aainstitutetext: Center for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japanbbinstitutetext:
Inamori Research Institute for Science,
620 Suiginya-cho, Shimogyo-ku,Kyoto 600-8411 Japanccinstitutetext: Kavli
Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8582, Japan

# A Half de Sitter Holography

Taishi Kawamoto a Shan-Ming Ruan a Yu-ki Suzuki a,b,c Tadashi Takayanagi
<EMAIL_ADDRESS><EMAIL_ADDRESS>yu-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A long-standing and intriguing question is: does the holographic principle
apply to cosmologies like de Sitter spacetime? In this work, we consider a
half dS spacetime wherein a timelike boundary encloses the bulk spacetime,
presenting a version of de Sitter holography. By analyzing the holographic
entanglement entropy in this space and comparing it with that in AdS/CFT, we
argue that gravity on a half dSd+1 is dual to a highly non-local field theory
residing on dSd boundary. This non-locality induces a breach in the
subadditivity of holographic entanglement entropy. Remarkably, this
observation can be linked to another argument that time slices in global de
Sitter space overestimate the degrees of freedom by redundantly counting the
same Hilbert space multiple times.

YITP-23-73

##  1 Introduction

The holographic duality has bestowed upon us a promising framework, enabling
the exploration of quantum gravity tHooft:1993dmi ; Susskind:1994vu . The most
outstanding example will be the AdS/CFT correspondence where quantum gravity
on $d+1$-dimensional Anti-de Sitter space (AdSd+1) becomes equivalent to
$d$-dimensional conformal field theory (CFTd) Maldacena:1997re ; Gubser:1998bc
; Witten:1998qj . Despite the resounding success attained in AdS/CFT, we are
still at a nascent stage in the development of the holographic duality
pertaining to gravity in de Sitter space. The potential dS holography promises
paramount application to realistic cosmological spacetime.

Figure 1: The Penrose diagram of dSd+1 bulk spacetime. The conformal time
$T\in[-\frac{\pi}{2},+\frac{\pi}{2}]$ is associated with the global time $t$
by $\cosh t=\frac{1}{\cos T}$. We introduce a timelike boundary at
$\theta=\theta_{0}$ which is described by a $d-$dimensional dS spacetime. The
dual bulk dSd+1 spacetime is given by the gray shaded region. Figure 2: The
left panel shows the geometry of time slices for a global dS3. In the right
panel we keep a half of dS3 space with a dS2 boundary (denoted by green
circle) at $\theta=\theta_{0}$.

Let us commence by exploring the explicit differences between dS holography
and AdS/CFT. The $d+1$-dimensional de Sitter space (dSd+1) can be described in
global coordinates as follows (see e.g., Spradlin:2001pw ; Galante:2023uyf
for comprehensive reviews):

$ds^{2}=-dt^{2}+\cosh^{2}td\Omega_{d}^{2},.$ (1.1)

whose Penrose diagram is shown in figure 1. Here, we have set the dS radius to
unity, and $d\Omega_{d}^{2}$ represents the metric of the unit $d$-dimensional
sphere:

$d\Omega_{d}=d\theta^{2}+\sin^{2}\theta d\Omega_{d-1}^{2}\,,$ (1.2)

The constant time slice in global dSd+1 coordinate is depicted in the left
diagram of figure 3. In the case of $d=2$, we simply write
$d\Omega_{d-1}^{2}=d\phi^{2}$ in this paper. As shown in the left panel of
figure 1, the conformal boundaries of global dS spacetime are spacelike
surfaces i.e., $S^{d}$, located at the future and past infinity $t=\pm\infty$.
The original dS/CFT correspondence postulates that the gravitational dynamics
in the dSd+1 bulk is dual to a Euclidean conformal field theory (CFT) on the
sphere Sd at future infinity $t\to\infty$ Spradlin:2001pw ; Witten:2001kn ;
Maldacena:2002vr . This proposal assumes that the quantum state is generated
by the Euclidean instanton using the Hartle-Hawking prescription and is then
continued to Lorentzian dS at $t=0$. However, the dual CFT becomes non-unitary
and exhibits numerous exotic characteristics. For instance, in the case of
dS${}_{4}/$CFT3, the 3D CFT dual to higher-spin gravity on dS4 is described by
the $SP(N)$ model, which incorporates ghost fields Anninos:2011ui ; Ng:2012xp
. Similarly, for dS${}_{3}/$CFT2, the 2D CFT dual to Einstein gravity on dS3
is obtained through an analytical continuation of current algebra or Liouville
CFT, with an imaginary central charge Hikida:2021ese ; Hikida:2022ltr ;
Chen:2022ozy ; Chen:2023prz . Furthermore, holography for dS2 has been
investigated in Maldacena:2019cbz ; Cotler:2019nbi ; Cotler:2023eza . Despite
the peculiar holographic properties, studies from the perspective of gravity
have also been developed, with a partial list of references including
Castro:2011xb ; Castro:2012gc ; Anninos:2020hfj ; Anninos:2021ihe ;
Castro:2023dxp .

One of the manifestations of the non-unitary nature inherent in the dS/CFT
correspondence is the computation of holographic entanglement entropy. In the
context of the AdS/CFT correspondence, the holographic entanglement entropy
$S_{A}$ for a subsystem $A$ in the dual CFT can be determined by evaluating
the area of the extremal surface $\Gamma_{A}$ in AdS, which ends on the the
boundary of the subsystem $A$ Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt :

$S_{A}=\frac{A(\Gamma_{A})}{4G_{\textrm{\tiny N}}}\,,$ (1.3)

where $G_{\textrm{\tiny N}}$ denotes the Newton constant and $A(\Gamma_{A})$
represents the area of the extremal surface $\Gamma_{A}$. In principle, this
geometric computation can be extended to various spacetimes, including de
Sitter space. However, in de Sitter space, the absence of a spacelike extremal
surface connecting two distinct points on Sd at future infinity leads to the
holographic entanglement entropy being complex-valued Narayan:2015vda ;
Sato:2015tta ; Hikida:2022ltr . In this context, the holographic entanglement
entropy is computed by using the timelike extremal surface in de Sitter space.
Subsequently, in Doi:2022iyj ; Doi:2023zaf , this complex-valued entropy was
appropriately interpreted as the pseudo-entropy Nakata:2021ubr ;
Mollabashi:2020yie ; Mollabashi:2021xsd , which generalizes the notion of
entanglement entropy to non-Hermitian density matrices (see Narayan:2022afv ;
Narayan:2023ebn  for closely related ideas). This consideration suggests a
connection between the emergent time coordinate in dS/CFT and the imaginary
part of the pseudo-entropy. Notably, an intriguing quantum entanglement
structure of dS holography has been recently proposed in Cotler:2023xku .

Several other approaches to de Sitter holography have been explored. One
notable example is the holography for de Sitter space in the static patch,
which has recently garnered significant attention and discussions from
multiple perspectives Susskind:2021omt ; Susskind:2021esx ;
Shaghoulian:2022fop ; Chapman:2021eyy ; Jorstad:2022mls ;
Chandrasekaran:2022cip ; Anegawa:2023wrk ; Franken:2023pni . Additionally,
another approach to dS holography has been investigated, based on the TTbar
deformation in AdS/CFT McGough:2016lol  and the dS/dS duality
Alishahiha:2004md , with studies conducted in Dong:2018cuv ; Gorbenko:2018oov
. An interesting dS/dS duality setup can also be found in Geng:2021wcq . The
quantum information structure in de Sitter has been analyzed by applying the
surface/state duality Miyaji:2015yva . Notably, it has been observed that the
state dual to the $t=0$ slice is maximally entangled.

In this paper, we propose a novel approach to dS holography that adheres to
the standard holographic formalism, wherein gravity in a given bulk space is
dual to a non-gravitational theory on its timelike boundary. However, since de
Sitter space lacks timelike boundaries, we introduce a procedure to create
one. Our proposal involves cutting a de Sitter space into a half at
$\theta=\theta_{0}$ by confining the sphere $S^{d}$ to a semi-sphere. More
generally, we can cut the bulk dS space by putting a boundary at
$\theta=\theta_{0}$, as depicted in figure 1 and 2. This resulting spacetime
is referred to as a half de Sitter space (or simply half dS). The boundary of
the $d+1$-dimensional half de Sitter space corresponds to $d$-dimensional
global de Sitter spacetime dSd, which is described by the metric:

$ds^{2}=-dt^{2}+\cosh t^{2}\sin^{2}\theta_{0}d\Omega^{2}_{d-1}\,.$ (1.4)

In this setup, we argue that gravity on a half dSd+1 is dual to a field theory
without gravity on dSd. We expect that such a field theory exhibits high non-
locality due to the finite geometric cut-off in dSd+1. It is worth noting that
this spacetime possesses the SO$(d,1)$ symmetry, which is a subgroup of the
original SO$(d+1,1)$ symmetry of dSd+1. Although our approach shares common
features with the TTbar approach Dong:2018cuv ; Gorbenko:2018oov  and
surface/state duality Miyaji:2015yva , our focus lies specifically on the half
dS space, as it contains a timelike boundary. Nevertheless, in principle, our
analysis of half dS holography can be extended to full dS geometry by
combining two copies of our holographic duality, as demonstrated in the case
of gluing AdS/CFT Kawamoto:2023wzj . To probe our holographic proposal, we
would investigate the holographic entanglement entropy (1.3) in dS holography
as a fundamental tool. Here we would like to emphasize that we stay with the
standard calculation of holographic entanglement entropy Ryu:2006bv ;
Ryu:2006ef ; Hubeny:2007xt , where we minimize the area, as opposed to the
different prescription of maximizing the area in the version of dS holography
discussed in Susskind:2021esx .

It is noteworthy that we can also apply the standard AdS/CFT correspondence to
study the field theory on dSd Hawking:2000da ; Maldacena:2012xp . In this
case, a CFT on dSd is dual to gravity in AdSd+1 whose conformal boundary is
given by dSd. Given the well-established nature of this holographic duality,
we will initially focus on this situation. Subsequently, we will delve into
the main subject of this paper, i.e., holography for the half dS. Throughout
this paper, we will consider two different prescriptions for both holography
in AdS and half dS, which are described below and depend on how we handle the
future infinity at $t\to\infty$.

Figure 3: The left panel shows a sketch of global de Sitter space, which has
only space-like boundaries at $t=\pm\infty$. The right one is a sketch of our
holographic setup which is obtained by cutting in half a de Sitter space.
Notice that this geometry has both the time-like and space-like boundary.

Case 1: Schwinger-Keldysh prescription without EOW

We apply the Hartle-Hawking prescription to construct the initial state at
$t=0$ using the Euclidean instanton geometry, which takes the form of a $d+1$
dimensional semi-sphere. Subsequently, we examine its Lorentzian time
evolution. The density matrix for this state at time $t$ is determined by the
corresponding geometry dictated by the Schwinger-Keldysh prescription, as
depicted in figure 4. For a detailed understanding of the Schwinger-Keldysh
prescription in holography and holographic entanglement entropy, we refer
readers to Skenderis:2008dg ; Dong:2016hjy . It is important to note that the
asymptotic infinity as $t\to\infty$ is absent in this particular setup. In
line with conventional Lorentzian holography in AdS, we anticipate that the
gravitational dynamics in half dSd+1 geometry is dual to a field theory
residing on its boundary, namely, dSd. We expect this field theory to exhibit
strong non-locality, primarily due to the fact that the boundary at
$\theta=\theta_{0}$ is not an asymptotic boundary. Unlike in the AdS/CFT
correspondence, the metric, in this case, does not exhibit divergent
behaviour, indicating that the dS boundary plays the role of a finite cutoff.

Case 2: Final state projection with EOW

On the other hand, if we consider the full geometry of a half dSd+1, we need
to impose a boundary condition at the infinity $t\to\infty$. If we impose the
Dirichlet b.c., then we expect a dual CFT lives there in addition to the dSd
boundary. To avoid this complicated situation, we focus on the other case:
Neumann boundary condition. Namely this means that the asymptotic boundary
$t=\infty$ is an end-of-the-world (EOW) brane. In the context of AdS/CFT, the
EOW brane is dual to the boundary conformal field theory (BCFT), whose
holographic duality is called the AdS/BCFT Takayanagi:2011zk ; Fujita:2011fp ;
Karch:2000gx . In our dS case, we expect that the dual theory is a non-local
field theory on dSd with a final state projection at future infinity
$t=\infty$.

In the presence of post-selection, a useful quantity is pseudo entropy
Nakata:2021ubr , which is a natural generalization of entanglement entropy
such that it depends on two different quantum states $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$. This quantity is defined as follows. We introduce the
reduced transition matrix

$\displaystyle\tau_{A}=\mbox{Tr}_{B}\left[\frac{|\psi_{1}\rangle\langle\psi_{2}|}{\langle\psi_{2}|\psi_{1}\rangle}\right].$
(1.5)

The pseudo entropy is defined by

$\displaystyle S_{A}=\mbox{Tr}\left[-\tau_{A}\log\tau_{A}\right].$ (1.6)

Note that this quantity in general takes complex values as $\tau_{A}$ is not
hermitian.

Interestingly, the gravity dual of this quantity is given by (1.3) when
$\Gamma_{A}$ is the minimal surface in an Euclidean time-dependent
asymptotically AdS Nakata:2021ubr . We can also regard the area of extremal
surface as a holographic pseudo entropy in Lorentzian AdS in the presence of
final state projection Akal:2021dqt . We will assume an extension of this
correspondence to de Sitter spaces in this paper. As studies of quantum many-
body systems and CFTs suggest Mollabashi:2020yie ; Mollabashi:2021xsd ;
Akal:2021dqt  the real part of pseudo entropy typically measures the amount of
quantum entanglement in the intermediate states between $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$.

