time. We also have a corresponding symmetry in Lorentzian
signature generated by a Killing or (conformal Killing) vector . The modular Hamiltonian
is then given in terms of the stress tensor as Kbdy = ER
is over a boundary space-like slice. When the theory has a gravity dual, the bulk state
is also invariant under a bulk Killing vector . In this subsection we will discuss (3.2) for
R ( :Tbdy), where the integral
states constructed around .
For this discussion it is useful to recall Wald's treatment of the rst law [25{27]
3If we merely de ne a surface by its coordinate location in the background solution, then a pure gauge
contribution. Here Alin is the rst order variation in the area due to a metric
uctuation
g. And
is any Cauchy slice in the entanglement wedge Rb. Equation (4.1) is a tautology,
it arises by integrating by parts the linearized Einstein tensor. It is linear in g and we can
write it as an operator equation by sending g !
small uctuations in the metric in the semiclassically quantized theory.
g^, where g^ is the operator describing
Linear order in the metric
Eg( g)
that [25{27]
For clarity we will rst ignore dynamical gravitons, and include them later (we would have
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nothing extra to include if we were in three bulk dimensions). We consider matter elds
with an o(G0N ) stress tensor in the bulk, assuming the matter stress tensor was zero on
the
background.4 Such matter elds produce a small change in the metric that can be
obtained by linearizing the Einstein equations around the vacuum. These equations say
= T mat, where T mat is the stress tensor of matter. Inserting this in (4.1) we nd
ER =
terms of the stress tensor due to the presence of a Killing vector with the right properties
at the entangling surface S
. Notice that we can disregard additive constants in both the
area and E , which are the values for the state . We only care about deviations from these
values. This is basically the inverse of the argument in [28]. This shows how (3.2) works in
this symmetric case. The term S^Wald like in (3.2) arises in some cases as we discuss below.
Let us now discuss the S^Wald like term. There can be di erent sources for this term.
A simple source is the following. The bulk entanglement entropy has a series of
divergences which include an area term, but also terms with higher powers of the curvature.
Depending on how we extract the divergences we can get certain terms with
nite
coefcients. Such terms are included in SWald like. A di erent case is that of a scalar eld
with a coupling
2(R
R0) where R is the Ricci scalar in the bulk, and R0 the Ricci
scalar on the unperturbed background, the one associated to the state . Then there exists
an additional term in the entropy of the form S^Wald like = 2
2. If we compute the
entropy as the continuum limit of the one on the lattice, then it will be independent of
. Under these conditions the bulk modular Hamiltonian is also independent of
and is
given by the canonical stress tensor, involving only rst derivatives of the eld. However,
the combination of Kbulk + S^Wald like = R
( :T grav( )), where T grav( ) is the standard
stress tensor that would appear in the right hand side of Einstein's equations. T grav( )
does depend on . The
dependent contribution is a total derivative which evaluates to
2
2 at the extremal surface. A related discussion in the eld theory context appeared
R
S
4This discussion can be simply extended when there is a non-zero but U(1)-symmetric background
stress tensor to obtain the bulk modular Hamiltonian.
We expect that we can view the propagating gravitons as one more eld that lives on the
original background, given by the metric g . In fact, we can expand Einstein's equations
in terms of g = g + g2 + h. Here h, which is of order pGN , represents the dynamical
graviton
eld and obeys linearized
eld equations.
g2 takes into account the e ects of
back-reaction and obeys the equation
E( g2)
= T grav(h) + T matter
where T grav(h) comes simply from expanding the Einstein tensor (plus the cosmological
constant) to second order and moving the quadratic term in h to the right hand side.
h obeys the homogeneous linearized equation of motion, so the term linear in h in the
equation above vanishes. We can now use equations (44-46) in [30], which imply that
(4.3)
(4.4)
(4.5)
(4.6)
Kbdy;1+2 = E1+2 =
A^lin(h + g2) + A^quad(h)
4GN
+ Ecan
where Kbdy;1+2 is the boundary modular Hamitonian (or energy conjugate to
translations) expanded to quadratic order in
uctuations.
