not exactly saturated was also given in Junge_2018 , as,

$\displaystyle\tilde{\mathcal{R}}_{\sigma,\mathcal{N}}(.)=\int dt\beta(t)$
$\displaystyle\sigma^{\frac{1-it}{2}}\mathcal{N}^{\dagger}\left([\mathcal{N}(\sigma)]^{\frac{-1+it}{2}}(.)[\mathcal{N}(\sigma)]^{\frac{-1-it}{2}}\right)\sigma^{\frac{1+it}{2}},$
(72)

where $\beta(t)=(\pi/2)(\cosh(\pi t)+1)^{-1}$. This twirled form of the Petz
map then provides a natural choice for a universal recovery map in holography,
where $t$ is interpreted as the boundary modular time. This was formalised in
the work of cotler2019_univR , which we briefly review here.

The starting point for approximate bulk reconstruction will be the approximate
form of the relative entropy equivalence from Jafferis_2016 .

$S(\rho_{A}|\sigma_{A})=S(\rho_{a}|\sigma_{a})+\mathcal{O}\left(\frac{1}{N}\right),$
(73)

where $\rho_{A}$, $\sigma_{A}$ are density operators on the boundary subregion
$A$ in Fig. 13 and $\rho_{a}$, $\sigma_{a}$ are density operators on the
corresponding entanglement wedge region $a$ in the bulk. In other words, the
JLMS equality conditions holds upto leading order in the CFT gauge group rank
$N$. Interpreting Eq. (71) as approximate saturation of the monotonicity of
the relative entropy requires the existence of a mapping
$\rho_{a}\rightarrow\rho_{A}$ of states from the entanglement wedge in the
bulk to the boundary subregion $A$. The AdS/CFT correspondence can be
described via an isometry $\mathcal{V}:\mathcal{C}\rightarrow\mathcal{H}$ from
the code subspace to the boundary CFT $\mathcal{H}$. The map $\mathcal{N}$ is
simply the partial trace operation, which traces out the complementary
boundary region $\bar{A}$. Thus, for any
$\rho_{a}\in\mathcal{B}(\mathcal{H}_{a})$, and any fixed, full-rank state
$\sigma_{\bar{a}}\in\mathcal{B}(\mathcal{H}_{\bar{a}})$, we have,

$\mathcal{N}(\rho_{a})=\mathrm{Tr}_{\bar{A}}[\mathcal{V}(\rho_{a}\otimes\sigma_{\bar{a}})\mathcal{V}^{\dagger}].$
(74)

Note that we only consider product density operators of the form
$\rho_{a}\otimes\sigma_{\bar{a}}$ on the codespace, where $\sigma_{\bar{a}}$
is a fixed state on $\mathcal{H}_{\bar{a}}$, so that the partial trace
operation truly becomes a mapping of states $\rho_{a}$ on the entanglement
wedge region (shaded region $a$ in Fig. 13) to states
$\mathrm{Tr}_{\bar{A}}[\mathcal{V}(\rho_{a}\otimes\sigma_{\bar{a}})\mathcal{V}^{\dagger}]$
on the boundary subregion $A$.

Now, consider the twirled Petz map corresponding to the map $\mathcal{N}$
defined in Eq. (74), $\tilde{\mathcal{R}}_{\sigma_{a},\mathcal{N}}$, defined
using a full-rank state $\sigma_{a}\in\mathcal{B}(\mathcal{H}_{a})$. This is a
map of the form
$\tilde{\mathcal{R}}_{\sigma_{a},\mathcal{N}}:\mathcal{B}(\mathcal{H}_{A})\rightarrow\mathcal{B}(\mathcal{H}_{a})$.
Finally, if we assume that the map $\mathcal{N}$ defined in Eq. (74)
approximately saturates the monotonicity inequality, Eq. (71) implies that the
twirled Petz map $\tilde{\mathcal{R}}_{\sigma_{a},\mathcal{N}}$ recovers any
$\rho_{a}\in\mathcal{B}(\mathcal{C})$ with a fidelity that is bounded by,

