[a]Victor Franken

# de Sitter Connectivity from Holographic Entanglement

###### Abstract

Considering two antipodal observers in de Sitter space, we illustrate how
spacetime connectivity between the holographic screens located on the
(stretched) horizons emerges from holographic entanglement. To do so, we
construct a covariant holographic entanglement entropy prescription in de
Sitter space, including quantum corrections. While this prescription is
inspired by the bilayer proposal, we argue that the monolayer proposal appears
to be inconsistent. Entanglement wedge reconstruction implies an extension of
static patch holography where the exterior region connecting the static
patches is included, and reconstructible from the two screens. A phase
transition occurs where there is an exchange of dominance between two
competing quantum extremal surfaces, leading to a transfer of the encoding of
the exterior region from one screen to the other. The effects in the bulk of
integrating out degrees of freedom from the screens are discussed.

##  1 Introduction

The AdS/CFT correspondence has led to great progresses in the understanding of
black hole information and the connection between spacetime and entanglement
(see e.g. [1, 2, 3, 4, 5, 6, 7, 8]). A typical example of this relation is the
emergence of spacetime from the entanglement between two disconnected
holographic CFTs in a thermofield-double (TFD) state. The TFD is a pure state
and tracing out one of the two sides produces the original CFT in a thermal
state. At small temperatures, corresponding to low entanglement between the
two CFTs, the gravity dual to the TFD state corresponds to two disconnected
spacetimes. On the other hand, at high temperatures where the two CFTs are
highly entangled, the dual spacetime consists of an eternal AdS black hole. As
pictured in figure 1, the two initial spacetimes are connected by an Einstein-
Rosen (ER) bridge in the latter case. The idea that spacetime connectivity
between two holographic theories emerges from entanglement is often associated
to the slogan “ER=EPR” [4].111For a proposition of formal definition of this
equivalence, see [8].

$\otimes$ (a)

(b)

Figure 1: Gravity dual to the thermofield-double state. (a) At low
temperature, the TFD is dual to two disconnected AdS spactimes. (b) At high
temperature, the bulk dual is a maximally extended AdS-Schwarzschild black
hole, with a wormhole connecting the two CFTs. The two initial spacetimes are
depicted in blue and the Einstein-Rosen bridge is depicted in red.

In parallel, a precise holographic dictionary between entanglement in the non-
gravitating theory and bulk quantities was developed by Ryu, Takayanagi, and
others [9, 10, 11, 12, 13]. This led to the development of subregion-subregion
duality, or entanglement wedge reconstruction, which allows to determine the
bulk subregion dual to a given subregion of the quantum theory [14, 15, 16].

In this paper we suppose that the holographic nature of gravity and the
notions of holographic entanglement entropy and entanglement wedge
reconstruction should be universal [17, 18, 19], at least in the semiclassical
limit $G_{N}\rightarrow 0$, and thus applicable to cosmological models, in
particular to the de Sitter spacetime. We interpret the area of extremal
surfaces in the bulk as first order contributions to the entropy of subsystems
in the non-gravitating dual quantum theory, and bulk fields semiclassical
entropy as second order contributions, see Section 3. These extremal surfaces
should satisfy the usual homology constraint [10] and their associated entropy
and entanglement wedge should satisfy weak and strong subadditivity [13] as
well as entanglement wedge nesting [20].

Although the task of finding an explicit holographic duality in de Sitter
space turns out to be complicated, partially for the lack of a string theory
description of de Sitter space, it has attracted a lot of attention lately
[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58].
In [56], based on a recent holographic conjecture in de Sitter space [24], we
considered the possibility to define a covariant holographic entropy
prescription in de Sitter space as well as its consequence concerning
entanglement wedge reconstruction.

The presence of holographic degrees of freedom on the cosmological horizon
(see Figure 2) leads to a number of constraints concerning the construction of
a holographic entanglement entropy formula. Two approaches to this problem
were considered by Susskind and Shaghoulian [46, 44, 45], named the monolayer
and bilayer proposal. We argue that generic assumptions make the monolayer
proposal inconsistent and propose a covariant prescription inspired by the
bilayer proposal. This exposes intricacies previously overlooked in the
monolayer and bilayer proposals of Susskind and Shaghoulian, resulting in
significant differences when quantum corrections are taken into account.
Moreover, it allows for an extension of the prescription to a general class of
closed FRW cosmologies [59]. Using this prescription we are able to study
entanglement wedge reconstruction which leads to the two main takeaways of
this work. First, the only consistent definition of extremal surfaces in de
Sitter space implies that the connectiveness of de Sitter space emerges from
the entanglement between the causal patches of two antipodal observers. In
particular, the spatial region connecting these two patches can be
reconstructed from the holographic degrees of freedom. The second one is that
there are two competing quantum extremal surfaces when computing the entropy
of one of the two screens, associated to two drastically different
entanglement wedges. When the first one dominates, the entanglement wedge of
one screen only covers its interior region, corresponding to the usual static
patch holography picture. When the second dominates, the entanglement wedge
also extends in the exterior and connects the two screens via a wormhole. The
dominant quantum extremal surface can undergo a phase transition from one to
the other, corresponding to a transfer of the encoding of the wormhole
connecting the two holographic screens. To conclude, we discuss the
implications in the bulk of integrating out degrees of freedom from the
screens, and how the encoding of the exterior region is lost when the screens
are pushed inside the static patches at a distance greater than the Planck
scale from the cosmological horizons. Further details on the following
discussions as well as additional results may be found in [56, 59].

##  2 A Holographic Description of de Sitter Space?

In this section we will motivate the holographic picture on which we will base
the rest of our paper. Roughly, we shall motivate that any physically
meaningful description of the universe is intrinsically observer dependent,
naturally leading to study the region causally connected to some observer
worldline. In an inflationary cosmology - de Sitter space - this region is
bounded by a cosmological horizon. We will then show how, analogously with the
AdS/CFT correspondence, holographic degrees of freedom located on this
boundary are expected to encode the complete state of the causal region of the
observer. The connectivity of the full universe can then be related to the
entanglement between the holographic degrees of freedom associated to causally
disconnected observers.

###  2.1 The Observer and the Static Patch

Since the 90’s, there have been strong indications that a quantum theory of
gravity should present holographic properties, meaning that the theory is
described by degrees of freedom located on the boundary of space [17, 18, 19].
This conjecture was explicitly realized by the AdS/CFT correspondence in the
context of string theory [60, 61]. One would naturally be interested to test
this correspondence in our own universe. However, both the inflationary period
and the late time universe are in an accelerated expansion phase, modeled by
the de Sitter space which has a closed topology, i.e. no boundaries. There are
three ways to interpret the absence of spatial boundary. The first one is to
conclude that the de Sitter space does not have any holographic description,
which would be highly unsatisfactory and inconsistent with the Bousso
covariant entropy conjecture [62]. The second is to seek for a modification of
the original formulation of holography, as Strominger did by proposing the
dS/CFT correspondence [63] with holographic degrees of freedom located at the
future infinity, timelike conformal boundary. The third one, which may be the
simplest one, would be to simply state that the de Sitter space does not have
any dynamical degree of freedom [64, 65]. Here we consider the latter as it is
well motivated by gravitational path integrals and swampland conditions [65,
66, 67, 7]. For example, the island formula [7, 68, 69] implies that there is
no way to entangle a closed universe with any complementary system [65],
leading to the conclusion that state of the universe is unique.

Another argument which suggests that the Hilbert space of de Sitter space is
trivial is that the algebra of observables in de Sitter space is trivial [70,
71]. The only possibility to get a sensible algebra is to include the observer
explicitly in the system, which effectively breaks the de Sitter symmetries as
it selects one particular static patch. This produces an algebra of type II1
which has a state of maximal entropy corresponding to an empty static patch
tensored with a simple model of observer (essentially a clock). The non-
triviality of the algebra in the presence of an observer is analogous to the
potentially very complex state that may be measured by an observer, despite
the triviality of the state of the full universe. This is somewhat reminiscent
of topological field theories, like Chern-Simons theory on a closed manifold,
whose Hilbert space is trivial.222We thank Lior Benizri and Jan Troost for
pointing this out to us. In such cases, degrees of freedom are effectively
added when considering subsystems, which has the effect of relaxing some of
the gauge invariance (the ones broken by the subsystem boundary).

Let us work under the assumption that the Hilbert space of de Sitter space has
a unique pure state. Since a closed universe cannot be observed from the
outside, an explicit and non-trivial description of de Sitter space should
include some observer. This observer can measure fields along its worldline,
and generally may interact with anything inside its causal patch. The
discussion above motivates the fact that it is natural to consider the static
patch of an observer in order to probe non-trivial behaviors of gravity in de
Sitter space. In particular, the static patch is self-contained in the sense
that an observer may neglect everything happening outside the cosmological
horizon. The region of the universe he observes is effectively bounded by its
cosmological horizon. So that, from his point of view, the observable universe
has a boundary on which it would seem sensible to place holographic degrees of
freedom. Moreover, it has been argued that the horizon of the observer, which
is the boundary of the static patch, is associated to an entropy proportional
to its area [72],

$S_{\rm GH}=\frac{A}{4G\hbar}.$ (1)

The fact that thermodynamic quantities can be associated to the horizon of the
static patch is indicative of an holographic behavior. In particular one could
expect that degrees of freedom located on the cosmological horizon should be
able to encode everything happening in some region of de Sitter space, but
which one? The discussion above indicates that an intuitive answer would be
the static patch, where the degrees of freedom describing it emerge from the
choice of worldline associated to the static patch observer. In the next
section, we formally motivate this using the Bousso bound [62].

