aainstitutetext: Institute for Theoretical Physics, KU Leuven, Celestijnenlaan
200D, B-3001 Leuven, Belgiumbbinstitutetext: Institute for Advanced Study,
Tsinghua University, Beijing 100084, Chinaccinstitutetext: Institute of
Physics, University of Amsterdam, Science Park 904, PO Box 94485, 1090 GL
Amsterdam, The Netherlandsddinstitutetext: Delta Institute for Theoretical
Physics, Science Park 904, PO Box 94485, 1090 GL Amsterdam, The Netherlands

# A teleportation protocol in Schwarzschild-de Sitter space

Sergio E. Aguilar-Gutierrez b Ricardo Espíndola c,d and Edward K. Morvan-
Benhaim<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a new information transfer protocol for de Sitter space, using
black holes as energy reservoirs. We consider antipodal observers in pure de
Sitter space in the Bunch-Davis state. They can store Hawking modes from the
cosmological horizon in a box. Alternatively, due to thermal fluctuations in
de Sitter space, pair-creation black holes can proceed and form the needed
energy reservoir. We focus on the Nariai case, corresponding to an equilibrium
state. Once it happens, energy pulses can be then released into the black
hole’s interior, opening a traversable wormhole. We provide bounds for the
amount of information that can be transferred. Specializing to
(1+1)-dimensions, we explore how the teleportation protocol leads to an
explicit geometric description of the information transmitted through an
island region. The protocol uncovers quantum information aspects of de Sitter
space, independent of a particular realization of de Sitter space holography.

##  1 Introduction

In recent years, traversable wormholes111See Kundu:2021nwp  for a recent
review. have provided key insights into how quantum information builds
spacetime in the context of the ER=EPR conjecture Maldacena:2013xja . To
generate a traversable wormhole in any crunching spacetime region, there needs
to be a violation of the averaged null energy condition (ANEC)
Freivogel:2019lej . This condition states that

$\int_{\Gamma}T_{\mu\nu}k^{\mu}k^{\nu}\mathrm{d}\lambda\geq
0\leavevmode\nobreak\ ,$ (1)

where $T_{\mu\nu}$ is the matter stress-energy tensor along complete achronal
null geodesics, $\Gamma$, described by the tangential vectors $k^{\mu}$ and
the affine parameter $\lambda$. This is essentially a statement about
causality in general relativity Morris:1988tu ; Friedman:1993ty ; Wall:2009wi
, which can be violated by quantum matter in curved spacetime Visser:1996iw ;
Visser:1995cc . Gao, Jafferis, and Wall (GJW) first proposed a protocol for
double-sided AdS black holes Gao:2016bin  that generates traversable
wormholes. Several improvements to this protocol have been made in the last
few years. Some examples include, stabilization of the wormhole with different
matter content Maldacena:2017axo ; Maldacena:2018lmt ; Maldacena:2018gjk ;
Maldacena:2020sxe ; Bintanja:2021xfs , and probing typical black hole
microstates deBoer:2018ibj ; DeBoer:2019yoe . Although the previous findings
rely on the AdS/CFT correspondence, there has been much interest in extending
holography to arbitrary types of spacetimes, perhaps even those pertinent for
early and late-time cosmology Hartman:2020khs ; Aguilar-Gutierrez:2021bns ;
Bousso:2022gth ; Espindola:2022fqb ; Ben-Dayan:2022nmb . In particular, there
is a recent proposal for information exchange in pure dS space by
Aalsma:2021kle  which proposes a protocol for generating traversable wormholes
in de Sitter space according to a pair of antipodal static patch observers.
One of the main technical difficulties in realizing the protocol in de Sitter
space is that one in principle lacks a holographic dual where we can prepare
the synchronization between the observers that will send light signals and
energy pulses. To overcome this issue, one can perform a state preparation
that maximally entangles the observers.

These developments have motivated us to propose a further adaptation when
black holes in de Sitter space can play a crucial role in the previous
protocol. The observers must collect radiation coming from the cosmological
horizon to later exchange energy pulses. This effectively requires them to
form energy reservoirs that can be then collapsed into black holes, as first
hinted in Aalsma:2021kle . We consider Schwarzschild-de Sitter (SdS) black
holes, which are some of the simplest solutions to Einstein equations in the
presence of a positive cosmological constant. In particular, we consider the
collapse into Nariai black holes, the only thermal equilibrium configuration
allowed in dS space, besides the vacuum. Alternatively, one can consider that
as the system evolves, pair creation of SdS black holes can proceed in the
original pure de Sitter spacetime.222See Ginsparg:1982rs  for the original
description as this semiclassical instability of de Sitter space, and
Bousso:1996au  for the pair creation rate from the no boundary proposal. See
also Draper:2022xzl ; Morvan:2022ybp ; Morvan:2022aon  for recent discussions
about a constrained instanton interpretation in this process. Given that our
aim is to build a traversable wormhole in the black hole interior, one needs
ANEC-violating matter. Each of the observers will release the energy they have
stored as positive energy pulses into the cosmological horizon, which will act
as a negative energy pulse directed towards the near Nariai black hole
interior. This allows for the formation of a traversable wormhole in both the
inflationary and the black hole regions, in which the observers are then able
to exchange light signals. We then provide bounds on the amount of information
that can be transmitted by the observers based on the size of the wormhole
opening.

One advantage of the above is that we do not require a particular realization
of de Sitter space holography333We refer the reader to Witten:2001kn ;
Strominger:2001pn ; Maldacena:2002vr  for pioneering work in this area, and
Galante:2023uyf  for a recent review.. Second, although we explicitly perform
the protocol in (1+3)-dimensions to describe how the observers can trap the
radiation; we expect that most of the arguments can be further generalized to
arbitrary dimensions.

However, the protocol so far discussed relies on a classical gravitational
background. We might also allow for semiclassical effects produced by bulk
matter fields. This can result in the formation of an entanglement island. We
explore the consequences of the island formula in this protocol. See
Penington:2019npb ; Almheiri:2019psf  for the original formulation of this
rule, and Almheiri:2019hni  for a pedagogical review. According to this rule,
the von Neumann entropy of a weakly gravitating region $\mathbf{R}$ is given
by the minimum value among all extrema of the generalized entropy of a region
$\mathbf{R}\cup\mathbf{I}$. In other words

$S(\mathbf{R})=\text{min\leavevmode\nobreak\ }\text{ext}_{\mathbf{I}}S_{\rm
gen}(\mathbf{R}\cup\mathbf{I})\leavevmode\nobreak\ ,$ (2)

where $\mathbf{I}$ is called the island region found by extremizing the
functional

$S_{\rm
gen}(\mathbf{R}\cup\mathbf{I})=\tfrac{A(\partial\mathbf{I})}{4G_{N}}+S_{\rm
matter}(\mathbf{R}\cup\mathbf{I})\leavevmode\nobreak\ .$ (3)

In this equation, $S_{\rm matter}(\mathbf{R}\cup\mathbf{I})$ denotes the von
Neumann entropy of the bulk matter fields, and $A(\partial\mathbf{I})$ is the
codimension-2 area functional evaluated at the endpoints of the region
$\mathbf{I}$.

The observer on the region $\mathbf{R}$ can perform a series of decoding
channels from the Hawking radiation, such as the Petz map Chen:2019gbt ;
Penington:2019kki ; Zhao:2020wgp ; Bak:2021qbo ; Vardian:2023fce ;
Bahiru:2022ukn , to reconstruct the entanglement wedge that includes the
island region. However, making meaningful operations on this information can
be too complex to carry out according to a low-energy observer Akers:2022qdl ;
Gyongyosi:2023sue . To overcome this issue, we will probe the island region
through our traversable wormhole protocol. In this case, information is sent
to the island region and later collected according to low-energy observers.
Although the island rule is less developed in the context of de Sitter space,
there are indications that islands, seeing as disconnected entanglement
wedges, will appear in general types of backgrounds Bousso:2022hlz ;
Bousso:2023sya .

To make an explicit realization of the above protocol, we employ the s-wave
dimensional reduction of a near Nariai black hole background, called the full-
reduction dS Jackiw–Teitelboim gravity (JT) gravity model. See Chen:2020tes ;
Piao:2023vgm ; Sybesma:2020fxg ; Goswami:2022ylc ; Aguilar-Gutierrez:2021bns ;
Aalsma:2021bit ; Levine:2022wos ; Balasubramanian:2020xqf ; Kames-King:2021etp
; Hartman:2020khs ; Baek:2022ozg ; Aalsma:2022swk ; Teresi:2021qff  for
different developments on the island rule in dS JT gravity. The antipodal
static patch observers prepare the shockwaves from the thermal bath. We then
locate a pair of detectors at $\mathcal{I}^{+}$ of the inflating regions,
representing meta observers, to collect Hawking modes coming from the black
hole, as shown in Figure 5. In this situation, the effect of the shockwave is
to shift the location of the island with respect to the one without the
backreaction. The antipodals observers are able to transmit information
probing the island region directly to the metaobservers.