$\displaystyle-2\log
F(\rho_{a},(\tilde{\mathcal{R}}_{\sigma_{a},\mathcal{N}}\circ\mathcal{N})(\rho_{a}))$
(75) $\displaystyle\leq$ $\displaystyle
S(\rho_{a}|\sigma_{a})-S(\mathcal{N}(\rho_{a})|\mathcal{N}(\sigma_{a})).$

This argument was then extended in cotler2019_univR  to show that the map
$\tilde{\mathcal{R}}_{\sigma_{a},\mathcal{N}}$ can recover for _all_ states,
extending the scope of this result beyond states that are factorised as
$\rho=\rho_{a}\otimes\sigma_{\bar{a}}$. Furthermore, since
$\tilde{\mathcal{R}}_{\sigma_{a},\mathcal{N}}$ recovers states on
$\mathcal{H}_{a}$ with high fidelity, it can be shown that the adjoint
$\tilde{\mathcal{R}}^{\dagger}_{\sigma_{a},\mathcal{N}}$ maps operators
$\phi_{a}$ with support on the entanglement wedge region $\mathcal{H}_{a}$ to
boundary operators $\mathcal{O}_{A}$ which are close in expectation values.
Finally, it was shown in cotler2019_univR  that an explicit formula for
operators $\mathcal{O}_{A}$ can be obtained by a specific choice of the fixed
states $\sigma_{a}$ and $\sigma_{\bar{a}}$, namely, the maximally mixed states
on $\mathcal{H}_{a}$ and $\mathcal{H}_{\bar{a}}$.

We conclude this section by noting a few more recent results that demonstrate
bulk reconstruction using variants of the Petz map. Moving away from the
twirled Petz map which involves an averaging over the modular time, it was
argued in chen2020_petz  that the standard form of the Petz map in Eq. (68)
suffices to achieve approximate bulk reconstruction. Their argument is based
on the original result of Barnum and Knill barnum_Knill2002 , which shows that
the Petz map is the near-optimal universal recovery map in terms of the
average entanglement fidelity between the ideal and noisy states. We restate
the main result of chen2020_petz  here, for completeness. Let
$\mathcal{M}_{a}$ be a subalgebra on the code space $\mathcal{C}$ with
dimension $d_{\rm code}$, and $\mathcal{N}$ be any noise map. Suppose there
exists an _optimal_ recovery map $\mathcal{R}_{\rm opt}$ such that

$\parallel(\mathcal{R}_{\rm
opt}\circ\mathcal{N})(\rho)-\rho_{a}\parallel_{1}<\delta,$

for all $\rho\in\mathcal{B}(\mathcal{C})$ and its projection $\rho_{a}$ onto
$\mathcal{M}_{a}$. Then, the Petz map

$\mathcal{R}_{\tau,\mathcal{N}}(.)\equiv\frac{1}{d_{\rm
code}}\mathcal{N}^{\dagger}\left[\mathcal{N}(\tau)^{-1/2}(.)\mathcal{N}(\tau)^{-1/2}\right],$

defined using the maximally mixed state $\tau$ on the codespace, satisfies,

$\parallel(\mathcal{R}_{\tau,\mathcal{N}})\circ\mathcal{N})(\rho)|_{a}-\rho_{a}\parallel_{1}\leq
d_{\rm code}\sqrt{8\delta},$

where $\parallel(.)\parallel_{1}$ denotes the $1$-norm or the trace-distance
and $(.)|_{a}$ denotes the projection onto the subalgebra $\mathcal{M}_{a}$.
In other words, so long as the error using the optimal recovery
$\mathcal{R}_{\rm op}$ is non-perturbatively small (in trace-norm), the error
after the Petz map recovery will also be non-perturbatively small upto a
factor $d_{\rm code}$.

More recently, it has been shown that a Petz-like map can be used to
demonstrate bulk reconstruction in random tensor network toy models
jia2020_petz_RTN ; Penington:2019kki  of holography. From a dynamical point of
view, there appears to be an interesting connection between the structure of
the twirled Petz map and the action of the modular Hamiltonian, as hinted in
cotler2019_univR  and also in sections 3.3.2 and 3.4. Exploring this
connection further might lead to further insights on the problem of bulk
reconstruction, and promises to be an exciting direction for future
investigations.