Figure 4: The geometry which describes the Schwinger-Keldysh contour of a half
dSd+1. The top and bottom region presents the Lorentzian and Euclidean
evolution, respectively. This is dual to a field living on dSd boundary which
is parametrized by the green surface. Figure 5: The geometry of a half dSd+1.
Notice that there are both timelike (green surface) and spacelike (purple
surface) boundaries.

This paper is organized as follows. In section 2, we study the AdS/CFT for
CFTs on de Sitter space. We calculate the entanglement entropy in both case 1
and case 2 and discuss physical interpretations. In section 3 we consider the
holography for a half de Sitter space in case 1. We calculate the holographic
entanglement entropy and study its property including the violation of
subadditivity. We will discuss its implication on the Hilbert structure of its
dual field theory. In section 4 we study the holographic duality for a half de
Sitter space in case 2, taking into account the presence of EOW brane. In
section 5 we summarize our conclusions and discuss future problems. In
appendix A, we show explicit calculations of geodesics in de Sitter space.

##  2 CFT on de Sitter Space from AdS/CFT

Before we work on the holography of de Sitter spaces, we would like to examine
the AdS/CFT correspondence with CFT living on the de Sitter space as a warm up
exercise. In this section, we focus on the holographic CFT2 living on dS2
spacetime whose metric is defined by

$\begin{split}ds^{2}_{\mathrm{dS}_{2}}=-dt^{2}+\cosh^{2}{t}d\phi^{2}=\frac{1}{\cos^{2}{T}}\left(-dT^{2}+d{\phi}^{2}\right)\,.\end{split}$
(2.1)

In the following dS metrics, we have chosen the de Sitter radius to be a unit,
and use the transformation between the global time $t$ and conformal time $T$
given by $\cosh{t}=\frac{1}{\cos{T}}$. We note that the conformal time is
compactified due to $T\in[-\frac{\pi}{2},\frac{\pi}{2}]$, while the global
time is not, i.e. $t\in(-\infty,+\infty)$.

For simplicity, we mainly consider AdS${}_{3}/$CFT2 where the CFT lives on
dS2. The holographic dual is described by the global AdS3,

$ds^{2}_{(g)}=-\cosh^{2}\rho d\tau^{2}+d\rho^{2}+\sinh^{2}\rho d\phi^{2}\,.$
(2.2)

where $\rho$ is the radial coordinate and $\tau$ is the global time coordinate
of AdS bulk spacetime. Since the extremal surface in AdS3 is nothing but a
geodesic, we can evaluate the area of the extremal surface using the geodesic
distance $D^{(g)}_{12}$ in the global AdS metric between two points
$(\rho_{1},\tau_{1},\phi_{1})$ and $(\rho_{2},\tau_{2},\phi_{2})$, as given by

$\cosh
D^{(g)}_{12}=\cos(\tau_{1}-\tau_{2})\cosh\rho_{1}\cosh\rho_{2}-\cos(\phi_{1}-\phi_{2})\sinh\rho_{1}\sinh\rho_{2}\,.$
(2.3)

###  2.1 de Sitter and Hyperbolic Slicing of AdS 3

Figure 6: Left: Penrose diagram of AdS3 with de Sitter slicing which is
defined in eq. (2.4). The solid curves denote the constant $\eta$ surfaces
with $\eta>0$ and the negative ones are described by the dashed curves. Right:
Penrose diagram of AdS3 with hyperbolic slicing as described by the metric
(2.8). The solid and dashed curves present constant $\mu$ surfaces with
positive and negative values, respectively.

To describe a CFT on dS2, we employ the de Sitter sliced AdS3 (refer to the
left plot in figure 6):

$ds^{2}_{(d)}=d\eta^{2}+\sinh^{2}\eta(-dt^{2}+\cosh^{2}td\phi^{2}),$ (2.4)

where $(t,\eta)$ is related to the global coordinates $(\tau,\rho)$ via

$\sinh\rho=\cosh t\sinh|\eta|,\ \ \ \ \tan\tau=\tanh\eta\,\sinh t\,$ (2.5)

The coordinate transformation (2.5) leads to the geodesic length between two
points $(t_{1},\eta_{1},\phi_{1})$ and $(t_{2},\eta_{2},\phi_{2})$ in the dS
sliced metric:

$\cosh
D^{(d)}_{12}=\cosh\eta_{1}\cosh\eta_{2}-\sinh\eta_{1}\sinh\eta_{2}\left(\cosh
t_{1}\cosh t_{2}\cos\left(\phi_{2}-\phi_{1}\right)-\sinh t_{1}\sinh
t_{2}\right)\,.$ (2.6)

The holographic CFT2 on dS2 with a physical metric (2.1) is living on the
conformal boundary of AdS3 which is defined by $\eta\to\infty$. We can fix the
UV cut-off $\epsilon$ in the CFT2 by taking the cut-off surface at
$\eta=\eta_{\infty}$ with

$e^{\eta_{\infty}}=\frac{1}{\tanh\frac{\epsilon}{2}}\approx\frac{2}{\epsilon}\,.$
(2.7)

For a later purpose it is also useful to consider a hyperbolic sliced AdS3
(refer to the right plot in figure 6), whose metric reads

$ds^{2}_{(h)}=-d\mu^{2}+\cos^{2}\mu(d\xi^{2}+\sinh^{2}\xi d\phi^{2})\,.$ (2.8)

The coordinate $(t,\eta)$ is related to that of the global AdS3 $(\tau,\rho)$
via

$\sinh\rho=\sinh\xi\cos\mu,\ \ \ \ \tan\tau=\cot\mu\,\cosh\xi,$ (2.9)

The geodesic length $D^{(h)}_{12}$ between two points at
$(\mu_{1},\xi_{1},\phi_{1})$ and $(\mu_{2},\xi_{2},\phi_{2})$ can be derived
from

$\cosh
D^{(h)}_{12}=\cos\mu_{1}\cos\mu_{2}\left(\cosh\xi_{1}\cosh\xi_{2}-\cos(\phi_{1}-\phi_{2})\sinh\xi_{1}\sinh\xi_{2}\right)+\sin\mu_{1}\sin\mu_{2}\,.$
(2.10)

It is also useful to note that the hyperbolic sliced metric and the de Sitter
sliced one are related by the following analytical continuation:

$\eta=i\left(\mu-\frac{\pi}{2}\right)\,,\qquad\xi=i\frac{\pi}{2}-t\,.$ (2.11)

Figure 7: The setup of computing entanglement entropy of CFT2 on dS2 via
AdS3/CFT2. We replace the region $t<0$ by the Euclidean instanton namely the
half sphere, following Hartle-Hawking prescription (blue shaded region). In
the case of Schwinger-Keldysh setup, we restrict the Lorentzian path-integral
to the region $0\leq t\leq t_{0}$ and glue the same path-integral at
$t=t_{0}$.

###  2.2 Case 1:Entanglement entropy in Schwinger-Keldysh prescription

In this section, we aim to compute the holographic entanglement entropy
Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt  for a CFT living on dS2 spacetime,
following the Schwinger-Keldysh prescription (case 1) of the CFT on a time-
dependent background. Refer to Maldacena:2012xp  for calculations of
holographic entanglement entropy for CFTs on de Sitter spaces in higher
dimensions.

Specifically, we assume that the quantum state at $t=0$ is generated by an
Euclidean path integral on a semi-sphere using the Hartle-Hawking
prescription, as depicted in figure 7. To define the entanglement entropy
$S_{A}$, we take the subsystem $A$ as an interval with endpoints
$(t_{0},\phi_{1})$ and $(t_{0},\phi_{2})$, as shown in figure 7.

To derive the holographic entanglement entropy, we construct the gravity dual
by gluing two copies of AdS3, each truncated at a time $t=t_{0}$, following
the holographic Schwinger-Keldysh prescription Skenderis:2008dg ; Dong:2016hjy
. Hence, there is no need to consider the treatment of the conformal boundary
of the dS2 at $t=\infty$. Utilizing the geodesic length given in eq. (2.6), we
can obtain the holographic entanglement entropy, viz,

$S^{\rm con}_{A}=\frac{c}{3}\log\left[\frac{2\cosh
t_{0}\sin\frac{|\phi_{1}-\phi_{2}|}{2}}{\epsilon}\right]\,,$ (2.12)

where we adopt the cutoff surface specified by eq. (2.7). Notably, this
entanglement entropy exhibits a linear growth as $S_{A}\simeq\frac{c}{3}t_{0}$
at late times. We attribute this contribution to a connected geodesic that
connects two boundary points. Thus, in the Schwinger-Keldysh setup, this
result (2.12) provides the final expression for the holographic entanglement
entropy.

###  2.3 Case 2: Holographic pseudo entropy with a final state projection

Figure 8: The holographic dual of CFT2 living on dS2 with a projection is
shown as the gray shaded region. The angular direction of AdS3 is not shown in
this figure. The solid black curve represents the two dimensional dS spacetime
where CFT2 lives. Left: The projection is performed at the infinity time
$t_{\textrm{\tiny P}}\to\infty$ with a dS2 brane (red curve) parametrized by a
constant $\eta_{\ast}$. Right: The projection at a finite time
$t_{\textrm{\tiny P}}$.  Figure 9: The plot of $S^{\rm con}_{A}$ (blue) and
$S^{\rm dis}_{A}$ (purple) as a function of time $t$. We set
$\eta_{\ast}=3,\phi_{2}-\phi_{1}=\frac{\pi}{5},\epsilon=\frac{1}{100}$ for
this numerical plot.

We now consider the setup in which there is a final state projection with a
boundary state $|\rm{B}\rangle$ at future infinity $t=\infty$, and interpret
the area of the extremal surface, i.e., $S_{A}$ as the holographic pseudo
entropy Nakata:2021ubr . To construct the gravity dual of the holographic CFT2
with a projection, we insert an end-of-the-world brane (EOW) brane on which we
impose Neumann boundary condition, i.e.,

$K_{ij}-Kh_{ij}+Th_{ij}=0\,,\qquad\text{with}\qquad K=2T\,,$ (2.13)

where $K_{ij}$ is the extrinsic curvature of the brane and the constant $T$ is
the brane tension. Since the bulk is AdS3 spacetime, one can show that the
solutions of eq. (2.13) have to be maximally symmetric spacetime as well. More
explicitly, one can obtain (see e.g., Kawamoto:2023wzj )

$R_{ij}[h]=h_{ij}\left(\varepsilon\,T^{2}-1\right)\,,\qquad\text{with}\quad
R[h]=2(\varepsilon\,T^{2}-1)\,,$ (2.14)

for a timelike brane with $\varepsilon=+1$ and a spacelike brane with
$\varepsilon=-1$, respectively. In order words, the intrinsic geometry of the
brane is nothing but two dimensional Minkowski ($|T|=1$), de Sitter ($|T|>1$),
AdS ($|T|<1$) spacetime or hyperbolic space ($\varepsilon=-1$). The de Sitter
EOW brane in AdS/BCFT was introduced in Akal:2020wfl  and was applied to the
calculation of holographic pseudo entropy in Akal:2021dqt  in the context of
black hole final state proposal Horowitz:2003he .

With taking the intersection of the brane and the dS2 boundary at
$t\to\infty$, we can find that there are two types of solutions of the brane:
a timelike dS brane and a spacelike hyperbolic brane. From the dS slicing and
hyperbolic slicing of AdS3, one can easily read the corresponding brane
profiles in global AdS3 (as shown in figure 6), i.e.,

$\begin{split}\text{dS${}_{2}$
brane:}\qquad\cosh\rho\cos\tau&=\pm\cosh\eta_{\ast}\,,\\\ \text{H${}_{2}$
brane:}\qquad\cosh\rho\cos\tau&=\sin\mu_{\ast}\,,\\\ \end{split}$ (2.15)

where $\eta_{\ast},\mu_{\ast}$ is a constant along the brane and is determined
by the tension which is given by the boundary entropy of the boundary state.
Furthermore, the parameter $\eta_{\ast}$ and $\mu_{\ast}$ should be determined
by the boundary condition at $t=\infty$.

We begin with the case with a dS2 EOW brane. Let us first note that the trace
of the extrinsic curvature of EOW brane can be expressed as follows:

$K\big{|}_{\eta=\eta_{\ast}}=-\frac{\partial_{\eta}\sqrt{-\gamma}}{\sqrt{-\gamma}}\Big{|}_{\eta=\eta_{\ast}}=-\frac{2\cosh\eta_{\ast}}{\sinh\eta_{\ast}}\,,$
(2.16)

which indicates that the tension and curvature of the EOW brane are given by
$T=-\coth\eta_{\ast}<0$ and $R=\frac{2}{\sinh^{2}\eta_{\ast}}$, respectively.
The corresponding boundary entropy associated with the boundary state
$\ket{B}$ projected at $t\to\infty$ can be calculated as Akal:2021dqt

$S_{\rm
bdy}=\frac{c}{6}\log\sqrt{\frac{1-\mathcal{T}}{1+\mathcal{T}}}=\frac{c}{6}\log\sqrt{\frac{|\mathcal{T}|-1}{|\mathcal{T}|+1}}-i\,\frac{\pi
c}{12}=-\frac{c}{6}\eta_{\ast}-i\,\frac{\pi c}{12}\,.$ (2.17)

which decodes the information of the boundary condition at $t\to\infty$.