The structure of the manuscript is as follows. In Sec. 2, we review some of
the basics of state preparation and SdS black holes to carry out our protocol.
We will then focus mostly on (3+1)-spacetime dimensions to make concrete
statements on how to trap the radiation. We find bounds on the amount of
information that can be transferred between static patch observers. Sec. 3 is
devoted to the (1+1)-dimensional formulation of the protocol from dS JT
gravity where we add metaobservers in the inflationary patch of the near
Nariai spacetime equipped with detectors. We show that the traversable
wormhole protocol can probe the island region according to the metaobservers.
In Sec. 4, we include a summary of our proposal and we conclude with some
future directions. For the convenience of the reader, App. A reviews the
Hayden-Preskill (HP) protocol to perform information recovery, which we
implement geometrically by means of our protocol. Meanwhile, App. B contains a
review of the s-wave reduction of SdS black holes in higher dimensions to
generate the full-reduction dS JT gravity theory. Lastly, App. C contains some
discussion about the small shift approximation in the shockwave analysis in
the weakly coupled gravity regime.

##  2 General protocol

###  2.1 State preparation

Our starting configuration follows the considerations in Aalsma:2021kle ;
Chandrasekaran:2022cip ; Witten:2023qsv  for constructing the Bunch-Davis
state of pure de Sitter space. We begin studying a single static patch region.
Take for example the south wedge, $S$. In the perturbative quantum gravity
regime, we can define a Hilbert space with energy eigenstates,
$\ket{E_{i}}_{S}\in\mathcal{H}_{S}$ with respect to a one-sided modular
Hamiltonian $H_{S}$. One can also define a natural thermal vacuum state, known
as the Bunch-Davis state by introducing a decoupled antipodal static patch
region. Let us denote it by the north, $N$, with energy eigenstates
$\ket{E_{i}}_{N}\in\mathcal{H}_{N}$ for the Hamiltonian $H_{N}$. Following the
Tomita-Takesaki theory takesaki2006tomita , we may generate the cyclic,
separating vacuum of the theory by means of the thermofield double
construction of the maximally entangled state on both $S$ and $N$,

$\ket{\Psi_{\rm
BD}}=\tfrac{1}{\sqrt{Z}}\sum_{i}\exp\quantity(-\tfrac{1}{2}\beta
E_{i})\ket{E_{i}}_{S}\otimes\ket{E_{i}}^{*}_{N}\in\mathcal{H}_{S}\times\mathcal{H}_{N}\leavevmode\nobreak\
,$ (4)

which is the maximally entangled state in this spacetime. Here the symbol $*$
denotes complex conjugation, and $\beta$ refers to the initial inverse
temperature of both patches. Such state is the vacuum of the time automorphism
generator, $H_{\rm grav}=H_{N}-H_{S}$, where the minus sign indicates the
difference in time orientations between antipodal patches; and importantly,
$H_{\rm grav}\ket{\Psi_{\rm BD}}=0$. This construction has been recently
discussed in the context of the algebra of observables for dS space in
Chandrasekaran:2022cip  in the presence of matter and weakly coupled gravity
(see Gomez:2022eui ; Seo:2022pqj ; Gomez:2023wrq ; Gomez:2023upk ;
Jensen:2023yxy ; Gomez:2023tkr ; Aguilar-Gutierrez:2023odp  for related
developments). The initial configuration is illustrated in Fig. 1.

Figure 1: de Sitter space as seen by a pair of antipodal observers following a
worldline $\gamma_{N}$ and $\gamma_{S}$ (in green) for the Bunch-Davis
preparation of state described as a thermofield double above, with respective
gravitational Hilbert spaces $\mathcal{H}_{N/S}$.

The pair of antipodal observers are then prepared in a maximally entangled
state, and their internal clocks are synchronized as we start the protocol.

###  2.2 Black holes in de Sitter space

The main motivation in this work is to incorporate the appearance of black
holes into the information exchange protocol in de Sitter space. We will allow
the observers to construct an energy reservoir from the Hawking radiation
coming out of the cosmological horizon in Fig. 1. The reservoir can then
collapse into a Schwarzschild-de Sitter (SdS) black hole connecting the
antipodal observers. We will specify the exact conditions for this to happen
in Sec. 2.4, after introducing some general notions in this section.
Alternatively, we may also consider the semiclassical instability process of
de Sitter space from pair production of SdS black holes Ginsparg:1982rs .

SdS black holes in $(d+1)$-dimensions are spherically symmetric vacuum
solutions to Einstein equations with a positive cosmological constant
$\Lambda>0$. These configurations can be described with the following metric
in static patch coordinates

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-f(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}\mathrm{d}\Omega_{d-1}^{2}\,,\quad\text{with}$
(5) $\displaystyle f(r)$ $\displaystyle=1-\frac{r^{2}}{\ell^{2}}-\frac{16\pi
G_{N}M}{(d-1)\Omega_{d-1}r^{d-2}}\,.$ (6)

Here, $M$ is the black hole mass parameter, $\ell^{2}=d(d-1)/(2\Lambda)$ is
the de Sitter curvature radius, $G_{N}$ is Newton’s constant and
$\Omega_{d-1}=2\pi^{d/2}/\Gamma(d/2)$ is the volume of a unit ($d-1$)-sphere.
In (5), we have performed a gauge fixing choice where the static patch time of
the $N$ and $S$ pole observers are described as $t=t_{S}=t_{N}$. On the other
hand, the parameter $M$ cannot be arbitrarily large, as this can result in
complex roots for the blackening factor $f(r)$, which would correspond to a
naked singularity. Such structures are forbidden in nature according to the
cosmic censorship conjecture; see Ong:2020xwv  for a review. Therefore, we
consider a range for the mass of SdS black holes $M\in[0,\,M_{\rm N}]$. For
$M=0$ one recovers empty de Sitter space, with a single event horizon located
at $r=\ell$. The maximal mass $M_{\rm
N}=\tfrac{d-1}{d}\tfrac{\Omega_{d-1}}{8\pi G_{N}}r_{\rm N}^{d-2}$ is referred
to as the Nariai mass and corresponds to the extreme case where the radii of
the black hole and de Sitter horizon coincide, $r_{\rm b}=r_{\rm c}=r_{\rm
N}=\ell\sqrt{\tfrac{d-2}{d}}$, known as the Nariai radius Nariai ;
Ginsparg:1982rs . The reader might check Morvan:2022ybp  for further details
on the quasilocal thermodynamics of these solutions. For our purposes, we
describe the temperature, $T$, and the thermodynamic entropy, $S$, of the
cosmological, $\rm c$, and black hole, $\rm b$, horizons by

$T_{\rm b,\,c}=\frac{\kappa_{\rm b,\,c}}{2\pi},\quad S_{\rm
b,\,c}=\frac{\mathcal{A}_{\rm b,\,c}}{4G_{N}}\leavevmode\nobreak\ ,$ (7)

where $\kappa_{\rm b,\,c}$, $\mathcal{A}_{\rm b,\,c}=\Omega_{d-1}r^{d-1}_{\rm
b,\,c}$ denote the surface gravities and areas of $\rm b$ and $\rm c$
respectively. For the surface gravity, we employ the same normalizations as in
Bousso:1996au  which guarantee that the black hole and cosmological horizons
are in thermal equilibrium for the Nariai configuration, such that the inverse
temperature (7) in the Nariai limit is $\beta_{\rm
N}=\tfrac{2\pi\ell}{\sqrt{d}}$.

###  2.3 Preparation of the black hole

We now discuss the conditions to produce the near Nariai black hole by
collecting radiation in a thermal bath in de Sitter space. Here we model the
bath as a Dyson sphere available to each of the static patch observers
collecting the radiation coming from the de Sitter horizon before sending the
message. When trapping all the radiation into a box of size $r_{\rm box}\leq
r_{\rm N}$, one can compute the time it would take to build a reservoir of any
energy $E_{\rm res}\leq M_{\rm N}$, using Stefan-Boltzmann law in
$(3+1)$-dimensions,

$\mathcal{P}_{\rm c}=\sigma\mathcal{A}_{\rm c}\,T_{\rm c}^{4}$ (8)

with $\mathcal{P}_{\rm c}=\mathrm{d}M/\mathrm{d}t$ the radiated power,
$\sigma$ the Stefan-Boltzmann constant, and $\mathcal{A}_{\rm c}$ the surface
area of the cosmological horizon. Inserting the $4D$ cosmological temperature,
as well as the mass written in terms of the horizon radius $r_{\rm c}$

$T_{\rm c}=\frac{r_{\rm c}^{2}/r_{\rm N}^{2}-1}{4\pi r_{c}\sqrt{1-(M/M_{\rm
N})^{2/3}}}\,,\quad M=\frac{r_{\rm c}}{2G_{N}}\left(1-\left(\frac{r_{\rm
c}}{\ell}\right)^{2}\right)\leavevmode\nobreak\ ,$ (9)

allows one to find

$\displaystyle\frac{\mathrm{d}r_{\rm c}}{\mathrm{d}t}$
$\displaystyle=\frac{G_{N}\sigma\left(\ell^{2}-3r_{\rm
c}^{2}\right)^{3}}{8\pi^{3}\left(2\ell^{2}r_{\rm c}-3\sqrt[3]{2}r_{\rm
c}^{5/3}\left(\ell^{2}-r_{\rm
c}^{2}\right){}^{2/3}\right)^{2}}\equiv\dot{r}_{\rm c}\,.$ (10)

The time $\tilde{\mathcal{T}}_{\rm N}$ necessary to fill up the whole de
Sitter space (till its radius matches the Nariai one) is therefore given by

$\tilde{\mathcal{T}}_{\rm N}=\int^{r_{\rm N}}_{\ell}\frac{\mathrm{d}r_{\rm
c}}{\dot{r}_{\rm c}}\approx\frac{30\sqrt{\pi}}{G_{N}}r_{\rm N}^{3}\,,$ (11)

where we inserted $\sigma=\frac{2\pi^{5}}{15}$. The evolution of the
cosmological and possible444In order to form a black hole, the energy density
must be sufficiently large. It would require careful tuning of the position of
the reflective boundary. However, we avoid any of those difficulties by
filling up a box of size $r_{\rm N}$, guaranteeing that we have a black hole
by the end of the process. black hole radii in this process are illustrated in
Fig. 3 (a).