###  4.4 A note on holography as a renormalization group flow

Holographic renormalization scheme which defines the dictionary between
boundary and bulk observables already makes it manifest that the radial
direction in the holographic bulk is dual to the energy scale/scale of
resolution in the dual field theory. Nevertheless, it does not in itself tell
us how local bulk operators such as the metric can be reconstructed as coarse-
grained operators in the dual field theory. One can say that there is a
passive and an active point of view of bulk reconstruction. In the active
point of view advocated in the HKLL and the more refined JLMS procedures, we
extrapolate the bulk operator to the boundary. On the other hand in the RG
flow point of view, we should coarse-grain the boundary operator under a
suitably defined RG flow such that it mimics the geometric radial flow, and
then demonstrate the emergence of local bulk operators from these flowed
operators. The coarse-graining is more specifically an evolution under a
sequence of CP (completely positive) unital maps of the dual field theory
operators.131313A CP unital map is the dual of a CPTP (completely positive
trace preserving) map which evolves density matrices to density matrices. Here
dual implies the map which takes us to the Heisenberg picture from the
Schrodinger picture. The unital map preserves the identity. Interestingly, the
real space RG has been already discussed explicitly from the quantum error
correction perspective furuya2020 ; ghodrati2021_modFlow  and the Petz map
also naturally emerges in this context. The passive RG flow perspective also
holds an enormous promise to reveal novel principles of bulk emergence as we
discuss below. It is not a mere reinterpretation of the active (HKLL/JLMS)
perspective since the latter is manifestly non-local while the RG flow tames
non-locality in a controlled way.

Various proposals for the RG flow have been advocated Lee:2009ij ; Lee:2013dln
; Behr:2015aat ; Behr:2015yna ; Mandal:2016rmt ; Sathiapalan:2017frk  based on
fundamental insights developed in Heemskerk:2009pn ; Heemskerk:2010hk . Here
we will focus on the highly efficient RG flow construction developed in
Behr:2015aat ; Behr:2015yna  based on earlier works in the context of the
fluid/gravity correspondence Kuperstein:2011fn ; Kuperstein:2013hqa . For a
review see Mukhopadhyay:2016fre . In this approach bulk locality is manifest
especially through the Ward identities. We outline this approach here in the
language of quantum error correction.

We first choose a code subspace. Let this be the space of all solutions of
pure gravity with a negative cosmological constant. This class of states can
be charactized by the expectation value of a specific single-trace operator,
namely the energy-momentum tensor, i.e. $\langle t_{\mu\nu}\rangle$ since the
expectation values of multi-trace operators factorize in the large $N$ limit
and other single-trace operators have vanishing or fixed expectation
values.141414To be precise one needs to also specify intial conditions to
describe the full geometry. However, $\langle t_{\mu\nu}\rangle$ is all we
need to find the geometry in a radial tube in Fefferman-Graham or Eddington-
Finkelstein coordinates when the boundary metric (dual to the physical metric
in which the dual field theory lives) is also specified. A better way to
implement this would be to invoke a Borel resummation of the derivative
expansion at late time in $\langle t_{\mu\nu}\rangle$ Heller:2013fn . Then the
initial conditions are encoded in the Stokes parameters.

The aim is to define a sequence of CP unital maps parametrized by a coarse-
graining scale $\Lambda$ under which

$t_{\mu\nu}\rightarrow t_{\mu\nu}(\Lambda).$

Since other single-trace operators do not play a role in this subspace, most
generally we should obtain

$\displaystyle t_{\mu\nu}(\Lambda)$ $\displaystyle=$ $\displaystyle
t_{\mu\nu}+a_{1}\frac{1}{\Lambda^{2}}\Box
t_{\mu\nu}+\frac{1}{\Lambda^{4}}(a_{2}t_{\mu}^{\,\,\rho}t_{\rho\nu}+a_{3}\eta_{\mu\nu}t_{\alpha\beta}t^{\alpha\beta}+a_{4}\Box^{2}t_{\mu\nu}+\cdots)$
(76) $\displaystyle+\mathcal{O}\left(\frac{1}{\Lambda^{6}}\right).$

Above $a_{i}$s are appropriate numerical constants which are determined by the
specific CP unital map (the insight that single trace operators should mix
with multi-trace operators under the RG flow even in the large $N$ limit is
from Heemskerk:2009pn ). Essentially we put in all possible multi-trace
operators on the right hand side except those which vanish due to the CFT Ward
identities:

