CPHT-RR067.112023, November 2023

On the Quantum Bousso Bound in
de Sitter JT gravity

Victor<EMAIL_ADDRESS>François
<EMAIL_ADDRESS>
1 CPHT, CNRS, Ecole polytechnique, IP Paris,
F-91128 Palaiseau, France

2 Department of Physics, University of Cyprus,
Nicosia 1678, Cyprus

Abstract

> We prove the validity of the Strominger-Thompson quantum Bousso bound in the
> infinite class of conformal vacua of de Sitter space in semiclassical JT
> gravity. The Bousso-Fisher-Leichenauer-Wall quantum Bousso bound follows
> from an analogous derivation, requiring only initial quantum non-expansion.
> In this process, we show that the quantity ${2\pi
> k^{\mu}k^{\nu}\braket{:T_{\mu\nu}:}-S^{\prime\prime}-\frac{6}{c}(S^{\prime})^{2}}$
> vanishes in any vacuum state, entailing a stronger version of Wall’s quantum
> null energy condition. We derive an entropy formula in the presence of a
> generic class of two reflecting boundaries, in order to apply our argument
> to the half reduction model of de Sitter JT gravity.

###### Contents

  1. 1 Introduction
  2. 2 JT gravity in de Sitter space
    1. 2.1 Two-dimensional de Sitter space
    2. 2.2 Semiclassical JT gravity
    3. 2.3 Entropy in the half reduction model
  3. 3 Classical Bousso Bound
    1. 3.1 The Bousso Bound and Energy Conditions
    2. 3.2 Proof of the Classical Bousso Bound
  4. 4 Strominger-Thompson Quantum Bousso Bound
    1. 4.1 Generalized Entropy and Quantum Energy Conditions
    2. 4.2 A sufficient condition
    3. 4.3 Proof of the Quantum Bousso Bound
    4. 4.4 Comparison to other Quantum Bousso Bounds
  5. 5 Conclusion
  6. A Entanglement entropy and reflecting boundaries
    1. A.1 Entanglement entropy without boundaries
    2. A.2 Entanglement entropy with reflecting boundaries
    3. A.3 Change of vacuum
  7. B Light-sheet, expansion and energy conditions

##  1 Introduction

In the context of black hole thermodynamics, the famous black hole entropy
formula111Throughout this paper, we will work in natural units, setting
$\hbar=1$.

$S_{\rm BH}=\frac{A}{4\hat{G}},$ (1.1)

where $A$ is the area of the black hole horizon and $\hat{G}$ the
gravitational Newton’s constant, must be generalized for the second law of
thermodynamics to hold [1, 2, 3, 4]. To this purpose, Bekenstein introduced
the notion of generalized entropy

$S_{\rm gen}=\frac{A}{4\hat{G}}+S_{\rm out},$ (1.2)

where $S_{\rm out}$ is the coarse-grained (or thermodynamic) entropy of matter
outside the black hole, which satisfies the generalized second law [5, 6]

${\rm d}S_{\rm gen}\geq 0.$ (1.3)

Building on this connection between spacetime geometry and information
(entropy), ’t Hooft and Susskind developed the holographic principle [7, 8],
associating the information contained in a region of spacetime to the area of
a lower dimensional surface. This led to the development of the AdS/CFT
correspondence [9], relating the gravitational theory in Anti de Sitter
spacetime with a conformal field theory living on the boundary of this
spacetime. Bousso then formalized the holographic principle by formulating a
covariant entropy bound [10, 11], whose statement goes as follows. Consider a
codimension $2$ spacelike surface $B$ and a codimension $1$ null hypersurface
$L$ emanating from it. If $L$ is of non-positive expansion, then the coarse-
grained entropy $S_{L}$ passing through $L$ is bounded from above by one
quarter the area of $B$, in Planck units. Later, the Bousso bound has been
generalized to the case where $L$ terminates on a second codimension $2$
surface $B^{\prime}$ [12]:

$S_{L}\leq\frac{1}{4\hat{G}}\Delta A,$ (1.4)

where $\Delta A$ is the difference of the areas of $B$ and
$B^{\prime}$.222These notions will be defined more carefully in Section 3 and
Appendix B.

More recently, Ryu and Takayanagi [13] provided a precise dictionary between
entropy and geometrical quantities in the context of AdS/CFT. The areas of
codimension $2$ extremal surfaces in AdSn+1 are associated with fine-grained
entropies in the CFT.333What we refer to as fine-grained entropy is the von
Neumann entropy, which is bounded from above by the usual thermodynamic
entropy. We will define this notion in Appendix A. Similarly to the
thermodynamic entropy of black holes, this relation must be corrected in
semiclassical regimes, by replacing the geometrical entropy of a codimension
$2$ surface $B$ with the fine-grained generalized entropy [14]:

$S_{\rm gen}(B)=\frac{A(B)}{4\hat{G}}+S(B),$ (1.5)

where $A(B)$ is the area of $B$ and $S(B)$ is the fine-grained entropy of the
bulk fields across $B$.

The notion of generalized entropy is particularly useful as it is divergence-
free. Both the Newton’s constant $\hat{G}$ and the fine-grained entropy are
cutoff-dependent. On the other hand, several pieces of evidence [15, 16, 17]
show that the generalized entropy is a cutoff-independent (divergence-free)
quantity. This highlights the idea that the generalized entropy contains
information about the complete theory of quantum gravity. From another
perspective, it has been shown in [18] that gravity and quantum field theory
must be coupled in order to define an algebra of observables in which entropy
is properly defined. In particular, semiclassical entropy in this theory is
given by the generalized entropy (1.5).

Motivated by these observations, Strominger and Thompson [19] conjectured that
applying the modification $A/4\hat{G}\rightarrow S_{\rm gen}$ to the classical
Bousso bound leads to a quantum version of the Bousso bound that holds at the
semiclassical level. They proved the validity of their conjecture in the
Russo–Susskind–Thorlacius (RST) model [20], a two-dimensional model of
evaporating black hole in AdS.

In this paper, we aim to study the Bousso bound and its quantum version in a
cosmological context. Indeed, the Bousso bound has been central in recent
developments of holography in cosmology [21, 22, 23, 24, 25, 26]. In
particular, one would like to understand in which regimes the classical bound
is violated [23, 27], while providing a proof of the quantum Bousso bound of
Strominger and Thompson in a cosmological context, in particular in de Sitter
(dS) space. This case of interest may differ from the commonly studied AdS
space, for instance by its topology and geometrical structure. To do so, we
consider conformal vacua of $(1+1)$-dimensional de Sitter using semiclassical
Jackiw-Teitelboim (JT) gravity [28, 29].

##### Outline

In Section 2, we introduce JT gravity in de Sitter space, which produces two
different two-dimensional models. The first one is a $\mathbb{Z}_{2}$-orbifold
of dS2, and the second one is a two-dimensional reduction of Schwarzschild-de
Sitter space. These models have been investigated in the literature [30, 31,
32, 33, 34, 35, 36, 37, 38, 39] to study islands and information recovery in
dS. We first review these models in classical gravity and then include
semiclassical corrections by coupling gravity to a two-dimensional CFT with
central charge $c$. In Section 2.3, we discuss the half reduction model with
reflective boundaries and provide the formula for the von Neumann entropy of a
Cauchy slice ending on one boundary. This result is important as it tackles an
issue that has been overlooked in the literature so far: in the half reduction
model, left and right-moving modes are correlated, modifying the usual entropy
formula. In particular, the fine-grained entropy of a Cauchy slice bounded by
a point and the boundary should not depend on the endpoint at the boundary.

The classical Bousso bound in the framework of JT gravity is introduced in
Section 3. We start by presenting the notions of light-sheet, expansion, and
energy conditions in JT gravity. Following the arguments of [19], we then
prove the Bousso bound in classical de Sitter JT gravity.

Moving to the semiclassical regime where the Null Energy Condition (NEC) may
be violated, we introduce the notion of quantum light-sheet, together with the
Quantum Null Energy Condition (QNEC) and the Quantum Bousso Bound (QBB) of
Strominger and Thompson in Section 4. We then prove the Strominger-Thompson
QBB in both JT gravity models of dS, in the infinite family of vacuum states
of the CFT. We achieve this in two steps. First, we prove a sufficient
condition for the bound: the normal ordered stress tensor
$\braket{:T_{\mu\nu}:}$ satisfies the inequality

$2\pi k^{\mu}k^{\nu}\braket{:T_{\mu\nu}:}\geq S^{\prime\prime},$ (1.6)

where $k^{\mu}$ is the tangent vector to the light-sheet and
$S^{\prime\prime}$ is the second derivative of the fine-grained entropy of a
Cauchy slice bounded by a point of the light-sheet. The second part of the
proof consists of showing that

$\mathcal{Q}\equiv 2\pi
k^{\mu}k^{\nu}\braket{:T_{\mu\nu}:}-S^{\prime\prime}-\frac{6}{c}(S^{\prime})^{2}=0$
(1.7)

in any conformal vacuum, by verifying that $\mathcal{Q}$ is invariant under a
change of vacuum state and that it vanishes in the Bunch-Davies state. In
other words, the sufficient condition is always satisfied. This last property
is a stronger version of the QNEC showed by Wall [40] in arbitrary two-
dimensional conformal vacua, $2\pi
k^{\mu}k^{\nu}\braket{T_{\mu\nu}}-S^{\prime\prime}-\frac{6}{c}(S^{\prime})^{2}\geq
0$. We then conclude by comparing the Strominger-Thompson bound to two other
proposals of QBB [41, 42], one of which emerges from a semiclassical version
of the focussing theorem: the Quantum Focussing Conjecture (QFC) [42]. In
particular, the Bousso-Fisher-Leichenauer-Wall bound [42] follows from an
analogous derivation, without requiring the QFC. Finally, possible future
works are briefly discussed in Section 5.

In Appendix A, we review fine-grained entropy in two-dimensional CFTs, and we
derive an entropy formula in the case where the system is spatially bounded by
reflecting boundaries. We also derive the transformation laws of the entropy
under a change of vacuum state. Lastly, the formalism of light-sheet,
expansion, and energy conditions in arbitrary dimension is reviewed in
Appendix B.

##  2 JT gravity in de Sitter space

In this section, we review the two-dimensional de Sitter JT gravity, in the
spirit of [30, 32, 33, 35, 36, 37, 38]. The de Sitter JT gravity action comes
from a spherical reduction, from $n+1$ to $2$ dimensions, of the Einstein-
Hilbert action with positive cosmological constant $\hat{\Lambda}$ [28, 29]:

$I_{\rm EH}=\frac{1}{16\pi\hat{G}}\int_{\hat{\mathcal{M}}}{\rm
d}^{n+1}X\sqrt{-\hat{g}}\left[\hat{R}-2\hat{\Lambda}\right]+\frac{1}{8\pi\hat{G}}\int_{\partial\hat{\mathcal{M}}}{\rm
d}^{n+1}Y\sqrt{-\hat{h}}\hat{K}.$ (2.8)

We denote by $\\{X^{M},\leavevmode\nobreak\ M=0,...,n\\}$ the coordinates on
the $(n+1)$-dimensional manifold $\hat{\mathcal{M}}$, $\hat{g}_{MN}$ the
metric tensor on $\hat{\mathcal{M}}$ and $\hat{g}$ its determinant, $\hat{R}$
the Ricci scalar and $\hat{G}$ the $(n+1)$-dimensional Newton’s constant. The
second term in (2.8) is the Gibbons-Hawking-York boundary term, defined on the
$n$-dimensional boundary $\partial\hat{\mathcal{M}}$ of $\hat{\mathcal{M}}$.
$\\{Y^{M},\leavevmode\nobreak\ M=0,...,n-1\\}$ are the coordinates on
$\partial\hat{\mathcal{M}}$, $\hat{h}_{MN}$ is the induced metric on
$\partial\hat{\mathcal{M}}$, with $\hat{h}$ its determinant and $\hat{K}$ the
trace of its extrinsic curvature.

The $(n+1)$-dimensional de Sitter space, dSn+1, is the maximally symmetric
solution with positive cosmological constant of the equations of motion
derived from (2.8). The radius of curvature $l_{n}$ of dSn+1 is related to
$\hat{\Lambda}$ by $\hat{\Lambda}=n(n-1)/(2l_{n}^{2})$. We then consider the
spherical reduction of the action (2.8) with the metric ansatz [38]

${\rm d}s^{2}=\hat{g}_{MN}{\rm d}X^{M}{\rm d}X^{N}=g_{\mu\nu}(x){\rm
d}x^{\mu}{\rm d}x^{\nu}+l_{n}^{2}\Phi^{2/(n-1)}(x){\rm d}\Omega_{n-1}^{2}.$
(2.9)

We will call $\mathcal{M}$ the two-dimensional manifold described by the
metric $g_{\mu\nu}$ with coordinates $\\{x^{\mu},\leavevmode\nobreak\
\mu=0,1\\}$, such that
$\hat{\mathcal{M}}=\mathcal{M}\crossproduct\mathbb{S}^{n-1}$. $\Phi$ is called
the dilaton, which encodes the size of the $(n-1)$-dimensional compact space
$\mathbb{S}^{n-1}$. The dimensional reduction of (2.8) then gives the two-
dimensional action

$\displaystyle I_{\rm EH}$ $\displaystyle=\frac{1}{16\pi
G}\int_{\mathcal{M}}{\rm d}^{2}x\sqrt{-g}\left[\Phi
R-2\hat{\Lambda}\Phi+\frac{(n-2)(n-1)}{l_{n}^{2}}\Phi^{\frac{n-3}{n-1}}+\frac{n-2}{n-1}\frac{(\nabla\Phi)^{2}}{\Phi}\right]$
$\displaystyle+\frac{1}{8\pi G}\int_{\partial\mathcal{M}}{\rm d}y\sqrt{-h}\Phi
K,$ (2.10)

where $R$ is the two-dimensional Ricci scalar, $\nabla$ the covariant
derivative compatible with the metric $g_{\mu\nu}$, and $K$ the trace of the
extrinsic curvature of the boundary $\partial\mathcal{M}$ of $\mathcal{M}$.
The two-dimensional Newton’s constant $G$ is given by

$\frac{1}{G}=\frac{S_{n-1}(l_{n})}{\hat{G}},$ (2.11)

with $S_{n-1}(l_{n})=2\pi^{n/2}\leavevmode\nobreak\ l_{n}^{n-1}/\Gamma(n/2)$
the surface area of the $(n-1)$-sphere of radius $l_{n}$. As we will discuss
in the next section, this dimensional reduction with $n=2$ and $n\geq 3$ leads
to two distinct versions of de Sitter JT gravity, respectively called half and
full reduction models. While having the same equations of motion, these two
models have very different effective geometries.