###  2.4 Shockwave preparation

To produce a traversable wormhole in a near Nariai black hole spacetime, the
distance between the observer and the horizon has to _grow_. For the inflating
patch, this implies that the cosmological horizon radius has to increase,
which is only achieved by an excess of positive energy leaking out of the
considered patch or equivalently increasing its negative energy density. This
implies that we send negative energy in the opposite direction, into the black
hole’s interior. See Fig. 2 for an illustration of this step in the
protocol.555Throughout the text, we employ the convention for the Penrose
diagrams with shockwave shifts in which light signals are discontinuous along
a path with the (cosmological or black hole) horizons remaining continuous.

Figure 2: Configuration where the antipodal observers (in green) produce a
thermal bath from Hawking radiation that they initially stored in the empty de
Sitter preparation of state. Each observer releases a positive energy pulse
(cyan) to the inflating region, which also results in a negative energy pulse
to the black hole interior (purple). This allows the observers to transmit
light signals (orange) to each other.

The deficit of positive or negative energy particles from one patch implies
that there would be an excess of opposite sign energy in the opposite
direction.

We can treat this step of the protocol as in the previous section, but now the
black hole and cosmological fluxes are not restricted by any reflective
boundary or Dyson sphere. For a general SdS black hole, the total power
passing through the cosmological horizon equals

$\mathcal{P}_{\rm tot}=\mathcal{P}_{\rm c}-\mathcal{P}_{\rm b}\,.$ (12)

The lifetime of a four-dimensional Nariai black hole $\mathcal{T}_{\rm N}$ can
be computed in a similar way as Sec. 2.3, resulting in

$\mathcal{T}_{\rm N}\approx\frac{\tilde{\mathcal{T}}_{\rm N}}{\sqrt{2}}\,.$
(13)

The respective evolution is shown in Fig. 3 (b).

(a)

(b)

Figure 3: The horizon radii $r_{h}$ as a function of the static patch time in
$d=3$. The cosmological radius $r_{\rm c}$ is shown in orange, and the black
hole horizon radius is in blue. (a) We start from empty de Sitter space and
consider absorbing boundary conditions for trapping the radiation. Using the
relation $\ell^{2}=r_{\rm b}^{2}+r_{\rm c}^{2}+r_{\rm b}r_{\rm c}$, allows us
to plot the black hole radius that could be formed with the gathered energy.
(b) We start with a Nariai black hole configuration and let the radiation
freely escape through the cosmological horizon.

In the strict Nariai limit $\mathcal{P}_{\rm tot}=0$. However, when the black
hole mass is very close to the Nariai mass $M=M_{\rm N}(1-\epsilon_{\rm N})$,
where $0<\epsilon_{\rm N}\ll 1$, one can find that at first order

$\displaystyle\mathcal{P}_{\rm tot}(M\rightarrow M_{\rm N})$
$\displaystyle=\mathcal{P}_{\rm tot}(M_{\rm N})+\epsilon_{\rm
N}\left(\frac{\partial\mathcal{P}_{\rm tot}}{\partial\epsilon_{\rm
N}}\right)_{\epsilon_{\rm N}=0}=\frac{\epsilon_{\rm N}}{3\pi^{3}\ell^{2}}\,.$
(14)

Then, the energy $E_{\rm N}$ released close to the Nariai limit when opening
the box for a duration $\delta\tau$ is given by

$E_{\rm N}=\frac{\epsilon_{\rm
N}}{3\pi^{3}\ell}\left(\frac{\delta\tau}{\ell}\right)\,.$ (15)

Such emission of energy can cause a run-away effect leading to the complete
evaporation of the black hole. Notice that imposing
$\delta\tau/\ell\equiv\epsilon_{\tau}\ll 1$, as we are considering a pulse,
the released energy ratio $E/M$ in the Nariai limit can be approximated by

$\frac{E_{\rm N}}{M_{\rm N}}\approx\epsilon_{\rm
N}\,\epsilon_{\tau}\left(\frac{\ell_{\rm pl}}{\ell}\right)^{2}\,.$ (16)

The result is an extremely small quantity, therefore justifying a shockwave
description.

###  2.5 GJW protocol for Nariai spacetimes

As the observers throw matter, they are able to generate a traversable
wormhole in the black hole interior, as well as in the inflating patch. In
that way, the observers can transmit light signals through the wormhole, as
shown in Fig. 2. Notice that we do not require a holographic theory in order
to establish this step in the protocol.

To make the previous statements more precise, we calculate the bounds below on
the amount of information transmitted during the process.

  * •

For the black hole patch, we focus on the static patch time that the message
spends behind the horizon of Fig 2, given by

$\Delta t=t_{\rm lc}+t_{\rm shift}+\delta t\,,$ (17)

where $t_{\rm lc}$ stands for the light-crossing time, $t_{\rm shift}$
corresponds to the time for gathering the required energy and $\delta t$ is
the time delay between collecting the energy and sending the message.

For the gathering time, $t_{\rm shift}$, we consider the reflection time for a
photon to go from the radius of the Nariai black hole ($r=r_{\rm N}$) to the
location of the static patch observer at $r=0$, which is given by:

$t_{\rm shift}\approx\ell\,\log{S_{\rm b}}\leavevmode\nobreak\ ,$ (18)

where $S_{\rm b}=S_{\rm N}/2$ is the Nariai black hole entropy. In the
following, we consider the minimum possible time for recovery of information,
where $t_{\rm lc}$ and $\delta t$ can be neglected. This implies that the time
difference before and after the absorption of photons, where the black hole
entropy increases due to the presence of the shockwave as $S_{\rm
b}\xrightarrow{}S_{\rm b}+S_{\rm Shock}$, is given by

$\Delta t\propto\log[S_{\rm b}+S_{\rm Shock}]\leavevmode\nobreak\ .$ (19)

In that case, the recovery time is given as

$\displaystyle\Delta\tau$ $\displaystyle\equiv\Delta t-\tau^{*}_{\rm
b}\propto\log\quantity[1+\frac{S_{\rm Shock}}{S_{\rm b}}]\leavevmode\nobreak\
,$ (20)

where $\tau^{*}_{\rm b}=\ell\log S_{\rm b}$ is the black hole scrambling time.

The condition for the wormhole to be traversable requires the recovery time to
be smaller than the original scrambling time ( $\Delta\tau<0$), such that

$-S_{\rm b}<S_{\rm Shock}<0\leavevmode\nobreak\ .$ (21)

The result is in agreement with the fact that the shock wave should have the
opposite sign than the energy of the black hole, as this kind of shock-wave
_reduces_ its entropy.

  * •

For the inflating patch, we may use the same expression as before, $\Delta
t=t_{\rm lc}+t_{\rm shift}+\delta t$, such that, neglecting the contribution
of $\delta t$, the recovery time is now given as follows

$\displaystyle\Delta t$ $\displaystyle=t_{\rm lc}+t_{\rm
shift}\propto\log[S_{\rm c}-S_{\rm Shock}]\leavevmode\nobreak\ .$ (22)

Therefore, the difference in emission times is then

$\displaystyle\Delta\tau$ $\displaystyle=\Delta
t-\tau^{*}_{c}\propto\log\quantity[1-\frac{S_{\rm Shock}}{S_{\rm
c}}]\leavevmode\nobreak\ .$ (23)

with $\tau^{*}_{c}=\ell\log S_{\rm c}$. We again notice that $\Delta\tau<0$ is
necessary for a traversable wormhole, which requires

$0<S_{\rm Shock}<S_{\rm c}\leavevmode\nobreak\ .$ (24)

This result reflects the fact that the shockwave has to increase the entropy
of the cosmological horizon in order to generate the traversable wormhole. It
also corroborates that the Gibbons-Hawking entropy is counting the degrees of
freedom laying behind the horizon.