$\partial^{\mu}t_{\mu\nu}=0,\quad\eta_{\alpha\beta}t^{\alpha\beta}=0.$ (77)

Owing to large-$N$ factorization of multi-trace operators, the evolution (76)
is effectively a classical equation. The point of the highly efficient RG flow
is that the coarse-graining which generates these unital maps should be done
with a very specific choice of the complementary subspaces which are traced
out such that we can define a metric

$g_{\mu\nu}(\Lambda)=g_{\mu\nu}[t_{\alpha\beta}(\Lambda)]$

in which we obtain a local Ward identity

$\nabla_{(\Lambda)}^{\mu}t_{\mu\nu}(\Lambda)=0$ (78)

at each scale $\Lambda$ with $\nabla_{(\Lambda)}$ being the covariant
derivative constructed from $g_{\mu\nu}(\Lambda)$. The Ward identity is
defined by considering $t^{\mu}_{\,\,\nu}(\Lambda)$ as the fundamental
variable and we lower and raise indices using $g_{\mu\nu}(\Lambda)$ and its
inverse respectively. In practice, we need to then restrict $a_{i}$s appearing
in (76) such that we can obtain $b_{i}$s defining $g_{\mu\nu}(\Lambda)$ via
the expansion

$\displaystyle g_{\mu\nu}(\Lambda)$ $\displaystyle=$
$\displaystyle\eta_{\mu\nu}+\frac{1}{\Lambda^{4}}t_{\mu\nu}+b_{1}\frac{1}{\Lambda^{6}}\Box
t_{\mu\nu}+$ (79)
$\displaystyle\frac{1}{\Lambda^{8}}(b_{2}t_{\mu}^{\,\,\rho}t_{\rho\nu}+b_{3}\eta_{\mu\nu}t_{\alpha\beta}t^{\alpha\beta}+b_{4}\Box^{2}t_{\mu\nu}+\cdots)+\mathcal{O}\left(\frac{1}{\Lambda^{10}}\right)$

with which (78) is satisfied. This is possible only for specific choices of
$a_{i}$s and hence the coarse-graining CP unital maps. The emergence of bulk
spacetime follows by considering the $D+1$-dimensional metric in the
Fefferman-Graham gauge after identifying $r$ with $\Lambda^{-1}$:

${\rm d}s^{2}=\frac{{\rm d}r^{2}+g_{\mu\nu}(r,x){\rm d}x^{\mu}{\rm
d}x^{\nu}}{r^{2}}.$ (80)

The metric above satisfies the $D+1$-dimensional Einstein’s equations or
appropriate classical gravity equations. We can always choose the $a_{i}$s in
(76) such that we obtain $g_{\mu\nu}(\Lambda=r^{-1})$ with which (78) is
satisfied and the metric (80) satisfies $D+1$-dimensional classical gravity
equations. It was shown in Behr:2015aat  that (78) is sufficient to guarantee
the emergence of a $D+1$-dimensional classical gravity because without the
emergence of a gauge (diffeomorphism) symmetry (78) cannot hold along the RG
flow. In fact the RG flow has an automorphism which is related to the residual
gauge symmetry of the Fefferman-Graham gauge that maps to conformal
transformations at the boundary. Via appropriate state-dependent conjugations
one can obtain the RG flow that reproduces the bulk metric in other gauges. In
the Fefferman-Graham gauge, the RG flow is manifestly state-independent
(except for a subtlety which we describe below).

The key point of highly efficient RG flow is that one should choose the
complementary subspace which is projected out such that the energy-momentum
exchanges with the complement can be absorbed into a redefinition of the
metric. One can generalize this to cases where other single-trace operators
have non-trivial expectation values simply by introducing sources for these
operators such that a general version of the local Ward identity (78) holds.
These scale-dependent sources then are the dual bulk fields after we identify
$r$ with $\Lambda^{-1}$.