###  2.2 The Holographic Principle and de Sitter Space

The geometry of de Sitter space and an example of global spacelike slice is
depicted in Figure 2. Every complete spacelike slice of $(n+1)$-dimensional de
Sitter space has the topology of a $n$-sphere. Under time evolution, this
$n$-sphere undergoes a phase of contraction followed by a phase of expansion,
reaching infinite size at time infinities $\mathcal{J}^{\pm}$.

$\mathcal{J}^{+}$Observer$\mathcal{J}^{-}$Observer$\Sigma_{A}$$A$$L$$H_{\rm
L}$$H_{\rm R}$$H_{\rm R}$$H_{\rm L}$$\Sigma$ Figure 2: Two antipodal observers
follow the antipodal trajectories defined by the left and right edge of the
diagram. The diagonal lines form the cosmological horizons $H_{\rm L}$ and
$H_{\rm R}$ for the two observers. They bound the two static patches $P_{\rm
L}$ and $P_{\rm R}$ depicted in blue. The dashed line is an arbitrary global
spacelike slice one which we place a sphere $A$ depicted by a black dot.
$\Sigma_{A}\in\Sigma$ is the red segment and the future lightsheet emanating
from $A$ is the dotted line. .

Let us consider an observer whose worldline follows a timelike geodesic in the
bulk. All geodesics are equivalent under the $SO(1,n+1)$ symmetry group of de
Sitter space, so we can choose a coordinate system in which the observer
worldline follows the left edge of the diagram. This corresponds to a static
observer at the north pole (or pode) of the $n$-sphere. The region of
spacetime that can interact with the observer is called the static patch
$P_{\rm L}$ and is delimited by a past and a future horizon, forming $H_{\rm
L}$. There is a unique worldline such that the static patch $P_{\rm R}$ of an
observer following this trajectory does not overlap with $P_{\rm L}$,
corresponding to the right edge of the Penrose diagram, describing a static
observer on the south pole, or antipode.

The static patch holography conjecture is motivated by the covariant entropy
bound and the holographic principle of Bousso [62, 19]. To understand this, we
start with an arbitrary Cauchy slice $\Sigma$. Let $A$ be a closed codimension
$2$ surface inside $P_{\rm L}$. The following discussion also applies to
$P_{\rm R}$ with associated horizon $H_{\rm R}$ by symmetry. The surface $A$
defines the boundary of two subsystems of $\Sigma$: one which is entirely
contained in $P_{\rm L}$, denoted $\Sigma_{A}$, and the complement of
$\Sigma_{A}$ with respect to $\Sigma$ which intersects $H_{\rm L}$ and $H_{\rm
R}$, see Figure 2. The Cauchy slice $\Sigma$ can evolve under unitary
evolution. In particular, one can consider the unitary transformations of
$\Sigma$ such that $A\in\Sigma$ is a fixed point. This defines the evolution
of $\Sigma_{A}$ with its boundary $A$ fixed. In particular, we can consider
the limit of future-directed unitary evolution of $\Sigma_{A}$ which goes to a
future-directed lightlike surface emanating from $A$ towards the pode, noted
$L$. $L$ may also be constructed by considering the congruence of future-
directed lightrays emanating from $A$ and going towards the pode. The second
law of thermodynamics implies that

$S_{\text{coarse}}(\Sigma_{A})\leq S_{\text{coarse}}(L),$ (2)

where $S_{\text{coarse}}$ describes the coarse-grained, or thermodynamic
entropy which corresponds to the number of degrees of freedom necessary to the
description of the quantum state of a system. The null congruence $L$ is
entirely contained in $P_{\rm L}$ by definition. Any null congruence contained
in $P_{\rm L}$ and directed towards the pode in $P_{\rm L}$ is a lightsheet,
i.e. is of non-positive expansion (see for example [56]). Therefore, the
covariant entropy bounds [62] applies and

$S_{\text{coarse}}(L)\leq\frac{\text{Area}(A)}{4G\hbar}.$ (3)

Combining equations (2) and (3) gives the inequality

$S_{\text{coarse}}(\Sigma_{A})\leq\frac{\text{Area}(A)}{4G\hbar}.$ (4)

Hence, a quantum theory of gravity in de Sitter space should account for the
property that the number of degrees of freedom needed to describe a spatial
region $\Sigma_{A}$ of the static patch is bounded from below by the area of
$A$ divided by $4G\hbar$. This bound makes very clear that any candidate of
underlying theory consistent with the covariant entropy bound should unify the
geometrical notions of gravity with the description of matter. The holographic
principle allows us to make the bound manifest and states that the region
$\Sigma_{A}$ can be described by a set of degrees of freedom localized on $A$
with less than one degree of freedom per Planck area.

The placement of $A$ as an holographic screen is arbitrary. However, the area
of closed surfaces on $\Sigma$ increase as we get closer to the cosmological
horizon,333Past the cosmological horizon, future directed lightrays emitted
towards the pode are no longer of non-positive expansion and equation (4) no
longer holds. such that pushing $A$ towards $H_{\rm L}$ maximizes the number
of degrees of freedom that are holographically encoded. Hence, on any
spacelike slice $\Sigma$, the surface with the greatest number of holographic
degrees of freedom is located at the cosmological horizon of the static patch,
and describes the part of $\Sigma$ in the static patch. We call this set of
degrees of freedom $\mathcal{S}_{\rm L}$. The previous argument applies to any
spacelike slice of the static patch, which leads to the static patch
holography conjecture:

_The quantum theory of gravity in the static patch of an observer in de Sitter
space is completely described by a quantum system defined on the cosmological
horizon of the observer[24]._444This can be regulated by placing the screen on
a stretched horizon close to the cosmological horizon, see Section 4.3.

A precise realization of this duality is still lacking although promising
results have been found in lower-dimensional de Sitter space [44, 45, 46, 47,
48, 49, 50, 51, 52]. We will not consider these issues here.

###  2.3 Bridging the Static Patches

Let us note $\mathcal{H}_{\rm L}$ the Hilbert space of the quantum system
located on the cosmological horizon $H_{\rm L}$ and describing $P_{\rm L}$.
The area of the de Sitter cosmological horizon is constant in time, in other
words any spacelike slice of $H_{\rm L}$ has the same area. The number of
degrees of freedom needed to describe the static patch is therefore constant
in time, and $\mathcal{H}_{\rm L}$ has a fixed dimension. We thus expect time
evolution of this quantum system to be unitary. As mentioned earlier, one can
consider the holographic dual to $P_{\rm R}$, with Hilbert space
$\mathcal{H}_{\rm R}$. The union of the two holographic theories describes the
two disconnected static patches, $P_{\rm L}\cup P_{\rm R}$ depicted in blue in
Figure 2. Here we would like to consider the effect of entangling the degrees
of freedom in $\mathcal{H}_{\rm L}$ and $\mathcal{H}_{\rm R}$.

This is reminiscent to the thermofield-double state in AdS, where two non-
interacting and disentangled holographic quantum theories are dual to two
disconnected bulk spacetimes. Once entangled, they form a state which is dual
to a spacetime in which the two original spacetimes are connected by a
wormhole. We have a very similar situation here with two quantum theories of
Hilbert spaces $\mathcal{H}_{\rm L}$ and $\mathcal{H}_{\rm R}$ independently
encoding $P_{\rm L}$ and $P_{\rm R}$, respectively. As in the AdS case, one
may wonder what the holographic dual to an arbitrary theory of Hilbert space
$\mathcal{H}_{\rm L}\otimes\mathcal{H}_{\rm R}$ is. Note that this region
should at least include the two static patches, encoded independently in
$\mathcal{H}_{\rm L}$ and $\mathcal{H}_{\rm R}$. When degrees of freedom on
both sides are entangled we expect the two bulk duals $P_{\rm L}$ and $P_{\rm
R}$ to be spatially connected by a spatial bridge. This is the case in pure de
Sitter space which is connected by a contracting or expanding region
connecting the two static patches (the white regions in Figure 2). In
particular, the static patch of an observer is in a thermal state [72] (see
section 4.1 for a reminder of this construction) such that its coarse-grained
entropy, given by the Gibbons-Hawking law $A/4G\hbar$, is equal to its
entanglement entropy. Hence the states of $\mathcal{H}_{\rm L}$ and
$\mathcal{H}_{\rm R}$ should be entangled.

We will confirm this intuition by studying entanglement wedge reconstruction
in de Sitter spacetime to show that the region between the two cosmological
horizons must be included in the holographic duality and that the entanglement
wedge of the holographic screens do cover this exterior region. A first step
in this direction is the definition of a covariant prescription to compute
entropies is the quantum dual theory. This task unravels subtleties that where
not present in the monolayer and bilayer proposals of Susskind and Shaghoulian
[46, 44, 45], and leads to non-trivial results when including quantum
corrections.