As argued in Freivogel:2019lej ; Freivogel:2019whb , regardless of the
protocol, the maximal amount of information that can possibly be transmitted
through the wormhole in this context is bounded by

$N_{\rm bits}\lesssim r_{\rm N}^{d-1}/G_{N}\approx S_{\rm
b}\leavevmode\nobreak\ ,$ (25)

given that we cannot transfer more bits of information than those available
from the surface area of the black hole horizon, as it gets destroyed during
the process. See App. A for the relation between the number of bits and the
criterion for successful information recovery. This suggests, that the amount
of information passing through the resulting wormhole throats in SdS space can
be estimated by considering the differences

$\displaystyle\Delta S_{\rm b}$ $\displaystyle=S_{\rm b}(M)-S_{\rm b}(M-E)$
(26) $\displaystyle\Delta S_{\rm c}$ $\displaystyle=S_{\rm c}(M-E)-S_{\rm
c}(M)\,,$ (27)

where $E=\int\mathcal{P}_{\rm tot}\leavevmode\nobreak\ \mathrm{d}t$ is the
total energy exchanged during the protocol. Notice that the maximal total
information that can be transmitted in this way

$\Delta S_{\rm tot}=\Delta S_{\rm b}+\Delta S_{\rm c}=S_{\rm N}/2+(S_{\rm
dS}-S_{\rm N}/2)=S_{\rm dS}$ (28)

when $E=M_{\rm N}$. Whereas the total entropy difference $\Delta S=S_{\rm
dS}-S_{\rm N}=S_{\rm N}/2$ in 4D. Furthermore, when assuming small energies
$E/M\equiv\epsilon$ and expanding at first order, we find that

$\displaystyle s_{\rm b}$ $\displaystyle\equiv\epsilon\left(\frac{\partial
S_{\rm b}}{\partial\epsilon}\right)_{\epsilon=0}=\epsilon M\frac{\partial
S_{\rm b}}{\partial M}=E\beta_{\rm b}$ (29) $\displaystyle s_{\rm c}$
$\displaystyle\equiv-\epsilon\left(\frac{\partial S_{\rm
c}}{\partial\epsilon}\right)_{\epsilon=0}=-\epsilon M\frac{\partial S_{\rm
c}}{\partial M}=E\beta_{\rm c}\,,$ (30)

where we normalized the Euclidean periodicities $\beta$ with respect to the
observers at $r_{\mathcal{O}}$. Notice that the wormhole sizes $s_{\rm b}\neq
s_{\rm c}$ for any $M$ except the Nariai mass $M_{\rm N}$, where $\beta_{\rm
b}=\beta_{\rm c}=2\pi r_{\rm N}$.

(a)

(b)

Figure 4: The cosmological entropy $S_{\rm c}$ (orange), black hole entropy
$S_{\rm b}$ (blue), and their sum, the total entropy (red), as a function of
time in $d=3$. (a) Empty de Sitter with absorbing boundary conditions from the
Dyson sphere to capture the energy. (b) Nariai spacetime where the energy flux
freely escapes through the cosmological horizon.

Figure 4 shows the evolution of the entropies as a function of time.

##  3 Information recovery in (1+1)-dimensions

In this section, we incorporate semiclassical effects into our protocol. For
that, we adopt dS JT gravity from the S-wave reduction of the near Nariai
black holes666The island rule has been derived in Svesko:2022txo  for the type
of model under consideration, albeit for static backgrounds. However, our
findings point out that the island still obeys maximin conditions
Akers:2019lzs , and the shockwave simply shifts its location.. The system is
defined by the action

$I_{\rm dS\,JT}=\tfrac{\phi_{0}}{16\pi
G_{2}}\int_{\mathcal{M}}\mathrm{d}^{2}x\sqrt{-{g}}R+\tfrac{1}{16\pi
G_{2}}\int_{\mathcal{M}}\mathrm{d}^{2}x\sqrt{-{g}}\quantity({R}-\tfrac{2}{\ell^{2}})\phi+\tfrac{1}{8\pi
G_{2}}\int_{\partial\mathcal{M}}\mathrm{d}y\sqrt{-{h}}\,\phi{K}+I_{\rm
matter}\leavevmode\nobreak\ ,$ (31)

where $\phi$ is the dilaton field, which describes scalar mode perturbations
around the Nariai black hole solution, $G_{2}$ the effective gravitational
coupling, and $I_{\rm matter}$ denotes the bulk CFT2 theory. See appendix B
for more details on the derivation of this theory from the dimensional
reduction of a near Nariai black hole. The equations of motion of followed by
varying (31) are given by

$\displaystyle-\nabla_{\mu}\nabla_{\nu}\phi+g_{\mu\nu}\square\phi+\frac{\phi}{\ell^{2}}g_{\mu\nu}$
$\displaystyle=8\pi G_{2}\langle T_{\mu\nu}\rangle\leavevmode\nobreak\ ,$ (32)
$\displaystyle R$ $\displaystyle=2/\ell^{2}\leavevmode\nobreak\ ,$

with $\expectationvalue{T_{\mu\nu}}$ the expectation value of the stress
tensor.

#### Islands before the shockwave

We proceed to prepare the protocol described in section 2. We place detectors
near the $\mathcal{I}^{+}$ region of the inflationary patches. In this
situation, metaobservers can collect Hawking modes in weakly gravitating
regions. See Fig. 5 for an illustration of the proposal.

Figure 5: Proposal for the teleportation protocol in the dS JT gravity. Same
color labeling as in Fig. 2. The signals sent through the wormhole arrive at a
pair of detectors, $\mathbf{R}$, in an interval bounded by the points
$[r,\,r^{\prime}]$ near $\mathcal{I}^{+}$ of the inflating patch. The
radiation is purified by the island, $\mathbf{I}$ (blue), bounded by the
points $[i,\,i^{\prime}]$. The axis for the Kruskal coordinates ($U,\,V$) used
below is also shown. The gray area represents the region covered by a single
Kruskal coordinate system, which includes the static patches and either the
black hole or the inflating regions.

In the configuration of interest, we evaluate the generalized entropy (3)
considering that the state of the full Cauchy slice is pure at a given
time777Notice that in our protocol, only the global state of the bulk geometry
together with the energy pulses is pure. This occurs since we prepared Bunch
Davis state in pure de Sitter space, and later trapped some of the matter
fields into a reservoir carried by the antipodal observers. from the
complement of the region, such that $S_{\rm
gen}(\mathbf{R}\cup\mathbf{I})=S_{\rm
gen}(\overline{\mathbf{R}\cup\mathbf{I}})$. We consider a completely symmetric
configuration for the endpoints of the island and those of the detectors in
$\mathbf{R}$. Let us introduce conformal vacuum coordinates $x^{\pm}$. The
metric has the form

$\mathrm{d}s^{2}=-\Omega(x^{+},\,x^{-})^{-2}\mathrm{d}x^{+}\mathrm{d}x^{-}\leavevmode\nobreak\
,$ (33)

where we denote $\Omega(x^{+},\,x^{-})$ as the conformal factor.

The von Neumann entropy for a single interval $[j,\,k]$ is given by

$S_{\rm
vN}([j,\,k])=\frac{c}{6}\log\left[\frac{(x^{+}_{j}-x^{+}_{k})(x^{-}_{j}-x^{-}_{k})}{\Omega(x^{+}_{j},\,x^{-}_{j})\Omega(x^{+}_{k},\,x^{-}_{k})\epsilon_{j}\epsilon_{k}}\right]\leavevmode\nobreak\
,$ (34)

where the factors $\epsilon_{j}$ and $\epsilon_{k}$ denote the usual short-
distance cutoffs of the von Neumann entropy. Meanwhile, the dilaton
$\phi(\partial I)$ gives the area contribution in (3),

$S_{\rm
gen}(\mathbf{R}\cup\mathbf{I})=\tfrac{\phi(\partial\mathbf{I})}{4G_{N}}+S_{\rm
matter}(\mathbf{R}\cup\mathbf{I})\leavevmode\nobreak\ .$ (35)

We use Kruskal coordinates ($V,\,U$) to denote the metric and the _sourceless_
dilaton

$\displaystyle\mathrm{d}s^{2}=-\tfrac{4\ell^{4}}{(\ell^{2}-UV)^{2}}\mathrm{d}U\,\mathrm{d}V\leavevmode\nobreak\
,$ (36)
$\displaystyle\phi(U,\,V)=\phi_{r}\tfrac{\ell^{2}+UV}{\ell^{2}-UV}\leavevmode\nobreak\
.$

Let $(U_{i},\,V_{i})$ denote one of the endpoints of $\mathbf{I}$, and
$(U_{r},\,V_{r})$ one of the endpoint in $\mathbf{R}$, as shown in Fig. 5.
Islands in the Bunch Davis vacuum dS JT gravity have been derived in many
parts of the literature in the limit when the spatial interval distance of
regions $\mathbf{R}$ and $\mathbf{I}$ is large compared to the relative
distance between separating them (see Hartman:2020khs ; Svesko:2022txo  for
example). We need an analytic continuation in the Kruskal coordinate system to
evaluate the von Neumann entropy given that $\mathbf{I}$ and $\mathbf{R}$ are
located in the black hole and inflating patches. Writing the metric (36) in
global conformal coordinates

$\mathrm{d}s^{2}=\ell^{2}\sec^{2}\sigma\leavevmode\nobreak\
(-\mathrm{d}\sigma^{2}+\mathrm{d}\varphi^{2})$ (37)

