# The Page curve and baby universes

Donald Marolfa Henry Maxfieldb Department of Physics, University of
California, Santa Barbara, CA 93106, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Black hole thermodynamics suggests that, in order to describe the physics of
distant observers, one may model a black hole as a standard quantum system
with density of states set by the Bekenstein-Hawking entropy
$S_{\mathrm{BH}}$. This idea has long been considered to be in strong tension
with Hawking’s prediction that radiation from black holes is nearly thermal,
and with low-energy gravity more generally. But the past two years have shown
that low-energy gravity _does_ offer a self-consistent description of black
hole evaporation consistent with the above idea, and which in particular
reproduces the famous Page curve. We provide a brief overview of this new
paradigm, focusing on Lorentz-signature asymptotically-flat spacetimes, and
emphasizing operationally-defined observables that probe the entropy of
Hawking radiation.

##  I Introduction

The past two years have witnessed a startling transformation in discussions of
black hole information. Before 2019, low-energy gravitational physics appeared
incontrovertibly to lead to information loss in the evaporation of black
holes, raising questions about pair-production rates for ‘remnants’ of black
hole evaporation, the status of the first and second laws of black hole
thermodynamics, and consistency with string theory and AdS/CFT. These tensions
suggested to many physicists that black holes might instead enjoy a feature
which we will call ‘Bekenstein-Hawking unitarity’ (BH unitarity): in order to
describe the physics of distant observers, one may model a black hole as a
standard quantum system with density of states set by the Bekenstein-Hawking
entropy $S_{\mathrm{BH}}$, with unitary evolution (up to couplings with its
environment). But BH unitarity appeared to require truly new physics coming
from some complete formulation of quantum gravity and which was not at all
apparent in the low-energy theory. See e.g. Mathur:2009hf ; Harlow:2014yka ;
Unruh:2017uaw ; Marolf:2017jkr  for reviews.

In striking contrast, it is now clear that low-energy gravity _does_ in fact
offer a self-consistent description of black hole evaporation in which BH
unitarity holds Penington:2019npb ; Almheiri:2019psf ; Penington:2019kki ;
Almheiri:2019qdq ; Marolf:2020xie ; Almheiri:2020cfm ; Marolf:2020rpm . The
key piece of evidence is a semiclassical calculation of operationally-defined
measurements of the entropy of radiation, involving ‘quantum-extremal islands’
and ‘replica wormholes’. This calculation reproduces the ‘Page curve’ shown in
figure 1, long regarded as a smoking gun for BH unitarity. As a result, the
outcomes of any operational tests performed on the radiation (as predicted by
low-energy gravity) are consistent with unitary evolution and density of
states $e^{S_{\mathrm{BH}}}$.

Figure 1: The Page curve for the entropy of Hawking radiation emitted before
time $u$ (solid curve). In Hawking’s original calculation the entropy of
radiation emitted during evaporation is always equal to the radiation’s
thermal entropy $S_{\mathrm{thermal}}$ (the always-increasing dotted curve).
But BH unitarity predicts that operational measurements of radiation emitted
during evaporation will be consistent with a density of states set by
$S_{\mathrm{BH}}\sim\frac{A}{4G}$ (dashed curve), and since to good
approximation the emitted radiation is entangled only with the black hole this
$S_{\mathrm{BH}}$ must also bound measurements of the radiation’s entropy. So
for BH unitarity to hold, as described by Page such measurements should lie
close to the solid curve defined at each retarded time $u$ by the lesser of
$S_{\mathrm{thermal}},S_{\mathrm{BH}}$.

Nonetheless, in this description Hawking’s original information loss
calculation still appears to be fundamentally correct, so that the von Neumann
entropy of the resulting density matrix does _not_ follow the Page curve. The
consistency of these two facts relies on ideas from the late 1980’s involving
spacetime wormholes and baby universes.

This note provides a brief overview of the new paradigm, summarising the
central ideas; we refer to Marolf:2020rpm  for more complete details and
references to the literature; see also the recent review Almheiri:2020cfm . We
will present these ideas in an asymptotically flat setting (though the
conclusions are no different with AdS asymptotics), formulating gravity in
terms of a path integral over Lorentz-signature metrics, and emphasizing the
operational tests to which we referred above.

##  II Summary

Entropy is not a directly observable quantity for a single system: it can only
be inferred from measurements performed on many copies of a system. We will
begin by introducing the ‘swap entropy’, an entropy-like quantity defined in
terms of such measurements. This is constructed to coincide with the von
Neumann entropy when we are given many uncorrelated copies of a state.
However, our later calculations will involve spacetime wormholes, which are
shortcuts between otherwise well-separated regions of spacetime and which for
us will connect the interiors of distant black holes. Such connections induce
unavoidable correlations between distant experiments, even those performed in
otherwise separate universes. As a result, taking such wormholes into account
one finds that the states of Hawking radiation emitted from identical black
holes will not in fact be independent. This then allows the swap entropy to
differ from the von Neumann entropy.