Similarly, the area is expanded
to linear and quadratic order. Finally, Ecan is the bulk canonical energy5 de ned by
Ecan = R !(h; L h)+matter contribution, where ! is the symplectic form de ned in [30].
From this expression we conclude that the modular Hamiltonian is the canonical energy
Kbulk = Ecan
We can make contact with the previous expression (4.2) as follows. If we include
the gravitons by replacing T mat ! T mat + T grav(h) in (4.2), then we notice that we get
Alin( g2), without the term Aquad(h). However, one can argue that (see eq. (84) of [30])
Z
( :T grav(h)) = Ecan(h) +
Aquad(h)
4GN
thus recovering (4.4).
for quantizing the graviton eld.
In appendix A we discuss in more detail the boundary conditions that are necessary
4.3
Quadratic order for coherent states
The problem of the gravity dual of relative entropy was considered in [12] in the classical
regime for quadratic uctuations around a background with a local modular Hamiltonian.
They argued that the gravity dual is equal to the canonical energy. Here we rederive their
result from (3.3).
ei R ^+ ^
We simply view a classical background as a coherent state in the quantum theory.
j i, where j i is the state associated to .
6 We see that in free eld theory
6Here
could be O(1=p
GN ) as long as the backreaction is small.
5This di ers from the integral of the gravitational stress tensor by boundary terms.
{ 8 {
we can view coherent states as arising from the action of a product of unitary operators,
one acting inside the region and one ouside. For this reason
nite coherent excitations
do not change the bulk von Neumann entropy of subregions, or
Sbulk = 0. Thus, the
contribution to the bulk relative entropy comes purely from the bulk Hamiltonian, which
we have argued is equal to canonical energy (4.5) . Therefore, in this situation we recover
the result in [12]
Sbdy( j ) = Sbulk( j ) =
Kbulk
Sbulk =
Kbulk = Ecanonical
(4.7)
5
The modular hamiltonian generates an automorphism on the operator algebra, the
modular ow. Consider the unitary transformation U(s) = eiKs. Even if the modular
hamiltonian is not technically an operator in the algebra, the modular
ow of an operator,
O(s)
U(s)OU( s), stays within the algebra. For a generic region, the modular ow
might be complicated, see [31] for some discussion about modular
ows for fermions in
1 + 1 dimensions. However, in our holographic context it can help us understand
subregionsubregion duality. In particular, it can help answer the question of whether the boundary
region R describes the entanglement wedge or only the causal wedge [32{35]. The
entanglement wedge is the causal domain of the spatial region bounded by the interior of S.
From (1.2), we have that
[Kbdy; ] = [Kbulk; ]
(5.1)
where
is any operator with support only in the interior of the entanglement wedge, and
where on the right-hand side we have suppressed terms subleading in GN . On the
lefthand side terms in Kbdy localized on S have dropped out, similarly as in (3.3). Thus the
boundary modular ow is equal to the bulk modular ow of the entanglement wedge, the
causal wedge does not play any role.
5.1
Smoothness of the full modular Hamiltonian in the bulk
If the global state is pure, one may also consider the ow generated by the total modular
operator, Kbdy;Total = Kbdy;R
KbdyR, which should be a smooth operator without any
ambiguities. It annihilates the global state. From our full formula for the bulk dual of the
modular Hamiltonian we see that Kbdy;Total = Kbulk;Total + o(GN ).
For problems that have a U(1) symmetry, such as thermal states and Rindler or
spherical subregions of CFTs, we know the full boundary modular Hamiltonian E . We can de ne
a time coordinate
which is translated by the action of E in the boundary theory. In these
situations the bulk state also has an associated symmetry generated by the Killing vector
. We can choose coordinates so that we extend
in the bulk and
simply translates
in the bulk. Then the bulk modular Hamiltonian is the bulk operator that performs a
translation of the bulk elds along the bulk
direction.
Let us now consider an eternal black hole and the thermo eld double state [36]. This
state is invariant under the action of HR
HL. Let us now consider the action of only
{ 9 {
HJEP06(21)4
HJEP06(21)4
(a)
(b)
modular Hamiltonian e itKbulk;R we get a new state on the horizontal line that has a singularity
at the horizon. (b) The area term introduces a kink, or a relative boost between the left and right
sides. Then the state produced by the full right side Hamiltonian is non-singular, and locally equal
to the vacuum state.