One needs to make a shift of coordinates $\varphi\xrightarrow{}\varphi+\pi$ to
connect points in the adjacent hyperbolic patches. This amounts to shifting
the point $(V_{r},\,U_{r})$ as

$V_{r}\xrightarrow{}-\tfrac{\ell^{2}}{V_{r}},\quad
U_{r}\xrightarrow{}-\tfrac{\ell^{2}}{U_{r}}\leavevmode\nobreak\ .$ (38)

For the region below the shockwave in Fig. 5, the generalized entropy can be
then found as

$S_{\rm
gen}=\frac{1}{2G_{2}}\quantity(\phi_{0}-\phi_{r}\frac{\ell^{2}+U_{i}V_{i}}{\ell^{2}-U_{i}V_{i}}+\frac{cG_{N}}{3})+\frac{c}{3}\log\frac{4\ell^{2}(\ell^{2}+U_{i}U_{r})(\ell^{2}+V_{i}V_{r})}{\epsilon_{i}\epsilon_{r}(\ell^{2}-U_{i}V_{i})(\ell^{2}-U_{r}V_{r})}\leavevmode\nobreak\
.$ (39)

The only island space-like separated from $(V_{r},\,U_{r})$ is given by

$\displaystyle V_{i}$
$\displaystyle=\tfrac{U_{r}\epsilon}{3}+\mathcal{O}(\epsilon^{2})\leavevmode\nobreak\
,\quad
U_{i}=\tfrac{V_{r}\epsilon}{3}+\mathcal{O}(\epsilon^{2})\leavevmode\nobreak\
,$ (40)

where $\epsilon\equiv\tfrac{G_{2}c}{\phi_{r}}$ is a small parameter in the
semiclassical regime, $\epsilon\ll 1$.

However, in the extermination of the generalized entropy (3) one must also
include the possibility for the island to be the empty set, $\emptyset$. In
that case, we can produce a Page curve transition between the trivial and non-
trivial island (40) saddles. One can derive that this transition occurs in
global coordinates (37) when the detector in the interval $[r,r^{\prime}]$ for
$\varphi_{r}=-\varphi_{r^{\prime}}\equiv\varphi_{\rm Page}$ and
$\sigma_{R}\simeq\tfrac{\pi}{2}$ constant in Fig. 5 is given by
Hartman:2020khs

$\tan\varphi_{\rm
Page}\propto\exp[\tfrac{3}{2G_{2}c}\quantity(\phi_{0}+\phi_{r}+\tfrac{cG_{2}}{3})]\leavevmode\nobreak\
.$ (41)

#### Adding the shockwaves

We allow the observers to release negative energy pulses toward the black hole
interior. Consider for instance with respect to the south pole observer in
Fig. 5. The stress tensor is given by

$\displaystyle\langle T_{VV}(V)\rangle$
$\displaystyle=\alpha\frac{\delta(V-V_{S})}{V_{S}}\leavevmode\nobreak\ ,$ (42)

which describes a shockwave moving towards the black hole interior along the
null ray $V=V_{S}$. As in Sec. 2 we employ $\alpha<0$ to form a traversable
wormhole on the black hole interior, and we consider that
$\absolutevalue{\alpha}\ll 1$. See further comments on the small shift
approximation in App. C.

We would like to compare the island location with the shockwave insertion in
the protocol, to inquire under what conditions the traversable wormhole enters
in causal contact with the island region. Given the stress tensor (42), the
dilaton solution in (36) would be modified according to the graviton equation
of motion of (31), resulting in

$\phi=\phi_{0}-\phi_{r}\frac{\ell^{2}+U_{i}V_{i}}{\ell^{2}-U_{i}V_{i}}+\frac{cG_{2}}{3}-\frac{8\pi
G_{2}\alpha}{V_{S}}\frac{\ell^{2}-V_{S}U_{i}}{\ell^{2}-V_{i}U_{i}}(V_{i}-V_{S})\theta(V_{i}-V_{S})\leavevmode\nobreak\
.$ (43)

We want to identify null congruences, which in JT gravity are described by
curves satisfying $\partial_{\pm}\Phi=0$ Aalsma:2021kle . We denote them as
follows

$\displaystyle\gamma:\quad V$
$\displaystyle=V_{S}\frac{2G_{2}\alpha}{\phi_{r}}+\mathcal{O}(\alpha^{3})\leavevmode\nobreak\
,$ (44) $\displaystyle\delta^{+}:\quad U$
$\displaystyle=\frac{\ell^{2}}{V_{S}}\frac{\phi_{r}}{2G_{2}\alpha}+\mathcal{O}(\alpha),$
(45) $\displaystyle\delta^{-}:\quad U$
$\displaystyle=-\frac{\ell^{2}}{V_{S}}\frac{2G_{2}\alpha}{\phi_{r}}+\mathcal{O}(\alpha^{3})\leavevmode\nobreak\
.$ (46)

The apparent horizon is the outermost surface, with $\partial_{+}\Phi=0$ and
$\partial_{-}\Phi<0$ Aalsma:2021bit , which is allowed by both of the curves
$\delta^{\pm}$. The curve describing the null shift for $\alpha<0$, which we
implement in the protocol to violate ANEC, corresponds to the one that reduces
the size of the black hole, which is $\delta^{-}$ for $V>V_{S}$. This means
that the null shift of the apparent horizon to leading order in $\alpha$ is
given by (46).

In our conventions, the horizon in the Penrose diagrams remains fixed while
the light signals are shifted. As the observer in the north pole sends a light
signal to the causal diamond of the island $\mathbf{I}$ in Fig. 5, it will get
shifted by (46), and it will appear at the location $\mathbf{R}$ of the
adjacent hyperbolic slice. The signal sent along $U=U_{\rm ls}$ enters the
inflating patch along the line:

$U=U_{\alpha}\equiv\frac{\ell^{2}}{U_{\rm ls}}+\frac{8\pi
G_{2}\absolutevalue{\alpha}}{4\pi\phi_{r}}\leavevmode\nobreak\ ,$ (47)

following the analytic continuation (38). Then, the light ray would intercept
the causal diamond of the island, found in (40), as long as
$U_{r}<U_{\alpha}$.

### Repeating the protocol

As we noticed in (16), the energy released as shockwaves in our protocol is
generically small, as it is suppressed by a $\ell_{\rm pl}^{2}$ factor. The
static patch observers can then decide on performing multiple shockwave
protocols, as long as it obeys the bound found in (28). In this situation, one
must account for backreaction on the island location. Given the presence of
the stress tensor in (42), one must find the map to vacuum coordinates
$x^{\pm}$ in (34). We can find such a map using the conformal anomaly of the
stress tensor

$\expectationvalue{T_{VV}(V)}=-\tfrac{c}{24\pi}\quantity{x^{+},\,V}.$ (48)

Taking the Ansatz

$x^{+}=V+\alpha\,f(V),\quad x^{-}=U\leavevmode\nobreak\ ,$ (49)

for $\alpha\ll 1$ we obtain the following relation at leading order in
$\alpha$

$-\tfrac{c}{24\pi}f^{\prime\prime\prime}(V)=\tfrac{1}{V_{S}}\delta(V-V_{S})\leavevmode\nobreak\
,$ (50)

where the solution is found to be

$f(V)=-\tfrac{12\pi}{c\leavevmode\nobreak\
V_{S}}(V-V_{S})^{2}\Theta(V-V_{S})\leavevmode\nobreak\ .$ (51)

Consider the vacuum coordinate above, for the von Neumann entropy in (34) and
the backreacted dilaton in (43). We will be interested in the regime

$\epsilon\ll\absolutevalue{\alpha}\ll 1\leavevmode\nobreak\ .$ (52)

The extremization of the generalized entropy (3) leads to the shifted island
location

$\displaystyle V_{i}$ $\displaystyle=\tfrac{U_{r}\epsilon}{3}+16\pi
G_{2}\alpha\tfrac{V_{S}\epsilon}{c}+\mathcal{O}(\epsilon^{2},\leavevmode\nobreak\
\alpha^{2}\epsilon)\leavevmode\nobreak\ ,$ (53) $\displaystyle U_{i}$
$\displaystyle=\tfrac{V_{r}\epsilon}{3}-16\pi
G_{2}\alpha\tfrac{\epsilon}{cV_{S}}+\mathcal{O}(\epsilon^{2},\leavevmode\nobreak\
\alpha^{2}\epsilon)\leavevmode\nobreak\ ,$ (54)

for which, the von Neumann entropy collected on $\mathbf{R}$ due to the island
is given as

$S_{\rm
isl}=\tfrac{\phi_{0}}{G_{2}}+\tfrac{c}{6}\quantity(1-\tfrac{3}{\epsilon}+2\log\tfrac{4\ell^{4}}{\epsilon_{i}\epsilon_{r}(\ell^{2}-U_{r}V_{r})})+16\pi
G_{2}\alpha+\tfrac{\epsilon\quantity(cV_{S}V_{r}U_{r}+48\pi
G_{2}\alpha\quantity(V_{S}^{2}V_{r}-\ell^{2}U_{r}))}{9\ell^{2}V_{S}}\leavevmode\nobreak\
.$ (55)

The result agrees with Hartman:2020khs ; Svesko:2022txo  when $\alpha=0$.