Due to the appearance of the EOW brane in the bulk, we must account for the
disconnected geodesic that connects a endpoint
$(t_{0},\eta_{\infty},\phi_{i})$ of boundary interval $A$ to a point
$(t_{\ast},\eta_{\ast},\phi_{\ast})$ on the de Sitter EOW brane. Thanks to the
rotation invariance, it is straightforward to get the extremal surface is
given by $\phi_{\ast}=\phi_{i}$. Furthermore, we can fix the coordinate values
of $t_{\ast}$ on the brane by ensuring that the geodesic length remains
stationary. For the de Sitter EOW brane, the disconnected contribution for
pseudo entropy can be derived from eq. (2.6) as follows:

$S^{\rm
dis}_{A}=\frac{c}{3}(\eta_{\infty}-\eta_{\ast})=\frac{c}{3}\log\frac{2}{\epsilon}-\frac{c}{3}\eta_{\ast}\,$
(2.18)

where the constant part is the real part of the boundary entropy defined in
eq. (2.17).

If we compare the connected contribution $S^{\rm con}_{A}$ and the
disconnected part $S^{\rm dis}_{A}$, we find that in the early time $S^{\rm
con}_{A}$ is favored and holographic pseudo entropy of the interval $A$ grows
linearly in time. However, $S^{d\rm dis}_{A}$ is always favored in the late
time and holographic pseudo entropy becomes a constant. This phase transition
is plotted in figure 9. At $\eta_{\ast}=0$, this phase transition happens when

$\cosh t_{0}\cdot\sin\frac{|\phi_{1}-\phi_{2}|}{2}=1.$ (2.19)

This describes the null geodesic in dS2 as sketched in figure 14, which is the
same as (3.10).

For the hyperbolic EOW brane, the disconnected PE is estimated from (2.6) as
follows:

$S^{\rm
dis}_{A}=\frac{c}{3}\log\frac{2}{\epsilon}+i\frac{c}{3}\left(\frac{\pi}{2}-\mu_{*}\right),$
(2.20)

where $\mu_{*}$ takes the range $-\frac{\pi}{2}\leq\mu_{*}\leq\frac{\pi}{2}$.

Before we go on, we would like to mention a possibility that this holographic
calculation of pseudo entropy may actually be interpreted as genuine
entanglement entropy. This is because the global AdS3 has the periodicity in
the time direction $\tau$. This may imply the periodicity for the time
evolution by $\pi$ of the boundary state $e^{-i\pi
H}|\text{B}\rangle\propto|\text{B}\rangle$. Even though this is suggested by
the classical geometry, we are not completely sure if this is true at the
quantum level which is dual to the full dynamics of the CFT. We leave this
issue for a future problem.

###  2.4 Holographic pseudo entropy under final state projection at
$t=t_{\textrm{\tiny P}}$

For later purpose, it is useful to consider a CFT on dS2 with a final state
projection at a finite time $t=t_{\textrm{\tiny P}}$. In the
AdS${}_{3}/$BCFT2, this is dual to inserting the EOW brane earlier as depicted
in the right panel of figure 8. It is obvious that $S_{A}$ vanishes at
$t=t_{\textrm{\tiny P}}$. Therefore the time evolution of $S_{A}$ looks like a
Page curve as sketched in the right panel of figure 10.

Since the brane profile is not completely covered by the dS slicing
coordinates, it is more convenient to work on the global coordinates (2.2). We
denote the points on the dS boundary as $(\tau_{0},\rho_{\infty},\phi_{0})$.
The dS boundary is thus given by

$\cosh\rho_{\infty}\cos\tau_{0}=\text{constant}=\cosh\eta_{\infty}\,,$ (2.21)

where $\eta_{\infty}$ determines the position of the dS boundary.
Corresponding, the point on the brane is referred to as
$(\tau_{*},\rho_{*},\phi_{*})$. Since the brane is still represents a two-
dimensional dS spacetime, we can parametrize the brane profile as

$\cosh\rho_{*}\cos\left(\tau_{*}+\tau_{\rm
shift}\right)=\text{constant}=\cosh\eta_{\ast}\,,$ (2.22)

where $\eta_{\ast}$ is related to the tension of the brane. Since we
parametrize the projection time is $t=t_{\textrm{\tiny P}}$ or
$T=T_{\textrm{\tiny P}}$ in terms of the time coordinate on the dS boundary,
we can find that this fix the brane parameter $\tau_{\rm shift}$ as

$\tau_{\rm shift}=\arccos\left(\frac{\cosh\eta_{\ast}\cos
T_{0}}{\cosh\eta_{\infty}}\right)-\tau{\textrm{\tiny
P}}=\arccos\left(\frac{\cosh\eta_{\ast}\cos
T_{0}}{\cosh\eta_{\infty}}\right)-\arctan\left(\tanh\eta_{\infty}\tan
T_{\textrm{\tiny P}}\right)\,,$ (2.23)

with using the coordinate transformations for the points on dS boundary as

$\begin{cases}\cosh\rho=\sqrt{\cosh^{2}\eta+\sinh^{2}\eta\tan^{2}T}\,,\\\
\tan\tau=\tanh\eta\tan T\,.\\\ \end{cases}$ (2.24)

On the other hand, the geodesic distance $D_{0b}$ from the dS boundary to dS
brane is derived as

$\cosh
D_{0b}=\cos(\tau_{0}-\tau_{*})\cosh\rho_{\infty}\cosh\rho_{*}-\cos(\phi_{0}-\phi_{*})\sinh\rho_{\infty}\sinh\rho_{*}.$
(2.25)

Extremization over the spatial direction simply results in
$\phi_{0}=\phi_{*}$, which is expected by symmetries. Furthermore, we need to
find that maximal value over the timelike direction by solving

$\frac{\partial
D_{0b}}{\partial\tau_{*}}=0\,,\quad\text{with}\quad\rho_{*}=\rho_{*}(\tau_{*})\,,$
(2.26)

which reduces to

$\cosh\eta_{\ast}\sinh\rho_{\infty}\tan\left(\tau_{*}+\tau_{\rm
shift}\right)-\frac{\sin\left(\tau_{*}+\tau_{\rm
shift}\right)}{\cos\left(\tau_{0}+\tau_{\rm
shift}\right)}\cosh\rho_{\infty}\sqrt{\cosh^{2}\eta_{\ast}-\cos^{2}\left(\tau_{0}+\tau_{\rm
shift}\right)}=0\,.$ (2.27)

The advantage of working on global coordinates is obvious in the above
extremization equation. After straightforward algebras, one can derive the
solutions as

$\begin{split}\cos\left(\tau_{*}+\tau_{\rm
shift}\right)&=\cosh\eta_{\ast}\sqrt{\frac{\cosh^{2}\rho_{\infty}\sin^{2}(\tau_{0}+\tau_{\rm
shift})-\sinh^{2}\rho_{\infty}}{\cosh^{2}\rho_{\infty}\sin^{2}(\tau_{0}+\tau_{\rm
shift})-\sinh^{2}\rho_{\infty}\cosh^{2}\eta_{\ast}}}\,,\\\
\cosh\rho_{*}&=\sqrt{\frac{\cosh^{2}\rho_{\infty}\sin^{2}(\tau_{0}+\tau_{\rm
shift})-\sinh^{2}\rho_{\infty}\cosh^{2}\eta_{\ast}}{\cosh^{2}\rho_{\infty}\sin^{2}(\tau_{0}+\tau_{\rm
shift})-\sinh^{2}\rho_{\infty}}}\,.\\\ \end{split}$ (2.28)

Substituting the above solutions to the expression for the distance of
geodesics, we can obtain the distance of the extremal geodesic between the dS
boundary and brane, i.e., the area of the disconnected HRT surface.

We are interested in the case where the dS boundary is taken as the cut-off
surface located at $\eta_{\infty}\sim\frac{2}{\epsilon}$. In this limit, we
can rewrite the global coordinates on the boundary as

$\begin{split}\tau_{0}&=T_{0}\,,\qquad\rho_{\infty}\approx\log\left(\frac{2}{\epsilon\cos
T_{0}}\right)\,,\qquad\tau_{\rm shift}=\frac{\pi}{2}-T_{\textrm{\tiny
P}}\,.\\\ \end{split}$ (2.29)

The extremal point on the brane reduces to

$\begin{split}\sin\left(T_{\textrm{\tiny
P}}-\tau_{*}\right)&=\cosh\eta_{\ast}\sqrt{\frac{\sin^{2}(T_{\textrm{\tiny
P}}-T_{0})}{\cosh^{2}\eta_{\ast}-\cos^{2}(T_{\textrm{\tiny P}}-T_{0})}}\,,\\\
\cosh\rho_{*}&=\sqrt{1+\frac{\sinh^{2}\eta_{\ast}}{\sin^{2}(T_{\textrm{\tiny
P}}-T_{0})}}\,,\\\ \end{split}$ (2.30)

and the corresponding geodesic distance is given by

$D_{0b}\approx\log\left(\frac{2}{\epsilon}\frac{\cos(T_{0}+\tau_{\rm
shift})}{\cos T_{0}}e^{-\eta_{\ast}}\right)\,.$ (2.31)

Thus, the holographic entanglement entropy from the disconnected HRT surface
is recast as

$S^{\rm dis}_{A}=\frac{2D_{0b}}{4G_{\textrm{\tiny
N}}}=\frac{c}{3}\log\left(\frac{2}{\epsilon}\frac{\sin\left(T_{\textrm{\tiny
P}}-T_{0}\right)}{\cos T_{0}}\right)-\frac{c}{3}\eta_{\ast}\,.$ (2.32)

In terms of the de Sitter time $t$, we finally find the following expression:

$S^{\rm dis}_{A}=\frac{c}{3}\log\left[\frac{2}{\epsilon}\cdot\frac{\sinh
t_{\textrm{\tiny P}}-\sinh t_{0}}{\cosh t_{\textrm{\tiny
P}}}\right]-\frac{c}{3}\eta_{\ast},$ (2.33)

where we have used $\sinh t_{\textrm{\tiny P}}=\tan T_{\textrm{\tiny P}}$. The
holographic pseudo entropy in the presence of final state projection is given
by the smaller one among the two:
$S_{A}=\mbox{min}[S^{\mathrm{con}}_{A},S^{\mathrm{dis}}_{A}]$.

###  2.5 Entanglement/Pseudo entropy of CFT2 on dS2

Figure 10: Time evolution of holographic entanglement entropy (denoted by the
black curve) with respect to the global time $t_{0}$. The projection time is
chosen as $t_{\textrm{\tiny P}}=6$ with $\eta_{\ast}=3$. We set
$\phi_{2}-\phi_{1}=\frac{\pi}{5},\epsilon=\frac{1}{100}$ for this numerical
plot.

It is intriguing to compare previous holographic results of entanglement
entropy and pseudo entropy with that derived in 2d CFT. Indeed, as in the
general replica method for 2d CFTs Holzhey:1994we ; Calabrese:2004eu , we can
easily compute the two-point function of twist operators of local CFT on the
$\mathrm{dS}_{2}$ by considering the Weyl scaling:

$\langle\sigma(t_{0},\phi_{1})\overline{\sigma}(t_{0},\phi_{2})\rangle_{\mathrm{dS}_{2}}=(\cos{T_{0}})^{2\Delta_{n}}\langle\sigma(T_{0},\phi_{1})\overline{\sigma}(T_{0},\phi_{2})\rangle_{\mathrm{cylinder}}\,.$
(2.34)

Here we have define the conformal coordinates $(T,\phi)$ by

$\begin{split}ds^{2}_{\mathrm{dS}_{2}}&=-dt^{2}+\cosh^{2}{t}d\phi^{2}=\frac{1}{\cos^{2}{T}}\left(-dT^{2}+d{\phi}^{2}\right)\,\\\
\end{split}$ (2.35)

with the transformation between the global time and conformal time given by
$\cosh{t}=\frac{1}{\cos{T}}$. The Euclidean part of the Hartle-Hawking state
is described by a half sphere whose metric reads

$ds^{2}_{\mathrm{HH}}=dt_{\textrm{\tiny E}}^{2}+\cos^{2}{t_{\textrm{\tiny
E}}}d\phi^{2}=\frac{1}{\cosh^{2}T_{\textrm{\tiny E}}}\left(dT_{\textrm{\tiny
E}}^{2}+d{\phi}^{2}\right)\,,$ (2.36)

where the Euclidean conformal time $T_{\textrm{\tiny E}}$ is defined by
$\cos{t_{\textrm{\tiny E}}}=\frac{1}{\cosh{T_{\textrm{\tiny E}}}}$ with
$-\frac{\pi}{2}\leq t_{\textrm{\tiny E}}\leq 0,-\infty\leq T_{\textrm{\tiny
E}}\leq 0$.

Figure 11: The conformal maps used for computing the pseudo entropy. The
projection is performed at a fixed time $t_{\textrm{\tiny P}}$ which is
indicated by the red circle.