After summarising the framework of semiclassical gravity to be used, we then
briefly describe recent computations of such swap entropies for radiation
emitted from evaporating black holes. While the formal von Neumann entropy
associated with a single black hole receives contributions only from a single
saddle-point (the ‘Hawking saddle’), the swap entropies also receive
contributions from replica wormholes. As a result, with standard initial
conditions the formal von Neumann entropy $S_{\mathrm{vN}}$ of radiation
emitted by a single black hole is indeed thermal
($S_{\mathrm{vN}}=S_{\mathrm{thermal}}$), while the swap entropy
$S^{\mathrm{swap}}$ follows the Page curve (solid curve in figure 1).

Finally, we explain the physics associated with the discrepancy between
$S_{\mathrm{vN}}$ and $S^{\mathrm{swap}}$ in terms of entanglement with baby
universes. This entanglement mediates strong correlations between the end
products of distinct experiments that form and evaporate otherwise-identical
black holes, even when the experiments are well-separated in space and time.
However, for robust reasons, the correlations between any set of evaporation
experiments must be entirely classical. In particular, one may describe these
correlations by declaring that identical evaporation experiments always lead
to identical pure states of Hawking radiation, but that the pure state is
drawn at random from an appropriate ensemble. But crucially, for a given
process that forms and evaporates black holes, all iterations of the process
are governed by the results of a _single_ random draw no matter how many times
the experiment is performed. This basic picture was described long ago
Coleman:1988cy ; Giddings:1988cx ; Giddings:1988wv ; Polchinski:1994zs . But
with the replica wormhole Page curve results for $S^{\mathrm{swap}}$, the
paradigm is now fully consistent with low-energy physics, and with a black
hole density of states given by the Bekenstein-Hawking entropy. In this way,
black hole evaporation is consistent with BH unitarity, but the detailed
dynamics (as predicted by low-energy gravity) is random, rather than uniquely
determined.

##  III Swap entropies

We have already stressed that we are most interested in operational
measurements performed by asymptotic observers. In general, an observer may
form and evaporate some number $n$ of black holes in widely separated regions
of spacetime, and make joint measurements on the resulting $n$ sets of Hawking
radiation which we will call ‘replicas’. The predictions for such experiments
are encoded in the expectation values
$\operatorname{Tr}(\mathcal{O}\rho^{(n)})$ of operators $\mathcal{O}$ acting
on the density matrix $\rho^{(n)}$ of the full $n$-replica system. In
particular, $\mathcal{O}$ need not factorise into a product of operators
acting on each individual replica.

If the $n$ processes of black hole formation and evaporation are identical and
uncorrelated, then $\rho^{(n)}$ will be the $n$-fold tensor product of a
single-replica state $\rho$: $\rho^{(n)}=\rho^{\otimes n}$. Under such
circumstances we can use observations on the $n$-replica state
$\rho^{(n)}=\rho^{\otimes n}$ to determine the degree to which the single-copy
sate $\rho$ is mixed.

A simple and useful such observation is the ‘swap test’ buhrman2001quantum ;
Hayden:2007cs  for $n=2$, a measurement of the swap operator $\mathcal{S}$
that acts by exchanging the two replicas:
$\mathcal{S}|\psi_{1}\rangle\otimes|\psi_{2}\rangle=|\psi_{2}\rangle\otimes|\psi_{1}\rangle$.
More generally, on $n$ replicas we can measure the operator $U_{\tau}$ which
enacts the cyclic permutation $\tau=(1\ 2\cdots n)$ on the replicas.111For
$n>2$, $U_{\tau}$ is not Hermitian, but it is normal (commutes with its
adjoint) so can be measured. We may encode the expectation values
$\operatorname{Tr}\left(U_{\tau}\rho^{(n)}\right)$ of these permutation
operators in ‘swap entropies’ $S_{n}^{\text{swap}}$, defined by

$S_{n}^{\text{swap}}(\rho^{(n)}):=-\frac{1}{n-1}\log\operatorname{Tr}\left(U_{\tau}\rho^{(n)}\right).$
(1)

The reason for this name arises from the uncorrelated-but-identical case
$\rho^{(n)}=\rho^{\otimes n}$ where $\rho^{(n)}$ consists of $n$ uncorrelated
copies of a state $\rho$. Then, $S_{n}^{\text{swap}}$ reduces to the Rényi
entropy defined by

$S_{n}(\rho):=-\frac{1}{n-1}\log\operatorname{Tr}\left(\rho^{n}\right).$ (2)

We may thus summarize this result by writing

$\rho^{(n)}=\rho^{\otimes n}\implies S_{n}^{\text{swap}}(\rho)=S_{n}(\rho).$
(3)

However, when we study the state of radiation $\rho^{(n)}$ from $n$ separate
black holes below, we will find reason to believe that $\rho^{(n)}$ is very
different from $\rho^{\otimes n}$, and in particular that it contains strong
correlations between the various sets of radiation collected. Then
$S_{n}^{\text{swap}}$ need not equal $S_{n}$, but will contain new physics
that we will interpret more fully below.