We now consider the situation illustrated in Fig. 5 where
$V_{r}U_{r}\approx\ell^{2}$ (corresponding to $\mathbf{R}$ being close to
$\mathcal{I}^{\pm}$). (55) to $\mathcal{O}(\alpha)$ in the regime (52) becomes

$S_{\rm isl}\simeq S_{\rm
BH}+\tfrac{c}{3}\log\frac{4}{\epsilon_{i}\epsilon_{r}\delta L^{2}}+16\pi
G_{2}\alpha\leavevmode\nobreak\ ,$ (56)

where $\delta L^{2}=\ell^{2}-U_{r}V_{r}$, and $S_{\rm
BH}=\tfrac{1}{2G_{2}}\quantity(\phi_{0}-\phi_{r})$ is the Bekenstein-Hawking
entropy of the Nariai black hole. Notice that (56) provides an information
bound on the amount of information that can be transmitted between the
observers through the island. The classical regime when $S_{\rm isl}\simeq
S_{\rm BH}$ agrees with (25).

As we repeat the procedure, the total null shift of the light rays intercepted
by the north pole observer in Fig. 5 would then correspond to (47) with
$\alpha\rightarrow\alpha^{\prime}<0$ for the net null shift, and the condition
of intercepting the island corresponds to $U_{r}<U_{\alpha^{\prime}}$.

##  4 Discussion and outlook

In summary, we proposed a new protocol to form traversable wormholes in
Schwarzschild-de Sitter black holes. Due to the lack of a holographic
description, a conceptual difficulty in the protocol arises. Consider
generating a traversable wormhole without coupling between antipodal static
patch observers in the near Nariai black hole. It might seem that one can
produce a signal between causally disconnected regions when there is no proper
synchronization of the observers to emit the shockwaves responsible for the
traversable wormhole. For this reason, we started from the Bunch-Davis state
of de Sitter space as the initial configuration. The observers are then
maximally entangled, and their clocks are synchronized. We also require that
the pair of observers store energy coming from the Hawking radiation of the
cosmological horizon, so they can later collapse the energy to form a near
Nariai black hole. We considered a symmetric protocol, where each of the
observers throws an excess of positive energy into the cosmological horizon,
which results in a negative energy excess into the black hole horizon. Then,
the violation of ANEC will follow, which forms a traversable wormhole in the
black hole interior, as well as in the inflationary region. Each observer can
then send light signals at the appropriate time, such that they cross the
newly formed wormhole. They are then able to apply an appropriate recovery
channel and decode the information by an appropriate protocol. In order to
quantify such recovery, we analyzed bounds on the amount of information that
can be transmitted through the wormhole. In the second part of the manuscript,
we incorporated semiclassical effects explicitly by establishing the protocol
in dS JT gravity and probing the island region. We considered how information
can be transmitted through a wormhole in the island according to metaobservers
in the inflating region. Notice that our protocol does not advocate a
particular notion of de Sitter holography and might as well provide new hints
towards a yet-to-be-known holographic dual theory to the SdS space-time. In
addition, when not restricted to a shockwave description ($E/M\ll 1$), the
maximal amount of bits that can be transferred from one static patch to the
other is equal to the total number of bits contained in empty de Sitter.
Naively, one could have guessed that solely the entropy of the black hole
would provide a bound on the possible information exchange ($\Delta
S_{b}=S_{N}/2$). However, releasing $M_{N}$ of energy will not only destroy
the Nariai black hole but extend the cosmological horizon as well ($\Delta
S_{c}=S_{dS}-S_{N}/2$).

We now comment on some extensions of our work.

### Other extensions with the island formula

One of the main drives in this work was incorporating semiclassical effects to
probe the island region with our protocol. This was explicitly performed in
$(1+1)$-dimensions, but we expect it can be similarly done in higher
dimensions. To study such effects in an explicit setting, perhaps double
holographic models with dS braneworlds Aguilar-Gutierrez:2023tic , would be a
fruitful direction. One might be able to incorporate quantum SdS black holes
Emparan:2022ijy ; Panella:2023lsi  in such a setting to treat quantum
backreaction effects exactly.

Moreover, the 2-dimensional analysis was based on the s-wave reduction of a
near Nariai black hole configuration, such that we do not have to deal with
conical defects Morvan:2022ybp  of general SdS spacetime. However, given the
instability of this configuration, one should in principle describe the
evaporation of arbitrary dS black holes from dS JT gravity. Modeling the
evaporation of dS black holes in dS JT gravity has been made in
Balasubramanian:2020xqf ; Baek:2022ozg . There has been a lot of progress in
information recovery during an adiabatic evaporation regime with free particle
bulk matter fields Hollowood:2021nlo  that could be extended to SdS space.

An alternative analysis of the islands could be persuaded with the set-up of
Aalsma:2022swk . In this model where the radiation is collected in a subregion
of asymptotically flat space coupled to the dS JT gravity model, instead of
considering the detectors near $\mathcal{I}^{+}$ of the inflating patches,
studied here. It might be possible to reproduce our protocol in this setting
with better analytic control over backreaction effects. In this model, one can
encounter the islands in the inflating patch of dS2 space due to backreaction
effects, which do not affect the observers in the asymptotically flat region.
Perhaps this setting would allow us to study a traversable wormhole on the
island in the inflating region which we have discussed in Sec. 3.

### States and algebra of observables

In the more general case with a Schwarzschild-de Sitter black hole, there is a
conceptual difficulty in properly defining states given that the cosmological
horizon and the black hole horizon are not in thermal equilibrium. In this
case, the construction of Hilbert spaces from Tomita-Takasaki theory can enter
into ambiguities for defining thermofield double states and studying the
algebra of semiclassical states on the SdS background. It would be very
interesting to make a rigorous treatment of the von Neumann algebra of these
types of black holes, as there are some expectations for generic closed
universes Jensen:2023yxy . For our protocol, it might be possible to make a
TFD-type of state preparation of modes coming from the near Nariai black hole
and cosmological horizons, as well as the modes contained in the thermal
baths. We leave a more rigorous treatment of the quantum states involved in
our protocol for future directions. Moreover, it would be interesting to study
whether the preparation of a vacuum state can be extended in the presence of
conical defects for general SdS black holes.

### Stretched horizon holography

Let us reiterate that our protocol does not employ a particular realization of
de Sitter space holography. However, stretched horizon holography
Susskind:2021omt  might allow for different modifications. In this framework,
the stretched horizon represents a spacetime region where the holographic dual
to de Sitter space is located. This is based on the observation of the
doubled-scaled SYK model might display some features of dS JT gravity
Susskind:2022bia ; Rahman:2022jsf ; Goel:2023svz  in certain regimes. This
approach mirrors several aspects of the AdS/CFT correspondence in that
observables can be gravitationally dressed with respect to the stretched
horizon.

It might be possible to perform the protocol in this framework following the
same steps as illustrated in the protocol above according to the initial state
preparation in empty de Sitter space888Notice that the worldline observers in
pure de-Sitter space Fig. 1 would also have access to an island near the
cosmological horizon Shaghoulian:2021cef , which has been noticed to have
different puzzling features, such as it moving backward in time
Sybesma:2020fxg  and violating entanglement wedge nesting Shaghoulian:2021cef
. and in transmitting information for SdS black holes. For probing the island
region, we might consider an alternative configuration to Fig. 5, which we
illustrate in Fig. 6. In this case, the pair of detectors are no longer
located at $\mathcal{I}^{+}$, but rather on the stretched horizon, extending
to the static patch observers. We expect the qualitative results would be just
as the found for the configuration in Fig. 5 for recovering information about
the island region.

Figure 6: Alternative realization of the protocol employing the stretched
horizon holographic proposal of Susskind:2021dfc .

It would also be interesting to study if a realization of the protocol could
be performed explicitly in a double-scaled SYK model in the limit that might
reproduce dS JT gravity Susskind:2022bia . In that case, one might be able to
adapt the realization of the GJW protocol in different contexts of the SYK
Maldacena:2017axo ; Maldacena:2018lmt ; Gao:2019nyj ; Gao:2023gta .

### Traversable wormhole production rates

Recently, Bintanja:2023vel  provided the false vacuum decay rate of a pair of
Reissner-Nordstrom (RN) AdS4 black holes into traversable wormholes connecting
the geometries. In the microcanonical ensemble, they find that

$\Gamma\sim\mathrm{e}^{\Delta S+2\beta(M_{\rm b}-M_{0})},$ (57)

where $\Delta S$ is the total entropy of the pair of RN AdS4 black holes,
$\beta$ refers to their (inverse) temperature, $M_{\rm b}$ the black hole
mass, and $M_{0}$ the mass of the traversable wormhole. It would be very
interesting to study this type of process in the context of our protocol.
Given that we required opposite energy pulses to produce the wormhole, we
expect the contribution of the traversable wormhole from the black hole and
the cosmological horizon patches will approximately cancel each other in the
near Nariai configuration, such that (57) might be modified to

$\Gamma\sim\mathrm{e}^{\Delta S+2\beta_{\rm N}M_{\rm N}}\leavevmode\nobreak\
,$ (58)

according to the static patch observers. This would reflect that thermal
fluctuations can also open traversable wormholes between the black hole and
inflating regions for Nariai spacetimes. However, the size of such wormholes
would not be significant enough to be of practical use. It might be useful to
study the quasi-local thermodynamics approach to cosmological and black hole
horizons Svesko:2022txo ; Banihashemi:2022jys ; Banihashemi:2022htw ;
Jacobson:2022gmo  to make this relation precise.