S
the right side boundary Hamiltonian HR.
7 It was argued in [37] that this corresponds
to the same gravity solution but where the origin of the time direction on the right side
is changed. This implies that the Wheeler de Witt patch associated to tL = tR = 0
looks as in
gure 2(b), after the action of e itHR On the other hand, if we consider the
bulk quantum
eld theory and we act with only the right side bulk modular Hamiltonian
Kbulk;R we would produce a state that is singular at the horizon. By the way, it is precisely
for this reason that algebraic quantum
eld theorists like to consider the total modular
Hamiltonian instead. It turns out that the change in the bulk state is the same as the one
would obtain if we were quantizing the bulk eld theory along a slice which had a kink as
shown in gure 2(b). Interestingly the area term in the full modular Hamiltonian (3.2) has
the e ect of producing such a kink. In other words, the area term produces a shift in the
coordinate, or a relative boost between the left and right sides [38]. The action of only
the area term or only KBulk;R would lead to a state that is singular at the horizon, but
the combined action of the two produces a smooth state, which is simply the same bulk
geometry but with a relative shift in the identi cation of the boundary time coordinates.8
Let us go back to a general non-U(1) invariant case. Since the bulk modular
Hamiltonian reduces to the one in the U(1)-symmetric case very near the bulk entangling surface
, we expect that the action of the full boundary modular Hamiltonian, including the
area term, will not be locally singular in the bulk | though it can be singular from the
boundary point of view due to boundary UV divergences.
5.2
Implications for entanglement wedge reconstruction
One is often interested in de ning local bulk operators as smeared operators in the
boundary. This operator should be de ned order by order in GN over a
xed background and
should be local to the extent allowed by gauge constraints. If we consider a t = 0 slice in
7Here left and right denote the two copies in the thermo eld double state.
8We thank D. Marolf for discussions about this point.
the vacuum state, then we can think of a local bulk operator (X) as a smeared integral
of boundary operators [39]
Z
bdy
(X) =
dxd 1dt G(Xjx; t)O(x; t) + o(GN )
(5.2)
One would like to understand to what extent this
operator can be localized to a subregion
in the boundary.
Given a region in the boundary R, we have been associating a corresponding region
in the bulk, the so-called entanglement wedge which is the domain of dependence of Rb,
D[Rb]. There is another bulk region one can associate to R, the causal wedge (with
spacelike slice RC ) which is the set of all bulk points in causal contact with D[R], [40]. RC is
generically smaller than Rb [35, 41].
In situations with a U(1) symmetry, such as a thermal state or a Rindler or spherical
subregion of a CFT, we have time-translation symmetry and a local modular Hamiltonian
that generates translations in the time . We can express bulk local operators in the
entanglement wedge (which coincides with the causal wedge) in terms of boundary operators
localized in D[R] [39, 42]9
(X) =
Z
R
dyd 1
Z
d G0(Xjy; )O(y; ) + o(GN ) ;
X 2 Rb
(5.3)
A natural proposal for describing operators in that case is that we can replace in (5.3)
by the modular parameter s. In other words, we consider modular ows of local operators
on the boundary, de ned as OR(x; s)
U(s)OR(x; 0)U 1(s)
A simple case in which Rb is larger than RC is the case of two intervals in a 1+1 CFT
such that their total size is larger than half the size of the whole system, see gure 3. Here,
it is less clear how to think about the operators in the entanglement wedge. We would
like to use the previous fact that the modular ow is bulk modular ow to try to get some
insight into this issue.
The modular ow in the entanglement wedge will be non-local, but highly constrained:
the bulk modular hamiltonian is bilocal in the elds [17]. If we have an operator near the
boundary of the causal wedge and modular evolve it, it will quickly develop a non zero
commutator with a nearby operator which does not lie in the causal wedge. Alternatively,
an operator close to the boundary of the entanglement wedge will have an approximately
local modular ow. It will follow the light rays emanating from the extremal surface and
it can be on causal contact with the operators in the causal wedge. See gure 3.