###  2.1 Two-dimensional de Sitter space

The dimensionally reduced action (2) simplifies considerably for $n=2$. In
this case, the scalar potential and the kinetic term of the dilaton vanish,
yielding the action

$I^{\rm half}_{\rm JT}=\frac{1}{16\pi G}\int_{\mathcal{M}}{\rm
d}^{2}x\sqrt{-g}\leavevmode\nobreak\ \Phi(R-2\Lambda)+\frac{1}{8\pi
G}\int_{\partial\mathcal{M}}{\rm d}y\sqrt{-h}\Phi K,$ (2.12)

with $\Lambda=1/l_{2}^{2}$. This is the action of the _half reduction model_
of de Sitter JT gravity. The situation for $n\geq 3$ is more involved. The
kinetic term for the dilaton can be removed by a suitable Weyl rescaling of
the metric
$g_{\mu\nu}\rightarrow\frac{1}{n}\Phi^{-\frac{n-2}{n-1}}g_{\mu\nu}$, but a
potential will still remain [38]. Expanding around the minimum of the
potential $\phi_{0}=1$ gives the action of the _full reduction model_ of de
Sitter JT gravity:

$\displaystyle I^{\rm full}_{\rm JT}=\frac{1}{16\pi G}\int_{\mathcal{M}}{\rm
d}^{2}x\sqrt{-g}\leavevmode\nobreak\
((\phi_{0}+\phi)R-2\Lambda\phi)+\frac{1}{8\pi G}\int_{\partial\mathcal{M}}{\rm
d}y\sqrt{-h}(\phi_{0}+\phi)K,$ (2.13)

with $\Lambda=1/l_{n}^{2}$. The dilaton $\Phi(x)=\phi_{0}+\phi(x)$ can be seen
as an expansion around the Nariai geometry $\Phi=\phi_{0}$, with $\phi$
describing the deviation (which must be small compared to $\phi_{0}$) away
from the Nariai solution, in analogy with AdS JT gravity [38]444The Nariai
geometry describes the largest black hole fitting inside de Sitter space [43].
For the full reduction model to explicitly derive from the Nariai geometry,
one needs to modify the ansatz (2.9) into  ${\rm d}s^{2}=\hat{g}_{MN}{\rm
d}X^{M}{\rm d}X^{N}=g_{\mu\nu}(x){\rm d}x^{\mu}{\rm
d}x^{\nu}+r_{N}^{2}\Phi^{2/(n-1)}(x){\rm d}\Omega^{n-1},$ (2.14) where
$r_{N}=\sqrt{\frac{n-2}{n}}l_{n}$ is the Nariai radius.. From the two-
dimensional point of view, $\phi_{0}$ is a topological term which does not
modify the equations of motion, so that the explicit form of $\phi$ is the
same in both models. However, it allows $\phi$ to take negative values,
changing the topology of the two-dimensional space.

The holographic principle assigns to any codimension $2$ surface an entropy
given by its area divided by $4\hat{G}$. A point in $\mathcal{M}$ corresponds,
from the $(n+1)$-dimensional point of view, to an $(n-1)$-sphere at constant
$x$, with area $S_{n-1}(l_{n})\Phi(x)$ as follows from the metric ansatz
(2.9). Using (2.11), we get

$\frac{S_{n-1}(l_{n})\Phi(x)}{4\hat{G}}=\frac{\Phi}{4G},$ (2.15)

which brings us to interpret $\Phi$ as the “area” of a point in JT gravity,
and therefore to impose $\Phi\geq 0$. In the half reduction, this leads to
$\phi\geq 0$, while in the full reduction, negative values of $\phi$ are
allowed as long as $\phi\geq-\phi_{0}$.

The two-dimensional bulk dynamics is usually studied in Kruskal coordinates
$(x^{+},x^{-})$ in the conformal gauge, where the metric reads

${\rm d}s^{2}=-e^{2\omega(x^{+},x^{-})}{\rm d}x^{+}{\rm d}x^{-}.$ (2.16)

The non-vanishing Christoffel symbols associated to this metric are
$\Gamma_{++}^{+}=2\partial_{+}\omega$, $\Gamma_{--}^{-}=2\partial_{-}\omega$,
while the Ricci scalar is given by
$R=8e^{-2\omega}\partial_{+}\partial_{-}\omega$. Varying the action with
respect to the dilaton yields the equation of motion for the metric:

$R=2\Lambda\quad\Leftrightarrow\quad\partial_{+}\partial_{-}\omega=\frac{\Lambda}{4}e^{2\omega},$
(2.17)

hence fixing the background geometry to be de Sitter. This is solved by

$e^{2\omega(x^{+},x^{-})}=\frac{4}{(1-\Lambda x^{+}x^{-})^{2}}.$ (2.18)

On the other hand, varying the action with respect to the inverse metric
yields the equation of motion for $\phi$,

$(g_{\mu\nu}\nabla^{2}-\nabla_{\mu}\nabla_{\nu}+g_{\mu\nu}\Lambda)\phi=0.$
(2.19)

In Kruskal coordinates and in the conformal gauge, the latter equation splits
into its off-diagonal and diagonal parts:

$\displaystyle 2\partial_{+}\partial_{-}\phi-\Lambda e^{2\omega}\phi$
$\displaystyle=$ $\displaystyle 0,$ (2.20)
$\displaystyle-\partial_{\pm}^{2}\phi+2\partial_{\pm}\omega\partial_{\pm}\phi$
$\displaystyle=$ $\displaystyle 0,$ (2.21)

Using (2.18), these equations are solved by

$\phi_{*}(x^{+},x^{-})=\phi_{r}\frac{1+\Lambda x^{+}x^{-}}{1-\Lambda
x^{+}x^{-}},$ (2.22)

with $\phi_{r}>0$ a constant. In the half reduction model, $\phi_{r}$ can be
normalized to one by matching the Wald entropy associated to the horizon with
the Gibbons-Hawking entropy of dS3 [38]. We will keep it general in this work.

While pure two-dimensional gravity is non-dynamical, the geometry of spacetime
in JT gravity is described by the dilaton field. In particular, the condition
$\Phi\geq 0$ implies

$-\frac{1}{\Lambda}\leq x^{+}x^{-}\leq\frac{1}{\Lambda},$ (2.23)

in the half reduction case, where $x^{+}x^{-}=1/\Lambda$ correspond to the
past and future null infinity $\cal{J}^{\pm}$ and $x^{+}x^{-}=-1/\Lambda$
correspond to the spatial boundaries.

$\cal{J}^{+}$$\Phi=+\infty,\ x^{+}x^{-}=1/\Lambda$Antipode$\Phi=0,\
x^{+}x^{-}=-1/\Lambda$$\Phi=+\infty,\
x^{+}x^{-}=1/\Lambda$$\cal{J}^{-}$Pode$\Phi=0,\
x^{+}x^{-}=-1/\Lambda$$\Phi=\phi_{r}$ Figure 1: Penrose diagram for two-
dimensional de Sitter space in the half reduction model. Any spacelike slice
is a segment whose boundaries are called pode and antipode; so that the full
dS2 spacetime in the half reduction has two timelike boundaries depicted by
the two vertical lines. The diagonal lines are the past and future
cosmological horizons for an observer at the pode and antipode, which delimit
the two static patches depicted in blue. Inside each static patch, the dilaton
varies from $\Phi=0$ on the pode/antipode to the constant positive value
$\Phi=\phi_{r}$ on the cosmological horizons. It diverges to
$\Phi\rightarrow+\infty$ on the past and future infinity $\cal{J}^{\pm}$.

From the higher dimensional perspective, these timelike boundaries correspond
to the poles of dS3, and we still call them poles (pode and antipode) in the
two-dimensional setup. Spatial slices are segments bounded by the poles, where
comoving observers are located. Each observer has an associated cosmological
horizon located at $x^{+}x^{-}=0$, which bounds their respective causal
patches. The Penrose diagram for two-dimensional de Sitter space in the half
reduction model is depicted in Figure 1, which can be seen as a
$\mathbb{Z}_{2}$-orbifold of dS2.555This comes from the condition $\Phi\geq 0$
[35]..

In the full reduction case, $x^{+}x^{-}$ is still bounded from above by
$1/\Lambda$, corresponding to $\cal{J}^{\pm}$. However, it is no longer
bounded from below, with $x^{+}x^{-}\rightarrow-\infty$ corresponding to the
black hole horizons. In fact, the full reduction model can be maximally
extended such that it is periodic in space and includes the black hole region
of the Schwarzschild-de Sitter space [38], see Figure 2. The Kruskal
coordinates $(x^{+},x^{-})$ cover the entire dS2 spacetime in the half
reduction model, and the blue and white region of Figure 2 in the full
reduction model. As we will see in the next section, coupling matter to JT
gravity induces a backreaction on the dilaton, hence changing the effective
geometry.

$\Phi=+\infty$$\Phi=+\infty$$\Phi=\phi_{0}-\phi_{r}$$\Phi=\phi_{0}+\phi_{r}$$\Phi=\phi_{0}$$\Phi=0$
Figure 2: Penrose diagram for two-dimensional de Sitter space in the full
reduction model. The left and right edges are identified, so that any complete
spacelike slice is topologically a circle. The two vertical dashed lines
depict the pode and the antipode where the dilaton takes the constant value
$\Phi=\phi_{0}$, and the two blue shaded regions their associated static
patches. The red shaded regions are the “black hole” interiors, with past and
future singularities depicted by the wavy lines, where $\Phi=0$. The white
regions are the exterior regions of de Sitter. The white and blue regions are
covered by Kruskal coordinates $x^{\pm}$. The geometry contains two horizons
associated with an observer sitting at the pode/antipode: a cosmological
horizon between the static patch and the de Sitter exterior region, where the
dilaton takes the constant value $\Phi=\phi_{0}+\phi_{r}$, and a black hole
horizon between the static patch and the black hole interior, where the
dilaton takes the constant value $\Phi=\phi_{0}-\phi_{r}$.

###  2.2 Semiclassical JT gravity

Semiclassical two-dimensional de Sitter JT gravity is obtained by coupling the
geometrical action $I_{\rm JT}$ introduced above to a $2d$ CFT with central
charge $c$, described by an action $I_{\rm CFT}$. In this paper, we work in
the semiclassical limit where $c\rightarrow\infty$, $G\rightarrow 0$, while
keeping $cG$ fixed. In the large $c$ limit, the backreaction is fully captured
by the (non-local) $1$-loop Polyakov action [44]

$I_{\rm Polyakov}=-\frac{1}{16\pi G}\int{\rm
d}^{2}x\sqrt{-g}\left[\frac{c}{48}R\frac{1}{\nabla^{2}}R\right].$ (2.24)

The total action is thus given by

$I=I_{\rm JT}+I_{\rm Polyakov}+I_{\rm CFT},$ (2.25)

where $I_{\rm JT}=I_{\rm JT}^{\rm half}$ or $I_{\rm JT}=I_{\rm JT}^{\rm
full}$, depending on the JT model under consideration. In this work, we will
consider a CFT composed of $c$ non-interacting scalar fields $\psi_{i}$,

$I_{\rm CFT}=\frac{1}{16\pi G}\int{\rm
d}x^{2}\sqrt{-g}\left[-\frac{1}{2}\sum_{i=1}^{c}\left(\nabla\psi_{i}\right)^{2}\right].$
(2.26)

The nature of the CFT action will not be of importance here, but will be
necessary in the introduction of entanglement entropies.

In the half reduction model, the conformal field theory is defined on a curved
background (2.16) with spatial boundaries $x^{+}x^{-}=-1/\Lambda$. In
particular, to provide a physically meaningful picture, consistent with the
higher dimensional one, we should impose reflecting boundary conditions:

$\left.\psi_{i}\right|_{x^{+}x^{-}=-1/\Lambda}=0.$ (2.27)

We will see in Section 2.3 that the importance of this boundary condition has
been overlooked in the past, and that it has crucial consequences when
computing entanglement entropies.

At the semiclassical level, the equation of motion for $\phi$ reads

$(g_{\mu\nu}\nabla^{2}-\nabla_{\mu}\nabla_{\nu}+g_{\mu\nu}\Lambda)\phi=8\pi
GT_{\mu\nu},$ (2.28)

or equivalently in conformal gauge:

$\displaystyle 2\partial_{+}\partial_{-}\phi-\Lambda e^{2\omega}\phi$
$\displaystyle=$ $\displaystyle 16\pi GT_{+-},$ (2.29)
$\displaystyle-\partial_{\pm}^{2}\phi+2\partial_{\pm}\omega\partial_{\pm}\phi$
$\displaystyle=$ $\displaystyle 8\pi GT_{\pm\pm}.$ (2.30)

The stress-energy tensor of the matter CFT can be written as

$T_{\mu\nu}=\tau_{\mu\nu}+T_{\mu\nu}^{\rm qu},$ (2.31)

where $\tau_{\mu\nu}$ is the (state-independent) flat space contribution to
the stress tensor, coming from the variation of $I_{\rm CFT}$666With the CFT
action (2.26), we have $\tau_{\mu\nu}=1/(16\pi
G)\sum_{i=1}^{c}\left(\nabla_{\mu}\psi_{i}\nabla_{\nu}\psi_{i}-\frac{1}{2}g_{\mu\nu}(\nabla\psi_{i})^{2}\right)$,
or in lightcone coordinates, $\tau_{\pm\pm}=(1/16\pi
G)\sum_{i=1}^{c}(\partial_{\pm}\psi_{i})^{2},\tau_{+-}=0$, but we will not
need this explicit expression in this paper., and $T_{\mu\nu}^{\rm qu}$ its
quantum mechanical part coming from the variation of the Polyakov action
$I_{\rm Polyakov}$. This term captures the (state-independent) conformal
anomaly that the stress-energy tensor acquires on a curved spacetime
background, which reads:

$g^{\mu\nu}\braket{T_{\mu\nu}}=\frac{c}{24\pi}R\leavevmode\nobreak\
\Longrightarrow\leavevmode\nobreak\
\braket{T_{+-}}=-\frac{c}{12\pi}\partial_{+}\partial_{-}\omega.$ (2.32)

In the same spirit as in [45], one can use (2.32) to solve the conservation
equation

$\nabla_{\mu}\braket{T^{\mu\nu}}=0,$ (2.33)

and get the components of the stress-energy tensor:

$\displaystyle\braket{T_{\pm\pm}(x^{\pm})}$ $\displaystyle=$
$\displaystyle\braket{\tau_{\pm\pm}}+\frac{c}{12\pi}\left(\partial_{\pm}^{2}\omega-\partial_{\pm}\omega\partial_{\pm}\omega\right)-\frac{c}{24\pi}t_{\pm}(x^{\pm}),$
(2.34) $\displaystyle\braket{T_{+-}(x^{+},x^{-})}$ $\displaystyle=$
$\displaystyle-\frac{c}{12\pi}\partial_{+}\partial_{-}\omega.$ (2.35)