#### Case 1: Schwinger-Keldysh description without EOW

First of all, it is known that the entanglement entropy for an interval in the
cylinder is shown as

$S_{A}^{\mathrm{cylinder}}=\frac{c}{3}\log{\left(\frac{2}{\epsilon}\sin{\frac{|\phi_{1}-\phi_{2}|}{2}}\right)}\,.$
(2.37)

where we have used the fact that the periodicity of the cylinder is $2\pi$.
Noting the contribution from Weyl scaling, we can obtain

$S_{A}^{\mathrm{dS}_{2}}=\frac{c}{3}\log{\cosh{t_{0}}}+\frac{c}{3}\log{\left(\frac{2}{\epsilon}\sin{\frac{|\phi_{1}-\phi_{2}|}{2}}\right)}\,,$
(2.38)

which reproduces the holographic result shown in eq. (2.12).

#### Case 2: Final projection at $t=t_{\textrm{\tiny P}}$

To compute the pseudo entropy of CFT2 living on $\mathrm{dS}_{2}$, we follow
the recipe illustrated in figure 11. Let us begin with the following two-point
function and its Weyl scaling:

$\begin{split}&\bra{\rm
B}e^{iH(T_{p}-T)}\sigma(T_{0},\phi_{1})\overline{\sigma}(T_{0},\phi_{2})\ket{\rm
HH}_{\mathrm{dS}_{2}}\\\ =&(\cos{T_{0}})^{2\Delta_{n}}\bra{\rm
B}e^{-iH(T_{\textrm{\tiny
P}}-T)}\sigma(T_{0},\phi_{1})\overline{\sigma}(T_{0},\phi_{2})\ket{0}_{\mathrm{cylinder}}\,,\end{split}$
(2.39)

where $|\rm B\rangle$ is the boundary state (Cardy state) in the boundary
conformal field theory (BCFT) Cardy:2004hm  and $|\rm HH\rangle$ denotes the
Hartle-Hawking state of dS2. We can first focus on the transition matrix,
i.e.,

$\bra{\rm B}e^{-iH(T_{\textrm{\tiny
P}}-T)}\sigma(T_{0},\phi_{1})\overline{\sigma}(T_{0},\phi_{2})\ket{0}_{\mathrm{cylinder}}\,,$
(2.40)

with ignoring the Weyl factor. The corresponding cylinder for evaluating the
transition matrix is composed of the infinitely long Euclidean cylinder
$-\infty\leq T_{\textrm{\tiny E}}\leq 0$ and also a finite Lorentzian cylinder
with $0\leq T\leq T_{\textrm{\tiny P}}$. By Wick rotation, we consider the
Euclidean cylinder $-\infty\leq T_{\textrm{\tiny E}}\leq T_{\textrm{\tiny
P}}^{E}=iT_{\textrm{\tiny P}}$. With denoting the complex coordinate for the
cylinder as

$w=T_{\textrm{\tiny E}}+i\phi\,,$ (2.41)

we can map the Euclidean cylinder to the upper half plane. The explicit
conformal map is given by

$\frac{e^{w}}{e^{T_{\textrm{\tiny P}}^{E}}}=\frac{z-i}{z+i}\Leftrightarrow
z=f(w)=-i\frac{e^{w-T_{\textrm{\tiny P}}^{E}}+1}{e^{w-T_{\textrm{\tiny
P}}^{E}}-1}\,.$ (2.42)

See the figure 11 for the illustration.

By assuming the mirror trick and the factorization, it is straightforward to
evaluate the transition (2.40). Similar to the connected and disconnected
geodesics, one can also obtain two distinct contributions, i.e.,

  1. 1.

Connected part:

$S^{\mathrm{con,cyl}}_{A}=\frac{c}{6}\log\frac{|f(w_{1})-f(w_{2})|^{2}}{\epsilon^{2}|f^{\prime}(w_{1})||f^{\prime}(w_{2})|}.$
(2.43)

After substituting

$w_{1}=i(T_{0}+\phi_{1}),w_{1}^{*}=i(T_{0}-\phi_{1}),w_{2}=i(T_{0}+\phi_{2}),w_{2}^{*}=i(T_{0}-\phi_{2})\,,$
(2.44)

we obtain the same formula (2.38) which it was derived in holographic
spacetime for Case 1.

  2. 2.

Disconnected part:

$S^{\mathrm{dis,cyl}}_{A}=\frac{c}{6}\log\frac{|f(w_{1})-\bar{f}(\bar{w}_{1})||f(w_{2})-\bar{f}(\bar{w}_{2})|}{\epsilon^{2}|f^{\prime}(w_{1})||f^{\prime}(w_{2})|}+2S_{\mathrm{bdy}}.$
(2.45)

Again, by substituting eq. (2.44), we can derive the contribution from the
disconnected part, viz,

$S^{\mathrm{dis,cyl}}_{A}=\frac{c}{6}\log{\frac{-4}{\epsilon^{2}}\sin^{2}{(T_{p}-T_{0})}}+2S_{\mathrm{bdy}}=\frac{c}{3}\log\left(\frac{2}{\epsilon}\sin{(T_{p}-T_{0})}\right)+\frac{\pi
i}{12}c+2S_{\mathrm{bdy}}\,.$ (2.46)

Finally, we need to consider the extra contribution from Weyl scaling and
rewrite the final answer as

$S_{A}^{\mathrm{dis,dS}}=\frac{c}{3}\log\left(\frac{2}{\epsilon}\frac{\sin{(T_{p}-T_{0})}}{\cos{T_{0}}}\right)+\frac{\pi
i}{12}c+2S_{\mathrm{bdy}}\,.$ (2.47)

Combining the two distinct results, we can also find that the pseudo entropy
present the same behavior as shown in figure 10. Especially, when the post-
selection is fixed at the future infinity, i.e., $T_{p}=\frac{\pi}{2}$, the
disconnected part reduces to a constant

$S_{A}^{\mathrm{dis,dS}}=\frac{c}{3}\log\left(\frac{2}{\epsilon}\right)+\frac{\pi
i}{12}c+2S_{\mathrm{bdy}}\,,$ (2.48)

As a summary, we find that pseudo entropy derived from boundary 2d CFT agree
with the holographic results for $S^{dis}_{A}$ (2.18) and (2.33) via the
prescription (2.17).

###  2.6 Holographic entanglement in higher dimensions

In this section, we study the holographic entanglement (pseudo) entropy in
higher dimensional setups, which can be found from the area of extremal
surfaces in the $(d+1)$-dim AdS space. The de Sitter sliced metric is given by

$ds^{2}=d\eta^{2}+\sinh^{2}\eta(-dt^{2}+\cosh^{2}t(d\phi^{2}+\sin^{2}\phi
d\Omega_{d-2})).$ (2.49)

#### Case 1: Schwinger-Keldysh without EOW

Figure 12: Holographic entanglement entropy $S^{\rm con}_{A}$ (2.56) and
$S^{\rm dis}_{A}$ (2.61) from Schwinger-Keldysh prescription without EOW
brane. We choose $d=3,\phi_{0}=\frac{\pi}{2},\eta_{\infty}=3$ and
$\eta_{*}=1$for both plots.

First let us evaluate the connected entropy $S^{con}_{A}$. We consider the
global AdS coordinates

$ds^{2}=d\rho^{2}-\cosh^{2}\rho d\tau^{2}+\sinh^{2}\rho(d\phi^{2}+\sin^{2}\phi
d\Omega_{d-2}^{2})$ (2.50)

and take the subsystem $A$ as

$0\leq\phi\leq\phi_{0},\quad\tau=\tau_{0},\quad\rho=\rho_{\infty}.$ (2.51)

In this coordinate, the extremal surface is labeled by $\phi=\phi(\rho)$. The
holographic entanglement entropy is the area of this surface divided by
$4G_{N}$

$S^{\rm con}_{A}=\frac{\mathrm{Vol}(\mathrm{S}^{d-2})}{4G_{N}}\int
d\rho\sqrt{1+\sinh^{2}\rho\phi^{{}^{\prime}2}}(\sinh\rho\sin\phi)^{d-2},$
(2.52)

where we set $\phi^{\prime}=\frac{\partial\phi}{\partial\rho}$. By taking the
variation, we obtain the Euler-Lagrange equation

$\phi^{\prime\prime}-\frac{d-2}{\tan\phi\sinh^{2}\rho}+\frac{d\phi^{\prime}}{\tanh\rho}-\frac{(d-2)\phi^{{}^{\prime}2}}{\tan\phi}+(d-1)\sinh\rho\cosh\rho\phi^{{}^{\prime}3}=0.$
(2.53)

We find a suitable solution $\tanh\rho\cos\phi=C,$ where the $C$ is
integration constant. We impose the boundary condition (2.51) and obtain the
solution

$\tanh\rho\cos\phi=\tanh\rho_{\infty}\cos\phi_{0}.$ (2.54)

Then, the holographic entanglement entropy reads

$S^{\rm
con}_{A}=\frac{\mathrm{Vol}(\mathrm{S}^{d-2})}{4G_{N}}\int_{1}^{\frac{\cosh\rho_{\infty}}{\cosh\rho_{\rm
min}}}du(u^{2}-1)^{\frac{d-3}{2}}\,,$ (2.55)

where $\rho_{\rm min}$ is defined by $\tanh\rho_{\rm
min}=\tanh\rho_{\infty}\cos\phi_{0}$. In particular, $S^{\rm con}_{A}$ for
$d=3$ reduces to

$\begin{split}S^{\rm
con}_{A}=\frac{\pi}{2G_{N}}\left(\frac{\cosh\rho_{\infty}}{\cosh\rho_{\rm
min}}-1\right)&=\frac{\pi}{2G_{N}}\left(\sqrt{\cosh^{2}t_{0}\sin^{2}\phi_{0}\sinh^{2}\eta_{\infty}+1}-1\right)\\\
&\approx\frac{\pi}{2G_{N}}\left(\frac{\cosh
t_{0}\sin\phi_{0}}{\epsilon}-1\right)\,,\\\ \end{split}$ (2.56)

We plot this as a function of $t_{0}$ and $\phi_{0}$ in figure 12.

#### Case 2: Final projection at $t=\infty$

Next, let us evaluate the disconnected extremal surfaces stretching between
the dS boundary and the dS EOW brane, which contributes to the holographic
pseudo entropy. Particularly, we consider the extremal surface connecting the
two points between $(\eta_{\infty},t_{0},\phi_{0},\Omega_{0})$ and
$(\eta_{*},t_{*},\phi_{*},\Omega_{0})$ and then maximize the area by taking
variation with respect to $t_{*}$. In the AdS3 case, the extremal surface
satisfies $t=$const and $\phi$=const. However, in higher-dimensional cases
this is no more true. Suppose that $\phi_{*}=\frac{\pi}{2}$ along the extremal
surface111This can be justified only in the $\phi=\frac{\pi}{2}$ case since we
have a $\mathbf{Z}_{2}$ symmetry $\theta\rightarrow\frac{\pi}{2}-\phi$. In
general the extremal surface does depend on $(\phi,t,\eta)$.. Then, its area
is given by

$S^{\rm dis}_{A}=\frac{\mathrm{Vol}(\mathrm{S}^{d-2})}{4G_{N}}\int
d\eta\sqrt{1-\sinh^{2}\eta t^{{}^{\prime}2}}(\sinh\eta\cosh t)^{d-2}.$ (2.57)

The Euler-Lagrange equation reads

$\frac{(d-2)\tanh
t}{\sinh^{3}\eta}+\frac{dt^{\prime}}{\tanh\eta\sinh\eta}-\frac{(d-2)\tanh
tt^{{}^{\prime}2}}{\sinh\eta}-(d-1)\cosh\eta
t^{{}^{\prime}3}+\frac{t^{\prime\prime}}{\sinh\eta}=0.$ (2.58)

A particular solution, which is the one at $\tau=$const. in the global AdS,
reads

$\tanh\eta\sinh t=\tanh\eta_{\infty}\sinh t_{0}=\tanh\eta_{*}\sinh t_{*}.$
(2.59)

We expect more general solutions and we need to maximize the areas of such
solutions with respect to the end points on the EOW brane. However, this is
highly complicated and we will focus on the above simple solution, which gives
the minimal value of S${}_{A}^{\rm dis}$.

By substituting this into the functional we obtain

$\displaystyle S_{A}^{\rm dis}$
$\displaystyle=\frac{\mathrm{Vol}(\mathrm{S}^{d-2})}{4G_{N}}\int
d\eta\sqrt{1-\sinh^{2}\eta t^{{}^{\prime}2}}(\sinh\eta\cosh t)^{d-2}$
$\displaystyle=\frac{\mathrm{Vol}(\mathrm{S}^{d-2})}{4G_{N}}\int^{\sqrt{1+\cosh^{2}t_{0}\sinh^{2}\eta_{\infty}}}_{\sqrt{1+\cosh^{2}t_{*}\sinh^{2}\eta_{*}}}dv(v^{2}-1)^{\frac{d-3}{2}}.$
(2.60)

Especially, in the $d=3$ case we find

$S_{A}^{\rm
dis}=\frac{\pi\sqrt{\tanh^{2}\eta_{\infty}\sinh^{2}t_{0}+1}}{2G_{N}}(\cosh\eta_{\infty}-\cosh\eta_{*})\simeq\frac{\pi}{2G_{N}}\cosh
t_{0}\left(\frac{1}{\epsilon}-\cosh\eta_{\ast}\right)\,,$ (2.61)

The results are plotted in the figure 12. Unlike the AdS3 case, both
$S_{A}^{\rm con}$ and $S_{A}^{\rm dis}$ diverge exponentially as a function of
$t_{0}$. Comparing two expressions in eqs. (2.56) and (2.61), we always have

$S_{A}^{\rm dis}\leq S_{A}^{\rm con}\,,$ (2.62)

due to $\cosh X\geq 1$.