So we see that to reconstruct the operator in the interior of the entanglement wedge,
one necessarily needs to understand better the modular ow. It seems natural to conjecture
that one can generalize (5.3) to two intervals (or general regions) by considering the
modular parameter instead of Rindler time, ie the simplest generalization of the AdS/Rindler
formula which accounts for the non-locality of the modular hamiltonian would be
(X) =
Z
R
dx
Z
dsG00(Xjx; s)O(x; s) ;
X 2 Rb
(5.4)
9It is sometimes necessary to go to Fourier space to make this formula precise [42, 43].
surface is the dotted black line, while the boundary of RC is the blue dashed line (color online). In
a), the shaded region denotes the de ning spatial slice Rb of the entanglement wedge. In b), the
shaded region is the de ning spatial slice RC of the causal wedge. The modular ow of an operator
close to the Ryu-Takayanagi surface will be approximately local, so that
1(s) will be almost local
and, after some s, it will be in causal contact with
C1. This ow takes the operator out of this
slice to its past or to its future. Alternatively, if we consider an operator near the boundary of the
causal wedge C2, it is clear that, under modular ow, [ C2(s); 2] 6= 0.
Here G00 is a function that should be worked out. It will depend on the bilocal kernel that
describes the modular Hamiltonian for free elds [17].
So we see that to reconstruct the operator in the interior of the entanglement wedge,
it is necessary to understand better modular ows in the quantum
eld theory of the bulk.
To make these comments more precise, a more detailed analysis would be required, which
should include a discussion about gravitational dressing and the constraints. We leave this
to future work.
Here we have discussed how the operators in the entanglement wedge can be though
of from the boundary perspective. However, note that from (3.2) (and consequently the
formula for the relative entropy), it is clear that one should think of the entanglement
wedge as the only meaningful candidate for the \dual of R", see also [32]. If we add some
particles to the vacuum in the entanglement wedge Rb (which do not need to be entangled
with Rb), the bulk relative entropy will change. According to (1.3), the boundary relative
entropy also changes and, therefore, state is distinguishable from the vacuum, even if we
have only access to R.
6
6.1
Comments and discussion
The relative entropy for coherent states
If we consider coherent states, since their bulk entanglement entropy is not changed, the
relative entropy will just come from the di erence in the bulk modular hamiltonian. Since
our formulation is completely general, one could in principle compute it for any reference
region or state and small perturbations over it.
A particularly simple case would be the relative entropy for an arbitrary subregion
between the vacuum and a coherent state of matter. To second order in the perturbation,
one only needs to work out how the modular hamiltonian for the free elds [17] looks like
for that subregion of AdS, and then evaluate it in the coherent state background.
Positivity of relative entropy and energy constraints
Our formula (3.3) implies that the energy constraints obtained from the positivity of the
relative entropy can be understood as arising from the fact that the relative entropy has
to be positive in the bulk.
Higher derivative gravity
Even though we focused on Einstein gravity, our discussion is likely to apply to other
theories of gravity. The modular hamiltonian will likely be that of an operator localized on the
entangling surface plus the bulk modular Hamiltonian in the corresponding entanglement
wedge. Thus the relative entropy will be that of the bulk. There could be subtleties that
we have not thought about.
6.4
Beyond extremal surfaces
A. Wall proved the second law by using the monotonicity of relative entropy [4, 5]. If we
consider two Cauchy slices
0
; t>0 outside a black hole, then Srel;t < Srel;0 is enough to
prove the generalized second law. Interestingly, section 3 of [30] shows the \decrease of
canonical energy": Ecan(t) < Ecan(0). The setup (Cauchy slices) that they both consider
is the same. Due to the connection between relative entropy and canonical energy, [12], we
expect a relation between these two statements. This does not obviously follow from what
we said due to the following reason.