The functions $t_{\pm}(x^{\pm})$ are functions of integration that arise in
the integration of the continuity equation (2.33). As we will see in the
following, they are state-dependent and characterize the choice of vacuum.
Using the classical solution (2.18), these expressions simplify to

$\displaystyle\braket{T_{\pm\pm}(x^{\pm})}$ $\displaystyle=$
$\displaystyle\braket{\tau_{\pm\pm}}-\frac{c}{24\pi}t_{\pm}(x^{\pm}),$ (2.36)
$\displaystyle\braket{T_{+-}(x^{+},x^{-})}$ $\displaystyle=$
$\displaystyle-\frac{c}{12\pi}\frac{\Lambda}{(1-\Lambda x^{+}x^{-})^{2}}.$
(2.37)

In terms of the normal ordered stress tensor
$:T_{\pm\pm}(x^{\pm}):\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
T_{\pm\pm}(x^{\pm})-\tau_{\pm\pm}(x^{\pm})$, we thus have

$\braket{:T_{\pm\pm}(x^{\pm}):}=-\frac{c}{24\pi}t_{\pm}(x^{\pm}).$ (2.38)

These expressions are only valid in Kruskal coordinates, as the stress-energy
tensor transforms non-trivially under conformal transformation
$x^{\pm}\rightarrow y^{\pm}(x^{\pm})$. In particular, $\tau_{\pm\pm}$
transforms as a rank-$2$ tensor,

$\tau_{\pm\pm}(y^{\pm})=\left(\frac{{\rm d}x^{\pm}}{{\rm
d}y^{\pm}}\right)^{2}\tau_{\pm\pm}(x^{\pm}),$ (2.39)

while the functions $t_{\pm}$ obey the anomalous transformation law:

$t_{\pm}(y^{\pm})=\left(\frac{{\rm d}x^{\pm}}{{\rm
d}y^{\pm}}\right)^{2}t_{\pm}(x^{\pm})+\\{x^{\pm},y^{\pm}\\},$ (2.40)

where $\\{x^{\pm},y^{\pm}\\}$ is the Schwarzian derivative defined by

$\\{x^{\pm},y^{\pm}\\}=\frac{\dddot{x}^{\pm}}{\dot{x}^{\pm}}-\frac{3}{2}\left(\frac{\ddot{x}^{\pm}}{\dot{x}^{\pm}}\right)^{2},$
(2.41)

with $\dot{x}^{\pm}={\rm d}x^{\pm}/{\rm d}y^{\pm}$. This yields the following
transformation law for the stress tensor:

$\braket{T_{\pm\pm}(y^{\pm})}=\left(\frac{{\rm d}x^{\pm}}{{\rm
d}y^{\pm}}\right)^{2}\braket{T_{\pm\pm}(x^{\pm})}-\frac{c}{24\pi}\\{x^{\pm},y^{\pm}\\}.$
(2.42)

A _vacuum state_ , defined as a state where $\braket{:T_{\pm\pm}(y^{\pm}):}=0$
in some coordinate system $y^{\pm}$, therefore corresponds to a state in which
the function $t$ vanishes in the coordinates $y^{\pm}$. From this, one sees
that $\tau_{\pm\pm}$ corresponds to the vacuum expectation value of
$T_{\pm\pm}(y^{\pm})$ in the vacuum state defined with respect to the
coordinates $y^{\pm}$, where $t_{\pm}(y^{\pm})=0$. All vacuum states are
related by conformal transformations of the lightcone coordinates in which
they are defined. We also refer to them as _conformal vacua_.

###  2.3 Entropy in the half reduction model

Let us consider a two-dimensional CFT with central charge $c$, on a curved
background with the metric ${\rm d}s^{2}=-e^{2\omega_{x}}{\rm d}x^{+}{\rm
d}x^{-}$. In the absence of spatial boundaries, the von Neumann entropy of a
Cauchy slice $\Sigma$, in the vacuum state defined in the coordinates
$x^{\pm}$, is given by [46]:

$S_{x}(\Sigma)=\frac{c}{6}\left(\omega_{x}(x_{1})+\omega_{x}(x_{2})\right)+\frac{c}{12}\ln\left[\frac{(x_{2}^{+}-x_{1}^{+})^{2}(x_{2}^{-}-x_{1}^{-})^{2}}{\delta_{1}^{2}\delta_{2}^{2}}\right],$
(2.43)

where the index $x$ refers to the choice of vacuum state,
$(x_{1}^{+},x_{1}^{-})$ and $(x_{2}^{+},x_{2}^{-})$ denote the coordinates of
the two endpoints of $\Sigma$, and $\delta_{1}^{2}$, $\delta_{2}^{2}$ are
boost-invariant UV cutoffs at the two endpoints of $\Sigma$, see Appendix A.
In the dS2 case we are interested in, $\omega_{x}$ is given by (2.18), and
this expression applies in the full reduction model.

Here, we would like to highlight the fact that this formula is not valid in
the half reduction model with reflective boundary conditions. First, we expect
the von Neumann entropy of a Cauchy slice covering the full spacetime to
vanish, as it is associated with a pure state. Second, the entropy of a slice
bounded by a point $P$ and one of the boundaries should not depend on the
endpoint at the boundary. Indeed, the reflection of right and left-moving
modes on the boundaries ensures that all Cauchy slices joining the point $P$
and the boundary are crossed by the same modes, i.e. they have the same
entropy. In particular, these slices are related by unitary evolution and
share the same causal “diamond”, as depicted by the blue shaded triangle in
Figure 3. The entropy formula (2.43) does not satisfy either of these two
conditions.

Considering reflecting boundary conditions at two spatial boundaries
parameterized by an arbitrary function $x^{-}=f_{x}(x^{+})$, the entanglement
entropy $S(\Sigma_{P})$ associated to a point $P$ located at $(x^{+},x^{-})$
has been computed in Appendix A.2 and is given in equation (A.9).777The
function $f_{x}$ is restricted to be a $C^{1}$ involution on $\mathbb{R}_{*}$,
with $0<{\rm d}f_{x}/{\rm d}x^{+}<\infty$ and $x^{+}f_{x}(x^{+})\leq 0$, see
Appendix A.2. In the half reduction model, the function $f_{x}$ is given by
$f_{x}(x^{+})=-1/(\Lambda x^{+})$, which gives for the vacuum state defined in
the Kruskal coordinates $x^{\pm}$:

$\displaystyle
S_{x}(\Sigma_{P})=\frac{c}{6}\left(\omega_{x}(x_{1})+\omega_{x}(x_{2})\right)+\frac{c}{12}\ln\left[\frac{(1+\Lambda
x^{+}x^{-})^{2}}{\Lambda\delta^{2}}\right].$ (2.44)

This vacuum state is called the Bunch-Davies vacuum. It is analogous to the
Minkowski vacuum in flat space or the Hartle-Hawking state of a black hole in
thermal equilibrium with its Hawking radiation. Inserting the explicit
expression (2.18) of $\omega_{x}$, we find a relation between the entropy and
the dilaton solution in the absence of matter (2.22):

$\displaystyle
S_{x}(\Sigma_{P})=\frac{c}{12}\ln\left[\frac{4}{\Lambda\delta^{2}}\frac{(1+\Lambda
x^{+}x^{-})^{2}}{(1-\Lambda
x^{+}x^{-})^{2}}\right]=\frac{c}{6}\ln\frac{\phi_{*}}{\phi_{r}}+\text{constant}.$
(2.45)

The reflective boundaries ensure that the state is symmetric with respect to
the $x^{+}$ and $x^{-}$ coordinates, and that the entropy of $\Sigma_{P}$ does
not depend on the position of its endpoint on the boundary, see Figure 3.

$\Sigma_{P}$$P$$A$ Figure 3: Penrose diagrams of the half reduction model,
with reflective boundary conditions at the spatial boundaries. They imply that
the causal “diamond” of a Cauchy slice $\Sigma_{P}$ ending on one boundary is
a triangle, depicted by the blue shaded region, and that $S(\Sigma_{P})$ is
independent of the location of $A$ along the left edge of the triangle.
Examples of radiation emanating as left-moving modes from $\cal{J}^{-}$ are
depicted. The red one crosses $\Sigma_{P}$ as a left-moving mode, while the
blue one crosses $\Sigma_{P}$ as a right-moving mode.

This formula can be generalized to any vacuum state, as shown in equation
(A.15) of Appendix A.3.

##  3 Classical Bousso Bound

Let us consider an $(n+1)$-dimensional spacetime $\hat{\mathcal{M}}$ and an
arbitrary codimension $2$ spacelike surface $B$ in $\hat{\mathcal{M}}$. A
light-sheet of $B$ is a codimension $1$ surface generated by light rays which
begin at $B$, extend orthogonally away from $B$ and are of non-positive
expansion. Considering the light rays to end at another codimension $2$
spacelike surface $B^{\prime}$, we denote $L(B-B^{\prime})$ the part of the
light-sheet emanating from $B$ and ending on $B^{\prime}$. The _Bousso bound_
, or _generalized covariant entropy bound_ , then states that the coarse-
grained entropy $S_{L(B-B^{\prime})}$ passing through $L(B-B^{\prime})$
satisfies the inequality [10, 11, 12]

$S_{L(B-B^{\prime})}\leq\frac{1}{4\hat{G}}(A(B)-A(B^{\prime})),$ (3.46)

where $\hat{G}$ is the Newton’s constant in $\hat{\mathcal{M}}$, and $A(B)$
and $A(B^{\prime})$ denote the area of the codimension $2$ surfaces $B$ and
$B^{\prime}$.

The Bousso bound has first been proven in 4 dimensions in [12] under either
one of two sets of assumptions. Later, it has been shown that it can actually
be derived from simpler assumptions [47]. In the following Section 3.1, the
two-dimensional version of the Bousso bound in de Sitter JT gravity is stated
and then proven in Section 3.2, following the strategy of [19]. We refer to
Appendix B for the definitions of the notions of light-sheet, expansion, and
Bousso bound in arbitrary dimension together with their dimensional reduction,
although this section can be followed without it. We assume that the classical
part $\braket{\tau_{\mu\nu}}$ of the stress-energy tensor satisfies the _Null
Energy Condition_ (NEC):

$k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}\geq 0,$ (3.47)

for any null vector $k^{\mu}$. This condition is satisfied by most classical
matter fields, and in particular by the classical CFT action (2.26) (see
footnote 6); though we will see that it can be broken once quantum effects are
taken into account.

###  3.1 The Bousso Bound and Energy Conditions

In this section, we recall the two-dimensional Bousso bound in JT gravity,
where the dilaton acts as the area of $0$-dimensional surfaces. For earlier
work on the two-dimensional Bousso bound, see [48].

As discussed at the beginning of Section 2, de Sitter JT gravity comes from
the dimensional reduction of the $(n+1)$-dimensional Einstein-Hilbert action.
In particular, the half reduction model emerges from dS3, and the full
reduction model emerges from $(n+1)$-dimensional Schwarzschild-de Sitter
space, with $n\geq 3$. Considering the metric ansatz (2.9), the dimensional
reduction of $L(B-B^{\prime})$ down to two dimensions gives a single light ray
bounded by two points $B$ and $B^{\prime}$, which we will still denote by
$L(B-B^{\prime})$. We call $\lambda$ the affine parameter along
$L(B-B^{\prime})$, normalized such that $\lambda=0$ on $B$ and $\lambda=1$ on
$B^{\prime}$. The future directed vector normal (and tangent) to
$L(B-B^{\prime})$ is defined by

$k^{\mu}=\pm\left(\frac{{\rm d}}{{\rm d}\lambda}\right)^{\mu}.$ (3.48)

The “$+$” sign is considered for a future-directed light-sheet, and the “$-$”
sign for a past-directed light-sheet, so that $k$ is always future-directed.
In this paper, the prime will denote the derivative with respect to $\lambda$,
i.e. ${}^{\prime}\equiv{\rm d}/{\rm d}\lambda=k^{\mu}\nabla_{\mu}$. $\lambda$
being an affine parameter, $k^{\mu}$ must satisfy the geodesic equation [49]:

$k^{\mu}\nabla_{\mu}k^{\nu}=0.$ (3.49)

The geodesic always propagates either along $x^{+}$ or $x^{-}$, such that
$k^{\mu}=(k^{+},0)$ or $k^{\mu}=(0,k^{-})$ respectively. This yields the
differential equation

$\displaystyle\partial_{\pm}k^{\pm}+2\partial_{\pm}\omega k^{\pm}$
$\displaystyle=0,$ (3.50)

which is solved by

$\displaystyle k^{\pm}$ $\displaystyle=Ce^{-2\omega},$ (3.51)

where the constant $C$ can be normalized to $1$ by an appropriate rescaling of
$\lambda$.

A priori there is no notion of expansion of a unique light ray, as it is
always trivially equal to zero. However, the dilaton encodes the size of the
compactified space, which varies along the two-dimensional space. To compute
the expansion parameter in JT gravity, we go back to its definition in terms
of the extrinsic curvature, see Appendix B and equation (B.4), to take into
account the compact space. With the ansatz (2.9), the extrinsic curvature of
the $n$-dimensional congruence of geodesics reduces to [49]

${K}^{(n)}=K^{(1)}+\frac{1}{\Phi}k^{\mu}\nabla_{\mu}\Phi,$ (3.52)

where $K^{(1)}$ is the extrinsic curvature of the $1$-dimensional light ray in
the two-dimensional spacetime $\mathcal{M}$. Since the extrinsic curvature of
a unique light ray vanishes, we get the expression of the expansion parameter
in JT gravity,

$\theta=\frac{1}{\Phi}k^{\mu}\nabla_{\mu}\Phi=\frac{1}{\Phi}\frac{{\rm
d}\Phi}{{\rm d}\lambda}=\frac{{\rm d}\ln\Phi}{{\rm d}\lambda}.$ (3.53)

We define a _light-sheet_ in de Sitter JT gravity as a light ray satisfying
the condition

$\Phi^{\prime}=\frac{{\rm d}\Phi}{{\rm d}\lambda}\leq 0$ (3.54)

everywhere. In Appendix B, we recall the classical focussing theorem in
arbitrary dimension, which states that light rays are focused under the
evolution of a null congruence if the NEC is satisfied. In order to state a
two-dimensional version of the classical focussing theorem in de Sitter JT
gravity, we use (3.53) to compute

$\frac{{\rm d}\theta}{{\rm d}\lambda}=\frac{1}{\Phi}\frac{{\rm
d}^{2}\Phi}{{\rm d}\lambda^{2}}-\theta^{2}.$ (3.55)

From the classical equation of motion (3.64), one sees that ${\rm
d}^{2}\Phi/{\rm d}\lambda^{2}=-8\pi G\leavevmode\nobreak\
k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}$ for any null vector $k^{\mu}$, which is
non-positive by the NEC (3.47). Since $\Phi$ is positive,
$\Phi^{\prime\prime}/\Phi$ is non-positive, and we get the _Classical
Focussing Theorem_ :

$\frac{{\rm d}\theta}{{\rm d}\lambda}\leq 0.$ (3.56)

This inequality tells us that along a single light ray, the function
$\Phi^{\prime}/\Phi$ is decreasing. Therefore, if the NEC is satisfied, we can
define a light-sheet as a light ray where $\Phi^{\prime}\leq 0$ only
initially, the classical focussing theorem then implying $\Phi^{\prime}\leq 0$
all along the light ray.