#### More on Case 2: Final projection at $t=t_{\textrm{\tiny P}}$

Figure 13: The time evolution of $S^{\rm con}_{A}$ (2.56) and $S^{\rm
dis}_{A}$ (2.64). We take $\eta_{\infty}=3$, $\phi_{0}=\frac{\pi}{2}$ ,
$\tau_{\rm shift}=\frac{\pi}{6}$ and $\cosh\eta_{c}=0.2$. At early times, the
connected part is dominated. After the phase transition, the disconnected part
is dominated and approaches zero at the projection time
$t_{0}=t_{\textrm{\tiny P}}$ , where the two branes intersect.

Next, we consider the final projection at finite time. For this, we shift the
projection brane along the global time as in the section 2.4. If we shift the
brane backward in $\tau_{\rm shift}$, the projection brane is defined by
$\cosh\rho\cos(\tau+\tau_{\rm shift})=\cosh\eta_{c},$ where $\eta_{c}$ is
related with the tension of the dS brane. The special solution to the extremal
surface condition is already obtained in the previous section as (2.59) for
$\phi_{0}=\frac{\pi}{2}$. Therefore the intersection of the extremal surface
and the projection brane defines the end point of the surface
$\tanh\eta_{*}\sinh t_{*}=\tanh\eta_{\infty}\sinh t_{0}$ and
$\cosh\rho_{*}\cos(\tau_{*}+\tau_{\rm shift})=\cosh\eta_{c}.$

After short algebra we obtain

$\cosh\eta_{*}=\frac{\cosh\eta_{c}}{\cos\tau_{\rm
shift}-\tanh\eta_{\infty}\sinh t_{0}\sin\tau_{\rm shift}},$ (2.63)

which gives for $d=3$

$S^{\rm
dis}_{A}=\frac{\pi\sqrt{\tanh^{2}\eta_{\infty}\sinh^{2}t_{0}+1}}{2G_{N}}\left(\cosh\eta_{\infty}-\frac{\cosh\eta_{c}}{\cos\tau_{\rm
shift}-\tanh\eta_{\infty}\sinh t_{0}\sin\tau_{\rm shift}}\right)\,.$ (2.64)

The figure 13 shows the plots. As in the AdS3 case, the disconnected entropy
reaches zero at the projection time $t_{\textrm{\tiny P}}$.

##  3 Holography for a half dS without EOW brane (Case 1)

Now we would like to move on to the our main target: holography with a
positive cosmological constant. In order to interpret the holography in de
Sitter space as in the standard framework where the bulk gravity is dual to a
quantum system on its time-like boundary, we focus on a half of de Sitter
space defined by restricting the global dSd+1 given by the metric (1.1) and
(1.2) to the region

$\displaystyle 0\leq\theta\leq\theta_{0},$ (3.1)

as depicted in figure 2 and figure 3. The standard idea of holographic
principle predicts that the bulk gravity on the half dSd+1 (3.1) is dual to a
certain quantum system on its boundary, namely, dSd at $\theta=\theta_{0}$,
given by the metric (1.4). We assume the range
$0\leq\theta_{0}\leq\frac{\pi}{2}$. If we choose $\theta_{0}=\frac{\pi}{2}$ in
particular, it is exactly a half of the original dS.

Below we will study how the holography for a half dS looks like by analyzing
the holographic entanglement entropy. For simplicity we will mainly focus on
the three dimensional space i.e. dS3. In this case, the holographic
entanglement entropy $S_{A}$, defined for an interval $A$, can be computed
from the geodesic length which connects two end points of $A$. This
consideration also suggests the range $0\leq\theta_{0}\leq\frac{\pi}{2}$. This
is because the geodesics which compute the holographic entanglement entropy
are not included in the half dS if $\theta_{0}\geq\frac{\pi}{2}$, even if the
subsystem is very small.

###  3.1 Geodesic Length in dS3

Figure 14: The Penrose diagram of dS2 boundary. The colorful curves denote
various spacelike geodesics on dS2 where the null geodesic is presented by the
green dashed curves.

To prepare for our analysis of holographic entanglement entropy $S_{A}$, let
us see the behavior of the geodesics in a dS3 whose metric reads

$\displaystyle ds^{2}=-dt^{2}+\cosh^{2}t(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (3.2)

If we choose two points $P_{1}$ and $P_{2}$ on dS3:

$P_{1}=(t_{1},\theta_{1},\phi_{1})\,,\qquad
P_{2}=(t_{2},\theta_{2},\phi_{2})\,,$ (3.3)

then the geodesic distance between $P_{1}$ and $P_{2}$, denoted by $D_{12}$,
can be found as

$D_{12}=\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos(\phi_{1}-\phi_{2})\right)\cosh
t_{1}\cosh t_{2}-\sinh t_{1}\sinh t_{2}\,.$ (3.4)

Especially for the geodesic which connected two points $\phi=\phi_{1}$ and
$\phi=\phi_{2}$ at the boundary $\theta=\theta_{0}$ and at the same time
$t=t_{0}$, we have:

$\displaystyle\cos
D_{12}=\left(\cos^{2}\theta_{0}+\sin^{2}\theta_{0}\cos(\phi_{1}-\phi_{2})\right)\cosh^{2}t_{0}-\sinh^{2}t_{0}.$
(3.5)

In order for a space-like geodesic to be present between the two points, the
following condition should be satisfied:

$\displaystyle\cos^{2}\theta_{0}+\sin^{2}\theta_{0}\cos(\phi_{1}-\phi_{2})\geq
1-\frac{2}{\cosh^{2}t_{0}}.$ (3.6)

The boundary of this boundary looks like a past light-cone of the future
infinity as depicted in figure 15. We write the value of
$\Delta\phi=\phi_{1}-\phi_{2}$ which saturates this bound as
$\Delta\phi_{\mathrm{max}}$. The space-like geodesic exists only when
$\Delta\phi\leq\Delta\phi_{\mathrm{max}}$. For explicit construction of such a
geodesic refer to the appendix A.

Figure 15: Plots of the boundaries beyond which the space-like geodesic does
not exist for $\theta=\frac{\pi}{2}$ (left), $\theta_{0}=\frac{\pi}{4}$
(middle) and $\theta_{0}=\frac{\pi}{8}$ (right). The hoirzontal and vertical
coordinates are $\phi$ and $T$ in the range $-\pi/2\leq\phi\leq\pi/2$ and
$0\leq T\leq\pi/2$, respectively. We consider the spacelike geodesic which
connects two points with $\phi_{1}=-\phi_{2}$ for each time. The blue curves
are the boundaries for the dS3 geodesic length $D_{12}$, while the red ones
are those for the dS2 metric (1.4). Both of which coincide only when
$\theta=\frac{\pi}{2}$.

Figure 16: The behaviors of the length $D_{12}$ of space-like geodesics. The
left panel shows $D_{12}$ as a function of $|\phi_{1}-\phi_{2}|$ for
$\theta_{0}=\frac{\pi}{8}$ at various fixed times $t_{0}$. Notice that at the
latter two times, the space-like geodesic ceases to excite for a large values
of $\Delta\phi=|\phi_{1}-\phi_{2}|$. The right panel shows $D_{12}$ as a
function of $|\phi_{1}-\phi_{2}|$ for the maximal half dS3i.e.,
$\theta_{0}=\frac{\pi}{2}$ at fixed times. The horizontal red dotted line
describes $D_{12}=\pi$ and vertical dashed lines are referred as the critical
size $\Delta\phi_{\mathrm{max}}$.

It is clear that for two different points $\phi_{1}\neq\phi_{2}$, this
inequality gets violated at enough later time. On the other hand, if $t_{0}$
and $\theta_{0}$ satisfy

$\displaystyle\cos(2\theta_{0})\geq 1-\frac{2}{\cosh^{2}t_{0}},$ (3.7)

then for any values of $\phi_{1}$ and $\phi_{2}$, the space-like geodesic
which connects the two points does exist. The geodesic length $D_{12}$ as a
function of $\Delta\phi$ at $\theta_{0}=\frac{\pi}{8}$ and
$\theta_{0}=\frac{\pi}{2}$ is depicted in figure 16.

When the bound (3.6) is violated, the geodesic length gets complex valued such
that its real part is $\pi$:

$\displaystyle
D_{12}=\pi+i\text{arccosh}\left[\sinh^{2}t-\left(\cos^{2}\theta_{0}+\sin^{2}\theta_{0}\cos(\phi_{1}-\phi_{2})\right)\cosh^{2}t_{0}\right].$
(3.8)

The imaginary contribution comes from the time-like geodesic and the final
real part $\pi$ does from the geodesic in an Euclidean space i.e., a semi-
sphere (see Hikida:2022ltr ).

Consider the maximal case $\theta_{0}=\frac{\pi}{2}$. In this case, due to the
Z2 symmetry $\theta\to\pi-\theta$, the geodesics which connect two points on
the boundary $\theta=\theta_{0}=\frac{\pi}{2}$ are all within the boundary
dS2, as shown in figure 14. At $t_{0}=0$, there is always a geodesic which
connects two points on the boundary and the geodesic length is simply found to
be

$\displaystyle D_{12}=\Delta\phi.$ (3.9)

However for $t_{0}\neq 0$, a space-like geodesic does not exist for
$\Delta\phi=|\phi_{1}-\phi_{2}|>\Delta\phi_{\mathrm{max}}$, where the bound is
explicitly given by

$\Delta\phi_{\mathrm{max}}=2\left(\frac{\pi}{2}-\arccos\left(\frac{1}{\cosh
t_{0}}\right)\right)=\pi-2\arctan\left[\sinh t_{0}\right]\,.$ (3.10)

This bound coincides with the boundary of the past light cone of a point at
$t=\infty$ and is identical to the condition (2.19) found in our previous
analysis of AdS/CFT, shown in the left panel of figure 15. These behaviors of
geodesic length are plotted in the right panel of figure 16.

The profile of the geodesic for $\Delta\phi\leq\Delta\phi_{\mathrm{max}}$ is
sketched in the left panel of figure 17. When
$\Delta\phi>\Delta\phi_{\mathrm{max}}$, the geodesic length gets complex
valued and can be interpreted as the union of time-like geodesic and space-
like one as depicted in the right panel of figure 17. Note that this
transition at $\Delta\phi=\Delta\phi_{\mathrm{max}}$ is peculiar to our dS
setup and can not be seen in the AdS setups.

Figure 17: Sketches of connected geodesics $\Gamma_{A}$ in the boundary dS2
for $\theta=\frac{\pi}{2}$. In the left panel, we present the space-like one
with $\Delta\phi\leq\Delta\phi_{\mathrm{max}}$. In the right panel, we plotted
what geodesic length computes for $\Delta\phi>\Delta\phi_{\mathrm{max}}$. It
is a union of time-like and space-like geodesic.

###  3.2 Holographic Entanglement Entropy and Violation of Subadditivity

Now let us consider the calculation of the holographic entanglement entropy
(1.3) in Case 1 (figure 4), namely Schwinger-Keldysh dS geometry, where there
is no EOW brane. In this case, $S_{A}$ for an interval $A$, can be computed
from the connected geodesic as

$\displaystyle S_{A}=\frac{D_{12}}{4G_{N}},$ (3.11)

whose length was already studied in the previous subsection.

Consider the holographic entanglement entropy at a fixed time $t=t_{0}$ for an
interval $A$ defined by $\phi_{1}\leq\phi\leq\phi_{2}$. For a generic choice
of the boundary $\theta_{0}$, the holographic entanglement entropy computed as
the geodesic length, takes real and positive values only for
$\Delta\phi\leq\Delta\phi_{\mathrm{max}}$, as depicted in figure 16. For the
maximal value $\Delta\phi=\Delta\phi_{\mathrm{max}}$ we find $D_{12}=\pi$,
which is a half of the length of the de Sitter horizon and thus $S_{A}$
becomes a half of de Sitter entropy $S_{A}=\frac{1}{2}S_{\rm dS}$.

In terms of the time evolution, for an earlier time $0\leq t\leq
t_{\mathrm{max}}$, $S_{A}$ is well-defined for any value of $\Delta\phi$,
where $t_{\mathrm{max}}$ is the time when (3.7) is saturated i.e.

$t_{\mathrm{max}}=\text{arccosh}\left(\frac{1}{\sin\theta_{0}}\right)=\log\left(\cot\left(\frac{\theta_{0}}{2}\right)\right)\,.$
(3.12)

However, for $t>t_{\mathrm{max}}$, the behavior of holographic entanglement
looks confusing partly because it takes a complex value for
$\Delta\phi>\Delta\phi_{\mathrm{max}}$ and also because it is a convex
function of $\Delta\phi$ even for $\Delta\phi\leq\Delta\phi_{\mathrm{max}}$ as
can be seen from figure 16. The latter fact shows the violation of (strong)
subadditivity. Indeed, the second derivative of the geodesic length

$\frac{d^{2}D_{12}}{d\Delta\phi d\Delta\phi}=\frac{\sqrt{2}\cosh
t\sin\theta_{0}\sin\left(\frac{\theta_{0}}{2}\right)\left(\cosh^{2}t\sin^{2}\theta_{0}-1\right)}{\left(1+\cos
D_{12}\right)^{3/2}},$ (3.13)

becomes positive when $t>t_{\mathrm{max}}$. In terms of the conformal time $T$
($\cosh t\equiv\frac{1}{\cos T}$), the critical time is rewritten as

$T_{\mathrm{crt}}=\frac{\pi}{2}-\theta_{0}\,,$ (3.14)

which is nothing but the intersection between the boundary $\theta=\theta_{0}$
and the cosmological horizon located at $T=\frac{\pi}{2}\pm\theta$.