Here we limited our discussion to the entanglement wedge. In other words, we are
always considering the surface S to be extremal. We expect that the discussion should
generalize to situations where the surface S is along a causal horizon. The question is:
what is the precise boundary dual of the region exterior to such a horizon? Even though
we can think about the bulk computation, we are not sure what boundary computation
it corresponds to. A proposal was made in [44], and perhaps one can understand it in
that context.
entanglement.
Being able to de ne relative entropies for regions which are not bounded by minimal
surfaces is also crucial to the interesting proposal in [45] to derive Einstein's equations
from (a suitable extension to non-extremal surfaces of) the Ryu-Takayanagi formula for
6.5
Distillable entanglement
In the recent papers [46, 47] it was argued that for gauge elds, only the purely quantum
part of the entanglement entropy corresponds to distillable entanglement. The \classical"
piece that cannot be used as a resource corresponds to the shannon entropy of the center
variables of [14]. Our terms local in S are the gravitational analog of this classical piece
and one might expect that a bulk observer with access only to the low-energy e ective eld
theory can only extract bell pairs from the bulk entanglement. This seems relevant for the
AMPS paradox [48{50].
Acknowledgments
We thank H. Casini, M. Guica, D. Harlow, T. Jacobson, N. Lashkari, J. Lin, D. Marolf, H.
Ooguri, M. Rangamani, and A. Wall for discussions. The work of D.L.J. is supported in
part by NSFCAREER grant PHY-1352084 and a Sloan Fellowship. A.L. was supported in
part by the US NSF under Grant No. PHY-1314198. J.M. was supported in part by U.S.
Department of Energy grant de-sc0009988 and by the It from Qubit grant from the Simons
Foundation. J.S. was supported at KITP in part by the National Science Foundation under
Grant No. NSF PHY11-25915, and is now supported in part by the Natural Sciences and
Engineering Research Council of Canada and by grant 376206 from the Simons Foundation.
A
A.1
Subregions of gauge theories
U(1) gauge theory
The problem of de ning the operator algebra of a subregion of a gauge theory was
considered in [14]. It was shown that for a lattice gauge theory there are several possible
de nitions of the subalgebra. It was further found that the subalgebra can have a center,
namely some operators that commute with all the other elements of the subalgebra. In
this case we can view the center as classical variables. Calling the classical variables xi,
then for each value of xi we have a classical probability pi and a density matrix i for each
irreducible block. The relative entropy between two states is then
S( j ) = H(pjq) + X piS( ij i)
i
(A.1)
where pi; qi are the probabilities of variables xi in the state
and
respectively. H is the
classical (Shanon) relative entropies of two probability distributions, H = Pi pi log(pi=qi).
In the continuum we expect that the relative entropy is nite and independent of the
microscopic details regarding the precise de nition of the algebra [
2
].
These microscopic details have a continuum counterpart. When we consider a region
R we would like to be able to de ne a consistent quantum theory within the subregion.
In particular, imagine that we consider all classical solutions restricted to the subregion.
Then we de ne a presymplectic product between two such solutions, which we will use to
quantize the gauge orbits. This presymplectic product should be gauge invariant so that
it does not depend on the particular representative. Let us consider a free Maxwell eld.
The presymplectic product is given by integrating
Z
(A1; A2) =
!(A1; A2) =
Z
(A1 ^ F 2
A ^ F 1)
2
(A.2)
where A1 = A1 dx is a gauge eld con guration. Here we imagine that both A1 and A2
are solutions to the equations of motion.
is any spacelike surface.
Demanding gauge invariance amounts to the statement
0 =
(A; d ) =
Z
^ F
(A.3)
where @ is the boundary of the spacelike surface. We have used the equations of motion
for F and integrated by parts. In order to make this vanish we need some boundary
conditions. In particular, let us concentrate on the boundary conditions required at the
boundary of
corresponding to the boundary of a region S = @ . One possible boundary
condition is to set Ai = Aicl for components along the surface, where Aicl is a classical gauge
eld on the surface. In this case, it is natural to set
= 0 on the surface. We can quantize
the problem for each xed Aicl and then integrate over all Aicl. These values of Aicl are the
\center" variables xi in the above discussion. This is called the \magnetic" center, since
the gauge eld Aicl de nes a magnetic eld F = dAcl on the surface.