To state and prove the Bousso bound in de Sitter JT gravity, we consider the
_hydrodynamic approximation_ , where the matter entropy has a phenomenological
description in terms of a local entropy current $s^{\mu}$. The entropy flux
$s$ passing through any point of the light-sheet is then the projection of
$s^{\mu}$ onto the normal vector $k^{\mu}$:

$s=-k^{\mu}s_{\mu}.$ (3.57)

The total matter entropy passing through $L(B-B^{\prime})$ is therefore the
integral of $s$ over $L(B-B^{\prime})$,

$S_{L(B-B^{\prime})}=\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
s(\lambda).$ (3.58)

The generalized covariant entropy bound in de Sitter JT gravity then reads:

$\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
s(\lambda)\leq\frac{1}{4G}\left(\Phi(0)-\Phi(1)\right),$ (3.59)

where the “areas” of the two endpoints of the light-sheet are given by the
value of dilaton. This can be derived by dimensional reduction of the
$(n+1)$-dimensional Bousso bound, as described in Appendix B. It has been
shown [47, 19] that the classical Bousso bound can be proven in $4$ dimensions
using the two conditions (B.12) and (B.14). They correspond to the validity
conditions for the hydrodynamic approximation, and their dimensional reduction
yields

$\displaystyle\Phi(\lambda)\left|\left(s(\lambda)\Phi^{-1}(\lambda)\right)^{{}^{\prime}}\right|$
$\displaystyle\leq 2\pi k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}(\lambda),$ (3.60)
$\displaystyle s(0)$ $\displaystyle\leq-\frac{1}{4G}\Phi^{{}^{\prime}}(0).$
(3.61)

The first one implies the NEC in two dimensions,
$k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}\geq 0$, while the second one implies
that the light-sheet is initially non-expanding. Using the classical focusing
theorem, this corresponds to imposing the non-expansion of the light ray.

###  3.2 Proof of the Classical Bousso Bound

In order to prove the classical Bousso bound in de Sitter JT gravity, we first
rewrite the entropy condition (3.60) as

$\displaystyle s^{{}^{\prime}}(\lambda)$ $\displaystyle\leq$ $\displaystyle
2\pi
k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}(\lambda)+s(\lambda)\frac{\Phi^{{}^{\prime}}(\lambda)}{\Phi(\lambda)}$
(3.62) $\displaystyle\leq$ $\displaystyle 2\pi
k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}(\lambda),$ (3.63)

where the second inequality follows from $\Phi^{{}^{\prime}}(\lambda)\leq 0$
along a light-sheet, while $s(\lambda)$ and $\Phi(\lambda)$ are both positive.
On the other hand, for any null vector $k^{\mu}$, the equation of motion
(2.28) gives:

$8\pi G\leavevmode\nobreak\
k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}=-k^{\mu}k^{\nu}\nabla_{\mu}\nabla_{\nu}\Phi=-\Phi^{{}^{\prime\prime}},$
(3.64)

so that

$s^{{}^{\prime}}(\lambda)\leq-\frac{1}{4G}\Phi^{{}^{\prime\prime}}(\lambda).$
(3.65)

From now on, we work in the conformal gauge (2.16), and denote by
$(x^{+},x^{-})$ the Kruskal coordinates of an arbitrary point of the light-
sheet with affine parameter $\lambda$. The initial point of the light-sheet,
$\lambda=0$, is noted $(x_{0}^{+},x_{0}^{-})$. The derivative with respect to
$\lambda$ is given by ${}^{\prime}\equiv d/d\lambda=\frac{\partial
x^{\pm}}{\partial\lambda}\partial_{\pm}$, with the $+$ or $-$ for a light-
sheet along $x^{+}$ or $x^{-}$ respectively, so that the entropy conditions
(3.65) and (3.61) are given by:

$\displaystyle\partial_{\pm}s\leavevmode\nobreak\ \frac{\partial
x^{\pm}}{\partial\lambda}$
$\displaystyle\leq-\frac{1}{4G}\partial_{\pm}\left(\partial_{\pm}\Phi\leavevmode\nobreak\
\frac{\partial x^{\pm}}{\partial\lambda}\right)\frac{\partial
x^{\pm}}{\partial\lambda},$ (3.66) $\displaystyle s(x_{0}^{\pm})$
$\displaystyle\leq-\frac{1}{4G}\partial_{\pm}\Phi(x_{0}^{\pm})\leavevmode\nobreak\
\frac{\partial x^{\pm}}{\partial\lambda}(x_{0}^{\pm}).$ (3.67)

In the following, we will consider the case of a future directed light-sheet
along $x^{+}$. Since $\partial x^{+}/\partial\lambda>0$ in this case, the
first entropy condition (3.66) gives:

$\partial_{+}s\leq-\frac{1}{4G}\partial_{+}\left(\partial_{+}\Phi\frac{\partial
x^{+}}{\partial\lambda}\right).$ (3.68)

Integrating this inequality between $x_{0}^{+}$ and $x^{+}>x_{0}^{+}$ yields

$s(x^{+})-s(x_{0}^{+})\leq-\frac{1}{4G}\leavevmode\nobreak\
\partial_{+}\Phi(x^{+})\leavevmode\nobreak\ \frac{\partial
x^{+}}{\partial\lambda}(x^{+})+\frac{1}{4G}\leavevmode\nobreak\
\partial_{+}\Phi(x_{0}^{+})\leavevmode\nobreak\ \frac{\partial
x^{+}}{\partial\lambda}(x_{0}^{+}),$ (3.69)

which can be further simplified using the second entropy condition (3.67)
into:

$s(x^{+})\leq-\frac{1}{4G}\partial_{+}\Phi(x^{+})\leavevmode\nobreak\
\frac{\partial x^{+}}{\partial\lambda}(x^{+}).$ (3.70)

From $s=-k^{+}s_{+}$ and $k^{+}=\frac{\partial x^{+}}{\partial\lambda}>0$ for
a future-directed light-sheet, this gives:

$-s_{+}(x^{+})\leq-\frac{1}{4G}\partial_{+}\Phi(x^{+}).$ (3.71)

One can thus write

$\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
s(\lambda)=\int_{x_{0}^{+}}^{x_{1}^{+}}{\rm
d}x^{+}\frac{\partial\lambda}{\partial
x^{+}}(-k^{+}s_{+})=\int_{x_{0}^{+}}^{x_{1}^{+}}{\rm
d}x^{+}(-s_{+})\leq-\frac{1}{4G}\int_{x_{0}^{+}}^{x_{1}^{+}}{\rm
d}x^{+}\partial_{+}\Phi,$ (3.72)

from which we find the classical Bousso bound, written in terms of the
$\lambda$ parameter:

$\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
s(\lambda)\leq\frac{1}{4G}(\Phi(0)-\Phi(1)).$ (3.73)

The derivation for a past-directed light-sheet along $x^{+}$, for which
$k^{+}=-\frac{\partial x^{+}}{\partial\lambda}=e^{-2\omega}$, or for a light-
sheet along $x^{-}$, can be carried out in a completely analogous way, leading
to the same result.

##  4 Strominger-Thompson Quantum Bousso Bound

In this section, we prove the Strominger-Thompson Quantum Bousso Bound (QBB)
in the framework of semiclassical de Sitter JT gravity introduced in Section
2. We first consider the semiclassical violations of the classical Bousso
bound and motivate the two quantum entropy conditions of [19]. We then
introduce the quantum version of the Bousso bound that was proposed by
Strominger and Thompson and proven in the two-dimensional RST model [20], and
derive a sufficient condition for the QBB to hold. Inspired by the work of
Wall on the generalized second law in $(1+1)$-dimensions [40], we study the
transformation laws of the entropy and stress tensor under a change of vacuum
state, and identify a quantity invariant under conformal transformations. This
is used to show that the sufficient condition is satisfied in any vacuum state
defined in arbitrary lightcone coordinates, hence establishing the proof of
the QBB in an infinite class of vacuum states.

###  4.1 Generalized Entropy and Quantum Energy Conditions

In our proof of the classical Bousso bound, we assumed the NEC in the
assumption (3.60). However, this local energy condition is not a fundamental
law of physics and it may be violated in the presence of matter, as well as
the Bousso bound [50, 51, 19]. It is for example the case when black holes
evaporate [52], and we expect a similar situation in de Sitter space. Indeed,
in the de Sitter JT gravity framework studied here, the total stress-energy
tensor is given by
$\braket{T_{\pm\pm}}=\braket{\tau_{\pm\pm}}-\frac{c}{24\pi}t_{\pm}$. While the
classical part $\braket{\tau_{\pm\pm}}$ satisfies the NEC, it might be
violated by the full $\braket{T_{\pm\pm}}$, depending on the functions
$t_{\pm}$, which might imply a violation of the classical focussing theorem
and the classical Bousso bound. This is the case for instance in the Unruh-de
Sitter vacua [31]. One would therefore like to define a quantum version of a
light-sheet, of the NEC, and of the Bousso bound that would be consistent at
the semiclassical level.

The starting point for such generalizations is the observation that a quantum
area (or generalized entropy) can be assigned to any point separating a
$1$-dimensional Cauchy slice into two portions. The “quantum area” of a point
$P(\lambda)$ along a light ray of affine parameter $\lambda$ is then given by

$A_{\rm qu}(\lambda)=\Phi(\lambda)+4GS(\lambda),$ (4.74)

where $S(\lambda)$ is the fine-grained entropy, as defined in equation (A.2),
of a Cauchy slice $\Sigma_{P}$ defined as the interior of $P(\lambda)$ 888In a
spacetime without boundary, this slice is defined such that
$\partial\Sigma_{P}=P$. Note that there is always two choices of $\Sigma_{P}$
depending on what we call the interior and exterior of $P$. However, the
definition of fine-grained entropy (A.2) implies that
$S\left(\overline{\Sigma_{P}}\right)=S(\Sigma_{P})$, so that both choices are
equivalent. Here we define $\overline{\Sigma_{P}}$, a complement of
$\Sigma_{P}$, such that there exists $\Sigma$ a global Cauchy slice whose
state is pure, with $\Sigma_{P}\cup\overline{\Sigma_{P}}=\Sigma$. In a closed
spacetime with boundary, we define $\Sigma_{P}$ such that
$\partial\Sigma_{P}=P\cup A$, where $A$ is an arbitrary point of the boundary,
with the same property of interior and exterior being interchangeable.. In
analogy with the classical case, we define a _quantum light-sheet_ in de
Sitter JT gravity as a light ray satisfying everywhere the condition

$A_{\rm qu}^{\prime}(\lambda)\leq 0.$ (4.75)

At the semiclassical level, a generalisation of the NEC has been proven in
[53, 54, 55]. This so-called _Quantum Null Energy Condition_ (QNEC) states
that:

$2\pi k^{\mu}k^{\nu}\braket{T_{\mu\nu}}\geq S^{\prime\prime}.$ (4.76)

This equation does not depend on $G$ and is unaffected by higher curvature
terms in the gravitational action, such that it derives from fundamental
principles in quantum field theory. In two dimensions, it was shown in [40]
that the QNEC implies the stronger statement

$2\pi k^{\mu}k^{\nu}\braket{T_{\mu\nu}}\geq
S^{\prime\prime}+\frac{c}{6}(S^{\prime})^{2},$ (4.77)

which follows from the transformation laws of these quantities under conformal
transformation.

The Bousso bound as well as the assumptions (3.60) and (3.61) must be modified
in order to hold at the semiclassical level. Motivated by the progresses in
black hole thermodynamics and holography due to the introduction of
generalized entropy (1.2), Strominger and Thompson [19] conjectured the
_Strominger-Thompson Quantum Bousso Bound_ :

$\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
s(\lambda)\leq\frac{1}{4G}(A_{\rm qu}(0)-A_{\rm qu}(1)).$ (4.78)

Other conjectured QBB were proposed [41, 42] and will be discussed in Section
4.4. At the semiclassical level, the classical stress tensor $\tau_{\mu\nu}$
must be replaced by the full stress tensor $T_{\mu\nu}$ which may violate the
NEC. Since the first condition (3.60) implies the NEC and the second one
(3.61) implies that the classical Bousso bound is satisfied at the beginning
of the light-sheet, the conditions (3.60) and (3.61) must be modified. Adapted
to the case of JT gravity, the semiclassical conditions in two dimensions are:

  * •

The first classical condition (3.63) is unchanged,

$\left|s^{{}^{\prime}}(\lambda)\right|\leq 2\pi
k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}(\lambda),$ (4.79)

without introducing the quantum mechanical part of the stress tensor, which
corresponds to imposing the classical condition in the non-backreacted
geometry. It is well defined since $\tau_{\mu\nu}$ always satisfies the NEC.

  * •

The modification $A\rightarrow A_{\rm qu}$ is applied to the second classical
condition (3.61), implying that the QBB is initially satisfied:

$s(0)\leq-\frac{1}{4G}A_{\rm qu}^{\prime}(0).$ (4.80)

Since $s(\lambda)$ is positive, this condition implies the quantum area to be
initially non-increasing. In particular, this is satisfied by a quantum light-
sheet, as defined in (4.75).

###  4.2 A sufficient condition

In order to investigate the QBB in de Sitter JT gravity, we start from the
equation of motion at the semiclassical level (2.30). For any null vector
$k^{\pm}$, this gives:

$8\pi
Gk^{\pm}k^{\pm}\left(\braket{\tau_{\pm\pm}}-\frac{c}{24\pi}t_{\pm}\right)=-\Phi^{\prime\prime}=-\partial_{\pm}\left(\partial_{\pm}\Phi\frac{\partial
x^{\pm}}{\partial\lambda}\right)\frac{\partial x^{\pm}}{\partial\lambda}.$
(4.81)

Inserting this equation into the first entropy condition (4.79) leads to the
inequality:

$\partial_{\pm}s\leavevmode\nobreak\ \frac{\partial
x^{\pm}}{\partial\lambda}\leq-\frac{\Phi^{\prime\prime}}{4G}+\frac{c}{12}k^{\pm}k^{\pm}t_{\pm},$
(4.82)

or equivalently, using the definition of the quantum area of a point (4.74):

$\partial_{\pm}s\leavevmode\nobreak\ \frac{\partial
x^{\pm}}{\partial\lambda}\leq-\frac{A_{\rm
qu}^{\prime\prime}}{4G}+S^{\prime\prime}+\frac{c}{12}k^{\pm}k^{\pm}t_{\pm}.$
(4.83)

As it can be seen from this last inequality, a sufficient condition for the
QBB (4.78) to be satisfied is

$S^{\prime\prime}+\frac{c}{12}k^{\pm}k^{\pm}t_{\pm}\leq 0.$ (4.84)

Written in terms of the normal ordered stress tensor, this condition is

$2\pi k^{\pm}k^{\pm}\braket{:T_{\pm\pm}:}\geq S^{\prime\prime},$ (4.85)

which closely resembles the QNEC (4.76). In particular, it becomes exactly the
QNEC in the limit $\tau_{\pm\pm}=0$. If the inequality (4.84) is true, one
gets

$\partial_{\pm}s\leavevmode\nobreak\ \frac{\partial
x^{\pm}}{\partial\lambda}\leq-\frac{A_{\rm qu}^{\prime\prime}}{4G},$ (4.86)

from which one can follow the same steps as in the proof of the classical
Bousso bound presented in Section 3.2.