## Acknowledgements

We thank Stefano Baiguera, Paul Balavoine, Ben Freivogel, Viktor Jahnke, Jan
Pieter van der Schaar, Andrea Legramandi, Raghu Mahajan, Zhenbin Yang, and
Ying Zhao for useful discussions. SEAG thanks the University of Amsterdam and
the Delta Institute for Theoretical Physics for their hospitality and support
during this project. The work of SEAG is partially supported by the KU Leuven
C1 grant ZKD1118 C16/16/005. EM is supported by Ama Mundu Technologies (Adoro
te Devote Grant 2019). RE is supported by the Dushi Zhuanxiang Fellowship and
acknowledges a Shuimu Scholarship as part of the Shuimu Tsinghua Scholar
Program.

##  Appendix A Hayden-Preskill protocol for near Nariai black holes

We refer the reader to Aalsma:2021kle  for an introduction to Hayden-Preskill
protocol Hayden:2007cs  for information recovery in pure de Sitter space, as
well as the original publication Hayden:2007cs  in the black hole context.

The basic principle of operation of this protocol is illustrated in Fig. 7.

Figure 7: Quantum circuit representation of the Hayden-Preskill protocol,
where $\ket{\psi_{\rm EB}}$ and $\ket{\psi_{\rm MN}}$ are initially maximally
entangled states. Figure based on Hayden:2007cs .

From purely quantum information arguments, we search for a criterion
establishing if the information transfer between the antipodal observers in
the near Nariai black hole spacetime has been successful.

We consider Fig. 7 where one of the observers, Alice possesses subsystem $M$
which is maximally entangled with a subsystem $N$ carried on by Charlie.
Meanwhile, an observer Bob has access to subsystem $E$ which is entangled with
a black hole, $B$. Alice waits until $E$ is maximally entangled with $B$, when
she throws $M$ into $B$. We model the absorption of the message as a random
unitary process $V^{B}$. This results in new subsystems, $B^{\prime}$
describing the black hole after absorption, and $R$ for the radiation. In that
process, $M$, the purifier of $N$, gets transferred from $B^{\prime}$

$\ket{\psi}=\tfrac{1}{\sqrt{B^{\prime}}}\sum_{i}\ket{i}_{B^{\prime}}\ket{0}_{R}\ket{i}_{N}\leavevmode\nobreak\
.$ (59)

to $R$

$\ket{\psi^{\prime}}=\tfrac{1}{\sqrt{B^{\prime}}}\sum_{i}\ket{0}_{B^{\prime}}\ket{i}_{R}\ket{i}_{N}\leavevmode\nobreak\
.$ (60)

where $\norm{B^{\prime}}$ refers to the Hilbert space dimension of B’. The
criterion in Hayden:2007cs  for successful message transmission is that

$\norm{\rho_{MB^{\prime}}-\rho_{M}\otimes\rho_{B^{\prime}}}\ll
1\leavevmode\nobreak\ ,$ (61)

where $\rho_{M}$, $\rho_{B^{\prime}}$, $\rho_{MB^{\prime}}$ refer to the
reduced density matrices by tracing out the subsystems $B^{\prime}N$, $NM$ and
$N$, and the norm is taken with respect to the Haar measure, defined as

$\int\mathrm{d}V^{B}\norm{\rho_{MB^{\prime}}-\rho_{M}\otimes\rho_{B^{\prime}}}=2^{-2C}\leavevmode\nobreak\
,$ (62)

where $C$ refers to the additional qubits in the radiation with respect to the
number of qubits in Alice’s message. This means that as soon as the black hole
has radiated more information than the one encoded in the original light
signal sent by Alice, Bob gains access to it.

We can directly apply this protocol to transmit light signals in the near
Nariai black hole background in Fig. 2. However, notice that in this case,
Alice does not need to wait until the Page time to throw $M$ into $N$, since
the preparation of the state from the BD vacuum already maximally entangles
the pair of antipodal observers, corresponding to the radiation received by
Bob, $R$, with the black hole $B$. As we saw in Sec. 2, we achieved
information recovery by transmitting signals across the traversable wormhole
produced in the near Nariai black hole over a time scale determined (20).
Meanwhile, if we perform the information transfer in the inflationary region,
the time scale for recovery is given by (23).

The reader is referred to Hayden:2007cs ; Aalsma:2021kle  for considerations
about the cloning paradox, and how they are avoided in the context of the HP
protocol for the black hole and pure de Sitter cases.

##  Appendix B S-wave reduction of SdS black holes

We follow the derivation of the full-reduction dS JT gravity model given
Svesko:2022txo . In our notation, hats are used for $(d+1)$-dimensional
quantities and unhatted for $(1+1)$-dimensional ones. Starting from Einstein
gravity in $(d+1)$-dimensions

$I=\tfrac{1}{16\pi
G_{N}}\int_{\hat{\mathcal{M}}}\mathrm{d}^{d+1}X\sqrt{-\hat{g}}(\hat{R}-2\Lambda)+\tfrac{1}{8\pi
G_{N}}\int_{\partial\hat{\mathcal{M}}}\mathrm{d}^{d}Y\sqrt{-\hat{h}}\,\hat{K}$
(63)

with a positive cosmological constant

$\Lambda=\frac{d(d-1)}{2\ell^{2}}>0\leavevmode\nobreak\ .$ (64)

The general solution is given by the SdS metric (5). We then consider
dimensional reduction with the metric Ansatz,

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=\hat{g}_{MN}\mathrm{d}X^{M}\mathrm{d}X^{N}$ (65)
$\displaystyle=g_{\mu\nu}(x)\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+r_{\rm
N}^{2}\Phi^{\tfrac{2}{d-1}}\mathrm{d}\Omega^{2}_{d-1}$

where $M,\,N=0,1,...,d$; $\mu,\,\nu=0,\,1,$ and $\Phi(x)$ is the dilaton. The
EFT coming from the top-down spherical reduction of SdS black holes is then
simply:

$I=\tfrac{1}{16\pi
G_{2}}\int_{\mathcal{M}}\mathrm{d}^{2}x\sqrt{-g}\quantity[\Phi
R+U(\Phi)]+\tfrac{1}{8\pi G_{2}}\int_{\partial\mathcal{M}}\sqrt{-h}\Phi K,$
(66)

with

$\displaystyle
U(\Phi)=\tfrac{d-1}{\ell^{2}}\quantity(\Phi^{-\tfrac{1}{d-1}}-\Phi^{\tfrac{1}{d-1}})\leavevmode\nobreak\
,$ (67) $\displaystyle
G_{2}=\frac{G_{N}}{\ell^{d-1}\Omega_{d-1}}\leavevmode\nobreak\ .$

The resulting equations of motion are

$\displaystyle(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\nabla^{2})\Phi+\tfrac{g_{\mu\nu}}{2}U(\Phi)=-8\pi
G_{2}T_{\mu\nu}^{(m)}$ (68) $\displaystyle
R=U^{\prime}(\Phi)\leavevmode\nobreak\ ,$ (69)

where we added a $(1+1)$-dimensional bulk-matter term $I_{m}$ generating the
matter stress-tensor $T_{\mu\nu}^{\rm(m)}$. We also notice that the minimum of
the potential (67) is located when $\Phi=1$, corresponding to the Nariai black
hole limit999Alternatively, we could set $\phi_{0}\neq 1$ and rescale $\ell$.,
for which

$\displaystyle U(\Phi=1)$ $\displaystyle=0\leavevmode\nobreak\ ,$ (70)
$\displaystyle\evaluated{\derivative{U}{\Phi}}_{\Phi=1}$
$\displaystyle=-\frac{2}{\ell^{2}}\leavevmode\nobreak\ .$

If we take $\Phi=\phi_{0}+\phi$, with $\phi_{0}=1$, we can perform a series
expansion for $\phi\ll 1$ and truncate the action (66) to first order in
$\phi$ we recover the so-called dS2 JT full-reduction model,

$I_{\rm dS\,JT}=I_{0}+\tfrac{1}{16\pi
G_{2}}\int_{\mathcal{M}}\mathrm{d}^{2}x\sqrt{-{g}}\quantity(\phi{R}-\tfrac{2}{\ell^{2}}\phi)+\tfrac{1}{8\pi
G_{2}}\int_{\partial\mathcal{M}}\mathrm{d}y\sqrt{-{h}}\,\phi{K}\leavevmode\nobreak\
,$ (71)

where $I_{0}=\tfrac{\phi_{0}}{16\pi
G_{2}}\int_{\mathcal{M}}\mathrm{d}^{2}x\sqrt{-{g}}R$ refers to the topological
term in the action. Notice that the addition of the dilaton mode breaks
asymptotic dS2 space isometries, namely reparametrizations of coordinates at
$\mathcal{I}^{+}$ Maldacena:2019cbz .