Finally, we will also find it convenient to consider the $n\to 1$ limit of the
swap entropy $S^{\text{swap}}:=\lim_{n\to 1}S_{n}^{\text{swap}}$, in analogy
with the representation of von Neumann entropy as the $n\rightarrow 1$ limit
of Rényis:

$S(\rho):=-\operatorname{Tr}\left(\rho\log\rho\right)=\lim_{n\to
1}S_{n}(\rho).$ (4)

For swap entropies this limit is somewhat formal222Though in many contexts it
can be defined by analytic continuation using Carlson’s theorem from complex
analysis, by using the resolvent method of Penington:2019kki , or by using the
methods of DHoker:2020bcv ., since the swap entropy is strictly speaking only
defined for integers $n\geq 2$. Nonetheless, there will be a natural way to
formulate the calculation of the swap entropy so that it applies at general
real $n$, and it will simplify in the $n\to 1$ limit. We interpret the result
as the von Neumann entropy that would be deduced from the observations of a
sophisticated experimentalist holding the state $\rho^{(n)}$ with many
replicas, after taking all experimental precautions to ensure that the
replicas are as uncorrelated as is possible.333This in fact requires
exponentially many copies of the system, $n\gg e^{S^{\text{swap}}}$ where
$S^{\text{swap}}$ is often comparable to the density of states of some
controlling subsystem.

The above construction may be interpreted as a physical version of the so-
called ‘replica trick’ often used to compute the von Neumann entropy of a
density matrix $\rho$. But our version is not a trick in any way. Each replica
is a physical system on which we can perform joint measurements. And potential
correlations between our replicas are not specified or forbidden by hand.
Instead, such correlations are determined by the dynamics and initial
conditions.

##  IV A first semiclassical contribution: the Hawking saddle

Now that we have described the observables of interest, we proceed to lay out
the framework we will use to calculate their expectation values.

We will consider states $\rho^{(n)}(u)$ defined by preparing $n$ identical
initial pure states of matter destined to collapse into black holes, allowing
them to evaporate, and collecting the Hawking radiation up to some time $u$.
If these $n$ black holes are sufficiently widely separated in space or time,
locality (cluster decomposition) would suggest that the $n$ states are
uncorrelated: we will see later that this expectation needs to be modified,
which is why we are belabouring the distinction between von Neumann entropy
and measurable quantities. We will idealise this distant separation by using
$n$ separate asymptotic boundaries, preparing the initial collapsing matter
state on $n$ separate copies of $\mathscr{I}^{-}$, and collecting Hawking
radiation on $n$ corresponding copies of $\mathscr{I}^{+}$ (up to retarded
time $u$).

Let us first consider the case $n=1$ of a single formation-evaporation
experiment. With a single black hole, we may compute the expectation values of
any operator $\mathcal{O}$ (supported on $\mathscr{I}^{+}$ before time $u$)
measuring the Hawking radiation. The knowledge of all such expectation values
is equivalent to the $n=1$ density matrix $\rho^{(1)}(u)$ of the Hawking
radiation, with expectation values given by
$\operatorname{Tr}(\rho^{(1)}(u)\mathcal{O})$. To compute these expectation
values, we use the in-in formalism in which the pure initial state
$|\mathrm{in}\rangle$ is described by the density matrix
$\rho_{\mathrm{in}}=|\mathrm{in}\rangle\langle\mathrm{in}|$, which involves
both a bra and a ket. The path integral then evolves both the bra and the ket
into the future in what one might call separate copies of the spacetime, for
the moment in fact defining two separate path integrals. We take the evolution
to stop at some achronal surface $\Sigma_{u}$ such that it would form a Cauchy
surface when taken together with the points on $\mathscr{I}^{+}$ before
retarded time $u$. We sew the two path integrals together along $\Sigma_{u}$
in order to trace over states that remain in the spacetime and have not yet
escaped to $\mathscr{I}^{+}$ before the retarded time $u$. Finally, we sum
over states on the portion of $\mathscr{I}^{+}$ before time $u$, weighted
according to the matrix elements of the operator $\mathcal{O}$ of
interest.444One might like to compute the matrix elements of the density
matrix $\rho^{(1)}(u)$ directly, by choosing $\mathcal{O}=|j\rangle\langle
i|$, where $|i\rangle$ runs over some complete basis of states on the relevant
portion of $\mathscr{I}^{+}$. We avoid this, because all such states would
have enormous energy at retarded time $u$ due to the absence of entanglement
present in any state close to the vacuum, rendering the calculation beyond
semiclassical control.