Here, we consider as an example the case of a past directed light-sheet along
$x^{+}$. Since $\partial x^{+}/\partial\lambda<0$, (4.86) gives:

$\partial_{+}s\geq-\frac{1}{4G}\partial_{+}\left(\partial_{+}A_{\rm
qu}\frac{\partial x^{+}}{\partial\lambda}\right).$ (4.87)

Integrating this inequality between $x^{+}$ and $x_{0}^{+}>x^{+}$ yields

$s(x_{0}^{+})-s(x^{+})\geq-\frac{1}{4G}\leavevmode\nobreak\ \partial_{+}A_{\rm
qu}(x_{0}^{+})\leavevmode\nobreak\ \frac{\partial
x^{+}}{\partial\lambda}(x_{0}^{+})+\frac{1}{4G}\leavevmode\nobreak\
\partial_{+}A_{\rm qu}(x^{+})\leavevmode\nobreak\ \frac{\partial
x^{+}}{\partial\lambda}(x^{+}),$ (4.88)

which can be further simplified using the second entropy condition (4.80) into

$-s(x^{+})\geq\frac{1}{4G}\partial_{+}A_{\rm qu}(x^{+})\leavevmode\nobreak\
\frac{\partial x^{+}}{\partial\lambda}(x^{+}).$ (4.89)

From $s=-k^{+}s_{+}$ and $k^{+}=-\frac{\partial x^{+}}{\partial\lambda}>0$ for
a past-directed light-sheet, this gives:

$-s_{+}(x^{+})\leq\frac{1}{4G}\partial_{+}A_{\rm qu}(x^{+}).$ (4.90)

One can thus write

$\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
s(\lambda)=\int_{x_{0}^{+}}^{x_{1}^{+}}{\rm
d}x^{+}\frac{\partial\lambda}{\partial
x^{+}}(-k^{+}s_{+})=\int_{x_{1}^{+}}^{x_{0}^{+}}{\rm
d}x^{+}(-s_{+})\leq\frac{1}{4G}\int_{x_{1}^{+}}^{x_{0}^{+}}{\rm
d}x^{+}\partial_{+}A_{\rm qu},$ (4.91)

from which we find the QBB:

$\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
s(\lambda)\leq\frac{1}{4G}(A_{\rm qu}(0)-A_{\rm qu}(1)).$ (4.92)

We thus showed that the inequality (4.84) is a sufficient condition for the
Strominger-Thompson QBB to hold in the de Sitter JT gravity framework.

###  4.3 Proof of the Quantum Bousso Bound

To show that the sufficient condition (4.84) is verified in any vacuum state,
we consider the transformation laws of $S$ and $t_{\pm}$ under a change of
vacuum, i.e. under a conformal transformation of the lightcone coordinates in
which the vacuum is defined. Under a change of coordinates $x^{\pm}\rightarrow
y^{\pm}(x^{\pm})$, the metric becomes

$\displaystyle{\rm d}s^{2}$ $\displaystyle=-e^{2\omega_{x}(x^{+},x^{-})}{\rm
d}x^{+}{\rm d}x^{-}=-e^{2\omega_{x}(x^{+},x^{-})}\frac{{\rm d}x^{+}}{{\rm
d}y^{+}}\frac{{\rm d}x^{-}}{{\rm d}y^{-}}{\rm d}y^{+}{\rm d}y^{-}$ (4.93)
$\displaystyle=-e^{2\omega_{y}(y^{+},y^{-})}{\rm d}y^{+}{\rm d}y^{-},$ (4.94)

from which we get the transformation of the conformal factor:

$\omega_{y}(y^{+}(x^{+}),y^{-}(x^{-}))=\omega_{x}(x^{+},x^{-})-\frac{1}{2}\ln\left[\frac{{\rm
d}y^{+}}{{\rm d}x^{+}}\frac{{\rm d}y^{-}}{{\rm d}x^{-}}\right].$ (4.95)

We start by studying the transformation law of the entropy $S(\Sigma)$ in the
presence of a boundary. To make the dependence in the boundary clear, we keep
the general formulation (A.9) where the boundary follows a trajectory
$x^{-}=f_{x}(x^{+})$, as described in Appendix A.2. The transformation law of
the entropy under a change of vacuum $x^{\pm}\rightarrow y^{\pm}(x^{\pm})$ is
(see equation (A.15)),

$\displaystyle S_{x}(\Sigma)\rightarrow S_{y}(\Sigma)$
$\displaystyle=\frac{c}{6}\omega_{x}(x^{+},x^{-})+\frac{c}{12}\ln\left[\dot{f}_{x}(f_{x}^{-1}(x^{-}))\frac{\left(y^{+}(x^{+})-y^{+}(f_{x}^{-1}(x^{-}))\right)^{2}}{\delta^{2}}\right]$
$\displaystyle-\frac{c}{12}\ln\left[\frac{{\rm d}y^{+}}{{\rm
d}x^{+}}(x^{+})\frac{{\rm d}y^{+}}{{\rm
d}x^{+}}\left(f_{x}^{-1}(x^{-})\right)\right],$ (4.96)

where $\dot{f}_{x}={\rm d}f_{x}/{\rm d}x^{+}$. From this relation, one can
then compute the transformation laws for the first and second derivatives of
the entropy. We take the case were the derivation is taken along a light-sheet
going in the $x^{+}$ direction, i.e. ${}^{\prime}\equiv{\rm d}/{\rm
d}\lambda=k^{+}\partial/\partial x^{+}$, giving

$S_{x}^{\prime}(\Sigma)\rightarrow
S_{y}^{\prime}(\Sigma)=\frac{c}{6}\left[\frac{\partial\omega_{x}}{\partial
x^{+}}-\frac{1}{2}\frac{\ddot{y}^{+}}{\dot{y}^{+}}+\frac{\dot{y}^{+}}{y^{+}-f_{y}^{-1}(y^{-})}\right]\frac{\partial
x^{+}}{\partial\lambda},$ (4.97)

where $\dot{y}^{+}={\rm d}y^{+}/{\rm d}x^{+}$. Taking the second derivative,
we get

$\displaystyle S_{x}^{\prime\prime}(\Sigma)$ $\displaystyle\rightarrow$
$\displaystyle
S_{y}^{\prime\prime}(\Sigma)=\frac{c}{6}\left[\frac{\partial^{2}\omega_{x}}{\partial
x^{+2}}-\frac{1}{2}\frac{\dddot{y}^{+}}{\dot{y}^{+}}+\frac{1}{2}\left(\frac{\ddot{y}^{+}}{\dot{y}^{+}}\right)^{2}-\frac{(\dot{y}^{+})^{2}}{(y^{+}-f_{y}^{-1}(y^{-}))^{2}}\right.$
(4.98) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\left.\frac{\ddot{y}^{+}}{y^{+}-f_{y}^{-1}(y^{-})}-2\frac{\partial\omega_{x}}{\partial
x^{+}}\left(\frac{\partial\omega_{x}}{\partial
x^{+}}-\frac{1}{2}\frac{\ddot{y}^{+}}{\dot{y}^{+}}+\frac{\dot{y}^{+}}{y^{+}-f_{y}^{-1}(y^{-})}\right)\right]\left(\frac{\partial
x^{+}}{\partial\lambda}\right)^{2}.$

Combining the two expressions above, we obtain the transformation law for the
quantity $S^{\prime\prime}+\frac{6}{c}(S^{\prime})^{2}$ introduced by Wall
[40]:

$S_{x}^{\prime\prime}+\frac{6}{c}(S_{x}^{\prime})^{2}\rightarrow
S_{y}^{\prime\prime}+\frac{6}{c}(S_{y}^{\prime})^{2}=\frac{c}{6}\left[\frac{\partial^{2}\omega_{x}}{\partial
x^{+2}}-\left(\frac{\partial\omega_{x}}{\partial
x^{+}}\right)^{2}-\frac{1}{2}\\{y^{+},x^{+}\\}\right]\left(\frac{\partial
x^{+}}{\partial\lambda}\right)^{2},$ (4.99)

which can be rewritten as

$S_{x}^{\prime\prime}+\frac{6}{c}(S_{x}^{\prime})^{2}\rightarrow
S_{y}^{\prime\prime}+\frac{6}{c}(S_{y}^{\prime})^{2}=S_{x}^{\prime\prime}+\frac{6}{c}(S_{x}^{\prime})^{2}-\frac{c}{12}\\{y^{+},x^{+}\\}\left(\frac{\partial
x^{+}}{\partial\lambda}\right)^{2}.$ (4.100)

Similarly, we can compute the $t_{\pm}(x^{\pm})$ function in the vacuum
defined by $t_{\pm}(y^{\pm})=0$ using equation (2.40). As for the entropy, we
will make the vacuum state explicit by writing $t^{(y)}_{\pm}$ the $t_{\pm}$
function in the vacuum defined in coordinates $y^{\pm}$, i.e.
$t^{(y)}_{\pm}(y^{\pm})=0$. The transformation under a change of vacuum then
writes

$t^{(x)}_{\pm}(x^{\pm})=0\rightarrow
t^{(y)}_{\pm}(x^{\pm})=\left\\{y^{\pm},x^{\pm}\right\\}.$ (4.101)

From the transformation laws of $S$, its derivatives, and $t_{+}$, we find
that the quantity

$\mathcal{Q}=\frac{c}{12}k^{+}k^{+}t^{(x)}_{+}+S_{x}^{\prime\prime}+\frac{6}{c}(S_{x}^{\prime})^{2}$
(4.102)

is a scalar under conformal transformation. The choice of initial vacuum state
being arbitrary, we can drop the vacuum indices and write

$\mathcal{Q}=\frac{c}{12}k^{+}k^{+}t_{+}+S^{\prime\prime}+\frac{6}{c}(S^{\prime})^{2}.$
(4.103)

The argument above can be extended to a light-sheet propagating along the
$x^{-}$ direction. This may seem cumbersome, due to the dependence of
$\dot{f}_{x}(f_{x}^{-1}(x^{-}))$ in $x^{-}$. However, the derivation of (A.9)
in Appendix A can be modified to obtain a very similar formula depending on
$\dot{f}_{x}(f_{x}(x^{+}))$999To do so, one describes the state of fields as
right-moving modes on $H_{-}$ instead of left-moving modes on $H_{+}$, see
Appendix A., allowing to follow the same procedure for a light-sheet along
$x^{-}$.

At this point, we can notice that in the Bunch-Davies vacuum,
$t_{\pm}(x^{\pm})=0$ and

$S_{\rm BD}^{\prime\prime}+\frac{6}{c}(S_{\rm BD}^{\prime})^{2}=0,$ (4.104)

as follows from equation (2.45), so that $\mathcal{Q}=0$. Because
$\mathcal{Q}$ is a scalar under conformal transformations, this must be true
in any vacuum state. To conclude, we have shown that

$S^{\prime\prime}+\frac{c}{12}k^{\pm}k^{\pm}t_{\pm}=-\frac{6}{c}(S^{\prime})^{2}\leq
0$ (4.105)

in any vacuum state, hence completing the proof of the Strominger-Thompson
QBB. This inequality together with the NEC satisfied by the classical part
$\braket{\tau_{\pm\pm}}$ of the stress tensor implies the QNEC. In particular,
it implies the stronger inequality (4.77), showed to be equivalent to the QNEC
by Wall in two dimensions [40].

The transformation laws of $S$ and its derivatives can also be derived in the
case without any boundaries. However, there is a priori an ambiguity in the
definition of the slice $\Sigma_{P}$ associated to a point $P$. Without
matter, the full reduction model is either spatially periodic or infinitely
extended, and one would need at least two points to define the boundary of a
non-trivial Cauchy slice. Hence, we will always consider one of these two
points to be fixed and spacelike separated from the second one moving along
the light-sheet. This produces the same transformation laws as in the half
reduction model, so that $\mathcal{Q}$ is invariant. In particular,
$S^{\prime\prime}+6/c(S^{\prime})^{2}$ does not depend on the position of the
fixed point. Using the fact that $t_{\pm}^{(x)}(x^{\pm})=0$ and that the
entropy formula (2.43) also satisfies (4.104), one recovers the result that
$\mathcal{Q}=0$ is any vacuum state. This situation is depicted in the Penrose
diagram of Figure 4, in the case where the fixed point $A$ lies at
$x_{A}^{\pm}=\pm\infty$. This is a preferred point since it is the only one
spacelike separated to any light-sheet contained in the white and/or blue
regions of Figure 2. One may also be tempted to define the problem such that
both endpoints of the slice belong to two lightrays forming a disconnected
light-sheet. We will not treat this problem in this paper although we expect
it is well defined in JT gravity.

Finally, we may note that the backreaction of matter on the dilaton modifies
the effective geometry, which can remove the ambiguity due to spatial
periodicity/infinite extension in some vacuum states. For example, the dilaton
may diverge in some regions of the full reduction model, creating effective
boundaries. This is the case in the Unruh-de Sitter vacuum [31, 37, 35], in
which the backreacted geometry ends on the past cosmological horizon and black
hole horizons of one of the static patches[35], see Figure 4. This vacuum
state describes the evaporation of the de Sitter cosmological horizon, and is
analogous to the Unruh vacuum of a black hole. In this example, there is a
conformal boundary at $x^{+}\rightarrow+\infty$, which acts as a weakly
gravitating region where the radiation emanating from the past cosmological
horizon can be collected. In this type of situation, one places the fixed
point $A$ at the effective boundary as in Figure 4.