There are other possibilities, such as xing the electric eld, or \electric center", where
the perpendicular electric eld is xed.
These would correspond to speci c choices on the lattice. Since we expect that relative
entropy is a
nite and smooth function of the shape of the region, [14] has shown that the
detailed boundary condition does not matter, as long as we choose something that makes
physical sense. Recently, [18, 19] carried out explicitly the eld theory calculation, being
careful with the center variables.
A.2
Gravity
Here we consider the problem of de ning a subregion in a theory of Einstein gravity. We
consider only the problem at the quadratic level where we need to consider free gravitons
moving around a xed background (which obeys Einstein's equations). These gravitons can
be viewed as a particular example of a gauge theory. We can also compute the symplectic
form, as given in [27], and then impose that the symplectic inner product between a pure
gauge mode and another solution to the linearized equations vanishes. Here the gauge
transformations are reparametrizations, generated by a vector eld . Note that
is not a
killing vector, it is a general vector eld and it should not be confused with
discussed in
section 4. Writing the metric as g + g, where g is the background metric and g is a small
uctuation. Then the gauge transformation acts as g !
g + L g, where L is the Lie
derivative. Then, as shown in [30], there is a simple expression for the sympectic product
with a such a pure gauge mode
Z
!( g; L g) =
Q
: (g; g)
(A.4)
with Q and
(g; g) given in eqns (32) and (17) of [30].
We would like to choose boundary conditions on the surface which make the right hand
side zero. We choose boundary conditions similar to the \magnetic" ones above. Namely,
we x the metric along the entangling surface S to gij = ij . We treat ij as classical and
then integrate over it. This is enough to make all terms in (A.4) vanish. Let us be more
explicit. By a change of coordinates we can always set the metric to have the following
form near the entangling surface. For simplicity we write it in Euclidean space, but the
same is true in Lorentzian signature
ds2 = d 2 + [ 2 + o( 4)](d + aidyi)2 + hij dyidyj
(A.5)
here ai and hij can be functions of
and , with a regular expansion around
= 0. In
these coordinates the extremal surface S is always at
= 0, both for the original metric
and the perturbed metric. Extremality implies that the trace of the extrinsic curvature is
zero, or KA = hij @XA hij = 0, where XA = (X1; X2) = ( cos ; sin ). This is true for
the background and the
uctuations
KA = 0;
KA = 0
(A.6)
HJEP06(21)4
which ensures that even on the perturbed solution we are considering the minimal surface.
These conditions ensure that the splitting between the two regions is de ned in a gauge
invariant way.
We demand that all uctuations are given in the gauge (A.5). Thus, near
= 0, g
leads to ai and hij . We now further set a boundary condition that hij = ij where ij is
a classical function which we will later integrate over. For de ning the quantum problem
we will view it as being classical. We will quantize the elds in the subregion for xed
values of ij and then integrate over the classical values of ij .
With these boundary conditions we see that all terms in (A.4) vanish. In fact, (A.4),
has three terms10
Z
!( h; L g) =
hQ( )
i
(g; h)
ai i +
hii +
B AB
(A.7)
1
2
Since the uctuation of the metric is zero at the entangling surface, hij = 0, we see
that many terms vanish. In addition, since we are setting
hij = 0, it is also natural to
restrict the vector elds so that i = 0 on the surface. This ensures that the rst term
in (A.7) vanishes. Note that the middle term is related to the fact that the area generates
a shift in the coordinate . After all the area is the Noether charge associated to such
shifts [26, 27].
The extremality condition makes sure that we are choosing a (generically) unique
surface for each geometry. We then treat the induced geometry on the surface as a classical
variable, quantize the metric in the subregion, and then sum over this classical variable.
In this region, we seem to have a gauge invariant symplectic product.
We have not explicitly computed the entanglement entropy for gravitons with these
choices, but we expect that it should lead to a well de ned problem and that relative
entropies will be nite.
Open Access.
This article is distributed under the terms of the Creative Commons
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10We did not keep track of the numerical coe cients in front of each of the three terms.
=
Z
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