Figure 2: An artists impression of the path integral computation of
$\operatorname{Tr}(\mathcal{O}\rho^{(1)}(u))$. The bra and ket copies of the
spacetime pieces are sewn together along the blue partial Cauchy surface
$\Sigma_{u}$ which meets $\mathscr{I}^{+}$ at retarded time $u$, and along
$\mathscr{I}^{+}$ before that time after acting with the operator
$\mathcal{O}$ there.

It is useful that the diagram in figure 2 can be read at several levels.
First, one could take it to describe a calculation in quantum field theory
(including perturbative gravitons) on a spacetime with a fixed metric that
describes a collapsing black hole as in Hawking’s original calculation
Hawking:1974sw . Second, one could take it to describe a similar calculation
in which the spacetime metric is chosen to describe a black hole that first
collapses and then evaporates, with a black hole that slowly decays by
radiating energy to infinity according to semi-classical expectations. In such
a spacetime the black hole mass eventually shrinks down to become Planck scale
at some finite retarded time $u_{\mathrm{Planck}}$, after which the spacetime
will contain Planck-scale curvatures whose physics is not under semi-classical
control. But such concerns are avoided by our considering only the portion of
$\mathscr{I}^{+}$ before some finite retarded time $u$ so long as we choose
$u$ to be significantly less than $u_{\mathrm{Planck}}$. In either case, we
integrate over quantum fields with the stated boundary conditions at
$\mathscr{I}^{-}$ and $\mathscr{I}^{+}$, and which satisfy the desired sewing
condition on the bulk partial Cauchy surface shown in blue.

Finally, we can take figure 2 to describe semiclassical contributions to the
full quantum gravity description of our process. In practice, this means that
the metric must solve the semi-classical Einstein equations. These equations
are sourced by a quantum stress-tensor which includes the effect of
integrating out fields at one loop, to incorporate the back-reaction from
Hawking radiation. To be more precise, one might like to integrate over a
family of such geometries to incorporate quantum fluctuations in the rate of
radiation of energy, momentum and angular momentum.

With any of the above interpretations, the resulting expectation values
$\operatorname{Tr}(\rho^{(1)}(u)\mathcal{O})$ will be much as anticipated in
Hawking:1976ra . In particular, they will be consistent with a roughly
‘thermal’ state, though with a temperature that evolves adiabatically as a
function of $u$ in a manner dictated by conservation of energy and the
equation of state of the evaporating black hole. As a result, correlations
between Hawking photons become negligible at time separations greater than the
inverse of the instantaneous black hole temperature. The effects of this
adiabatic variation are small and, as a result, the von Neumann entropy
$S\left(\rho^{(1)}(u)\right)$ computed from such matrix elements will grow
monotonically in time, with the rate at each time again being determined by
the effective temperature. For brevity, we refer to this as giving a thermal
entropy $S_{\mathrm{thermal}}(u)$, as depicted in figure 1.

Thinking of figure 2 as describing a self-consistent solution of the
semiclassical physics that satisfies the desired boundary conditions raises
the question of whether there might be other such solutions. Indeed, the
semiclassical description of a fully quantum process often involves finding
several such solutions and summing the resulting contributions. In a path
integral description one refers to this as a sum over saddle points, and we
will use this terminology below. In particular, we will refer to contributions
of the form shown in figure 2 as Hawking saddles due to their similarity to
the descriptions of Hawking:1974sw  and Hawking:1976ra .

Having described the Hawking saddle for $\rho^{(1)}(u)$, we can immediately
see that taking $n$ such saddles gives a similar semiclassical contribution to
expectation values for $n$ identical black holes, computed from the $n$-copy
density matrix $\rho^{(n)}(u)$. Pictorially, we simply draw $n$ separate
copies (replicas) of figure 2 (perhaps with different operators $\mathcal{O}$
on the chosen portion of $\mathscr{I}^{+}$, or more generally an operator that
mixes the $n$ copies of the radiation) with no connections or identifications
between replicas.

Such collections of $n$ spacetimes will again be called Hawking saddles, and
if these are the only contributions then $\rho^{(n)}(u)$ is just a tensor
product of $n$ uncorrelated states $\rho^{(1)}(u)$ as one would naively
expect:

$\text{Hawking saddles give}\ \
\rho^{(n)}_{\text{Hawking}}(u)=\left(\rho^{(1)}(u)\right)^{\otimes n}.$ (5)

Indeed, by choosing our operator $\mathcal{O}$ to be a cyclic permutation
$U_{\tau}$, we may consider the Hawking saddles as contributions to the swap
entropies (1). The definition of swap entropies instructs us to sum over
boundary conditions on the various copies of $\mathscr{I}^{+}$, identifying
the state on the $k$th bra copy of $\mathscr{I}^{+}$ with the state on
$(k+1)$th ket copy as dictated by the cyclic permutation $U_{\tau}$ in (1).
Implementing these boundary conditions on the Hawking saddle, we arrive at a
path integral as depicted in figure 3.