##  3 Covariant Holographic Entropy in de Sitter Space

As we have yet no candidate of holographic dual of the static patch in
arbitrary dimension, it would be interesting to investigate the possibility to
study holographic entanglement entropy in this picture. The computations are
done exclusively in the bulk and may provide us with hints concerning the
nature of the holographic dual.555Similarly, one can study holographic
complexity with the same motivations, see e.g. [73, 43, 46, 47, 74, 75, 76,
77, 78, 79, 80] Shaghoulian and Susskind recently considered two prescriptions
that they call the monolayer and bilayer proposals. However, we still lacked
of a precise covariant procedure, which leads to a number of nontrivial
subtleties being missed.

We want to find a bulk prescription capable of computing entanglement
entropies of spatial subsystems of $H_{\rm L}\cup H_{\rm R}$, corresponding to
a subsystem $\mathcal{A}$ of the holographic screens $\mathcal{S}_{\rm
L}\cup\mathcal{S}_{\rm R}$. Here we differentiated between the spatial
subregion $A$ with the subsystem $\mathcal{A}$ consisting of degrees of
freedom located on $A$. However, as holographic degrees of freedom are located
on $H_{\rm L}\cup H_{\rm R}$ with a specific density of degrees of freedom per
unit area, we consider that specifying one defines uniquely the other. The
monolayer and bilayer proposals are motivated by the bit thread formulation of
holographic entanglement entropy [81, 82] and by wedge holography [83, 84, 85,
86, 87]. In these prescriptions, de Sitter space is divided in three regions -
the two static patches (in blue in Figure 2) and the exterior region between
them (in white in Figure 2) - and extremal surfaces are computed in each
region separately. The two prescriptions go as follows:

$\bullet$ Monolayer proposal: The entanglement entropy of a subregion $A$ of
the two screens and its complement is $1/(4G\hbar)$ times the area of an
extremal surface homologous666If there are multiple such extremal surfaces, we
choose the one with the minimal area. to $A$, and lying between the two sets
of degrees of freedom; in this case between the two cosmological horizons.

$\bullet$ Bilayer proposal: The entanglement entropy of a subregion $A$ of the
two screens and its complement is $1/(4G\hbar)$ times the sum of the areas of
the extremal surfaces homologous to $A$ in each of the three subregions of the
bulk, that is, in the exterior and the two interior regions.

Here we argue briefly that the monolayer proposal appears inconsistent with
the construction of the holographic description of de Sitter space presented
above. For a detailed discussion, see [56]. The holographic description
considered here starts from two complementary static patches encoded
respectively on their cosmological horizons. In particular, we already
highlighted that whatever the state of two screen system $\mathcal{S}_{\rm
L}\cup\mathcal{S}_{\rm R}$ is, it should at least encode the two static
patches. The holographic entanglement entropy prescription should be
consistent with this. As announced in the introduction of this paper, we
suppose entanglement wedge reconstruction in the de Sitter background, with
the associate usual properties of entanglement wedge nesting and homology
constraint:

Entanglement wedge reconstruction: The bulk domain of dependence of the
spacial slice whose boundary is the union of extremal surfaces in the exterior
and the two interior regions (or only in the exterior region for the monolayer
proposal) is reconstructible from the holographic subsystem $A$. This bulk
domain of dependence is referred to as the entanglement wedge of $A$, noted
$\mathcal{E}_{A}$. For $\mathcal{E}_{A}$ to be well defined, the covariant
entropy prescription should define a set of surfaces, homologous to $A$ in
their respective region, that are spacelike separated from any point of the
holographic screen and whose union is the boundary of a spacial region of the
bulk.

By definition, the extremal surface always lies in the exterior region when
considering the monolayer proposal. Hence, $\mathcal{E}_{A}$ lies entirely in
the exterior region, and therefore never extends into the static patches. In
particular, this implies that $\mathcal{E}_{\mathcal{S}_{\rm
L}\cup\mathcal{S}_{\rm R}}$ does not cover the static patches, which is a
contradiction. We will therefore discard the monolayer proposal in the rest of
this work. We will give additional evidence against the monolayer proposal
later in this work. In particular, the monolayer proposal fails to recover the
prediction that the full de Sitter space is in a pure state [65, 66, 67, 7,
68], and it also fails to apply to situations where the holographic screens
are pushed on stretched horizon inside the static patch, contrary to the
bilayer proposal. In [56], we carried out the definition of a covariant
entropy prescription for both proposals, and showed additional inconsistent
results in the monolayer case when considering non-trivial subsystems of the
holographic screens.

One may argue that the absence of extremal surfaces in the static patch does
not necessarily imply that entanglement wedges cannot cover them. However, in
the absence of extremal surface in the static patches there is no preferred
location for the entanglement wedge in these regions, such that the only other
sensible possibility would be to state that the entanglement wedge of a
subsystem $A$ of a screen always cover the associated static patch in addition
to the region already defined. However this would also be contradictory as it
leads to the conlusion that the entanglement wedge of a spatial subsystem of
the horizon whose size goes to zero covers the full static patch. It would be
interesting to explore the possibility that the monolayer proposal of [46]
gets supplemented by a consistent set of rules concerning the structure of the
entanglement wedges, allowing them to extend in the interior regions. However,
it is difficult to see how such a consistent set of rules could be formulated,
since in all other well understood examples in the literature, the
entanglement wedge structure depends on the location of the extremal surfaces,
and the monolayer proposal does not involve any extremal surfaces in the
interior regions.

Following the HRT proposal of AdS/CFT, we would like to specify the bilayer
proposal and define it in a covariant way. The prescription goes as follow
[56]:

($i$) Consider an arbitrary Cauchy slice $\Sigma$ of dSn+1.

($ii$) Define the screens $\mathcal{S}_{\rm L}$ and $\mathcal{S}_{\rm R}$ as
the intersections of $\Sigma$ with $H_{\rm L}$ and $H_{\rm R}$, the
cosmological horizons of the pode and the antipode, respectively.

($iii$) Divide $\Sigma$ into 3 parts:

$\Sigma_{i}=\Sigma\cap P_{i},\quad\Sigma_{\rm
E}=\Sigma\backslash\\{\Sigma_{\rm L}\cup\Sigma_{\rm R}\\},$ (5)

with $i=L,R$. Any Cauchy slice $\Sigma^{\prime}$ of dSn+1 passing through
$\mathcal{S}_{\rm L}$ and $\mathcal{S}_{\rm R}$ is equivalent to $\Sigma$, in
the following sense. Decomposing in a similar way
$\Sigma^{\prime}=\Sigma^{\prime}_{\rm L}\cup\Sigma^{\prime}_{\rm
E}\cup\Sigma^{\prime}_{\rm R}$, the causal diamonds of $\Sigma^{\prime}_{i}$
and $\Sigma_{i}$ in region $i$ are the same, for all $i\in\\{\rm L,E,R\\}$.
See Figure 3.

($iv$) Consider any subregion $A$ of $\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm
R}$. We first look for a surface $\chi_{\rm E}$ of minimal extremal area that
is homologous to $A_{\rm E}=A$ in the exterior region and is lying on a Cauchy
slice $\Sigma^{\prime}_{\rm E}$. We also look for a surface $\chi_{\rm L}$
($\chi_{\rm R}$) of minimal extremal area, homologous to $A_{\rm
L}=A\cap\mathcal{S}_{\rm L}$ ($A_{\rm R}=A\cap\mathcal{S}_{\rm R}$) in the
left (right) causal patch and lying on a Cauchy slice $\Sigma^{\prime}_{\rm
L}$ ($\Sigma^{\prime}_{\rm R}$). For all $i\in\\{\rm L,E,R\\}$, recall that
the codimension $2$ surface $\chi_{i}$ being homologous to the codimension $2$
surface $A_{i}$ means that there exists a codimension $1$ surface
$\mathcal{C}_{i}$ such that $\partial\mathcal{C}_{i}=\chi_{i}\cup
A_{i}$.777This implies in particular that $\chi_{i}$ is anchored on $A_{i}$,
i.e. $\partial\chi_{i}=\partial A_{i}$. Among all possible choices of surfaces
$\mathcal{C}_{i}$, a particular one, $\mathcal{C}^{\prime}_{i}$, lies on
$\Sigma_{i}^{\prime}$.

($v$) At next-to-leading order in $G\hbar$, the Von Neumann entropy of the
holographic dual subsystem on $A$, that is the entanglement entropy between
$A$ and its complement in $\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm R}$, is

$S(A)=\min\text{ext}_{\chi_{i}}\left[\sum_{i}{\frac{{\rm
Area}(\chi_{i})}{4G\hbar}}\\\ +S_{\rm
semicl}\left(\bigcup_{j}{\cal{C}}^{\prime}_{j}\right)\right]+\mathcal{O}\left((G\hbar)^{1}\right),$
(6)

where $S_{\rm semicl}\left(\bigcup_{j}{\cal{C}}^{\prime}_{j}\right)$ is the
semiclassical entropy of bulk fields on the union of
$\mathcal{C}^{\prime}_{i}$ with $i=L,E,R$.888The necessity of joint
extremization was already pointed out at the classical level in [44, 45]. This
choice is always minimal compared to the sum of the entropy of bulk fields on
the three slices, by weak subadditivity. This is reminiscent of the island
formula [88].