$\Sigma_{P}$$P$$B^{\prime}$$B$$A$ Figure 4: Penrose diagram for two-
dimensional de Sitter space in the full reduction model. A portion of a light
ray bounded by two points $B$ and $B^{\prime}$ is depicted by the dashed line,
as well as a Cauchy slice $\Sigma_{P}$ bounded by a point $P$ of the light ray
and the point $A$ at spatial infinity. In the Unruh-de Sitter vacuum, the
dilaton diverges on the past cosmological horizon and on the black hole
horizon of the static patch of the antipode, eliminating the gray shaded
regions from the backreacted solution. The past horizon emits radiation (in
blue) crossing $\Sigma_{P}$ as right-moving modes, and ending up in the weakly
gravitating region at $x^{+}\rightarrow\infty$.

###  4.4 Comparison to other Quantum Bousso Bounds

Following the proposal of Strominger and Thompson, other conjectures were made
to include semiclassical corrections in the Bousso bound. The most recent one
was obtained by Bousso, Fisher, Leichenauer, and Wall (BFLW) as a corollary of
the Quantum Focussing Conjecture (QFC) [42], a semiclassical extension of the
classical focussing theorem. This QBB states that the entropy of the light-
sheet $L(B^{\prime}-B)$ can be defined as the difference between the fine-
grained entropies of Cauchy slices bounded by the codimension $2$ surfaces $B$
and $B^{\prime}$ respectively, and applies to quantum light-sheets instead of
classical light-sheets. On the other hand, Bousso, Casini, Fisher, and
Maldacena (BCFM) [41] showed that in the weak gravity limit,
$S_{L(B-B^{\prime})}$ may be defined as a difference of fine-grained entropies
of an arbitrary state and the vacuum, with both states restricted to
$L(B-B^{\prime})$. In this section, we review how the BFLW quantum Bousso
bound can be derived from the QFC and note that it follows from our proof of
the Strominger-Thompson bound. Finally, we recall the BCFM quantum Bousso
bound and compare it with the two others.

####  4.4.1 BFLW Quantum Bousso Bound from the QFC

In two dimensions, a quantum expansion $\Theta$ can be defined by:

$\Theta=\frac{1}{\Phi(\lambda)}\frac{{\rm d}}{{\rm d}\lambda}A_{\rm
qu}(\lambda),$ (4.106)

which reduces to the classical expansion (3.53) in the classical
limit.101010Reinserting factors of $\hbar$ in the quantum area gives $A_{\rm
qu}=\Phi+4G\hbar S$, hence yielding the classical result
$\Theta\rightarrow\theta$ in the limit $\hbar\rightarrow 0$. The _Quantum
Focussing Conjecture_ in two dimensions then states that the quantum expansion
cannot increase along any light ray [42]:

$\frac{{\rm d}\Theta}{{\rm d}\lambda}\leq 0.$ (4.107)

Since $\Phi>0$, if $A_{\rm qu}^{\prime}$ is initially non-positive, the QFC
implies that $A_{\rm qu}^{\prime}(\lambda)\leq 0$ all along the light ray. If
the QFC is satisfied, the definition (4.75) of a quantum light-sheet can
therefore be restricted to $A_{\rm qu}^{\prime}(0)\leq 0$ only initially, the
QFC then implying that $A_{\rm qu}^{\prime}(\lambda)\leq 0$ at all later
points $\lambda\geq 0$ of the light ray.

As a supporting evidence for the QFC, it was noted in [42] that it implies a
QBB. Indeed, the property $A_{\rm qu}^{\prime}(\lambda)\leq 0$, following from
$A_{\rm qu}^{\prime}(0)\leq 0$ and the QFC, gives in particular

$0\leq\frac{1}{4G}(A_{\rm qu}(0)-A_{\rm qu}(1)),$ (4.108)

which is cutoff-independent. This can be seen as a quantum version of the
Bousso bound (3.59), where $S_{L(B^{\prime}-B)}$ has been identified with
$S(1)-S(0)$:

$S(1)-S(0)\leq\frac{1}{4G}(\Phi(0)-\Phi(1)).$ (4.109)

This bound and the Strominger-Thompson bound have similar forms. Though, they
are not equivalent since the Strominger-Thompson bound is a stronger
inequality relying on stronger conditions valid only in the hydrodynamic
limit, which breaks down at small scales.

In fact, formally removing the coarse-grained entropy flux from the conditions
(4.79) and (4.80) gives

$\displaystyle k^{\mu}k^{\nu}\braket{\tau_{\mu\nu}}$ $\displaystyle\geq 0,$
(4.110) $\displaystyle A_{\rm qu}^{\prime}(0)$ $\displaystyle\leq 0,$ (4.111)

from which one can follow through our proof and recover the BFLW quantum
Bousso bound, without relying on the hydrodynamic limit nor on the QFC.

####  4.4.2 BCFM Quantum Bousso Bound

Similarly to the BFLW quantum Bousso bound, the BCFM quantum Bousso bound [41]
proposes a modification of the left-hand side of (3.59) in order to define a
cutoff-independent quantity. In their work, the entropy of the light-sheet
$S_{L(B^{\prime}-B)}$ is defined as the difference $\Delta S$ between the
fine-grained entropies of the state under consideration and the vacuum,
restricted to $L(B^{\prime}-B)$. A proper definition of $\Delta S$ may be
found in [56, 41]. Moreover, the BCFM bound applies to light-sheets in their
classical sense, contrary to the Strominger-Thompson and BFLW bounds that
apply to quantum light-sheets. The quantity $\Delta S$ is divergence-free and
the QBB is proven for any portion of the light-sheet. In particular, $\Delta
S$ is well defined for light-sheets of arbitrary sizes, whereas the
hydrodynamic limit used in the Strominger-Thompson bound breaks down at small
scales.

This definition of entropy on the light-sheet is only valid in the weak
gravity limit, i.e. when the spacetime geometry in the presence of matter is
well approximated by the vacuum geometry. Outside this limit, the backreaction
of matter on the spacetime geometry is non-negligible and $\Delta S$ is not
well defined. Indeed, the meaning of “same” light-sheets in two different
geometries is unclear. We do not consider this version of the bound in this
work. Further studies of the BCFM bound may be interesting since it is
expected to not be equivalent to the BFLW bound [42], as $\Delta S$ is
intrinsically local to the light-sheet while the BFLW bound is fundamentally
non-local and relies on a different notion of light-sheet (quantum light-
sheet).

##  5 Conclusion

In this work, we have proven the validity of the Strominger-Thompson quantum
Bousso bound in de Sitter JT gravity coupled to a CFT with central charge $c$.
The proof applies both to the half and full reduction models and relies on a
general sufficient condition, which is checked by deriving a stronger version
of the QNEC. The BFLW quantum Bousso bound follows directly from our argument,
without assuming the QFC. Here we comment briefly on possible future works.

The derivation of the strong version of the QNEC does not rely on the de
Sitter background and should be valid in any system for which the
transformation rules (A.12) or (A.15) apply. We expect that our arguments may
easily extend to AdS JT gravity.

As mentioned above, we do not prove the Quantum Focussing Conjecture in this
work. The verification of the BFLW quantum Bousso bound provides good evidence
that the QFC should be valid in the models studied here, but one would be
interested in an explicit proof of this fundamental property.

Finally, we derived an entropy formula in the presence of reflective
boundaries. In the presence of these boundaries, left and right-moving modes
are correlated. As a consequence, the entropy of a Cauchy slice bounded by a
point and the boundary cannot depend on the endpoint at the boundary. It would
be interesting to consider the problem of islands and information recovery in
the half reduction model using this formula. It may also be used in an attempt
to compute holographic entanglement entropies in the framework of de Sitter
holography.

## Acknowledgements

We are grateful to Alex Kehagias, Andrew Svesko, and Watse Sybesma for
interesting discussions during early stages of this research. We thank Hervé
Partouche and Nicolaos Toumbas for helpful discussions and feedback. F.R.
would like to acknowledge the hospitality of the Ecole Polytechnique, while
V.F. would like to thank the University of Cyprus for hospitality, where early
stages of this work have been done. This work is partially supported by the
Cyprus Research and Innovation Foundation grant EXCELLENCE/0421/0362.

##  Appendix A Entanglement entropy and reflecting boundaries

In this Appendix, we aim to provide a general entanglement entropy formula in
a two-dimensional curved background, with two reflective boundaries. We assume
massless scalar fields and follow closely the prescription of [46].

Considering a Cauchy slice $\Sigma_{\mathcal{M}}$ of the two-dimensional
spacetime $\mathcal{M}$ described by a state $\ket{\Psi}$, the state of the
fields in a subsystem $\Sigma\subset\Sigma_{\mathcal{M}}$ is described by the
density matrix

$\rho=\tr_{\bar{\Sigma}}{\ket{\Psi}\bra{\Psi}},$ (A.1)

where $\tr_{\bar{\Sigma}}$ traces out the degrees of freedom on
$\bar{\Sigma}=\Sigma_{\mathcal{M}}\backslash\Sigma$. The entanglement entropy
(or fine-grained entropy) of $\Sigma$ is defined as

$S(\Sigma)=-\tr_{\Sigma}\rho\ln\rho,$ (A.2)

where $\tr_{\Sigma}$ traces over the degrees of freedom on $\Sigma$.

###  A.1 Entanglement entropy without boundaries

Let us consider a Cauchy slice $\Sigma$ in a curved two-dimensional spacetime
with metric (2.16) and without boundaries, and review some results of [57,
46]. For free massless scalars, the right-moving and left-moving modes are
uncoupled such that we can consider them separately. We consider left-moving
modes and note the coordinates of the endpoints of $\Sigma$ as
$(x_{1}^{+},x_{1}^{-})$ and $(x_{2}^{+},x_{2}^{-})$, such that left-moving
modes along the interval $[x_{1}^{+},x_{2}^{+}]$ all pass through $\Sigma$. By
constructing a complete set of left-moving modes localized inside and outside
of the interval, one can then trace on the vacuum state built on the $x^{\pm}$
coordinates. The entropy has logarithmic UV divergences at each endpoint of
the interval, arising from short wavelength fluctuations around these
endpoints. They can be regulated by excluding contributions within a short
distance $\varepsilon_{{\rm L}}(x_{1})$ and $\varepsilon_{{\rm L}}(x_{2})$
around $x_{1}^{+}$ and $x_{2}^{+}$, respectively. Similarly, an infrared
cutoff $L$ is introduced, such that $x^{+}$ is restricted to the range
$[-L,L]$. When the size of the interval $|x_{2}^{+}-x_{1}^{+}|$ is small
enough compared to the infrared cutoff, we get

$S(\Sigma)=\frac{c}{12}\ln\left[\frac{(x_{2}^{+}-x_{1}^{+})^{2}}{\varepsilon_{{\rm
L}}(x_{1})\varepsilon_{{\rm L}}(x_{2})}\right].$ (A.3)

The cutoffs $\varepsilon_{{\rm L}({\rm R})}$ are expressed in terms of the
$x^{\pm}$ coordinates. One can rewrite these cutoffs in terms of cutoffs
$\delta_{{\rm L}({\rm R})}$ in inertial coordinates according to

$\varepsilon_{{\rm L}({\rm
R})}(x^{+},x^{-})=e^{-\omega(x^{+},x^{-})}\leavevmode\nobreak\ \delta_{{\rm
L}({\rm R})},$ (A.4)

so that the product of the cutoffs for left and right-moving modes is boost-
invariant. Adding the contribution from the right-moving modes leads to the
total entropy of $\Sigma$,

$S(\Sigma)=\frac{c}{6}\left(\omega(x_{1})+\omega(x_{2})\right)+\frac{c}{12}\ln\left[\frac{(x_{2}^{+}-x_{1}^{+})^{2}(x_{2}^{-}-x_{1}^{-})^{2}}{\delta_{1}^{2}\delta_{2}^{2}}\right],$
(A.5)

where $\delta_{i}^{2}=\delta_{{\rm L}}(x_{i})\delta_{{\rm R}}(x_{i})$, for
$i=1,2$. In two-dimensional space, $\delta_{i}^{2}$ can be seen as a numerical
constant [46].

In such spacetime without boundaries, the right and left-moving modes are
completely independent. But in a model with reflecting boundaries like the
half reduction model of de Sitter JT gravity, the reflection of right-moving
modes into left-moving modes and vice-versa induces correlations between them.
A study of entanglement entropy in the presence of a single moving mirror
following the timelike trajectory $x^{+}-x^{-}=$ constant has been provided in
[46]. In the next section, we discuss a more general case where the spacetime
contains two reflecting boundaries following more general trajectories.

###  A.2 Entanglement entropy with reflecting boundaries

Let us consider the Kruskal coordinates $(x^{+},x^{-})$ and the metric (2.16)
in the conformal gauge. Such set of coordinates naturally splits the spacetime
into four distinct regions. Inspired by the half reduction model of de Sitter
JT gravity, we consider the case where the spacetime contains two spatial
reflecting boundaries lying entirely in the regions $x^{+}x^{-}\leq 0$. We
parameterize their trajectories through a function111111One could find a
change of coordinates $y^{\pm}(x^{\pm})$ such that the boundary is described
by $y^{+}-y^{-}=$ constant. However, choosing a system of coordinates in (A.3)
corresponds to selecting a specific vacuum, defined in these coordinates. It
is therefore convenient to have the freedom to choose the coordinate system in
which we define the vacuum. $x^{-}=f(x^{+})$, hence satisfying
$x^{+}f(x^{+})\leq 0$, and impose in addition the following conditions:

  1. 1.

Introducing an infrared cutoff $L$, the left boundary is bounded by the points
of $x^{\pm}$ coordinates $(-L,0)$ and $(0,L)$, while the right boundary is
bounded by the points $(0,-L)$ and $(L,0)$. We thus have $f:I\rightarrow I$,
with $I=(-L,0)\cup(0,L)$.

  2. 2.

All modes emanating from any Cauchy slice bounded by the points $(-L,0)$ and
$(0,-L)$ reflect once and only once on a boundary, which is equivalent to have
$f$ a bijection from $I$ to $I$.

  3. 3.

The trajectory must be continuous and differentiable, which is equivalent to
impose $f$ to be a $C^{1}$ function.

  4. 4.

The trajectory must be timelike to define a spatial boundary, which is
equivalent to have $0<\dot{f}<+\infty$, where $\dot{f}\equiv{\rm d}f/{\rm
d}x^{+}$.

  5. 5.

We restrict ourselves to states that have symmetric left and right-moving
radiation, and in general to systems symmetric under $x^{+}\leftrightarrow
x^{-}$. This implies that $f$ is an involution, i.e. $f=f^{-1}$.

This construction ensures that, when a mode reflect on a boundary:

  * •

Left-moving modes propagating along $x^{-}$ at fixed $x^{+}=x_{0}^{+}$
transform into right-moving modes propagating along $x^{+}$ at fixed
$x^{-}=f(x_{0}^{+})$.

  * •

Right-moving modes propagating along $x^{+}$ at fixed $x^{-}=x_{0}^{-}$
transform into left-moving modes propagating along $x^{-}$ at fixed
$x^{+}=f(x_{0}^{-})$.