But if we use only these Hawking saddles, the resulting swap entropy
calculation is identical to a calculation of the von Neumann entropy of
$\rho^{(1)}(u)$. Consistent with (3) and (5), if such saddles dominate we find
$S^{\mathrm{swap}}_{n}(u)\approx S_{n}\left(\rho^{(1)}(u)\right)$ where the
right-hand-side is just the $n$th Rényi entropy $S_{\mathrm{thermal},n}(u)$ of
thermal radiation at $\mathscr{I}^{+}$ (again with appropriately adiabatically
varying temperature $T(u)$). Taking the limit $n\rightarrow 1$ then yields
$S^{\mathrm{swap}}\approx S_{\mathrm{thermal}}$, the rising dotted curve in
figure 1.

Figure 3: A saddle for $S^{\text{swap}}_{2}$ formed by sewing two copies of
the Hawking saddle figure 2 together along the parts of $\mathscr{I}^{+}$
before retarded time $u$. These identifications are shown in green and red.

##  V Replica wormholes

We now come to the main point. While the Hawking saddles may seem natural, the
results of Penington:2019kki ; Almheiri:2019qdq  strongly indicate that other
saddles will also contribute to our swap entropies, and that the contributions
of these new saddles will be important after the Page time. The new saddles
are known as replica wormholes.

It is useful to explain replica wormholes in the language of the gravitational
path integral. Here we take the perspective that the path integral sums over
bulk spacetimes that satisfy some given set of boundary conditions dictated by
the physics one wishes to study. For example, for a fixed initial state on
$\mathscr{I}^{-}$, our $n=2$ swap entropy is associated with gluing four
copies of $\mathscr{I}^{+}$ together as shown in figure 3. The expectation
value of $S^{\text{swap}}_{2}$ is then given by summing over bulk spacetimes
compatible with the given boundary conditions.

Now, one can and should ask what class of bulk spacetimes should be included
in this sum. For example, should we restrict to smooth spacetimes? Or to
spacetimes with appropriate causal structures? Our central assumption
(motivated further below) is that this class should include ‘replica wormhole’
spacetimes of the form depicted in figure 4.

Figure 4: A replica wormhole for $S^{\text{swap}}_{2}$. The boundary
conditions are identical to those in figure 3, but the identifications differ
in the bulk due to the presence of islands (${\cal I}$). On the islands the
bulk spacetimes are sewn together in the same pattern that the swap boundary
conditions impose on the early part of $\mathscr{I}^{+}$, effectively
‘undoing’ the swap on this part of the geometry (where most of the bulk
entropy resides).

Such spacetimes may be constructed from the Hawking spacetimes discussed above
by a ‘cut-and-paste’ procedure, whereby we change the manner in which bra and
ket spacetimes along the future Cauchy surface $\Sigma_{u}$. Specifically, we
may imagine starting with a geometry as in figure 3, cutting each copy of
$\Sigma_{u}$ along some so-called _island_ $\mathcal{I}\subset\Sigma_{u}$, and
then sewing the replicas together cyclicly along $\mathcal{I}$ as shown for
$n=2$ in figure 4. That is, we choose to swap the islands in precisely the
same pattern that we swap the parts of $\mathscr{I}^{+}$ before retarded time
$u$. The result is a single connected spacetime satisfying the swap boundary
conditions.

The replica wormhole geometries are locally identical to the more conventional
in-in geometries considered above, except at the boundary of the island where
we change the identification between replicas. This locus is a codimension-two
surface $\gamma=\partial\mathcal{I}$, which we call the _splitting surface_.
Along this surface, we must introduce a new sort of singularity, which is
however of a rather mild type.

In more detail, the geometry in a neighbourhood of the splitting surface
$\gamma$ is a Lorentzian analogue of a conical singularity. To explain this,
it is useful to first recall certain features of conical singularities in
Euclidean-signature spacetimes. In particular, a conical excess of magnitude
$2\pi(n-1)$ can be created by taking $n$ copies of a smooth spacetime, cutting
them all open along a common half-line, and then pasting them together
cyclicly to form a connected geometry. This process is the Euclidean version
of the cutting and gluing along the island $\mathcal{I}$ which produced figure
4. The $n=2$ version of this construction in shown in figure 5. Such an excess
introduces a delta-function Ricci tensor at the conical defect which then
integrates to give a finite contribution to the Euclidean Einstein-Hilbert
action $\int\\!\\!\sqrt{g}R$. Though we focus on Lorentz-signature spacetimes
here, it is worth commenting that in Euclidean signature such conical
spacetimes are just limits of smooth spacetimes in which a finite amount of
Einstein-Hilbert action becomes concentrated at the defect. The Lorentzian
splitting surface $\gamma$ present in the replica wormholes similarly
contributes a finite action Louko:1995jw ; Colin-Ellerin:2020mva .