To see why the convex entropy function violates the subadditivity, first note
that $S_{A}$ is a function of the length $y=L(A)$ of the interval $A$, owing
to the translational invariance. It is obvious from the convex nature
$\partial^{2}_{y}S_{A}(y)>0$ that we have $S_{A}(2y)>2S_{A}(y)$, which
violates the subadditivity $S_{A}+S_{B}\geq S_{AB}$. Of course, this shows
that the strong subadditivity, which is expressed as $S_{AB}+S_{BC}\geq
S_{ABC}+S_{B}$, is broken. Refer to figure 18 for plots of the regions where
the subadditivity is satisfied, whose further interpretation will be given
later. This behavior is a complete contrast to that of AdS/CFT, where the
strong subadditivity is always satisfied Headrick:2007km ; Wall:2012uf .

Figure 18: We plotted the behavior of subadditivity violation for
$\theta_{0}=\frac{\pi}{4}$ (left) and $\theta_{0}=\frac{\pi}{2}-0.01$ (right).
We consider the subsystem $A$ and $B$ such that $A$ connects the two points
$(\phi,T)=(0,T_{0})$ and $(1/2,T_{0}+\Delta T)$ and $B$ does the two points
$(1/2,T_{0}+\Delta T)$ and $(1,T_{0})$. $T$ is introduced by $\cos
T=\frac{1}{\cosh t}$. The horizontal and vertical coordinates describe $T_{0}$
and $\Delta T$. We colored the region where the subadditivity holds i.e.
$S_{A}+S_{B}-S_{AB}\geq 0$.

###  3.3 Static Time Slices

To study the nature of subadditvity violation, let us first focus on the
maximal case $\theta_{0}=\frac{\pi}{2}$. In this case, it is useful to
consider the static coordinate $(\chi,s)$ of dS2 introduced as follows

$\displaystyle\cos\chi=\cos\phi\cosh t,\ \ \ \ \ \cosh
s=\frac{\sin\phi}{\sqrt{\frac{1}{\cosh^{2}t}-\cos^{2}\phi}},$ (3.15)

which leads to the metric

$\displaystyle ds^{2}=d\chi^{2}-\sin^{2}\chi ds^{2}.$ (3.16)

Then the geodesic length in the bulk dS3 which connects between two points
$(\chi_{1},s_{0})$ and $(\chi_{2},s_{0})$ on the boundary dS2 at the same
constant time $s_{0}$ is simply given by $D_{12}=|\chi_{1}-\chi_{2}|$. This
means that the holographic entanglement entropy on the constant time slice in
this static coordinate is linear about the subsystem size $|A|$ and saturates
the subadditivity. This clearly shows that if we consider time slices other
than $s=$const. , including the constant $t$ slices, the strong subadditivity
gets violated because the subsystem size $|A|$ gets shorter as

$\displaystyle|A|=\int^{\chi_{2}}_{\chi_{1}}d\chi\sqrt{1-\sin^{2}\chi\left(\frac{ds}{d\chi}\right)^{2}}\leq\chi_{2}-\chi_{1}=D_{12}.$
(3.17)

This analysis shows that at $\theta_{0}=\frac{\pi}{2}$, the static slice
$s=$const. is the only time slice which is consistent with the Hilbert space
structure of dual unitary quantum systems.

We can provide another feature of this restriction from the spacetime
structure of de Sitter space. Consider a Wheeler-DeWitt patch in the bulk dS3
for a generic time slice of the boundary dS2. Since space-like geodesics which
connects two boundary points are all on the boundary dS2, the geodesics are
outside of the Wheeler-DeWitt patch, except when the time slice is the
constant $s$ slice, as depicted in figure 19 by comparing this with that in
AdS. In a sensible holography we expect that any bulk counterpart dual to
objects in the boundary theory at a specific time will be within its Wheeler-
DeWitt patch.

For generic values of $\theta_{0}$, the situation gets less sharp but looks
qualitatively similar. This can be seen from the left panel of figure 18. As
the time evolves following constant $t$ (or equally $T$) slices, the region
which satisfies the subadditivity gets squeezed into the future direction and
eventually disappears. As $\theta_{0}$ gets closer to $\frac{\pi}{2}$, the
region which satisfies the subadditivity becomes very narrow as in the right
panel of figure 18.

Figure 19: Sketches of Wheeler-DeWitt patches (blue regions) and the geodesics
(red curves) in AdS3 (left), in dS3 at $t=0$ (middle) and in dS3 at $t>0$
(right).

###  3.4 Time slices and dual Hilbert space

Having in mind the analysis of holographic entanglement entropy, we would like
to consider the holographic dual interpretation. For simplicity let us focus
on the case $\theta_{0}=\frac{\pi}{2}$. From the Hilbert space viewpoint of
its dual quantum system, assuming that it is unitary, we can only allow the
static time slice (constant $s$ one) on the boundary dS2 as we have seen in
the previous subsection. Owing to the $SO(2,1)$ symmetry of dS2, we can boost
and rotate the canonical static time slice $t=0$ by this symmetry as depicted
in the left panel of figure 20. Below we call these $SO(2,1)$ transformation
of $t=0$ slice nice slices. Note that this family of nice time slices do not
include $t=t_{0}>0$ slices but does the constant $s$ ones in the coordinate
(3.16). Therefore we expect that all nice slices correspond to an identical
Hilbert space ${\cal H}_{\mathrm{dS}}$ of a quantum system dual to a half dS3.
Since the dual state looks maximally entangled due to the fact that the
extremal surface $\Gamma_{A}$ is included in the boundary dS2, the dimension
of ${\cal H}_{\mathrm{dS}}$ should be given by $S_{dS}=\frac{2\pi}{4G_{N}}$,
i.e. the de Sitter entropy of the dS3 gravity.

This argument looks very similar to the surface/state duality proposed in
Miyaji:2015yva , which argues that a codimension two space-like surface
$\Sigma$ in general gravitational spacetimes is holographically dual to a
certain quantum state $|\Phi_{\Sigma}\rangle$. The difference is that in the
present paper we put a real boundary by restricting the dS3 to the half dS3 so
that it has a genuine boundary given by dS2 at $\theta=\theta_{0}$. However,
it is natural that there is a holographic duality even before we put the time-
like boundary. In this context, our lesson from this present analysis is that
in order for the surface/state duality works well we need to require that the
extemal surface $\Gamma_{A}$ should be within the Wheeler-DeWitt patch of
$\Sigma$.

We can decompose a constant $t=t_{0}>0$ slice into multiple nice time slices
as depicted in the right panel of figure 20. As $t_{0}$ gets larger, we need
more nice time slices to cover it as can be seen from that fact that the
length of $t=t_{0}$ slice exponentially grows $\sim e^{t_{0}}$. This shows
that the constant $t$ slice (except $t=0$ one) overestimates the dimension of
Hilbert space. Indeed this is obvious from the fact that to cover the
$t=t_{0}$ slice we need many nice slices whose Hilbert spaces clearly have
overlaps as they cover the $t=0$ slice many times. Thus we cannot expect that
such a slice describes a well-defined Hilbert space, which is consistent with
our observation of the subadditivity violation. The division of the constant
$t$ slice into subsystems does not correspond to the factorization of a
Hilbert space. This behavior is quite different from that in local quantum
field theories, where the choice of time slice on a given causal diamond gives
the identical subsystem of a Hilbert space. In our de Sitter holography, even
though the segment $A$ on the constant $t$ slice and $\Gamma_{A}$ on the nice
slice both live on the same causal diamond as in the left panel of figure 17,
only the latter has a well-defined meaning as a proper subsystem of ${\cal
H}_{\mathrm{dS}}$.

Figure 20: Sketches of nice time slices (constant $s$ slices) in the boundary
dS2 which describe the Hilbert space of dual quantum system. In the left panel
we showed these slices in a global dS2. In the right panel, we showed that a
constant $t$ slice (red curve) is decomposed into multiple constant $s$ slices
(purple, blue and green curves).

###  3.5 Non-locality and Holographic Entanglement Entropy

Then it is natural to ask why only nice slices (constant $s$ slices) can
describe a proper Hilbert space. Even though we do not give a conclusive
answer to this question, we would like to suggest that the non-local nature of
dual quantum system plays a crucial role. First of all, it is obvious that the
way we put the boundary of a half dS3 leads to a finite cut off in the dual
field theory. Also the volume law entanglement which we observe for nice
slices is typical for vacuum states in highly non-local field theories
Li:2010dr ; Shiba:2013jja . Moreover, we can even find that a growth of
entanglement entropy, defined by the replica method, can exceed the volume law
i.e. $S_{A}\sim|A|^{p},\ \ p>1$ in highly non-local field theories, which
shows the violation of subadditivity. Indeed, as shown in Li:2010dr  via the
replica method, a non-local free scalar field theory in $d$ dimensions with
the action

$S=\int dx^{d}\phi(x)e^{(-\partial_{x}^{2})^{q}}\phi(x)\,,$ (3.18)

leads to the entanglement entropy whose UV divergence scales as

$S_{A}\propto\left(\frac{L_{A}}{\epsilon}\right)^{d-2+2q}\,,$ (3.19)

where $L_{A}$ is the linear size of the subsystem $A$. Thus, $S_{A}$ grows
faster than the volume law if $q>\frac{1}{2}$. Here note that the entanglement
entropy defined from the replica method based on Euclidean path-integral is
not guaranteed to satisfy the subadditivity, though the entanglement entropy
defined from a quantum state in a Hilbert space automatically satisfies the
subadditivity.

In our holographic dual of a half dS3, it is natural to expect that a similar
non-local field appears and this may explain the violation of subadditivity on
generic time slices. As the value of $\theta_{0}$ gets smaller, this non-local
effects get milder and the range of time slices which satisfy the
subadditivity, gets broader as depicted in figure 18.

In this way, even though the proper Hilbert space interpretation becomes
difficult for generic time slices including constant $t$ slices, we may be
able to formally define the entanglement entropy by the replica method, which
allows the violation of strong subadditivity. Notice that in such a non-local
field theory, Hamiltonian cannot be defined in a standard way as the action
involves derivatives with respect to the Euclidean time whose order is higher
than two. If this interpretation is true, we should be able to define $S_{A}$
even when $A$ is so large that the two end points of $A$ cannot be connected
by a space-like geodesic. If we apply the original holographic entanglement
entropy formula (3.11), we find that it is complex valued

$D_{12}=\pi+i\text{arccosh}\left[\sinh^{2}t-\cos(2\theta_{0})\cosh^{2}t_{0}\right]\,.$
(3.20)

See Hikida:2022ltr ; Chapman:2022mqd  for more studies about complex geodesics
in dS space. As depicted in the right panel of figure 17, the real part comes
from the geodesic in the Euclidean instanton (semi-sphere) and the imaginary
part comes from the time-like geodesic in the de Sitter space. However, if we
consider the holographic calculation of entanglement entropy via the replica
method (for the derivation of the covariant HEE in AdS/CFT see Dong:2016hjy ),
there are two candidates of extremal surface: one is in the bra geometry and
the other is the ket geometry as in figure 21. This prescription was analyzed
in Chen:2020tes  for a bra-ket wormhole in AdS/CFT.

Figure 21: Sketches of calculation of holographic entanglement entropy in the
Schwinger-Keldysh geometry. The sum of the contributions associated with two
extremal surfaces $\Gamma^{(1)}_{A},\Gamma^{(2)}_{A}$ gives rise to the real
entanglement entropy.

The correct result is given by the sum of two contributions:

$\displaystyle Z^{(n)}_{\mathrm{tot}}=e^{(1-n)S}+e^{(1-n)S^{*}},$ (3.21)

where $n$ is the replica number. $S_{A}$ is given by

$\displaystyle S_{A}=\lim_{n\to 1}\frac{1}{1-n}\log
Z^{(n)}_{\mathrm{tot}}\simeq\frac{S+S^{*}}{2}=\mbox{Re}[S].$ (3.22)

Thus we find that the actual $S_{A}$ is given by the real part:

$\displaystyle S_{A}=\frac{\pi}{4G_{N}}=\frac{1}{2}S_{\mathrm{dS}},$ (3.23)

where $S_{\mathrm{dS}}=\frac{2\pi}{4G_{N}}$ is the de Sitter entropy.

In this way, we find that the saturation behavior i.e. $S_{A}$ grows
monotonically for $0\leq\Delta\phi\leq\Delta\phi_{\mathrm{max}}$ and it takes
the constant value $\frac{1}{2}S_{\mathrm{dS}}$ for
$\Delta\phi\geq\Delta\phi_{\mathrm{max}}$. For $\Delta\phi>\pi$, the standard
holographic prescription guarantees the relation $S_{A}=S_{A^{c}}$, where
$A^{c}$ is the complement of $A$. Explicit plots are shown in figure 22.