($vi$) The part in region $i$ of the “entanglement wedge” of the dual system
living on $A$ is the causal diamond of $\mathcal{C}^{\prime}_{i}$. The causal
diamond $D(\mathcal{C})$ of a spacelike region $\mathcal{C}$ is defined as the
set of points $p$ of the bulk such that all timelike of lightlike curves that
pass through $p$ must pass through $\mathcal{C}$. The entanglement wedge,
called $\mathcal{E}(A)$, is therefore

$\mathcal{E}(A)=D(\mathcal{C}^{\prime}_{\rm L})\cup
D(\mathcal{C}^{\prime}_{\rm E})\cup D(\mathcal{C}^{\prime}_{\rm R}).$ (7)

$A$$\chi_{\rm L}$$\chi_{\rm E}$$\chi_{\rm R}$ Figure 3: The Cauchy slice
$\Sigma$ is the black curve and is cut in three parts by the horizons. $S_{\rm
L}$ and $S_{\rm R}$ lie at the intersection between the black curve and the
two diagonal lines. The red dot is a subregion of $S_{\rm L}\cup S_{\rm R}$
The region spanned by equivalent slices $\Sigma^{\prime}$, equal to
$D(\Sigma_{\rm L})\cup D(\Sigma_{\rm E})\cup D(\Sigma_{\rm R})$ is depicted in
blue. Quantum extremal surfaces of each region are represented by black dots.
All three belong to a Cauchy slice $\Sigma_{i}^{\prime}$ depicted by a dashed
line. The $\mathcal{C}_{i}$ are depicted by the red segments on
$\Sigma^{\prime}$. The entanglement wedge associated to the choice of extremal
surfaces $\chi_{i}$ is the union of the three red regions.

It was pointed out in [56] that for some examples of subsystem $A$, there may
be no slice $\Sigma_{i}^{\prime}$ in some region $i\in\\{L,E,R\\}$ such that
there exist $\chi_{i}\in\Sigma_{i}$ an homologous extremal surface. However,
whether extremal surfaces exist or not, non-extremal surfaces lying on the
boundary of the causal diamond of $\Sigma_{i}$ can be seen as solutions of an
extremization problem in an enlarged domain that has no boundary. In other
words, we relax the definition of extremal surface
$\delta\text{Area}(\chi_{A})/\delta\chi_{A}=0$ and supplement the area
functional with Lagrange multipliers that impose the surface to lie inside a
given region. Here we give an example of such implementation and refer to [56,
59] for more details and more complicated examples. We consider closed
codimension $2$ surfaces in a $(n+1)$-dimensional de Sitter background and
assume that the $SO(n)$ symmetry is not spontaneously broken when extremizing
the area functional. In other words, we look for $S^{n-1}$ spheres of extremal
area in de Sitter space. The $(n+1)$-dimensional de Sitter admits the
conformal metric

$ds^{2}=\frac{1}{cos^{2}(\sigma)}\left(-d\sigma^{2}+d\theta^{2}+\sin^{2}(\theta)d\Omega_{n-1}^{2}\right),$
(8)

where $\sigma\in(-\pi/2,\pi/2)$ and $\theta\in[0,\pi]$, which is explicitly
$SO(n)$-symmetric. In these coordinates, every pair of coordinates
$(\sigma,\theta)$ parameterize a $(n-1)$-sphere corresponding to a point in
the Penrose diagram. The area of these spheres is given by

$\text{Area}(\sigma,\theta)=\omega_{n-1}\left(\frac{\sin\theta}{\cos\sigma}\right)^{n-1}$
(9)

where $\omega_{n-1}$ is the volume of the sphere of radius $1$. It is easy to
show that the only extremal $S^{n-1}$, for which
$\partial\text{Area}/\partial\sigma=\partial\text{Area}/\partial\theta=0$, is
located at $(0,\pi/2)$. these coordinates correspond to the bifurcate horizon,
the intersection between the cosmological horizons of the two observers.
Hence, if we were to look for an extremal closed surface in a region
$\mathcal{R}$ that does not contain the bifurcate horizon, there would be no
solution to the extremization problem. Let us note $f_{i}(\sigma,\theta)\geq
0$ the set of inequations that define $\mathcal{R}$. We supplement the area
functional with terms proportional to Lagrange multipliers:

$\widehat{\text{Area}}(\sigma,\theta,\nu_{i},a_{i})=\text{Area}(\sigma,\theta)+\sum_{i}\nu_{i}\left(f_{i}(\sigma,\theta)-a_{i}^{2}\right).$
(10)

$\widehat{\text{Area}}$ is defined in a domain without boundary as
$\sigma,\theta,\nu_{i},a_{i}$ all span $\mathbb{R}$. Solving the problem of
extremizing $\widehat{\text{Area}}$ over all these variables give the
$S^{n-1}$ with extremal area with respect to $\mathcal{R}$, that is either a
true extrema inside $\mathcal{R}$ (associated with $\nu_{i}=0$ for all $i$),
or the sphere with the smallest or biggest area, or saddle points, inside
$\mathcal{R}$ which will lie somewhere on the boundary $\partial\mathcal{R}$
(associated with $a_{i}=0$ for at least one $i$). If there are multiple such
solutions, we are instructed to take the one with smallest area.

##  4 Entanglement Wedge Reconstruction

We now apply our covariant prescription to simple subsystems: The one-screen
system and the two-screen system. In all instances, we always consider
$SO(n)$-symmetric slices $\Sigma$ such that $\mathcal{S}_{\rm L}$ and
$\mathcal{S}_{\rm R}$ are spherical screens. Let us denote $\sigma_{\rm L}$
and $\sigma_{\rm R}$ the conformal time of $\mathcal{S}_{\rm L}$ and
$\mathcal{S}_{\rm R}$, respectively. One can re-express this in global time
$\tau$ using

$\cosh\tau=\frac{1}{\cos\sigma}.$ (11)

Since the screens lie on a Cauchy slice we always have $\tau_{\rm L}\tau_{\rm
R}\geq 0$, or $\sigma_{\rm L}\sigma_{\rm R}\geq 0$. While we do expect each
screen to encode its own static patch, it is not clear a priory if and how the
exterior region is encoded in some of these subsystems. We end with a
discussion on stretched horizons and the effects of integrating out degrees of
freedom from the holographic screens.

###  4.1 The Pair of Holographic Screens is Pure and Encodes Everything

The two-screen system is the simplest one to compute. We define
$A=\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm R}$ and from our definitions of
last section,

$A_{\rm L}=\mathcal{S}_{\rm L},\quad A_{\rm E}=\mathcal{S}_{\rm
L}\cup\mathcal{S}_{\rm R},\quad A_{\rm R}=\mathcal{S}_{\rm R}.$ (12)

All these surfaces are closed. Hence, in each region $i=L,E,R$ we are looking
for an extremal surface without boundary lying on a Cauchy slice
$\Sigma^{\prime}_{i}$ as defined in the third step of our procedure. The empty
surface defined by the empty set of points is the subset of any Cauchy slice
$\Sigma^{\prime}_{i}$ and has minimal extremal area by definition. It
satisfies the homology constraint in every region,

$A_{i}\cup\varnothing=\partial\mathcal{C}^{\prime}_{i},$ (13)

where $\mathcal{C}^{\prime}_{i}$ is any of the Cauchy slices
$\Sigma^{\prime}_{i}$ as defined in the third step of our procedure. This is
verified by definition of $\Sigma^{\prime}_{i}$. Hence, the entropy of the
two-screen system, consisting of some spatial slice of the full dual theory,
is given by

$S(\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm
R})=0+\mathcal{O}\left((G\hbar)^{1}\right),$ (14)

where we used $\text{Area}(\varnothing)=0$,
$\Sigma^{\prime}=\Sigma^{\prime}_{\rm L}\cup\Sigma^{\prime}_{\rm
E}\cup\Sigma^{\prime}_{\rm R}$, and $S_{\rm
semicl}\left(\Sigma^{\prime}\right)=0$ since $\Sigma^{\prime}$ is a complete
Cauchy slice. In other words, the full holographic system has zero entropy at
next-to-leading order. Following the covariant prescription, the part of the
entanglement wedge in each region $i=L,E,R$ is given by the causal diamond of
$\mathcal{C}_{i}^{\prime}=\Sigma^{\prime}_{i}$, such that

$\mathcal{E}(\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm R})=D(\Sigma_{\rm L})\cup
D(\Sigma_{\rm E})\cup D(\Sigma_{\rm R}),$ (15)

where we used $D(\Sigma^{\prime}_{i})=D(\Sigma_{i})$. This region is spanned
by equivalent complete slices $\Sigma^{\prime}$. In other words,
$\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm R}$ encodes all complete Cauchy
slices passing through $\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm R}$. The fact
that the entanglement wedge of the two holographic screens covers the exterior
region in addition to their static patches leads to an extension of the
statement of static patch holography:

#### Doubled static patch holography

_The full de Sitter spacetime can be encoded holographically in terms of a
theory on the two cosmological horizons. 999Or on stretched horizons separated
from the cosmological horizon by less than a Planck length, see Section 4.3.
The states of the interior regions are encoded on the screens at the
corresponding horizons, as in static patch holography. Additionally, the
exterior region between the cosmological horizons emerges from the
entanglement between the screens at the horizons and can also be reconstructed
from the holographic degrees of freedom._

This story is reminiscent of the double-sided black hole in AdS as already
highlighted in Section 2.3. Both screens have Hilbert space
$\mathcal{H}_{L(R)}$ and encode their respective static patch $P_{L(R)}$. Once
we consider the full theory of Hilbert space $\mathcal{H}_{\rm
L}\otimes\mathcal{H}_{\rm R}$, the entanglement between the two sets of
degrees of freedom can be seen from the bulk point of view as the spatial
connectivity between the two patches $P_{L(R)}$. In other words, the
entanglement wedge of the union of the two screens is bigger than the union of
their independent entanglement wedges, and it covers the full exterior region
that bridges the two static patches. This enrichment of the bulk geometry
emerges directly from entanglement in the dual theory.