This coarse-graining was implemented in the hydrodynamic sector in
Behr:2015yna . To do this one assumes that $\langle t_{\mu\nu}\rangle$ is
given in terms of constitutive relations via hydrodynamic variables which can
be expanded in the derivative expansion. In this case, we need to sum to all
orders in $\Lambda^{-1}$ at a fixed order in derivatives. The RG flow is
implemented by coarse-graining of the hydrodynamic variables with the scale
itself being a local functional of these variables (see Fig. 15). In this
case, the RG flow (76) reduces simply to first order ODEs which evolve
infinite number of transport coefficients. This reproduces Einstein’s
equations (or other classical gravity equations). To see this we simply need
to follow the procedure in Kuperstein:2011fn ; Kuperstein:2013hqa  and rewrite
the gravity equations (in the derivative expansion) as a first order flow of
transport coefficients.

Figure 15: The highly efficient RG flow in the hydrodynamic sector can be
simply constructed as evolution of the velocity field $u^{\mu}$ and the
temperature field $T$ with the coarse-graining length scale (identified with
the bulk radial coordinate $r$) via the constitutive relation defining the
stress-tensor. The effective background metric is evolved such that the stress
tensor is locally conserved. This implies that the scale of coarse-graining
$\Lambda$ becomes dependent on $u^{\mu}(r)$ and $T(r)$ in a very specific way
so that the violation of the local energy-momentum conservation can be
absorbed into a re-definition of the metric. The highly efficient RG flow
simply reduces to the first order flow equations for the transport
coefficients. These have unique solutions requiring that at the end-point
corresponding to the location of the horizon we obtain an incompressible non-
relativistic Navier-Stokes fixed point. This leads to values of transport
coefficients at the boundary which give a dual spacetime without naked
singularities at the future horizon. Figure from Kuperstein:2013hqa .

This raises a profound question. In the fluid/gravity correspondence
Bhattacharyya:2007vjd ; Baier:2007ix , one explicitly solves the equations of
gravity in derivative expansion and obtains the transport coefficients in the
UV by requiring that the dual solution has a regular future horizon. However
in the RG flow we do not see the metric directly but only
$t_{\mu\nu}(\Lambda)$. How do we then determine the transport coefficients?
This was solved in Kuperstein:2013hqa . It was shown that there is a
particular scale $\Lambda_{h}$ where there is naively a singularity in the RG
flow with the pressure and most transport coefficients blowing up – this scale
corresponds to the horizon in the dual geometry. However, if one redefines
time and the scale in a universal way (analogous to taking the near-horizon
limit), then the singularity actually transforms to a fixed point described by
non-relativistic incompressible Navier-Stokes equations with a single
parameter, namely the shear-viscosity. This is however possible only if the
transport coefficients do not blow up faster than certain bounds as $\Lambda$
approaches $\Lambda_{h}$. These conditions then fix the integration constants
in the ODEs describing the RG flow of the transport coefficients, and we
precisely obtain those values of the transport coefficients (shear-viscosity
at first order and five other transport coefficients at second order etc) at
the boundary computed in Policastro:2001yc ; Bhattacharyya:2007vjd ;
Baier:2007ix  which lead to a regular future horizon. Essentially the
regularity of spacetime is simply a consequence of the regularity of the
highly efficient RG flow which in the hydrodynamic limit is simply the
condition that it ends at the scale of the mean-free path at a fixed point
described by the non-relativistic incompressible Navier-Stokes equations
(governed just by an effective shear viscosity).151515In Kuperstein:2013hqa ,
it was shown that the counterterms which remove the UV divergences are also
fixed by the Navier-Stokes fixed point.

The highly efficient RG flow can lead to a new way to understand bulk
emergence. The RG flow can be naturally viewed as a encoding of infrared
(coarse-grained) physics into microscopic degrees of freedom. The highly
efficient RG flow principle of preservation of effective Ward identities (78)
then leads to emergence of bulk fields. It might seem that at least in the
Fefferman-Graham gauge this RG flow (76) is manifestly state-independent.
However, this is not completely true because one needs to impose infrared
boundary conditions which depends on the code subspace. In the case of the
hydrodynamic sector this is specified by the Navier-Stokes fixed point.
However, within the code subspace the RG flow can be cast in a state-
independent form. This is precisely how the reconstruction of the black hole
interior in the dual field theory should be state-dependnet as we will discuss
in section 5.2. The boundary operator which will reconstruct the bulk operator
in the black hole interior will depend on the choice of the code subspace but
not on the specific state in this code subspace. This motivates a study of the
highly efficient RG flow beyond the hydrodynamic sector.