Figure 5: Making radial cuts in two disks and sewing them together with the
identifications shown in red and green results in a conical excess of $2\pi$.

Since these replica wormhole configurations have finite gravitational
action,555Including quantum corrections, we should interpret the gravitational
action as a renormalised effective action, whereby a cutoff-dependent
contribution from the path integral over fluctuations is cancelled by
appropriate counterterms. we take the position that it is reasonable to
include them in the path integral, even if such spacetimes may have properties
that are both novel and unfamiliar. This is particularly justified in light of
the classic result that smooth configurations have measure zero in the natural
sense associated with quantum field theory path integrals. This result is
deeply related to the familiar strong UV effects (and the associated
divergences) in standard quantum field theory. The same logic then suggests
that the gravitational path integral must include singular configurations as
well, and finite action provides a reasonable principle for which to include.

The claim now is that spacetimes of replica wormhole topology provide new
saddles that contribute to $S^{\text{swap}}_{n}$ in the context of evaporating
black holes. For $n>1$ such saddles necessarily require a complex metric,
since the replica wormholes with Lorentzian metric as described cannot solve
the semiclassical equations of motion at the singular splitting surface. In
Euclidean signature, a saddle would require altering the metric to smooth out
the conical defect at $\gamma$, but the same smoothing cannot be achieved in
Lorentzian signature without complexifying the metric. The need for complex
saddles should not surprise or worry us, since asymptotic evaulation of
oscillatory integrals generically requires deforming the contour of
integration to pass through a saddle-point in a complexified domain, and such
deformations are required to describe any quantum mechanical tunnelling
process as a saddle-point contribution to a real-time path integral.

While the existence and importance of saddles with such topology may seem
surprising, and while a full computation at finite $n$ for Lorentz-signature
evaporating black holes has yet to be performed, there is strong evidence for
this claim from the following known results. First, one can indeed find
saddles with non-trivial splitting surfaces when one imposes similar boundary
conditions in contexts where the effects of bulk quantum fields can be ignored
Colin-Ellerin:2020mva . Second, at least for $n=2$, analogous Euclidean-
signature saddles have been constructed that include back-reaction from
quantum fields Mirbabayi:2020fyk . Third, formal calculations at non-integer
$n$ allow one to study the limit $n\rightarrow 1$. For $n-1\ll 1$, one finds a
replica wormhole saddle precisely when the $n=1$ Hawking saddle contains a so-
called Quantum Extremal Surface (QES). In such a case, the effective action
(i.e., minus the logarithm of the path integral) exceeds the action of the
$n=1$ Hawking saddle by $(n-1)S_{\mathrm{gen}}(\gamma)$, where
$S_{\mathrm{gen}}(\gamma)=\frac{A(\gamma)}{4G}+S_{\mathrm{matter}}(\gamma)$ is
the generalized entropy of the QES Lewkowycz:2013nqa ; Faulkner:2013yia ;
Dong:2016hjy ; Dong:2017xht . Here, $A(\gamma)$ is the area of the splitting
surface, and $S_{\mathrm{matter}}(\gamma)$ is the von Neumann entropy of
matter fields on $\Sigma_{u}\setminus\mathcal{I}$ in the state described by
the $n=1$ saddle. Finally, evaporating black holes do in fact contain a non-
trivial QES $\gamma(u)$ located very close to the surface where the past light
cone of retarded time $u$ intersects the apparent horizon666Recall that the
apparent horizon is the surface where the outgoing null expansion vanishes. of
the evaporating black hole Penington:2019npb ; Almheiri:2020cfm .

The swap entropy that results from approximating the path integral by the
replica wormhole saddle is easy to calculate using the above results for the
limit $n\rightarrow 1$. The result is the ‘quantum extremal surface formula’,

$S^{\mathrm{swap}}=\min\operatorname*{extr}_{\gamma}S_{\mathrm{gen}}(\gamma).$
(6)

The $\operatorname*{extr}_{\gamma}$ denotes extremisation of the generalised
entropy with respect to variations of the splitting surface $\gamma$, which is
the criterion for finding a saddle point; a resulting stationary surface is a
QES. If there is more than one QES, $\min$ denotes choosing the QES with the
minimal value of $S_{\mathrm{gen}}(\gamma)$, which follows from choosing the
dominant saddle point.