Left screenRight screen$\otimes$ (a)

(b)

Figure 4: Extended static patch holography: (a) We consider initially two
independant static patches, independently encoded on two holographic screens
of Hilbert space $\mathcal{H}_{\rm L}$ and $\mathcal{H}_{\rm R}$. (b) Once we
take into account the entanglement between these two theories, described by a
Bunch-Davies state on a Hilbert space $\mathcal{H}_{\rm
L}\otimes\mathcal{H}_{\rm R}$, the full bulk dual theory appears to be the
complete de Sitter space. This picture is analogous to the AdS black hole
presented in Figure 1.

Similarly to the double-sided AdS black hole, the state of both the dual
quantum theory and the state of the de Sitter space can be described by a
thermofield-double state called the Bunch-Davies state [89], constructed as
follows. We consider a geodesic in de Sitter space, which is arbitrary and
equivalent to any other one thanks to the isometry group of de Sitter space,
and place the worldline of an observer along this geodesic. This worldline
defines a static patch. Among the Killing vectors of de Sitter space, there is
one that is timelike, future-directed, and defined everywhere in the static
patch. This killing vector maps the worldline to itself, by shifting the
proper time on the worldline. The associated conserved charge $H$ is the
Hamiltonian of the static patch.101010In [71], this Hamiltonian is
supplemented by the Hamiltonian of the observer itself. Let us write
$\ket{\Psi_{i}}$ and $E_{i}$ are the eigenvectors and eigenvalues of $H$. We
know that the static patch is in a thermal state of inverse temperature
$\beta$:

$\rho=\frac{1}{Z}e^{-\beta H}.$ (16)

We consider two copies of such static patch, with Hilbert spaces
$\mathcal{H}_{\rm L}$ and $\mathcal{H}_{\rm R}$ and associated eigenstates
$\ket{\Psi_{i}}_{\rm L}$ and $\ket{\Psi_{i}}_{\rm R}$. Each of these theories
independently describes the static patch, they correspond to the left picture
of Figure 4 where the two static patches are completely independent.

We now construct the thermofield-double state whose Hilbert space is
$\mathcal{H}_{\rm L}\otimes\mathcal{H}_{\rm R}$ and define the Hamiltonian

$H_{\rm BD}=H_{\rm L}-H_{\rm R},$ (17)

with $H_{\rm L}$ and $H_{\rm R}$ two copies of the original Hamiltonian.
Considering physical states of $\mathcal{H}_{\rm L}\otimes\mathcal{H}_{\rm R}$
with vanishing eigenvalues ensures that both static patches have the same
time. This can be seen as one of the static patch playing the role of a clock
for the other [51]. With this constraint in mind, one constructs the
thermofield-double state,

$\ket{\Psi_{\rm BD}}=\frac{1}{\sqrt{Z}}\sum e^{-\frac{1}{2}\beta
E_{i}}\ket{\Psi_{i}}_{\rm L}\otimes\ket{\Psi_{i}}_{\rm R},$ (18)

which satisfies the constraint $H_{\rm BD}\ket{\Psi_{\rm BD}}=0$. Tracing out
the degrees of freedom from $\mathcal{H}_{\rm L}$ ($\mathcal{H}_{\rm R}$), one
recovers the thermal density matrix (16) associated with $P_{\rm L}$ ($P_{\rm
R}$). The Bunch-Davies state corresponds to the full de Sitter space as in the
right panel of Figure 4. In this picture, the two static patches in blue are
still encoded in the Hilbert spaces $\mathcal{H}_{\rm L}$ and
$\mathcal{H}_{\rm R}$, while the exterior region in red is encoded in
$\ket{\Psi}_{\rm BD}$, emerging from the entanglement between
$\mathcal{H}_{\rm L}$ and $\mathcal{H}_{\rm R}$. It was argued in [70] that in
a quantum description of de Sitter space, the static patches must be maximally
entangled which is associated to $\beta\rightarrow 0$. This might seem
counter-intuitive as we know that the de Sitter background can be associated
with a finite temperature $1/\beta_{dS}=2\pi$ where the de Sitter radius has
been fixed to $1$. But it was argued in [90] that it is possible to define a
temperature-like quantity, named tomperature, which stays finite as the global
temperature goes to infinity.

The Bunch-Davies state is pure, which is consistent with the discussion of
Section 2.1 and is consistent with the idea that the state of the whole
universe should be unique and pure. The entropy of the pair of holographic
screens is therefore expected to be pure not only at next-to-leading order but
to all orders. In case one would doubt that the exterior region is encoded on
the screens, one could argue that when the screens coincide at the bifurcate
horizons there is no exterior region such that the two-screen system must be
in a pure state. Indeed, there exist Cauchy slices $\Sigma^{\prime}$ such that
$\Sigma^{\prime}_{\rm L}$ and $\Sigma^{\prime}_{\rm R}$ join at the bifurcate
horizon, such that $\Sigma^{\prime}_{\rm E}=\varnothing$. Moreover, the
evolution operator along the screens should be linear (possibly unitary),
which cannot transform a pure state into a mixed state. Hence, whether we
believe or not that the exterior region is also encoded on the screens, the
entropy of the two-screen system must vanish to all orders:

$S(\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm
R})=0+\mathcal{O}\left((G\hbar)^{n}\right),$ (19)

for any $n\geq 0$. The discussion above is another argument against the
monolayer proposal. In the monolayer proposal, extremal surfaces do not extend
in the static patch such that we do not expect the entanglement wedge of the
screens to extend in them either (see Section 3). Beyond the fact that this is
inconsistent with static patch holography, one may note that this implies that
non vanishing $\mathcal{O}((G\hbar)^{0})$-terms would appear in
$S(\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm R})$ in the monolayer proposal. In
particular, the correction term in equation (6) would be modified to
$S_{\text{semicl}}(\mathcal{C}^{\prime}_{\rm E})$ which does not vanish in
general.

###  4.2 Single-Screen System: A Phase Transition of
$\mathcal{E}(\mathcal{S}_{\rm L})$

In this section we consider the simplest non-trivial subsystem of the full
holographic system, that is $\mathcal{S}_{\rm L}$ or $\mathcal{S}_{\rm R}$.
Here we discuss $\mathcal{S}_{\rm L}$. Since the two-screen system has zero
entropy,

$S(A)=S(\bar{A}),$ (20)

where $\bar{A}$ is the complementary subsystem with respect to
$\mathcal{S}_{\rm L}\cup\mathcal{S}_{\rm R}$. Hence our computations for any
susbsystem directly transfer to its complement. In particular, we can also
expect $\chi_{\rm E}(A)=\chi_{\rm E}(\bar{A})$.

In the left static patch, $A_{\rm L}=\mathcal{S}_{\rm L}$ and $\varnothing$ is
the minimal extremal surface with $\mathcal{C}^{\prime}_{\rm
L}=\Sigma^{\prime}_{\rm L}$ for the same reason as in the previous section.
The part of the entanglement wedge lying in the left static patch therefore
covers $\Sigma^{\prime}_{\rm L}$, which is exactly what we expect from static
patch holography. On the other hand, $A_{\rm R}=\varnothing$ which is
homologous to $\varnothing$ with $\mathcal{C}^{\prime}_{\rm R}=\varnothing$.
The empty surface $\varnothing$ being of extremal and vanishing area, it is
the minimal extremal homologous surface in the right static patch. The causal
diamond of $\varnothing$ being $\varnothing$, the entanglement wedge of
$\mathcal{S}_{\rm L}$ does not extend in the right static patch.