Figure 22: Plots of holographic entanglement entropy $S_{A}$ on a constant $t$
time slice based on the prescription (3.22) for $d=2$. We plotted $S_{A}$ as a
function of $\Delta\phi$ in the range $0\leq\Delta\phi\leq 2\pi$ for
$\theta_{0}=\frac{\pi}{2}$ (left) and $\theta_{0}=\frac{\pi}{8}$ (right). The
blue and orange curves correspond to $t=0$ and $t=2$. Note that we have the
critical time $t_{\mathrm{max}}=0$ for $\theta_{0}=\frac{\pi}{2}$ and
$t_{\mathrm{max}}\simeq 1.61$ for $\theta_{0}=\frac{\pi}{8}$.

###  3.6 Overcounting of de Sitter Hilbert Space

Figure 23: Penrose diagram of dS2 spacetime which corresponds to the dS
boundary located at $\theta=\theta_{0}=\frac{\pi}{2}$ in dS3 bulk spacetime.
The green dashed lines denote the null limit of geodesics. On the time slice
$T=T_{0}$ or $t=t_{0}$, we divide the system into six parts.

In previous analyses regarding the holographic entanglement entropy of a
single interval, it was observed that the entropy is bounded by half of the dS
entropy, i.e.,

$S_{A}\leq\frac{1}{2}S_{\mathrm{dS}}\,,$ (3.24)

as illustrated in figure 22. This outcome is expected since the corresponding
bulk dual represents only half of the dS spacetime. Generalizing this
calculation to cases where subsystem $A$ comprises multiple disconnected
intervals is also possible. Since this calculation follows the standard
prescription Headrick:2010zt , explicit results will not be presented in
detail. However, it is important to note that the holographic entanglement
entropy of a subregion consisting of multiple intervals is not bounded by the
dS entropy $S_{\rm dS}$ at late times, as the sum of lengths of disconnected
geodesics can be very large. Focusing on the case with the boundary dS space
located at $\theta=\theta_{0}=\frac{\pi}{2}$, it has been derived in eq. (A.4)
that the entropy of a single interval reaches the maximal value
$\frac{1}{2}S_{\mathrm{dS}}$ when its length corresponds to the critical size
$\Delta\phi_{\rm max}$. It should be noted that the maximum size
$\Delta\phi_{\rm max}(t_{0})$ decreases to zero as the boundary time $t_{0}$
increases. Consequently, let us consider a time slice at $t=t_{0}$ which can
be divided into $2N$ identical intervals, where
$\frac{\pi}{N}\geq\Delta\phi_{\rm max}(t_{0})$. Figure 23 provides an explicit
example with $N=3$. By considering a subsystem with $N$ intervals, one can
find that the holographic entanglement entropy is shown as

$S_{A_{1}\cup A_{2}\cup\cdots\cup A_{N}}=N\times S_{A_{i}}=\frac{N}{2}S_{\rm
dS}\geq S_{\rm dS}\,.$ (3.25)

This value can even approach infinity as $N$ tends to infinity, while
$\lim\limits_{t_{0}\to\infty}\Delta\phi_{\rm max}\to 0$. It might be
questioned whether this violates the entropy bound of de Sitter space, whose
Hilbert space is expected to be finite. However, as observed from the
violation of (strong) subadditivity, we expect that a generic time slice does
not correspond to a pure state in a single de Sitter Hilbert space. Instead,
the unbounded entropy of multiple intervals, as expressed in eq. (3.25), can
be interpreted as the result of overcounting the dimension of the de Sitter
Hilbert space. As shown in figure 23, each critical interval $A_{i}$ at
$t=t_{0}$ could be understood as a maximally entangled state in a single de
Sitter Hilbert space. Back to the original time at $t=0$, those single Hilbert
spaces are overlapping with each other along this time slice. The recent paper
Cao:2023gkw  discusses the tensor network representation of dS space with
overlapping qubits. However, we note that the studies in Cao:2023gkw  focus on
bulk dS spacetime which is different from our dS boundary picture as shown in
figure 23.

###  3.7 Higher dimensional half de Sitter space

It is straightforward to generalize our analysis for a half dS3 to a half
dSd+1 for $d\geq 3$. Below we will provide analysis for
$\theta_{0}=\frac{\pi}{2}$ in the global dSd+1 described by (1.1) and (1.2).
The time-like boundary of this space is given by dSd described by the global
metric

$\displaystyle ds^{2}=-dt^{2}+\cosh^{2}t(d\phi^{2}+\sin^{2}\phi
d\Omega^{2}_{d-2}),$ (3.26)

where $d\Omega^{2}_{d-1}$ is the metric of the unit sphere Sd-2. As in the
previous case of $d=2$, we would like to argue that gravity on a half dSd+1 is
dual to non-local field theory on the dSd.

To examine this duality we would like to calculate the holographic
entanglement entropy. This is given by the area of extremal surface
$\Gamma_{A}$, which ends on the boundary of the subsystem $A$, by (1.3).

For this we choose the subsystem $A$ to be a disk on the boundary dSd:

$\displaystyle t=t_{0},\ \ 0\leq\phi\leq\phi_{1}.$ (3.27)

In this case, we can find the profile of the extremal surface $\Gamma_{A}$
which ends on the boundary $\phi=\phi_{1}$ of the subsystem $A$ (3.27) as

$\displaystyle\frac{\cos\phi}{\tanh t}=\frac{1+L^{2}}{1-L^{2}}.$ (3.28)

This surface can be simply obtained by mapping the extremal surface

$t^{2}-\sum_{i=1}^{d-1}(x_{i})^{2}=L^{2},$ (3.29)

in the Poincare dSd given by the metric

$\displaystyle ds^{2}=\frac{-dt^{2}+\Sigma_{i=1}^{d-1}dx_{i}^{2}}{t^{2}}.$
(3.30)

The constant $L$ is related to the choice of the subsystem $A$ via

$\displaystyle\frac{1+L^{2}}{1-L^{2}}=\frac{\cos\phi_{1}}{\tanh t_{0}}.$
(3.31)

The profile of the extreme surface (3.28) is plotted in figure 24.

This $d-1$ dimensional extremal surface $\Gamma_{A}$ is wrapped on Sd-2. The
induced metric on $\Gamma_{A}$ reads

$\displaystyle
ds^{2}=\frac{4L^{2}(1+L^{2})^{2}}{\left((1+L^{2})^{2}-(1-L^{2})^{2}\cos^{2}\phi\right)^{2}}d\phi^{2}+\frac{(1+L^{2})^{2}\sin^{2}\phi}{(1+L^{2})^{2}-(1-L^{2})^{2}\cos^{2}\phi}d\Omega^{2}_{d-2}.$

This surface $\Gamma_{A}$ is space-like when $0<L^{2}\leq 1$. It becomes
light-like when $L^{2}=0$ and time-like when $L^{2}<0$.

Figure 24: Profiles of extremal surfaces (3.28) in global dS. The horizontal
and vertical coordinate are $\phi$ and $T$, respectively. The blue, yellow,
green ,and red curves describe the extremal surface for $L^{2}=1,1/4,0$ and
$-1/4$, respectively.

The area of $\Gamma_{A}$ is computed as

$\displaystyle
A(\Gamma_{A})=\mathrm{Vol}(\mathrm{S}^{d-2})\int^{\phi_{1}}_{0}d\phi\frac{2L(1+L^{2})^{d-1}(\sin\phi)^{d-2}}{\left((1+L^{2})^{2}-(1-L^{2})^{2}\cos^{2}\phi\right)^{\frac{d}{2}}}.$
(3.33)

Since this is a function of $\phi_{1}$ and $t_{0}$, we write this as
$A(\phi_{1},t_{0})$. It is straightforward to see that at $t_{0}=0$ and
$\phi_{1}=\frac{\pi}{2}$ we have

$\displaystyle A\left(\frac{\pi}{2},0\right)=\frac{1}{2}\mbox{Vol}(S^{d-1}).$
(3.34)

This means that the corresponding holographic entanglement entropy coincides
with the half of de Sitter entropy $\frac{1}{2}S_{dS}$. The behavior of
$A(\phi_{1},t_{0})$ is plotted in figure 25. In general it is a monotonically
increasing function of $t_{0}$ and $\phi_{1}$ and gets finally saturated to
the maximal value $\frac{1}{2}S_{dS}$. Note that when $L^{2}$ becomes negative
i.e when $\Gamma_{A}$ gets time-like, we choose the other extremal surface
which goes past and which is wrapped on the semi-sphere in the Euclidean part
as in right panel of figure17. Thus as in the same argument of (3.23), the
holographic entanglement entropy is given by the real part of $A(\Gamma_{A})$,
namely $\frac{1}{2}S_{\rm dS}$.

Figure 25: The behaviors of the holographic entanglement entropy for a half
dS4 ($d=3$) in case 1. In the left panel, we plotted
$\frac{1}{2\pi}A(\Gamma_{A})$ as a function of $\phi_{1}$ at $t=0$ (blue),
$t=1/2$ (orange) and $t=1$ (green). In the right panel, we showed
$\frac{1}{2\pi}A(\phi_{1},t_{0})$ as a function of $t_{0}$ for
$\phi_{1}=\pi/8$ (blue), $\phi_{1}=\pi/3$ (orange), $\phi_{1}=\pi/2$ (green).

##  4 Holography for a half dS with EOW brane (Case 2)

Now we would like to study another setup of holography for a half de Sitter
space in the presence of EOW brane (case 2). We regard the future boundary
$t=\infty$ of the de Sitter space as an EOW brane where we impose the Neumann
boundary condition for the gravitational field. In the dual field theory, we
have the final state projection as in the case of AdS/CFT analysis done in
section 2.3.

To obtain the properties of the gravity dual, we would like to analyze the
holographic entanglement entropy given by the area of extremal surface as in
(1.3) and (3.11). However, since the field theory sides has the final state
projection, it should properly be regarded as the holographic pseudo entropy
Nakata:2021ubr . For simplicity focus on the maximal case
$\theta_{0}=\frac{\pi}{2}$ in dSd+1 for simplicity. We argue gravity in the
half dSd+1 with the EOW brane at $t=t_{P}$ is dual to a non-local field theory
on its boundary dSd with a finial state projection at $t=t_{p}$. The new
aspect here is that the space-like extremal surface can end on the boundary of
the EOW brane.

Consider the case where the EOW brane is situated at the future infinity
$t_{p}\to\infty$, which is dual to the final state projection at the future
infinity.

Figure 26: Sketches of geodesics in dS2 which are used to calculation the
holographic pseudo entropy in a half dS3. When the subsystem $A$ is small
$\Delta\phi\leq\Delta\phi_{\mathrm{max}}$, the geodesic $\Gamma_{A}$ is space-
like and connected shown in the left panel. On the other hand when $A$ gets
larger $\Delta\phi>\Delta\phi_{\mathrm{max}}$, $\Gamma_{A}$ consists of two
disconnected time-like geodesics as in the right panel.

First let us analyze the holographic pseudo entropy in this setup for $d=2$.
When the interval $A$ is small such that
$\Delta\phi<\Delta\phi_{\mathrm{max}}$, the holographic pseudo entropy $S_{A}$
is given by the length of connected space-like geodesic
$D^{\mathrm{con}}_{12}$ as we had in the case 1. However, for
$\Delta\phi>\Delta\phi_{\mathrm{max}}$, when a connected space-like geodesic
does not exist, $S_{A}$ can be computed from the two disconnected geodesics
which connect one of the end points of $A$ with a point on the EOW brane as
depicted in figure 26.

Note also that such a disconnected geodesic becomes time-like for
$\Delta\phi>\Delta\phi_{\mathrm{max}}$, which shows that the holographic
pseudo entropy is pure imaginary. This is the crucial difference from the
analysis of $S_{A}$ in the case 1. In the case 2, in the presence of EOW
brane, the geodesics $\Gamma_{A}$ can end on it and the real part of $S_{A}$
for this contribution (see the right panel of figure 26) becomes smaller than
the connected one (see the right panel of figure 17) which has a positive real
part.

In summary, the behavior of $S_{A}$ for $d=2$ is identical to that in section
3.5 for $\Delta\phi<\Delta\phi_{\mathrm{max}}$, while we have
$\mathrm{Re}[S_{A}]=0$ for $\Delta\phi>\Delta\phi_{\mathrm{max}}$:

$\displaystyle\mbox{For}\ 0\leq\Delta\phi<\Delta\phi_{\mathrm{max}}:\ \ \ \
S_{A}=\frac{D^{\mathrm{con}}_{12}}{4G_{N}},$ $\displaystyle\mbox{For
}\Delta\phi>\Delta\phi_{\mathrm{max}}:\ \ \ \ \mbox{Re}[S_{A}]=0.$ (4.1)

It is straightforward to generalize this to higher dimensions by employing the
results of $A(\phi_{1},t_{0})$ in case 1 presented in section 3.7. When
$\phi_{1}$ is smaller than the saturation value in case 1, $S_{A}$ takes the
same value as that in case 1. For large values of $\phi_{1}$, we have
Re$[S_{A}]=0$. An explicit plot for $d=3$ is shown in figure 27.

Figure 27: The behaviors of the holographic pseudo entropy for a half dS4
($d=3$) in case 2. In the left panel, we plotted $\frac{1}{2\pi}A(\Gamma_{A})$
as a function of $\phi_{1}$ at $t=0$ (blue), $t=1/2$ (orange) and $t=1$
(green). In the right panel, we showed $\frac{1}{2\pi}A(\Gamma_{A})$ as a
function of $t_{0}$ for $\phi_{1}=\pi/8$ (blue), $\phi_{1}=\pi/3$ (orange),
$\phi_{1}=\pi/2$ (green).