##  5 Decoding the black hole interior

###  5.1 How the islands emerge from replica wormholes

The black hole interior poses the most formidable challenge in the
understanding of bulk reconstruction. If the evolution of the black hole can
be described by an unitary theory, we should expect that after the Page time
(defined below), the information of the interior should begin to leak out into
the Hawking radiation. This is a consequence of Page’s theorem Page_1993 ;
Page_2013  which states that for a typical pure state in a bipartite system
with one system being much smaller than the other, the trace distance between
the density matrix of the smaller subsystem and the microcanonical ensemble is
of the order of $e^{-S/2}$ with $S$ being the number of qubits in the larger
one. At initial stages, the black hole is much larger than its Hawking
radiation, so the entanglement entropy of the latter should grow. The Page
time occurs when the black hole and the emitted radiation have the same coarse
grained entropy161616The coarse-grained entropy is defined as follows.
Consider expectations values of simple operators (averaged over certain time-
scales) and construct all density matrices which reproduce them. The coarse-
grained entropy is the entropy of such a density matrix which has maximal von-
Neumann entropy. Since the argument of Page uses typicality, the Page time is
appropriately defined via the coarse-grained entropy which is determined
mostly by the size of the Hilbert space.. After the Page time, the
entanglement entropy of the radiation should decrease as it becomes the larger
subsystem. The already emitted Hawking quanta should also be purified by the
subsequently emitted Hawking quanta. Thus its total entanglement entropy
should follow the Page curve – initially it should grow linearly in time
following Hawking’s original computation but then it should fall back to zero
after Page time as it gets purified.

The crucial question is that if the semi-classical description is valid for
the effective field theory (EFT) observables, then what exactly goes wrong
with the semi-classical computation of the entanglement entropy of the Hawking
radiation after the Page time. The question is further sharpened by the
Almheiri-Marolf-Polchinski-Sully (AMPS) Almheiri:2012rt  paradox which points
that such assumptions would lead to violation of the monogamy of entanglement
because the Hawking quanta are maximally entangled with their infalling
counterparts according to the EFT computations, but they also have to be
maximally entangled with the quanta emitted before Page time in order to
purify it and validate the unitarity of the evolution. The strong sub-
additivity property of the entanglement entropy would prevent simultaneous
maximal entanglement of a system with two other systems PhysRevA.69.022309 .
We will discuss more aspects of this paradox later.171717The AMPS paradox
builds on the discussions by Mathur Mathur:2009hf  which introduced the strong
subadditivity originally to examine if small semi-classical corrections can
restore unitarity in Hawking radiation. Many aspects of the AMPS paradox were
discussed also in PhysRevLett.110.101301 .

It is to this question (paradox) that the AdS/CFT correspondence has recently
produced some remarkable new insights along with a deeper understanding of
which part of the black hole interior is encoded in a given subregion of the
Hawking radiation system. The main lesson is that the EFT in the semi-
classical black hole geometry is indeed unimpeachable in its usual domain of
validity set by an energy scale, however there are other subleading Euclidean
saddles in the path integral for Rènyi entropies which give sufficient
contribution after Page time to restore unitarity. Furthermore, these saddles
imply that the fine-grained entropy181818The fine-grained entropy can be
defined like the coarse-grained entropy above but taking into account more
operators that probe the short distance structure of the density matrix.
However, we should keep in mind that we are already doing an averaging which
allows a tensor factorization between the black hole and radiation system to
emerge. A very concrete discussion on this issue can be found in Ghosh:2021axl
within the context of the semi-classical approximation itself. We will have a
more elaborate discussion on this in the context of microstate models in
Section 5.3.2. of Hawking radiation (obtained from the Rènyi entropies) would
involve contributions from islands (which are regions of bulk spacetime
diconnected from the asymptotic boundary and including portions of the black
hole interior as already introduced in the context of explicit models in
section 3.5) after Page time as these are connected to the already emitted
quanta via wormholes. This implies the ER= EPR mechanism Maldacena_2013  of
resolution of AMPS paradox with wormholes implying that after Page time
operators in the interior also affect this far away radiation, and therefore
the early radiation and the interior are not separable systems. We will
discuss later that if one has to resolve this issue by explicitly following
the state in real time we will need self-averaging and complexity. The crucial
inputs in both cases would be the island reconstruction for which the
framework of operator error correction would play an important role.