Since the two interior regions do not contribute to the entropy, the entropy
of $\mathcal{S}_{\rm L}$ entirely comes from the exterior region, as one would
expect from the ER=EPR correspondence. Here, $A_{\rm E}=\mathcal{S}_{\rm L}$
has no boundary but the empty surface is not homologous since there is not any
$\Sigma^{\prime}_{\rm E}$ that contains a spatial system only bounded by
$\mathcal{S}_{\rm L}$. Hence, we look for a non-trivial $SO(n)$-symmetric
surface in the exterior that has minimal extremal area.111111We suppose that
the $SO(n)$ symmetry is not spontaneously broken during the extremization of
the area functional. While the bifurcate horizon is the only extremal surface
in the exterior region, every $SO(n)$-symmetric slice of the horizon has the
same area as the bifurcate horizon, see Figure 5. In [56], we showed that all
such surfaces in the exterior region are solutions of the extremization
problem (10) and should be considered as candidates in our holographic entropy
prescription. At leading order, these surfaces give degenerate results:

$S(\mathcal{S}_{\rm L})=\frac{\text{Area}(\mathcal{S}_{\rm
L})}{4G\hbar}+\mathcal{O}((G\hbar)^{0}),$ (21)

Hence we recover the result of [24, 44, 45] that the entanglement entropy of
one screen is equal to the Bekenstein-Hawking formula at leading order.

However, these degenerate extremal surfaces lead to completely different
entanglement wedges. In particular, extremal surfaces lying on $H_{\rm L}$
lead to an entanglement wedge that does not extend in the exterior but only in
the interior and along the horizon. On the other hand, extremal surfaces lying
on $H_{\rm R}$ lead to an entanglement wedge that extends in the exterior
region. The most extreme case being when the extremal surface lies on
$\mathcal{S}_{\rm R}$. In this case, $\mathcal{C}^{\prime}_{\rm
E}=\Sigma^{\prime}_{\rm E}$ and the entanglement wedge connects the two
holographic screens.

This degeneracy may be lifted in two ways. The first one is to consider
classical perturbations of the metric and was studied in [59]. The second one,
which we consider in this work, is to take into account the next-to-leading
order contribution to the entropy. The entropy of $\Sigma^{\prime}_{\rm L}$
decreases as $\mathcal{S}_{\rm L}$ evolves in (absolute value of) time along
$H_{\rm L}$. Both screens $\mathcal{S}_{\rm L}$ and $\mathcal{S}_{\rm R}$ are
always favored in the extremization of the generalized entropy (6), as they
are the $SO(n)$-symmetric slices of the horizons at the highest global time
$|\tau|$ (and hence the minimal semiclassical correction) in the region
spanned by $\Sigma^{\prime}_{i}$.121212As already noted, $SO(n)$-symmetric
surfaces are associated with a unique value of $\tau$ and $\theta$. The
procedure then simply instructs us to choose the one with the smallest
semiclassical correction, i.e. with the highest value of $|\tau|$. Hence,

$\chi_{\rm E}=\left\\{\begin{array}[]{ll}\mathcal{S}_{\rm L}&\mbox{if
}|\tau_{\rm L}|>|\tau_{\rm R}|\\\ \mathcal{S}_{\rm R}&\mbox{if }|\tau_{\rm
L}|<|\tau_{\rm R}|\end{array}\right.,\qquad\mathcal{E}(\mathcal{S}_{\rm
L})=\left\\{\begin{array}[]{ll}D(\Sigma_{\rm L})&\mbox{if }|\tau_{\rm
L}|>|\tau_{\rm R}|\\\ D(\Sigma_{\rm L})\cup D(\Sigma_{\rm
E})&\mbox{if}|\tau_{\rm L}|<|\tau_{\rm R}|\end{array}\right.,$ (22)

with a phase transition at $\tau_{\rm L}=\tau_{\rm R}$. In this case, higher
order corrections are needed to lift the degeneracy. As for the entropy, we
get

$S(\mathcal{S}_{\rm L})=\frac{\text{Area}(\mathcal{S}_{\rm
L})}{4G\hbar}+\left\\{\begin{array}[]{ll}S_{\text{semicl}}(\Sigma^{\prime}_{\rm
L})&\mbox{if }|\tau_{\rm L}|>|\tau_{\rm R}|\\\
S_{\text{semicl}}(\Sigma^{\prime}_{\rm R})&\mbox{if }|\tau_{\rm L}|<|\tau_{\rm
R}|\end{array}\right.,$ (23)

where we used $S_{\text{semicl}}(\Sigma^{\prime}_{\rm
R})=S_{\text{semicl}}(\Sigma^{\prime}_{\rm L}\cup\Sigma^{\prime}_{\rm E})$.
The entanglement wedges and extremal surfaces are depicted in Figure 5.

$\mathcal{S}_{\rm L}$$\chi_{\rm E}$ (a) $|\tau_{\rm L}|>|\tau_{\rm R}|$

$\mathcal{S}_{\rm L}$$\chi_{\rm E}$ (b) $|\tau_{\rm L}|<|\tau_{\rm R}|$

Figure 5: The sceen $\mathcal{S}_{\rm L}$ of which we compute the entanglement
entropy is depicted by a red point while its entanglement wedge
$\mathcal{E}(\mathcal{S}_{\rm L})$ is the red region. The complementary screen
$\mathcal{S}_{\rm R}$ is the blue dot. The extremal surfaces in the two
interior regions are $\varnothing$ while the non-trivial extremal surface in
the exterior, $\chi_{\rm E}$, is depicted by a black dot. The light blue
region indicates the region spanned by all $\Sigma^{\prime}$ and which is not
contained in $\mathcal{E}(\mathcal{S}_{\rm L})$. This blue region also
corresponds to $\mathcal{E}(\mathcal{S}_{\rm R})$. The thick magenta line
corresponds to the union of all candidate extremal surfaces with minimal area
$\text{Area}(\mathcal{S}_{\rm L})$.

We thus showed that, while the full holographic system encodes the whole de
Sitter space, there is always one of the two screens that contains enough
information to encode both the interior of his static patch and the exterior
region. The complementary screen then only encodes its own interior. The
screen that encodes the exterior is the one with the smallest entanglement
entropy. Another way to see this is to consider the Quantum Bousso Bound (QBB)
of Strominger and Thompson [91]. While the classical Bousso Bound relates the
number of degrees of freedom of a region to an area, the Strominger-Thompson
QBB which was shown to be valid in JT gravity [92] associates these degrees of
freedom to a quantum area

$A_{\text{qu}}(\chi)=\text{Area}(\chi)+4G\hbar
S_{\text{semicl}}(\Sigma_{\chi}),$ (24)

where $\Sigma_{\chi}$ is a spacelike slice bounded by $\chi$. From this point
of view, we can see the screen with the smallest entanglement entropy as the
screen with the biggest quantum area and therefore containing the largest
number of degrees of freedom, thus dominating the competition for the encoding
of the exterior region.

At $\tau_{\rm L}=\tau_{\rm R}$, there is a phase transition where the screen
containing the greatest number of degrees of freedom switch from one to the
other. This results in an exchange of dominance between the two competing
quantum extremal surfaces. This phase transition reflects into the
entanglement wedge picture as the encoding of the exterior region transfers
from one screen to the other. In [8], this type of transition in AdS/CFT was
identified with the transfer of an emergent type III1 subalgebra whose
reconstruction complexity is exponential in Newton’s constant, called complex
factor, from one screen subsystem algebra to the other. It would be very
interesting to understand how the analysis of [8] would apply in this context.

###  4.3 Pushing the Screens Inside the Static Patches

Lastly, we would like to comment on the effect of pushing the holographic
screens inside the the static patch. Let us work in static patch coordinates,

$ds^{2}=-(1-r^{2})dr^{2}+\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega_{n-1}^{2},$ (25)

where we have set the de Sitter radius to $1$. We consider the stretched
horizon as the timelike hypersurface defined by $r=1-\epsilon$. In general
stretched horizons are defined such that $\epsilon\sim l_{\rm Pl}$ with
$l_{\rm Pl}$ the Planck length. But we would like to relax this definition for
the moment to consider hypersurfaces lying further from the cosmological
horizon. A screen placed on the stretched horizon therefore evolves along the
hyperbola $r=1-\epsilon$ and its area remains constant, decreased by a factor
$(1-\epsilon)^{n-1}$ compared to the parent screen on the cosmological
horizon. The temperature measured in the static patch is given by

$T_{\rm dS}=\frac{1}{2\pi\sqrt{1-r^{2}}},$ (26)

in units of the de Sitter radius. The stretched horizon therefore acts as a
cutoff, with the de Sitter temperature varying from $1/(2\pi)$ along the
observer trajectory, to a temperature scaling as $\epsilon^{-\frac{1}{2}}$.

In Section 2.2, we argued that putting the holographic screen on the
cosmological horizon maximizes the number of holographic degrees of freedom.
But one could decide to stay away from the mathematical horizon and place the
holographic screen on the hypersurface $r=1-\epsilon$ in the static patch, as
in [24]. This is usually the prescription followed when using static patch
holography as it plays the role of a cutoff. However, in the context of
entanglement wedge reconstruction, this leads to radical issues. For example,
applying blindly our covariant prescription would lead to the same conclusion
that the region between the two stretched horizon are reconstructible from the
holographic degrees of freedom located on them. However, at a rate of $1$
degree of freedom per Planck area, the number of holographic degrees of
freedom decreases to zero as the stretched horizons is pushed towards the
podes, i.e. as $\epsilon\rightarrow 1$. This seems paradoxical.