In particular, we are able to express all modes either as right-moving modes
emanating from the interval $[-L,L]$ on the $x^{-}$ axis, or as left-moving
modes emanating from the interval $[-L,L]$ on the $x^{+}$ axis.

Similarly to the spatial boundaries, we parameterize $\mathcal{J}^{\pm}$ as a
spacelike curve described by $x^{-}=g(x^{+})$, with $g:I\rightarrow I$ a
$C^{1}$ involution such that $g(x^{-})x^{-}\geq 0$ and $-\infty<\dot{g}<0$.
From now on, we take the limit $L\rightarrow\infty$, which will be justified
at the end of this section. In Figure 5,

$x^{-}=g(x^{+})$$x^{-}=f(x^{+})$$x^{-}=f(x^{+})$$x^{-}$$x^{+}$$x^{+}$$x_{P}^{+}$$f^{-1}(x^{-})$$x^{-}$$\Sigma$
(a) Case $x^{-}\geq 0$. The left-moving modes passing through $\Sigma$ are the
left-moving modes emanating from the interval $[x_{P}^{+},x^{+}]$ depicted in
red. The right-moving modes passing through $\Sigma$ are the left-moving modes
emanating from the interval $[f^{-1}(x^{-}),x_{P}^{+}]$ depicted in blue.
Tracing over every modes on $\Sigma$ is therefore equivalent to tracing over
left-moving modes on the interval $[f^{-1}(x^{-}),x^{+}]$.

$x^{-}=g(x^{+})$$x^{-}=f(x^{+})$$x^{-}=f(x^{+})$$x^{-}$$x^{+}$$x^{+}$$x_{P}^{+}$$f^{-1}(x^{-})$$x^{-}$$\Sigma$
(b) Case $x^{-}\leq 0$. The left-moving modes passing through $\Sigma$ are the
left-moving modes emanating from (or which will cross) the interval
$[x_{P}^{+},x^{+}]$ depicted in red. The right-moving modes passing through
$\Sigma$ are the left-moving modes emanating from (or which will cross) the
interval $(-\infty,x_{P}^{+}]\cup[f^{-1}(x^{-}),+\infty)$ depicted in blue.
Tracing over every modes on $\Sigma$ is therefore equivalent to tracing over
left-moving modes on the interval $(-\infty,x^{+}]\cup[f^{-1}(x^{-}),+\infty)$
or on the complementary interval $[x^{+},f^{-1}(x^{-})]$.

Figure 5: Cauchy slice $\Sigma$ bounded by a point $(x_{P}^{+},f(x_{P}^{+}))$
on the left boundary of the spacetime and a point $(x^{+},x^{-})$ in the bulk.
Considering reflective boundary conditions at the timelike boundaries of the
spacetime induces correlations between left and right-moving modes passing
through $\Sigma$.

the timelike and spacelike boundaries are depicted with examples of reflecting
light rays, and we introduce the following notation to describe segments of
the cosmological horizons:

$\displaystyle\begin{aligned} H_{-}^{-}&=\\{(x^{+},x^{-})|x^{+}=0,x^{-}\leq
0\\},\\\ H_{+}^{-}&=\\{(x^{+},x^{-})|x^{-}=0,x^{+}\leq 0\\},\end{aligned}$
$\displaystyle\begin{aligned} H_{-}^{+}&=\\{(x^{+},x^{-})|x^{+}=0,x^{-}\geq
0\\},\\\ H_{+}^{+}&=\\{(x^{+},x^{-})|x^{-}=0,x^{+}\geq 0\\},\end{aligned}$
(A.6)

so that $H_{-}=H_{-}^{-}\cup H_{-}^{+}$ is the horizons along $x^{-}$ and
$H_{+}=H_{+}^{-}\cup H_{+}^{+}$ is the horizon along $x^{+}$.

From our construction, we can consider right and left-moving modes at
$\mathcal{J}^{-}$ and note that all the right-moving modes pass through the
half line $H_{-}^{-}$ while all the left-moving modes pass through the half
line $H_{+}^{-}$. Because of the reflecting boundary conditions, all right-
moving (left-moving) modes passing through $H_{-}^{-}$ ($H_{+}^{-}$) also
cross the half line $H_{+}^{+}$ ($H_{-}^{+}$) as left-moving (right-moving)
modes. In other words, all modes coming from $\mathcal{J}^{-}$ cross $H_{-}$
as right-movers and $H_{+}$ as left movers, such that we can express the
quantum state of the fields as left-moving (right-moving) states at $H_{-}$
($H_{+}$). From here we will always refer to the state of the fields as left-
moving states on $H_{+}$. The condition that $f$ is an involution ensures that
the following procedure is independent of the reference axis.

Now, consider a subsystem $\Sigma$ of some Cauchy slice
$\Sigma_{\mathcal{M}}$, bounded by the left boundary of $\mathcal{M}$ at
$(x_{P}^{+},f(x_{P}^{+})$) and some point $(x^{+},x^{-})$. As mentioned above,
right and left-moving modes on $\Sigma$ are correlated and tracing over right
and left-movers on $\Sigma$ is equivalent to tracing over left-movers on some
interval of $H_{+}$. First, all left-moving modes on $\Sigma$ are the same
modes as the left-moving modes on the interval $[x_{P}^{+},x^{+}]$ of $H_{+}$.
As for right-movers on $\Sigma$, the projection slightly varies depending on
the position of the endpoint $(x^{+},x^{-})$:

  * •

When $x^{-}\geq 0$, the right-moving modes on $\Sigma$ are the left-moving
modes in the interval $[f^{-1}(x^{-}),x_{P}^{+}]$ of $H_{+}$.121212We keep the
notation $f^{-1}$ for clarity, although we imposed $f=f^{-1}$. Therefore,
tracing over every modes on $\Sigma$ is equivalent to tracing over left-moving
modes on the interval $[f^{-1}(x^{-}),x^{+}]$ of $H_{+}$, see Figure 5(a).

  * •

When $x^{-}\leq 0$, the right-moving modes on $\Sigma$ are the same as the
left-moving modes in the subset
$(-\infty,x_{P}^{+}]\cup[f^{-1}(x^{-}),\infty)$ of $H_{+}$. Therefore, tracing
over every modes on $\Sigma$ is equivalent to tracing over left-moving modes
on the interval $(-\infty,x^{+}]\cup[f^{-1}(x^{-}),\infty)$ of $H_{+}$, see
Figure 5(b). Since the left-movers on $H_{+}$ should be in a pure state, this
is equivalent of tracing over the complement of this subset, i.e.
$[x^{+},f^{-1}(x^{-})]$.

Assuming $|x^{+}-f^{-1}(x^{-})|\ll L$, we can use equation (A.3) to get

$S(\Sigma)=\frac{c}{12}\ln\left[\frac{(x^{+}-f^{-1}(x^{-}))^{2}}{\delta
x^{+}\leavevmode\nobreak\ \delta f^{-1}(x^{-})}\right],$ (A.7)

where $\delta x^{+}$ and $\delta f^{-1}(x^{-})$ are the UV cutoffs at the
endpoints of the interval of $H_{+}$. In other words, we have smeared over the
regions $[x^{+}-\delta x^{+},x^{+}]$ and $[f^{-1}(x^{-}),f^{-1}(x^{-})+\delta
f^{-1}(x^{-})]$ on $H_{+}$ in the case $x_{-}\geq 0$, and similarly when
$x_{-}\leq 0$. We would like to relate these quantities to the invariant
cutoff $\delta^{2}$, i.e. to the UV cutoff associated to the position of
$(x^{+},x^{-})$. Since the left-moving modes on $[x^{+}-\delta x^{+},x^{+}]$
are the same as the left-moving modes on the same interval at $x^{+}$ fixed,
we write $\delta x^{+}=\varepsilon_{{\rm L}}(x^{+},x^{-})$. To relate the
interval $[f^{-1}(x^{-}),f^{-1}(x^{-})+\delta f^{-1}(x^{-})]$ on $H_{+}$ to an
interval around $(x^{+},x^{-})$, we need to take into account the reflection
on the left boundary. Considering the reflection of two light rays coming from
the two endpoints of the interval, smearing over the left-moving modes on
$[f^{-1}(x^{-}),f^{-1}(x^{-})+\delta f^{-1}(x^{-})]$ on $H_{+}$ is equivalent
to smearing over the right-moving modes on
$[f(f^{-1}(x^{-})),f(f^{-1}(x^{-})+\delta f^{-1}(x^{-}))]$ at fixed $x^{+}$
around the point $(x^{+},x^{-})$. At first order in $\delta f^{-1}(x^{-})$,
this interval is $[x^{-},x^{-}+\delta f^{-1}(x^{-})\dot{f}(f^{-1}(x^{-}))]$,
from which we can identify the cutoff for the right movers at the point
$(x^{+},x^{-})$ to be:

$\varepsilon_{{\rm R}}(x^{+},x^{-})=\delta f^{-1}(x^{-})\leavevmode\nobreak\
\dot{f}(f^{-1}(x^{-})).$ (A.8)

It is now possible to write $\delta x^{+}\leavevmode\nobreak\ \delta
f^{-1}(x^{-})$ in terms of an affine proper distance, and we conclude that

$S(\Sigma)=\frac{c}{6}\omega(x^{+},x^{-})+\frac{c}{12}\ln\left[\dot{f}(f^{-1}(x^{-}))\frac{(x^{+}-f^{-1}(x^{-}))^{2}}{\delta^{2}}\right],$
(A.9)

where $\delta^{2}=\delta_{{\rm L}}\delta_{{\rm R}}$ is the cutoff in inertial
coordinates at the endpoint of $\Sigma$.

For this formula to be well defined, we need to motivate the assumption
$|x^{+}-f^{-1}(x^{-})|\ll L$. As $L\rightarrow\infty$, this is equivalent to
imposing $-L\ll f^{-1}(x^{-})\ll L$. For $f$ to break this condition, one
would need to localize $x^{-}$ with an infinite precision as
$L\rightarrow\infty$, which cannot be done in the presence of the UV cutoff
$\delta$.

###  A.3 Change of vacuum

As noted in [46], the entropy formula (A.3) is only valid in the vacuum state
defined with respect to the coordinates $x^{\pm}$, with the associated metric
${\rm d}s^{2}=-e^{2\omega_{x}(x^{+},x^{-})}{\rm d}x^{+}{\rm d}x^{-}$, where we
made the coordinate system explicit in the notation $\omega_{x}$. In general,
the vacuum state may be defined with respect to an arbitrary set of lightcone
coordinates $y^{\pm}(x^{\pm})$, in which the metric is ${\rm
d}s^{2}=-e^{2\omega_{y}(y^{+},y^{-})}{\rm d}y^{+}{\rm d}y^{-}$. Under this
change of coordinates, the endpoints of $\Sigma$ become
$(y^{+}(x_{1}^{+}),y^{-}(x_{1}^{-}))$ and
$(y^{+}(x_{2}^{+}),y^{-}(x_{2}^{-}))$. By diffeomorphism invariance, the
Cauchy slice defined by this interval in coordinates $y^{\pm}$ is physically
identical to the Cauchy slice defined by the endpoints $(x_{1}^{+},x_{1}^{-})$
and $(x_{2}^{+},x_{2}^{-})$ in the original coordinates. For this reason, we
call them both $\Sigma$, independently of the system of coordinates in which
it is described.

To write the entropy contribution from the left-moving modes in a vacuum
defined in coordinates $y^{\pm}$, one writes the size of the interval
$[x_{1}^{+},x_{2}^{+}]$ in $y^{\pm}$ coordinates [46],

$S_{y}(\Sigma)=\frac{c}{12}\ln\left[\frac{(y^{+}(x_{2}^{+})-y^{+}(x_{1}^{+}))^{2}}{\varepsilon_{{\rm
L}}(y(x_{1}))\varepsilon_{{\rm L}}(y(x_{2}))}\right],$ (A.10)

where the index $y$ denotes the vacuum in which the entropy is computed. As we
change the coordinate system, the cutoffs $\varepsilon_{{\rm L}({\rm R})}$
must change according to:

$\varepsilon_{{\rm L}({\rm
R})}(y^{+},y^{-})=e^{-\omega_{y}(y^{+},y^{-})}\leavevmode\nobreak\
\delta_{{\rm L}({\rm R})}.$ (A.11)

Therefore, the entropy $S_{y}(\Sigma)$ of $\Sigma$ in the vacuum state defined
in coordinates $y^{\pm}$ is given by [46]:

$\displaystyle\begin{split}S_{y}(\Sigma)&=\frac{c}{6}\left(\omega_{y}(y(x_{1}))+\omega_{y}(y(x_{2}))\right)+\frac{c}{12}\ln\left[\frac{(y^{-}(x^{-}_{2})-y^{-}(x^{-}_{1}))^{2}(y^{+}(x^{+}_{2})-y^{+}(x^{+}_{1}))^{2}}{\delta_{1}^{2}\delta_{2}^{2}}\right]\\\
&=\frac{c}{12}\left[2\left(\omega_{x}(x_{1})+\omega_{x}(x_{2})\right)+\ln\left[\left.\frac{{\rm
d}x^{+}}{{\rm d}y^{+}}\right|_{x_{1}^{+}}\left.\frac{{\rm d}x^{+}}{{\rm
d}y^{+}}\right|_{x_{2}^{+}}\left.\frac{{\rm d}x^{-}}{{\rm
d}y^{-}}\right|_{x_{1}^{-}}\left.\frac{{\rm d}x^{-}}{{\rm
d}y^{-}}\right|_{x_{2}^{-}}\right]\right.\\\
&+\left.\ln\left[\frac{(y^{+}(x^{+}_{2})-y^{+}(x^{+}_{1}))^{2}(y^{-}(x^{-}_{2})-y^{-}(x^{-}_{1}))^{2}}{\delta_{1}^{2}\delta_{2}^{2}}\right]\right],\end{split}$
(A.12)

where we have used (4.95). The formulas above should be applied to spacetimes
that have infinite extends in both $x^{+}$ and $x^{-}$ (or $y^{+}$ and
$y^{-}$) directions. In particular it does not take into account any possible
boundaries.