Now let us interpret the behavior of $S_{A}$ in terms of dual field theory on
dSd. For $\Delta\phi<\Delta\phi_{\mathrm{max}}$, $S_{A}$ is identical to that
in case 1, its interpretation is the same. Thus, we find the violation of
subadditivity and this is expected to be due to the non-local nature of the
dual field theory. For $\Delta\phi>\Delta\phi_{\mathrm{max}}$, we find a
behavior special to case 2 that the real part of $S_{A}$ does vanish.

This transition at $\Delta\phi=\Delta\phi_{\mathrm{max}}$ may be natural from
the viewpoint of the dual field theory on dSd. To see this, let us remember
that the pseudo entropy is defined from the transition matrix (1.5), which
depends on the two states $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$. In our
setup $|\psi_{1}\rangle$ is the state which is obtained by the forward time
evolution of Hartle-Hawking state, while $|\psi_{2}\rangle$ is the one created
by the backward time evolution of the final state. It is also useful to note
that the phenomenological observation Mollabashi:2020yie ; Mollabashi:2021xsd
; Akal:2021dqt  implies that the real part of pseudo entropy typically
measures the amount of entanglement in the intermediate state between the two
states, though the interpretation of imaginary part is not well understood at
present. Now the final state is defined by imposing a boundary condition at
$t=t_{p}=\infty$ and the analysis of the space-like EOW brane Akal:2020wfl
suggests that the state has no real space entanglement as in the standard
boundary state Miyaji:2014mca . At the time slice $t=t_{0}$, we can have such
an disentangling effect from a point at future infinity when two points are
separated more than $2(t_{p}-t_{0})$ assuming the light like propagation of
physical signals on dSd. Indeed this border is $\Delta\phi_{\mathrm{max}}$.
Therefore it is natural that the real part of pseudo entropy gets vanishing
for $\Delta\phi>\Delta\phi_{\mathrm{max}}$. On the other hand, for
$\Delta\phi<\Delta\phi_{\mathrm{max}}$, such two points do not feel the
existence of the EOW brane and thus the result is the same as that in case 1.

In a similar way, we can analyze the holographic pseudo entropy when the final
projection is inserted at a finite time $t=t_{p}$. Obviously at $t=t_{p}$, we
have $S_{A}=0$. Moreover, the real part of $S_{A}$ gets vanishing when the
geodesic $\Gamma_{A}$ becomes light-like. This can happen $t<t_{p}$ if the
subsystem $A$ is large enough, in which case a imaginary part of $S_{A}$
starts to be non-zero until $t=t_{p}$ as $\Gamma_{A}$ becomes time-like, as
depicted in figure 28 for plots. It is curious to note that this page curve
like behavior of Re$S_{A}$ is qualitatively similar to the one found for the
AdS/CFT setup in eq. (2.33).

Figure 28: The behaviors of the holographic pseudo entropy for a half dS3
($d=2$, left) and a half dS4 ($d=3$, middle and right) in case 2 with the EOW
brane at $t_{p}=1$. We plotted $\frac{1}{\pi}A(\Gamma_{A})$ (for $d=2$, left)
and $\frac{1}{2\pi}A(\Gamma_{A})$ (for $d=3$, middle) as a function of time
$t_{0}$. We chose $\phi_{1}=\frac{\pi}{8}$ (blue), $\phi_{1}=\frac{\pi}{6}$
(orange), $\phi_{1}=\frac{\pi}{4}$ (green), $\phi_{1}=\frac{\pi}{3}$ (red). In
the right panel, we plotted Re$\left[\frac{1}{2\pi}A(\Gamma_{A})\right]$
(blue) and Im$\left[\frac{1}{2\pi}A(\Gamma_{A})\right]$ (orange) as a function
of time for $\phi_{1}=\frac{\pi}{3}$ for $d=3$. Note that the real part starts
vanishing when the geodesic becomes null and we have both the real and
imaginary part vanishing for $t>t_{p}(=1)$.

##  5 Summary and Discussions

The primary objective of this paper is to investigate the holographic duality
involving gravity in de Sitter spaces. Unlike many other approaches, our focus
centers on a half de Sitter space, achieved by introducing a timelike boundary
in global dS spacetime. Within this framework, we propose that the gravity on
a $d+1$-dimensional half de Sitter space is dual to a non-local field theory
residing on its $d$-dimensional boundary.

Before delving into our investigation of de Sitter holography, we conducted an
analysis of the holographic duality linking gravity in a $d+1$-dimensional AdS
and a CFT living on a $d$-dimensional dS. This particular scenario can be
regarded as a special case within the framework of the AdS/CFT correspondence.
Of course, its validity and fundamental computational methods are well-
established. We examined two distinct setups, referred to as the Case 1 and
the Case 2, respectively. In the Case 1, the quantum state in the dual CFT is
described by the Schwinger-Keldysh formalism, whereas in the Case 2, we
consider the dual CFT incorporating a final state projection, as illustrated
in figure 4 and figure 5. In particular, the gravity dual in the Case 2 is
given by adding an end-of-the-world brane (EOW brane) on an AdS geometry.

In these setups, we evaluated holographic entanglement entropy, which is
determined by the area of an extremal surface. In the Case 1, we observed that
the holographic entanglement entropy for a subsystem of fixed size
consistently increases as the size of de Sitter space is inflating with
respect to the global time. Conversely, in the Case 2, it initially presents
growth but eventually decreases to zero. At the time $t=t_{\textrm{\tiny P}}$,
corresponding to the implementation of the final state projection, the
entanglement entropy vanishes. It is important to note that the extremal
surface area we computed should be more appropriately interpreted as the
pseudo entropy due to the presence of the final state projection.
Additionally, we also confirmed that independent CFT calculations in CFT2
reproduce the results which agree with that from gravity dual.

With the AdS/CFT results as our foundation, we proceeded to examine holography
for gravity in a half de Sitter space. We focused on two setups, namely the
Case 1 and the Case 2, and investigated the behaviour of holographic
entanglement entropy. Note that we employed the standard calculation of
holographic entanglement entropy Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt ,
where we minimize the area, as opposed to the prescription in Susskind:2021esx
where the area is maximized. Remarkably, we discovered that the properties of
holographic entanglement entropy in a half de Sitter space diverge from those
in the AdS/CFT correspondence. Notably, connecting two arbitrary points in a
global de Sitter space using a spacelike geodesic is not always possible.
Consequently, for a two-dimensional dS space, which serves as the timelike
boundary of a three-dimensional half de Sitter space, there is typically no
spacelike geodesic linking the endpoints of an interval. As a result, the
definition of holographic entanglement entropy, denoted as $S_{A}$, in a
conventional sense is problematic. Similarly, we observed the same limitation
for extremal surfaces in higher dimensions ($d>2$).

However, a resolution to this issue arises when we consider both timelike and
spacelike geodesics within a Hartle-Hawking contour, as illustrated in the
right panel of figure 17. The joint geodesics allow us to connect two
endpoints beyond critical size. Consequently, the holographic entanglement
entropy, denoted as $S_{A}$, acquires a complex value. In the Case 1, we
contend that the appropriate holographic entanglement entropy can be obtained
by taking its real part, which is nothing but half of the de Sitter entropy
$\frac{1}{2}S_{dS}$. In the Case 2, with the presence of the EOW brane, we
argue that the holographic pseudo entropy can be computed by utilizing
timelike geodesics terminating on the EOW brane, as depicted in the right
panel of figure 26.

Furthermore, even within parameter regions where a spacelike geodesic exists,
the holographic entanglement entropy generally exhibits super-extensive
behaviour relative to the subsystem size. Consequently, the corresponding
function describing holographic entanglement entropy violates the (strong)
subadditivity property. Notably, we have discovered that this issue is
resolved solely by focusing on the time slices associated with the static
coordinate in the case of the maximal half de Sitter space
($\theta_{0}=\frac{\pi}{2}$). On these particular time slices, the holographic
entanglement entropy $S_{A}$ adheres to the volume law. In this regard, we
anticipate that the quantum state represented by the half de Sitter space
manifests maximal entanglement specifically on these special time slices. This
implies that we can establish a well-defined Hilbert space solely for static
time slices. Conversely, a generic time slice in the boundary de Sitter space
spans the same Hilbert space as the static time slice multiple times,
resulting in an overcounting of the genuine degrees of freedom (refer to
figure 20). Additionally, it is worth noting that when subadditivity is
violated, the extremal surface extends beyond the Wheeler-DeWitt patch, as
depicted in figure 19.

Although providing a Hilbert space interpretation for generic time slices,
including the constant time slice of the global coordinate, presents
challenges, we propose a potential understanding of the entanglement entropy
by employing the replica method and defining the area of extremal surface
$S_{A}$ as the entropy. Notably, in highly non-local field theories, the
realization of super-extensive entanglement entropy can be realized. It is
important to note that in such non-local field theories, a standard Hilbert
space cannot be defined due to the appearance of infinitely many time
derivatives in the action. Applying this perspective, the holographic
entanglement entropy in the Case 1 exhibits initial growth, followed by
saturation at $\frac{1}{2}S_{dS}$, as depicted in figure 22 and figure 25.
Note that this saturation was absent in the evolution of $S_{A}$ derived in
the Case 1 in the framework of the AdS/CFT correspondence and that it shows
that the entropy is bounded in de Sitter space.

In the Case 2, we observe that the real part of the holographic pseudo entropy
exhibits initial growth and eventually vanishes at the critical time, while
the imaginary part remains non-zero, as depicted in figure 27 and figure 28.
We identify that the real part of $S_{A}$ becomes zero when the subsystem $A$
surpasses the light cone (refer to the right panel of figure 26). This
behaviour arises due to the influence of the EOW brane, where the reflection
of two points on $A$ by the EOW brane boundary condition affects $S_{A}$. The
EOW brane boundary state possesses vanishing quantum entanglement, thereby
diminishing the overall quantum entanglement.

There are several promising avenues for future investigation. Firstly, it
would be intriguing to explore alternative frameworks that provide a more
manageable description of the “overcounting” phenomenon in the Hilbert space
dual to a half de Sitter space in global coordinates. One possible approach
could involve qubit systems or tensor networks, offering a more controllable
perspective. Additionally, it is of great interest to extend our analysis to
holography in more generic asymptotically dS spacetimes, such as de Sitter
black holes, as well as various cosmological models. Lastly, a pivotal and
profound question lies in understanding how the creation and evolution of the
universe can be described in terms of the dual field theory, utilizing the
insights gained from the holographic duality.

## Acknowledgements

We are grateful to Takato Mori, Yusuke Taki and Zixia Wei for useful
discussions. This work is supported by the Simons Foundation through the “It
from Qubit” collaboration and by MEXT KAKENHI Grant-in-Aid for Transformative
Research Areas (A) through the “Extreme Universe” collaboration: Grant Number
21H05187. TT is also supported by Inamori Research Institute for Science, and
by JSPS Grant-in-Aid for Scientific Research (A) No. 21H04469. SMR is also
supported by JSPS KAKENHI Research Activity Start-up Grant NO. 22K20370. YS is
supported by Grant-in-Aid for JSPS Fellows No.23KJ1337. T. K. is supported by
Grant-in-Aid for JSPS Fellows No. 23KJ1315.

##  Appendix A Explicit Space-like Geodesics in dS3

Here we study the connected geodesic anchored on the boundaries of an interval
$A$ in dS3. This is the geodesic which connects
$(t,\theta,\phi)=(t_{0},\frac{\pi}{2},-\phi_{0}+\frac{\pi}{2})$ and
$(t,\theta,\phi)=(t_{0},\frac{\pi}{2},\phi_{0}+\frac{\pi}{2})$ in the
coordinate of dS3 (3.2). We assume $t_{0}\geq 0$ without losing generality.
The length $L$ of a curve is

$\displaystyle
L=\int^{\phi_{0}+\frac{\pi}{2}}_{-\phi_{0}+\frac{\pi}{2}}d\phi\sqrt{\cosh^{2}t-\left(\frac{dt}{d\phi}\right)^{2}}.$
(A.1)

This leads to the differential equation

$\displaystyle\frac{dt}{d\phi}=\cosh
t\sqrt{1-\frac{\cosh^{2}t}{\cosh^{2}t_{*}}},$ (A.2)

where the middle point $\phi=\frac{\pi}{2}$ is the turning point and we set
$t=t_{*}$ at this point. By integrating this, we find

$\displaystyle\phi_{0}=\frac{\pi}{2}-\arctan\left[\frac{\cosh t_{*}\sinh
t_{0}}{\sqrt{\cosh^{2}t_{*}-\cosh^{2}t_{0}}}\right],$ $\displaystyle
L=\pi-2\arctan\left[\frac{\sinh
t_{0}}{\sqrt{\cosh^{2}t_{*}-\cosh^{2}t_{0}}}\right].$ (A.3)

Note that assuming $t_{0}>0$, there is an upper bound of $\phi_{0}$, reached
at $t_{*}\to\infty$, which we call $\phi_{\max}$:

$\displaystyle\phi_{\mathrm{max}}=\frac{\pi}{2}-\arctan\left[\sinh
t_{0}\right].$ (A.4)

In this maximal case, we find $L|_{\phi=\phi_{*}}=\pi$, which is a half of de
Sitter horizon length. This maximal value $\phi=\phi_{\mathrm{max}}$
corresponds to the limit where the original space-like geodesic gets light-
like as depicted in the left panel of figure 17. Therefore we cannot find an
appropriate space-like geodesic when the subsystem $A$ is larger than this
maximal size.

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