In this context, we can think of the Hawking wormhole for the swap entropy as
a replica wormhole for which the splitting surface is trivially given by the
empty set $\gamma=\varnothing$, with
$S_{\mathrm{gen}}(\varnothing)=S_{\mathrm{thermal}}(u)$ given by the von
Neumann entropy of Hawking radiation as computed above, with no area
contribution. In contrast, to good approximation the non-trivial QES gives
$S_{\mathrm{gen}}(\gamma(u))\sim\frac{A(u)}{4G}$, where $A(u)$ is the area of
the evaporating black hole apparent horizon at retarded time $u$, since the
entropy of quantum fields on $\Sigma_{u}\setminus\mathcal{I}$ is relatively
small.

It now remains only to minimise over these two possibilities, choosing the
dominant saddle between the Hawking spacetimes above and the new replica
wormholes. But the two generalised entropies are precisely the two curves
shown in figure 1 that took part in our discussion of the Page curve. As a
result, by selecting the minimal value we find a phase transition precisely at
the Page time, and indeed, as advertised we find that the swap entropy is
given precisely by the Page curve of figure 1.

##  VI Baby universes and superselection sectors

We argued above that $S^{\mathrm{swap}}$ for multiple copies of the Hawking
radiation receives contributions from replica wormholes that cause it to
differ from the von Neumann entropy $S\left(\rho^{(1)}\right).$ Such a
discrepancy requires the radiation from our various black hole evaporation
experiments to be correlated. But what is the mechanism that produces such
correlations, and what form do they take?

The Hawking and replica wormhole saddles suggest a direct answer to this
question. It is clear that they naturally entangle the Hawking radiation with
whatever degrees of freedom remain inside the black hole and do not escape to
$\mathscr{I}^{+}$. This is of course what leads to the usual statement that
the state $\rho^{(1)}$ of Hawking radiation on $\mathscr{I}^{+}$ is highly
mixed.

However, in the replica wormhole saddles the island $\mathcal{I}$ inside a
given black hole (in the bra spacetime) can be glued to the island
$\mathcal{I}$ inside another black hole (in the ket spacetime). The result is
a modification of the inner product on these black hole interiors, so that we
do not have a separate Hilbert space factor for the interior of each black
hole, but only a single overall Hilbert space that describes the interiors of
all black holes that happened to form.

In part for historical reasons, this Hilbert space is called the ‘baby
universe Hilbert space’. One motivation for this name can be seen by
considering a black hole that evaporates entirely, to leave behind an
essentially flat Minkowski spacetime containing Hawking radiation. After the
complete evaporation the black hole interior may be thought of as a closed
‘baby universe’, no longer attached to the ‘parent’ asymptotically flat
universe from which it was born. While this description requires some
assumptions about the physics once the black hole becomes of Planckian size,
we emphasise that it is only a pedagogical crutch, and no such assumptions are
required for our conclusions Marolf:2020rpm .

Due to the contribution of replica wormholes, the presence of such baby
universes may alter the Hawking radiation produced from any given black hole,
and in turn an experiment on a black hole may alter the state of baby
universes. This dependence on the state of baby universes highlights an
implicit assumption in our discussions above, namely a choice of initial state
of closed universes. It merely happens that our discussion was conducted in
the absence of initial closed universes (called the Hartle-Hawking baby
universe state $|\mathrm{HH}\rangle_{\mathrm{BU}}$ in Marolf:2020xie ;
Marolf:2020rpm , after noting that this is equivalent to having no boundaries
besides those explicitly required for the calculation in question). By
modifying the initial state of baby universes, we may obtain different
predictions for subsequent experiments. Furthermore, such modifications of the
baby universe state occur naturally whenever an observation of the sort
discussed above is made.

We thus conclude that entanglement between Hawking radiation and the baby
universe Hilbert space induces correlations between the widely-separated
evaporation experiments required by our swap entropies, explaining the
discrepancy between von Neumann entropy and swap entropy. But such
correlations which do not decay with spacetime separation would appear to
violate cherished notions of locality, perhaps allowing causality-violating
transfer of information between replicas. Fortunately, this should be
forbidden by arguments going back to the 1980’s Coleman:1988cy ;
Giddings:1988cx ; Giddings:1988wv , which limit the dependence between
separate experiments to classical correlations.