Here, we would like to argue that while static patch holography stays
consistent when the holographic screen is pushed inside the static patch, the
bilayer proposal must be reconsidered. Pushing the screens in the static
patches effectively integrates out holographic degrees of freedom, as the area
of the screen decreases by a factor $(1-\epsilon)^{n-1}$. In particular, this
change of location can be seen as replacing the holographic theory by a cutoff
version of the initial one, where the degrees of freedom encoding the two
strip regions bounded by the stretched horizons and their associated
cosmological horizon have been integrated out.

Moreover, we expect that with this loss of degrees of freedom, the
entanglement between the two holographic theories is no longer sufficient to
reconstruct the exterior region between the two cosmological horizons. In
other words, the entanglement wedges of subsystems of this cutoff theory can
no longer extend in the region between the two holographic screens. This can
be seen as follows. As the strips regions are effectively removed from the
holographic duality, the screens can no longer be deformed in the exterior
region. Indeed, extremal surfaces should be seen as continuous deformations of
the screens subsystem into the holographic bulk region. But removing the two
strips completely disconnect the two static patches from the exterior, such
that extremal surfaces and entanglement wedges cannot extend in this region.
Hence, the entanglement entropy of one screen should only receive a
semiclassical contributions from the fields in its interior.

The two cutoff theories can be unitarily evolved from the pair of screens at
the symmetric slice $\tau=0$, as their area is constant in time. However, the
full holographic theory is not expected to be a pure state as the entropy of
the two-screen system is no longer expected to vanish to all orders in
$\hbar$, since there cannot be contributions from the bulk region between the
screens. The discussion above should be restricted to semiclassical gravity,
such that variations of the location of the holographic screens at the Planck
scale or below should not be taken seriously. In particular, a screen located
on the stretched horizon with $\epsilon$ scaling as $\mathcal{O}(\hbar^{a})$
with $a>1$ would not get its area decreased by more than one Planck area. In
other words, a stretched horizon sufficiently close to the cosmological
horizon will not have less holographic degrees of freedom. For this reason, we
restrict our conclusions concerning the loss of the degrees of freedom to
screens that are separated from the cosmological horizon by more than a Planck
length, further inside than the stretched horizon with $\epsilon\sim l_{\rm
Pl}$.

## Acknowledgements

I am grateful to Hervé Partouche, Nicolaos Toumbas, and in particular to Lior
Benizri and François Rondeau for helpful discussions and feedback on the
manuscript. This work is partially supported by the Cyprus Research and
Innovation Foundation grant EXCELLENCE/0421/0362.

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# A web-based gamification of upper extremity robotic rehabilitation

1st Payman Sharafianardakani 2nd Hadi Moradi, Fariba Bahrami  School of
Electrical and Computer Engineering, College of Engineering
University of Tehran
Tehran, Iran
<EMAIL_ADDRESS>School of Electrical and Computer Engineering,
College of Engineering
University of Tehran
Tehran, Iran
<EMAIL_ADDRESS><EMAIL_ADDRESS>3nd Malahat Akbarfahimi  School of
Rehabilitation Sciences Iran
University of Medical Sciences
Tehran, Iran
<EMAIL_ADDRESS>
###### Abstract

In recent years, gamification has become very popular for rehabilitating
different cognitive and motor problems. It has been shown that rehabilitation
is effective when it starts early enough and it is intensive and repetitive.
However, the success of rehabilitation depends also on the motivation and
perseverance of patients during treatment. Adding serious games to the
rehabilitation procedure will help the patients to overcome the monotonicity
of the treatment procedure. On the other hand, if a variety of games can be
used with a robotic rehabilitation system, it will help to define tasks with
different levels of difficulty with greater variety. In this paper, we
introduce a procedure for connecting a rehabilitation robot to several web-
based games. In other words, an interface is designed that connects the robot
to a computer through a USB port. To validate the usefulness of the proposed
approach, a researcher-designed survey was used to get feedback from several
users. The results demonstrate that having several games besides
rehabilitation makes the procedure of rehabilitation entertaining.

###### Index Terms:

Gamification, Serious games, Rehabilitation, Stroke, Web games, Upper
extremity

##  I Introduction

Stroke or Cerebrovascular was the second leading cause of mortality and
disability globally until 2012. Over 40% of stroke survivors encounter medium
to critical disorders that need intensive care, and 10% of them require long-
term care in nursing homes or other facilities [1]. After a stroke, Survivors
enter the rehabilitation process. The purpose of standard rehabilitation is to
help patients till recovering their lost ability promptly. A variety of
rehabilitation techniques are utilized to recover patients’ abilities, and one
of them is robotic rehab. Therefore, there are many studies investigating
rehabilitation robots ranging from exoskeletons to end-effector robots. The
choice of a robot or robotic technology depends on the type of disability.

Today, despite many advances in rehabilitation, many rehabilitation centers
use traditional and time-consuming methods. In addition to forcing
physiotherapists to monitor and intervene constantly, these methods also make
patients tired and reluctant to perform the assigned tasks. Thus, these tasks
seem meaningless and monotonous to patients. This reluctance causes patients
not to make good progress in recovering quickly. Due to the importance of this
issue, one of the proposed methods in rehabilitation or training is
gamification.

Using game mechanics and playful thinking with aesthetics to motivate, learn,
and solve problems is called gamification [2]. In other words, transforming an
action that has no conceptual connection with a game into an action associated
with the game is called gamification. For instance, gamification was made in a
study on the stairs of a metro station in Stockholm, Sweden. Each step was
arranged to the sound of a musical note. When people stepped on the steps,
they played the corresponding note. In the end, the results showed that 66%
more people used music stairs instead of escalators [3]. Serious games are
very similar to gamification, and you cannot find a boundary between them.
However, in general, serious games are full-blown games that offer education,
not just entertainment.

In this study, we focused to gamify upper extremity rehabilitation using a
robotic arm and available games on the web. The main advantage of our proposed
approach to the current systems is in using available games on the web by
connecting the robotic arm through a USB port to computers. This approach
reduces the cost of game development and increases the available options to
patients and therapists to fit every need.

##  II Related work

For many years gamification has been used in education, and results of that
have been demonstrated. The effects of gamification on students have shown
that they are more enthusiastic and motivated than before [2]. These results
cause to use of this method in other fields. One of these fields is the field
of rehabilitation. The use of gamification in rehabilitation robotics has
abundantly increased and we can see vestigial of gamification even in
commercialized robots. Robots such as Armeo Power and Armeo Spring from Hocoma
company [4], Inmotion ARM [5], and Robotherapist 2D [6], with the help of
games, encourage patients to perform rehabilitation movements. These games
motivate and help the treatment process to be noun-uniform. Housman et al.
[7], surveyed a group of patients and demonstrated that 90% of patients agree
with game-based robotic rehabilitation. Furthermore, LaPiana et al. [8] showed
that more than 90% of patients enjoy and engage with games in the
rehabilitation process. Also, Nguyen et al. [9] evaluated a rehabilitation
process with 22 subjects in 2 methods: with gamification and without
gamification. In the end, the result indicates a more favorable consequence on
the gamified application. Nevertheless, there was no statistical difference in
user engagement scores.

Gamification methods are diverse but generally, they can be categorized in
three methods: 1)Virtual reality (VR), 2) Augmented Reality (AR) 3) Common
gamification.

####  II-1 Virtual reality

VR is a simulation of a world apart from reality that allows the user to enter
a space different from his surroundings and have the desired experience; An
experience that can be fun and entertaining with the purpose of scientific,
educational, healthcare, and music. VR can be an advanced version of human-
computer interaction in which users are immersed in a virtual 3D environment
containing digital objects [10]. In VR, sometimes it is hard to differentiate
between the real and unreal world, and several devices such as earphones and
head-mounted shows (HMIDs) help to improve this experience. In the field of
stroke rehabilitation, VR has many benefits. For example, Dias et al. [11]
demonstrated that using VR helps people by increasing their motivation for
recovery through the use of a fun and comfortable environment. VR encourages
patients to maintain the therapy process and not lose motivation in the long
run. VR games are designed in such a way that patients are constantly
challenged and immersed to attempt harder. Although VR games have many
benefits, one of their main disadvantages is their expensive equipment.
Furthermore, some cases have also observed that the elderly do not show
interest in this method [12]. That is why we did not select this method since
our target group in this paper is older people.

####  II-2 Augmented reality

AR is a combination of virtual reality and physical reality or physical
objects. In AR, the user clearly sees the difference between the virtual and
real worlds and has more control over virtual objects by communicating with
real objects. AR also plays an essential role in the rehabilitation of stroke
patients. For example, Khademi et al. [13] utilized a haptic method with AR to
demonstrate improved hand stiffness. On the other hand, Mousavi et al. [14]
experimented with the two groups. The first group used AR and the second group
used a combination of mouse and computer. They eventually showed that motor
movements increased more in the AR group than in the mouse and computer-based
method. Although AR games have many benefits, the most popular augmented
reality headsets on the market today are bulky, expensive, and require a
computer connection, making the whole process limited and inconvenient for
people. Consumers may also utilize AR software on their smartphones or
tablets. However, showing graphics correctly in mobile AR is a huge challenge.
Electric interference, for example, can disrupt mobile sensors such as
accelerometers, which is frequent in metropolitan settings. Furthermore,
augmented reality apps may be extremely distracting and can cause physical
harm. Many individuals have been wounded while playing Pokemon Go, for example
[15]. Similarly, augmented reality apps might result in significant casualties
when employed in potentially dangerous locations such as busy roadways,
construction sites, and medical facilities.