A similar transformation law can be derived in the presence of spatial
boundaries. First, supposing that the boundary in coordinates $x^{\pm}$
follows a trajectory $x^{-}=f_{x}(x^{+})$, the spatial boundary in coordinates
$y^{\pm}$ is described by

$y^{-}=f_{y}(y^{+})=y^{-}(f_{x}(x^{+})).$ (A.13)

From this, we can compute the transformation law of the term
$\dot{f}_{x}(f_{x}^{-1}(x^{-}))$:

$\dot{f}_{x}(f_{x}^{-1}(x^{-}))\rightarrow\dot{f}_{y}(f_{y}^{-1}(y^{-}))=\dot{f}_{x}(f_{x}^{-1}(x^{-}))\frac{\frac{{\rm
d}y^{-}}{{\rm d}x^{-}}(x^{-})}{\frac{{\rm d}y^{+}}{{\rm
d}x^{+}}(f_{x}^{-1}(x^{-}))}.$ (A.14)

Hence, the entropy of a Cauchy slice $\Sigma$ bounded by a point
$(x^{+},x^{-})$ and the boundary $x^{-}=f_{x}(x^{+})$, on a curved background
with the metric ${\rm d}s^{2}=-e^{2\omega_{x}}{\rm d}x^{+}{\rm d}x^{-}$ and in
a vacuum state defined in coordinates $y^{\pm}(x^{\pm})$, is given by:

$\displaystyle\begin{split}S_{y}(\Sigma)&=\frac{c}{6}\omega_{x}(x^{+},x^{-})+\frac{c}{12}\ln\left[\dot{f}_{x}(f_{x}^{-1}(x^{-}))\frac{\left(y^{+}(x^{+})-y^{+}(f_{x}^{-1}(x^{-}))\right)^{2}}{\delta^{2}}\right]\\\
&-\frac{c}{12}\ln\left[\frac{{\rm d}y^{+}}{{\rm d}x^{+}}(x^{+})\frac{{\rm
d}y^{+}}{{\rm d}x^{+}}\left(f_{x}^{-1}(x^{-})\right)\right].\end{split}$
(A.15)

##  Appendix B Light-sheet, expansion and energy conditions

In this Appendix, we recall the formalism describing the classical Bousso
bound in the hydrodynamic approximation, following [12, 47]. We will first
consider an $(n+1)$-dimensional framework, and then carry out the dimensional
reduction yielding the classical Bousso bound in two-dimensional de Sitter JT
gravity, that is considered in Section 3.

We consider a light-sheet $L(B-B^{\prime})$, emanating and ending from the
codimension $2$ surfaces $B$ and $B^{\prime}$, respectively. We call $\lambda$
the affine parameter along each null geodesic generating $L(B-B^{\prime})$,
normalized such that $\lambda=0$ on $B$ and $\lambda=1$ on $B^{\prime}$. Since
${\rm d}/{\rm d}\lambda$ is null and orthogonal to the light-sheet by
construction, we define the future directed vector normal to $L(B-B^{\prime})$
by $k^{M}=\pm\left(\frac{{\rm d}}{{\rm d}\lambda}\right)^{M}$, with a “$+$”
sign for future-directed light-sheets and a “$-$” sign for past-directed
light-sheets. For any choice of affine parameter, the normal vector must
satisfy the geodesic equation

$k^{M}\hat{\nabla}_{M}k^{N}=0,$ (B.1)

with $\hat{\nabla}_{M}$ the covariant derivative compatible with the metric
$\hat{g}_{MN}$ of $\hat{\mathcal{M}}$. One can introduce the induced metric
$h_{ab}^{(n)}$ and the extrinsic curvature $K_{ab}^{(n)}$ of the
$n$-dimensional hypersurface $L(B-B^{\prime})$ of normal vector $k^{M}$,
respectively by:

$\displaystyle h_{ab}^{(n)}$ $\displaystyle=$ $\displaystyle\frac{\partial
X^{M}}{\partial y^{a}}\frac{\partial X^{N}}{\partial y^{b}}\hat{g}_{MN},$
(B.2) $\displaystyle K^{(n)}_{ab}$ $\displaystyle=$
$\displaystyle\frac{\partial X^{M}}{\partial y^{a}}\frac{\partial
X^{N}}{\partial y^{b}}\hat{\nabla}_{M}k_{N},$ (B.3)

where $\\{X^{M},\leavevmode\nobreak\ M=0,...,n\\}$ are a set of coordinates on
$\hat{\mathcal{M}}$ and $\\{y^{a},\leavevmode\nobreak\ a=0,...,n-1\\}$ a set
of coordinates on $L(B-B^{\prime})$. With these definitions, the expansion
parameter of the congruence is defined as the trace $h_{(n)}^{ab}K_{ab}^{(n)}$
of the extrinsic curvature tensor[49],

$\theta^{(n)}=\hat{\nabla}_{M}k^{M},$ (B.4)

where the superscript $(n)$ indicates that $\theta^{(n)}$ is the expansion
parameter of an $n$-dimensional null congruence. By definition of a light-
sheet, $\theta^{(n)}$ is non-positive everywhere on it,
$\theta^{(n)}(\lambda)\leq 0,\forall\lambda\in[0,1]$.

The hydrodynamic approximation relies upon a local entropy current
$\hat{s}^{M}$. The $n$-dimensional entropy flux $s^{(n)}$ on the light-sheet
is given by

$s^{(n)}=-k^{M}\hat{s}_{M}.$ (B.5)

The integral of $s^{(n)}$ over $L(B-B^{\prime})$ gives the total matter
entropy passing through $L(B-B^{\prime})$. Considering a coordinate system
$y=(y^{1},...,y^{n-1})$ on $B$, each null geodesic generating the light-sheet
is thus labelled by $y$, and $(y^{1},...,y^{n-1},\lambda)$ provides a
coordinate system on $L$. We denote by $h(y,\lambda)$ the determinant of the
induced metric on any fixed $\lambda$ cross-section of $L(B^{\prime}-B)$. It
is related to the determinant $h(y,0)$ of the induced metric on $B$ by

$h(y,\lambda)=\left[\mathcal{A}^{(n)}(y,\lambda)\right]^{2}\leavevmode\nobreak\
h(y,0),$ (B.6)

where $\mathcal{A}^{(n)}(y,\lambda)$ is the area decrease factor, defined in
terms of the expansion scalar $\theta(\lambda)$ of a given generator of $L$
by:

$\mathcal{A}^{(n)}\equiv\exp\left[\int_{0}^{\lambda}{\rm
d}\tilde{\lambda}\leavevmode\nobreak\ \theta^{(n)}(\tilde{\lambda})\right].$
(B.7)

Equivalently, $\theta^{(n)}$ is the logarithmic derivative of
$\mathcal{A}^{(n)}$, $\theta^{(n)}(\lambda)={\rm d}\ln\mathcal{A}^{(n)}/{\rm
d}\lambda$. By definition of a light-sheet, $\theta^{(n)}(\lambda)\leq 0$, or
equivalently ${\rm d}\mathcal{A}^{(n)}/{\rm d}\lambda\leq 0$,
$\forall\lambda$.

With these notations, the total entropy flux passing through the light-sheet
$L(B-B^{\prime})$ reads:

$\displaystyle\int_{L(B-B^{\prime})}s^{(n)}$ $\displaystyle=$
$\displaystyle\int_{B}{\rm d}^{n-1}y\int_{0}^{1}{\rm
d}\lambda\leavevmode\nobreak\ \sqrt{h(y,\lambda)}\leavevmode\nobreak\
s^{(n)}(y,\lambda)$ (B.8) $\displaystyle=$ $\displaystyle\int_{B}{\rm
d}^{n-1}y\sqrt{h(y,0)}\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
\mathcal{A}^{(n)}(y,\lambda)\leavevmode\nobreak\ s^{(n)}(y,\lambda).$

With this formal definition, the Bousso bound states that the coarse-grained
entropy (B.8) passing through $L(B-B^{\prime})$ satisfies the inequality

$\int_{L(B-B^{\prime})}s^{(n)}\leq\frac{1}{4\hat{G}}(A(B)-A(B^{\prime})).$
(B.9)

As shown in [12], a sufficient condition to prove the generalized covariant
entropy bound (B.9) is to derive the inequality

$\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\
\mathcal{A}^{(n)}(y,\lambda)\leavevmode\nobreak\
s^{(n)}(y,\lambda)\leq\frac{1}{4\hat{G}}(\mathcal{A}^{(n)}(0)-\mathcal{A}^{(n)}(1)),$
(B.10)

for each null generator of $L$, i.e. at fixed $y$. In the following we will
therefore drop the $y$ dependence of $\mathcal{A}^{(n)}$ and $s^{(n)}$. Note
that in JT gravity, the area decrease factor along the light ray reduces to

$\mathcal{A}(\lambda)=\frac{\Phi(\lambda)}{\Phi(0)}.$ (B.11)

This is obtained by inserting the relation (3.53) into the general definition
(B.7) of the area decrease factor.

The hydrodynamic approximation cannot be fundamental, since the entropy is a
non-local notion. It is therefore only valid in certain regimes and over
certain scales. In particular, it is a good approximation only if the size of
the system whose entropy is computed is much larger than some microscopic
length scale. To ensure its validity, two conditions can be imposed, that have
been shown to be sufficient to prove the Bousso bound [12, 47, 19]:

  1. 1.

The “ _gradient of entropy density_ ” condition:

$\left|k^{M}k^{N}\hat{\nabla}_{M}\hat{s}_{N}\right|\leq 2\pi
k^{M}k^{N}\braket{\hat{T}_{MN}},$ (B.12)

where $\braket{\hat{T}_{MN}}$ is the stress-energy tensor in
$\hat{\mathcal{M}}$. Using the property (B.1) of the vector $k^{M}$, this
inequality can be rewritten in terms of the entropy flux
$s^{(n)}=-k^{M}\hat{s}_{M}$:

$\left|(s^{(n)}(\lambda))^{\prime}\right|\leq 2\pi
k^{M}k^{N}\braket{\hat{T}_{MN}},$ (B.13)

where ${}^{\prime}\equiv{\rm d}/{\rm d}\lambda=k^{M}\hat{\nabla}_{M}$.

  2. 2.

The “ _isolated system_ ” condition:

$s^{(n)}(0)\leq-\frac{\theta^{(n)}(0)}{4\hat{G}}.$ (B.14)

The first condition is necessary for the whole phenomenological description of
entropy in terms of entropy densities to be valid. It requires that the
gradient of the entropy density is bounded by the energy density. This ensures
that $\hat{s}^{M}$ only varies significantly over length scales greater than
the largest wavelength of any of the modes that contribute to the entropy [12,
47]. Below this scale, there is no meaning for a variation of the entropy
density. Notably, this condition implies the NEC

$k^{M}k^{N}\braket{\hat{T}_{MN}}\geq 0.$ (B.15)

The second condition was first stated in [47] as $s^{(n)}(0)=0$. They argued
that it does not add any notable restriction to the system, since it should
always be possible to modify slightly the entropy current around the initial
surface without breaking the first condition nor changing the total integrated
entropy by a non-negligible amount. The modified condition (B.14),
corresponding to imposing that the bound is valid at the beginning of the
light-sheet, was used in [19] to prove the Bousso bound in the Callan-
Giddings-Harvey-Strominger (CGHS) model [58] with similar motivations.
Notably, this modification implicitly requires the light-sheet to be initially
non-expanding, i.e. $\theta^{(n)}(0)\leq 0$. Imposing these “isolated system”
conditions is equivalent to requiring that modes carrying a non-negligible
contribution to the entropy should be fully contained on the light-sheet, and
not spanned beyond the initial surface [47].

On the other hand, the Raychaudhuri equation for $n>1$ determines the
evolution of the expansion parameter $\theta^{(n)}$ along the congruence:

$\frac{{\rm d}\theta^{(n)}}{{\rm
d}\lambda}=-\frac{1}{n-1}(\theta^{(n)})^{2}-\hat{\sigma}_{MN}\hat{\sigma}^{MN}+\hat{\omega}_{MN}\hat{\omega}^{MN}-\hat{R}_{MN}k^{M}k^{N},$
(B.16)

where $\hat{\sigma}_{MN}$ and $\hat{\omega}_{MN}$ are the shear and twist
tensors respectively. For a congruence hypersurface orthogonal,
$\hat{\omega}_{MN}=0$. Assuming in addition that the NEC
$\braket{\hat{T}_{MN}}k^{M}k^{N}\geq 0$ is satisfied for any null vector
$k^{N}$, then the Einstein equations imply that $\hat{R}_{MN}k^{M}k^{N}\geq
0$, and the Raychaudhuri equation leads to the Classical Focussing Theorem:

$\frac{{\rm d}\theta^{(n)}}{{\rm d}\lambda}\leq 0.$ (B.17)

Physically, this tells us that the light rays are focused during the evolution
of the congruence. Since the two conditions (B.12) and (B.14) require the NEC
(B.15) and initial non-expansion, this motivates the fact that the Bousso
bound does not apply to null congruences of positive expansion. Moreover,
supposing the NEC, we can reduce the definition of a light-sheet to a null
congruence that is initially non-expanding.

The Bousso bound and the entropy conditions in the JT gravity setup are
obtained from their $(n+1)$-dimensional counterparts using the spherical
reduction (2.9). The two-dimensional entropy flux $s$ is related to the
$n$-dimensional entropy flux $s^{(n)}$ by

$s(\lambda)=S_{n-1}(l_{n})\Phi(\lambda)\leavevmode\nobreak\ s^{(n)}(\lambda),$
(B.18)

where $S_{n-1}(l_{n})=2\pi^{n/2}\leavevmode\nobreak\ l_{n}^{n-1}/\Gamma(n/2)$
is the surface area of the $(n-1)$-sphere of radius $l_{n}$. The total entropy
flux (B.8) passing through the light-sheet then rewrites as

$\displaystyle\int_{L(B-B^{\prime})}s^{(n)}$
$\displaystyle=l_{n}^{n-1}\Phi(0)\int{\rm d}^{n-1}\Omega\int_{0}^{1}{\rm
d}\lambda\leavevmode\nobreak\ s^{(n)}(\lambda)\frac{\Phi(\lambda)}{\Phi(0)}$
(B.19) $\displaystyle=S_{n-1}(l_{n})\int_{0}^{1}{\rm
d}\lambda\leavevmode\nobreak\ s^{(n)}(\lambda)\Phi(\lambda)$ (B.20)
$\displaystyle=\int_{0}^{1}{\rm d}\lambda\leavevmode\nobreak\ s(\lambda).$
(B.21)

Using equation (B.11), the dimensional reduction of the $(n+1)$-dimensional
Bousso bound (B.9) gives the Bousso bound in two-dimensional JT gravity given
in (3.59)131313Equivalently, we could have derived this inequality by
inserting the relations (B.11), (B.18) and (2.11) into the sufficient
condition (B.10) of [12], immediately yielding the Bousso bound in JT
gravity.. Similarly, the two entropy conditions (3.60) and (3.61) can be
obtained from their $(n+1)$-dimensional counterparts (B.12) and (B.14) using
(B.18) and

$T_{\mu\nu}=S_{n-1}(l_{n})\Phi\leavevmode\nobreak\ \hat{T}_{\mu\nu},$ (B.22)

where $T_{\mu\nu}$ and $\hat{T}_{\mu\nu}$ are the stress-energy tensors in
$(1+1)$ and $(n+1)$ dimensions respectively.

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