The essence of this argument is that any observable on a single black hole,
formalised by a set of boundary conditions on a single flat asymptotic region
(or an identified bra and ket copy of such to form a single boundary of an in-
in contour as above), may be considered as an operator on the baby universe
Hilbert space Marolf:2020xie ; Marolf:2020rpm . This encapsulates the idea
above that the an observation of Hawking radiation both depends on and affects
the state of baby universes. The set of all such operators generates an
algebra $\mathcal{A}_{\mathrm{asympt}}$ of asymptotic observables. Independent
observations on multiple black holes are described by a product of such
operators, and correlated observations (such as the measurement of the
permutation operators described above) are described by a sum of such
products. Furthermore, the path integral with a given set of boundary
components does not require us to specify any particular ordering of those
components. As such, the algebra $\mathcal{A}_{\mathrm{asympt}}$ is
commutative: the predictions for any two experiments on separate black holes
does not depend on the order in which they are performed. Using this mutual
commutativity, all the asymptotic observables may be simultaneously
diagonalised. This means that the baby universe Hilbert space is spanned by
mutually orthogonal simultaneous eigenstates $|\alpha\rangle_{\mathrm{BU}}$
called $\alpha$-states, in which all single black hole observables take
definite values. Hence, if baby universes are prepared in one of the special
states $|\alpha\rangle_{\mathrm{BU}}$, there can be no correlations of the
sort described above. In other words, the baby universe Hilbert space is a
direct sum of superselection sectors with respect to the algebra
$\mathcal{A}_{\mathrm{asympt}}$ of asymptotic observables.

Correlations between separate asymptotic regions can arise when the baby
universe state is in a superposition of many $\alpha$ states,
$|\psi\rangle_{\mathrm{BU}}=\sum_{\alpha}\sqrt{p_{\alpha}}e^{i\theta_{\alpha}}|\alpha\rangle_{\mathrm{BU}}$.
From this, the expectation value for any asymptotic observable is given by an
average over the definite values in each superselection sector, weighted by
$p_{\alpha}=|\langle\alpha|\psi\rangle_{\mathrm{BU}}|^{2}$. But this means
that all correlations can be described by a classical probability
distribution, with no interference between different superselection sectors,
and no dependence on the relative phases $\theta_{\alpha}$. This is in
particular true for the Hartle-Hawking baby universe state
$|\psi\rangle_{\mathrm{BU}}=|\mathrm{HH}\rangle_{\mathrm{BU}}$, corresponding
to an empty set of initial closed universes.

##  VII Conclusions

We may now interpret our calculations of the swap entropies in the light of
the general comments regarding baby universes. Both the original Hawking
evaporation computation and the calculation of swap entropies above were
performed in the baby universe Hartle-Hawking state. So both results should be
interpreted as an average over an ensemble of superselection sectors labelled
by orthonormal baby universe states $|\alpha\rangle_{\mathrm{BU}}$ with
weights given by $|\langle\alpha|\mathrm{HH}\rangle_{\mathrm{BU}}|^{2}$.

The swap entropy in any given superselection sector is equal to the von
Neumann entropy in that sector, since passing to a specific superselection
sector eliminates correlations and hence (3) applies. We interpret the swap
entropy obtained from the replica wormholes as an average over the von Neumann
entropy of Hawking radiation in each sector, and conclude that in typical
sectors, this entropy follows the Page curve.

In contrast, the computation of the entropy from the Hawking wormholes alone
is interpreted as the von Neumann entropy of the average state. This may be
much larger than the average entropy, since von Neumann entropy is not a
linear observable. The key point here is just the familiar statement that an
ensemble of pure states is generically highly mixed. But since any particular
asymptotic observer will obtain experimental outcomes consistent with a single
superselection sector, this large entropy can never have observable
consequences.

Furthermore, in appropriate contexts additional formal arguments require the
density of states in the asymptotically-flat sector of the theory associated
with any given superselection sector to be bounded by the Bekenstein-Hawking
entropy Marolf:2020xie . This guarantees (under the relevant assumptions) that
the Page curve for swap entropy bounds the entropy of Hawking radiation in
every superselection sector.

In summary, baby universes and superselection sectors are natural consequences
of the bulk gravitational path integral whose existence provides sufficient
physics to allow both BH unitarity and the original Hawking evaporation
calculation to hold without substantial modification. The calculation of the
Page curve from replica wormholes can be interpreted as directly supporting
this paradigm. This is achieved without introducing fundamentally new physics,
requiring only a conservative interpretation of well-established physical
laws. This same physics appears to also resolve at least certain other long-
standing issues in quantum gravity Hsin:2020mfa ; Chen:2020ojn , though there
is certainly much more to learn. Again, we refer the reader to the literature
Penington:2019kki ; Almheiri:2019qdq ; Marolf:2020xie ; Almheiri:2020cfm ;
Marolf:2020rpm  for more complete discussions of both related physics and open
issues. In particular, our brief introduction above should suffice to follow
the discussions in sections 7.2 and 7.3 of Marolf:2020rpm  without studying
the rest of that work in detail.

###### Acknowledgements.

### Acknowledgements

We thank Bob Wald for conversations motivating us to write this overview. We
also thank Sean Collin-Ellerin, Xi Dong, Steve Giddings, Mukund Rangamani,
Douglas Stanford, Zhencheng Wang and many others for useful conversations
about baby universes and superselection sectors. We are grateful for support
from NSF grant PHY1801805 and funds from the University of California. H.M.
was also supported in part by a DeBenedictis Postdoctoral Fellowship.

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