####  II-3 Common Gamification

CG is a method that is usually used on orthotic robots like exoskeletons and
end-effectors. In this method, the robots connect to a computer, and the games
simulate their movements. For instance, Inmotion Arm [5], developed by Bionic
and MIT, uses the CG method. Inmotion Arm is one of the most successful
examples in recent years and is widely used in 20 countries worldwide [5]. In
addition to having several games to motivate patients, the robot also has a
robust evaluation system that records patient movement feedback. One of these
games is Clock [16], which is very simple and helpful. The practical results
of this robot worldwide have shown that using the robot along with the CG
method will improve patients in the long run. On the other hand, Instead-
Technology in Robotherapist 2D did the CG method, And they used some games to
motivate patients. They also demonstrated that this using Robotherapist 2D
besides these games help patients to improve their upper-extremity motor
activity [16].

Examination of different methods shows us that each method has its own
essential features, and according to the type of rehabilitation, one of these
methods should be selected for gamification. However, it can be said that the
VR method and the CG method are more appropriate for the exoskeleton and end-
effector rehabilitation robots. However, the VR method is more expensive,
while we always have access to the desired equipment in the CG method. So in
this investigation, we finally chose the CG method.

As it was mentioned earlier, the first and foremost purpose of gamification is
to increase patients’ motivation and therapy effectiveness [17, 18, 19]. Using
games in the rehabilitation process makes rehabilitation tasks meaningful. On
the other hand, having a game to continue rehabilitation causes the treatment
process to be monotonous. So having multiple games is very critical. The
method presented in this article helps us access many games and ultimately
increases the variety in rehabilitation. However, the main advantage of our
approach is using simple available games. Furthermore, we chose the CG method
because we need a low-cost device, thus this method has low cost and helps
therapists have an effective rehabilitation. In this regard, therapists can
choose between the available games according to the patient’s ability and give
the patient the right to choose between several games. As a result, a
therapist has many options for effective treatment, and patients are satisfied
with the treatment process.

##  III Web-based games

As mentioned earlier, gamification as an effective method increases motivation
and makes the rehabilitation process enjoyable. One of the challenges in the
gamification of rehabilitation robots is the number of games used. Using a few
games makes the rehabilitation process monotonous and tedious for patients in
the long term. Our method to solve this challenge is to use web games. Web
games are actually a kind of web service. Web services allude to a supplier
that gives the ability to perform particular operations through a browser.
Consumers may reach services at any time and place through the internet and
web services and access locally configurable online features [1]. Web services
are divided into two kinds websites and web applications. A web application is
an application that can be accessed via the Internet anywhere, anytime. The
main concept of a web application is to overcome the limitations of local
gadgets utilizing remote computing power [1]. One of the demos that use remote
computing power is a web game. Web games are not similar to online games;
while online games have to be downloaded the game and installed on the local
computer for the game to play, web games are launched without installation in
a web browser [1]. These features make web games a good choice for our
rehabilitation robot. This article provides a way for a rehabilitation robot
with one or two degrees of freedom to connect to web games and organize a web-
based rehabilitation framework based on the features of these games.

Figure 1: The previous game used on the robot based on open source version of
the Angry Birds game

##  IV Methods

The method presented in this paper is to design and program the robotic arm
that can be connected through the USB port to any computer and control games
like a mouse. . Using this method has given us access to a large number of
games. In this way, we used the serial port to have a pure connection between
the computer and the robot. One of the most significant advantages of this
kind of connection is stability. Furthermore, this method is cheap because we
have access to computers everywhere we want. Consequently, with the pure
connection and many games, we have a comprehensive system that helps us
decrease monotonously in rehabilitation.

In the past, this robot worked with a few games that were designed exclusively
for it (Fig. 1). There are always two main problems in rehabilitation. 1. get
patients used to help 2. uniformity of treatment steps. For the first problem,
an active resistance control has been used to challenge patients more. While
using the method presented in this study has made the patient take more
initiative and think more to overcome the challenges, in the methods provided
for other robots, pre-determined goals were always set for patients, and
robots always help patients. This is like we are constantly cycling with
auxiliary wheels. Therefore, we will never be able to ride a bike
independently. For the second problem, we provide the method that we explained
in this paper.

Figure 2: User interface to setup the USB connection between the
rehabilitation robotic arm and a computer

###  IV-A User interface

For connecting to web games, the first issue is to connect easily. For this
purpose, we designed a user interface. We used the C# Windows form application
to have a helpful and straightforward user interface. All parts related to
connecting to a computer and storing the data collected from patients are
included in this code. As shown in Fig. 2, after connecting the robot to the
computer via USB, all port characteristics are automatically detected, and
this connection is established only by pressing the connect button. Also, by
pressing open game, one of the games opens. All these features make it easy to
use the robot and work with the game.

###  IV-B Robot description

The University of Tehran rehabilitation robot is a one-degree-of-freedom robot
with two encoders. These encoders show the position of the arm and motor. Our
recommended method is to convert the arm position to the mouse’s position on
the computer screen. In this method, we first positioned the encoders and then
mapped the range of the mouse on the screen. Fig. 3(a) shows the workspace of
the robot arm movement, which shows that this movement is limited.
Furthermore, to shoot in these games, we turned the button on the joystick to
the left click on the computer (Fig. 3(b)) Additionally, there is a
microswitch for safety. When the patients remove their hands from the
joystick, the robot and game will stop.

(a)

(b)

Figure 3:  (a) shows The robot’s workspace, and (b) The joystick including the
shoot button and the microswitch

###  IV-C Games

Each of these games has difficulty levels and becomes more difficult after
passing each stage. All games are a combination of visual and audio features
that excite the user and increase their motivation. Fig. 4 illustrates some of
these web games [20][21]. According to the occupational therapist, these games
not only motivate but also engage the user. Because the fact that you always
have to choose the best path creates a mental challenge for the user.

Figure 4:  Two web-based games used in our study connected to our
rehabilitation robot [20][21]

###  IV-D Experiment and results

Finally, we performed an experiment on normal subjects (Fig. 5). We had a
protocol, including a survey that subjects filled out at the end of the test.
In this evaluation, we had 15 subjects with the same features. All subjects
were in the range of 20 and 30 years old, and they were right-handed.
Moreover, they did not have the long antecedent of exercise by hand, and They
were not tired or lethargic before the test. Also, this experiment did not
have any sexuality limit.

Figure 5: A subject is playing with the robotic arm

####  IV-D1 Protocol

Type of Questions | Number of Questions | Very Satisfied | Satisfied | Natural | Unsatisfied | Very Unsatisfied
---|---|---|---|---|---|---
Robot Convenience | 3 | 37.78% | 42.22% | 13.33% | 6.66% | 0
Robot Safety | 1 | 53.33% | 33.33% | 13.33% | 0 | 0
Making Entertainment and Motivation with Games | 1 | 46.66% | 53.33% | 0 | 0 | 0
Concentration | 1 | 53.33% | 46.66% | 0 | 0 | 0
Simplicity in Use | 1 | 66.66% | 33.33% | 0 | 0 | 0
Tolerance for Tasks and the Degree of Difficulty | 1 | 40% | 26.66% | 26.66% | 6.66% | 0
Causes Pain and Fatigue | 1 | 33.33% | 53.33% | 13.33% | 0 | 0
TABLE I: Survey results

The protocol used in this experiment was to perform each person with two games
and each game with two different powers. These powers changed after passing
two levels in the game, and the whole test period was about 20 to 30 minutes.
Because our robot had an active resistance control, we wanted to test both
very high and very low power on subjects. As a result, the applied torques
were 8 Nm and 16 Nm for each game. Also, during this period, all the movement
data of the people were recorded. Finally, after working with the robot, the
people filled a questionnaire, and the results of this questionnaire were the
average of the answers in each section.

####  IV-D2 Results

In the end, we had a survey with a questionnaire table, which showed that more
than 90% of people were satisfied with multiple games with the robot. This
satisfaction results from playing two games for 20 minutes, and the question
analyzed the games in terms of entertainment and motivation. Furthermore, in
the focus discussion, all subjects believed that they had the necessary focus
during the experiment, and some games made them more focused.

According to Table 1, for this questionnaire, we use the Likert scale. We had
9 questions and table 1 shows the subject of every question. In the part
making entertainment and motivation, everyone believes that the games provide
entertainment and cause to increase motivation. Additionally, in the part of
concentration, everyone believes the games provided concentration during the
experiment.

##  V Conclusion

We believe that games can revolutionize rehabilitation systems overall and
stroke rehabilitation in specific. These games, along with entertainment,
increase patients’ motivation. Consequently, patients practice more and
recover faster. In this article, we provide a way to connect a large number of
games with a robot. As a result, we increased the variety of rehabilitation
and allowed the therapist to choose from the available games, whichever is
appropriate for the patient’s condition. Also, it can offer different games
and tasks for the patient on additional days of the week. Finally, in a survey
of ordinary people, we measured the accuracy of this gamification. The results
of this survey showed that the variety in rehabilitation is satisfactory for
many people.

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