##  Appendix C Shifts from metric gluing

In shenkbutter , the authors provide a convenient way to obtain the shift in
trajectory experienced by a probe message following a null-like geodesic. We
extend this formalism to Schwarzschild-de Sitter. Let us express Kruskal
coordinates, introduced in (36), in terms of static patch coordinates (5, 6):

$\displaystyle U_{\rm b,\,c}$
$\displaystyle=\mathrm{e}^{\frac{f^{\prime}(r_{\rm
b,\,c})}{2}\left(r_{*}(r)-t\right)}\leavevmode\nobreak\ ,$ (72) $\displaystyle
V_{b,c}$ $\displaystyle=-\mathrm{e}^{\frac{f^{\prime}(r_{\rm
b,\,c})}{2}\left(r_{*}(r)+t\right)}\,,$

centered at the horizons $r_{\rm b,\,\rm c}$ for $0\leq r<r_{\mathcal{O}}$ and
$r_{\mathcal{O}}\leq r<\infty$ respectively. Evaluating the tortoise
coordinate $\mathrm{d}r_{*}=\mathrm{d}r/f(r)$ results in

$\mathrm{d}s^{2}=\left\\{\begin{array}[]{ll}-\frac{4f(r)}{f^{\prime}(r_{\rm
b})}\mathrm{e}^{-f^{\prime}(r_{\rm b})r_{*}(r)}\mathrm{d}U_{\rm
b}\,\mathrm{d}V_{\rm
b}+r^{2}\mathrm{d}\Omega_{d-1}^{2}\equiv\mathrm{d}s^{2}_{\rm b}&\quad\,0\leq
r<r_{\mathcal{O}}\\\ -\frac{4f(r)}{f^{\prime}(r_{\rm
c})}\mathrm{e}^{-f^{\prime}(r_{\rm c})r_{*}(r)}\mathrm{d}U_{\rm
c}\mathrm{d}V_{\rm
c}+r^{2}\mathrm{d}\Omega_{d-1}^{2}\equiv\mathrm{d}s^{2}_{\rm
c}&\quad\,r_{\mathcal{O}}\leq r<\infty\end{array}\right.\,.$ (73)

When gluing two SdS space-times of different masses to each other, both the
black hole and cosmological line elements will experience shifts along
$U=U_{0}$ and $V=V_{0}$ respectively,

$\displaystyle\mathrm{d}s^{2}_{\rm b}$
$\displaystyle=\begin{cases}\mathrm{d}s^{2}_{\rm b}&\quad\,U\leq U_{0}\\\
\mathrm{d}\tilde{s}^{2}_{\rm b}&\quad\,U_{0}<U\end{cases}\leavevmode\nobreak\
,$ (74) $\displaystyle\mathrm{d}s^{2}_{\rm c}$
$\displaystyle=\begin{cases}\mathrm{d}s^{2}_{\rm c}&\quad\,V\leq V_{0}\\\
\mathrm{d}\tilde{s}^{2}_{\rm c}&\quad\,V_{0}<V\end{cases}\leavevmode\nobreak\
,$

where we simplified the notation to $(U_{\rm b,\rm c},\,V_{\rm b,\rm
c})\rightarrow(U,\,V)$ for more clarity. We focus on the black hole region
$0\leq r<r_{\mathcal{O}}$, and introduce a perturbation at $t=t_{0}$ forming a
shell propagating along the surface

$\displaystyle\tilde{U}_{0}$
$\displaystyle=\mathrm{e}^{\frac{\tilde{f}^{\prime}(\tilde{r_{\rm
b}})}{2}\left(\tilde{r}_{*}(r_{\mathcal{O}})-t_{0}\right)}\leavevmode\nobreak\
,$ (75) $\displaystyle U_{0}$
$\displaystyle=\mathrm{e}^{\frac{f^{\prime}(r_{\rm
b})}{2}\left(r_{*}(r_{\mathcal{O}})-t_{0}\right)}\,,$

such that the gluing/matching condition relating $\tilde{V}$ to $V$ can be
expressed as follows

$\displaystyle\tilde{U}_{0}\tilde{V}$
$\displaystyle=-\mathrm{e}^{\tilde{f}^{\prime}(\tilde{r}_{\rm
b})\tilde{r}_{*}(r)}\leavevmode\nobreak\ ,$ (76) $\displaystyle\quad U_{0}V$
$\displaystyle=-\mathrm{e}^{f^{\prime}(r_{\rm b})r_{*}(r)}\,.$

When working in the small energy limit $E/M=\epsilon\ll 1$, $\tilde{U}_{0}$
can be well approximated by $U_{0}$, allowing us to deduce that at first order
the shift along the $V$ coordinate can be expressed like so

$\displaystyle\tilde{V}$
$\displaystyle=V-\frac{\epsilon}{U_{o}}\left(\frac{\partial}{\partial\epsilon}\mathrm{e}^{\tilde{f}^{\prime}(\tilde{r}_{\rm
b})\tilde{r}_{*}(r)}\right)_{\epsilon=0}$ (77) $\displaystyle\equiv
V+\alpha\,.$

As was pointed out in shenkbutter , $\mathrm{e}^{f^{\prime}(r_{\rm
b})r_{*}(r)}$ can be compactly written (in $4D$) as

$\displaystyle\mathrm{e}^{f^{\prime}(r_{\rm b})r_{*}(r)}$
$\displaystyle=\left(\frac{r-r_{\rm b}}{\ell}\right)\left(\frac{r-r_{\rm
c}}{\ell}\right){}^{-\frac{r_{\rm c}\left(2r_{\rm b}+r_{\rm c}\right)}{r_{\rm
b}\left(r_{\rm b}+2r_{\rm c}\right)}}\left(\frac{r_{\rm b}+r_{\rm
c}+r}{\ell}\right){}^{\frac{r_{\rm c}^{2}-r_{\rm b}^{2}}{r_{\rm b}\left(r_{\rm
b}+2r_{\rm c}\right)}}$ (78) $\displaystyle\equiv(r-r_{\rm b})\,C_{\rm
b}(r,r_{\rm b})\,,$

and the shift $\alpha$ becomes

$\displaystyle\alpha$ $\displaystyle=\frac{G_{N}E\beta_{\rm b}}{2\pi r_{\rm
b}U_{0}}\frac{\partial}{\partial r_{\rm b}}\bigg{(}(r_{\rm b}-r)\,C_{\rm
b}(r,r_{\rm b})\bigg{)}\,,$ (79)

where we used $\tilde{M}=M(1-\epsilon)$ and normalized the temperature with
respect of the observer radius101010Using the normalized temperature in
$dM=TdS$ allows to write  $\frac{\partial r_{\rm h}}{\partial
M}=\frac{\partial r_{\rm h}}{\partial S_{\rm h}}\frac{\partial S_{\rm
h}}{\partial M}=\frac{\beta_{\rm h}G_{N}}{2\pi r_{\rm b}}\,.$ (80) .
Obviously, the same is happening in the cosmological region
$r_{\mathcal{O}}\leq r<\infty$ (along $V$), such that the coordinate shifts
can be expressed as

$\displaystyle\alpha_{\rm b}$ $\displaystyle=\frac{r_{\rm
b}}{2U_{0}}\frac{s_{\rm b}}{S_{\rm b}}\frac{\partial}{\partial r_{\rm
b}}\bigg{(}(r_{\rm b}-r)\,C_{\rm b}(r,r_{\rm b})\bigg{)}\leavevmode\nobreak\
,$ (81) $\displaystyle\alpha_{\rm c}$ $\displaystyle=\frac{r_{\rm
c}}{2V_{0}}\frac{s_{\rm c}}{S_{\rm c}}\frac{\partial}{\partial r_{\rm
c}}\bigg{(}(r_{\rm c}-r)\,C_{\rm c}(r,r_{\rm c})\bigg{)}\,,$

where we restored the indices and introduced the wormhole entropies $s_{\rm
b,\rm c}=E\beta_{\rm b,\rm c}$. Notably, in the Nariai limit $r_{\rm b,\rm
c}\rightarrow r_{\rm N}$ the function $C_{\rm N}(r,r_{\rm N})=(r-r_{N})^{-1}$,
such that the shifts tend to 0. However, for $M\neq M_{N}$ and a gluing far in
the past, the value for $r$ is getting pushed towards the respective horizon
radius. We then find

$\displaystyle\alpha_{\rm b}$ $\displaystyle=\frac{r_{\rm
b}}{2U_{0}}\frac{s_{\rm b}}{S_{\rm b}}C_{\rm b}(r_{\rm b},r_{\rm
b})\leavevmode\nobreak\ ,$ (82) $\displaystyle\alpha_{\rm c}$
$\displaystyle=\frac{r_{\rm c}}{2V_{0}}\frac{s_{\rm c}}{S_{\rm c}}C_{\rm
c}(r_{\rm c},r_{\rm c})\,.$

Notice again that a probe message sent towards the black hole will experience
a different shift than a probe message sent towards the cosmological horizon
when encountering shockwaves of the same magnitude $E$.

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