Figure 16: This figure is from Almheiri2020Islands . It depicts the
construction where a holographic system $B$ is coupled to a large system $R$
without gravity. The dynamics of $B$ is described by the evaporating black
hole with a constant curvature which is glued to flat Minkowski space in which
$R$ lives. $R$ collects the Hawking quanta of the evaporating black hole. The
red regions are the entanglement wedges of the interval(s) which contribute to
the von-Neumann entropy of the Hawking quanta in $R$ while the green regions
are those that contribute to $B$. (a) At early time, the entropy of the matter
fields in whole of the Cauchy slice contributes to $B$. (b) After Page time,
the interval exterior to the quantum extremal surface, whose entanglement
wedge is the island, contributes to the von-Neumann entropy of the Hawking
quanta in $R$.

The anti-de Sitter black hole cannot evaporate unless the boundary conditions
allow the quanta of bulk matter to escape out of the asymptotic region. The
understanding of how the black hole interior is encoded into these Hawking
quanta especially after the Page time can be achieved by coupling a
holographic system $B$ with another larger system $R$ living in one higher
dimension, and which may or may not be holographic. The role of $R$ is to
collect the Hawking quanta of the evaporating black hole in the dual geometry
that depicts the evolving system $B$ holographically. For tractability, we
also need the bulk matter to have large central charge $c$ so that we can
ignore quantum gravity fluctuations. What has been shown to be crucial is
that, after the Page time, the entanglement wedge of the quantum extremal
surface (QES) of the asymptotic boundary is encoded in $B$, while the wedge
including the exterior of the QES called the island and containing parts of
the black hole interior is encoded into $R$ as illustrated in Fig. 16 for
$0+1$-dimensional $B$ coupled to a $1+1$-dimensional $R$. To see this, we need
to consider the (pure) state of the quantum matter fields on a Cauchy slice of
the black hole geometry and $R$, and consider contributions of saddles that
contribute to the Rènyi entropy of the Hawking radiation in $R$ other than the
semi-classical black hole itself. These saddles connect the replicas of the
Rènyi entropy computation via wormholes, and are therefore called replica
wormholes. These wormholes connect the island $I$ including parts of the black
hole interior to the Hawking quanta in $R$, thus invalidating the tri-
partition in the AMPS paradox and resolving it.

Figure 17: A thermofield double version of the setup described in Fig. 16
taken from Almheiri2020Islands . The particles with the same color
(blue/green) are entangled according to effective field theory computations.
Here we consider the entanglement entropy of the Hawking quanta in the red
intervals at time $t=0$ and at $t=t_{0}$ with $t_{0}$ exceeding the Page time.
In (a) we see that if we do not include the island, then the von-Neumann
entropy of the red intervals ($R$ )should grow linearly with time and with the
slope determined by the rate of Hawking pair production. We see in (b) that if
one includes the island ($I$), then the matter contribution to the von-Neumann
entropy in $I\cup R$ cannot grow as the entangled pairs get collected in this
combined region.

We need to explain why such saddles which give exponentially small
contributions in the semi-classical limit to the Rènyi entropy should be
important at all. The leading saddle is the usual
$\widetilde{\mathcal{M}}_{n}/Z_{n}$ geometry discussed in Section 2 denoting
$n$ copies of the semi-classical black hole geometry quotiented by the cyclic
permutation symmetry. In the presence of gravity, we need to take into account
the other replica wormhole saddles too. The point is that the contribution
from the $\widetilde{\mathcal{M}}_{n}/Z_{n}$ saddle decreases exponentially
with time (consistent with the linear growth of the von-Neumann entropy in the
$R$ region) while the contribution from the other replica wormhole saddles
grow with time essentially due to the growth of the island. Around the Page
time, these effects compete with each other, and after sufficiently long time
the replica wormholes give the leading contribution to the Rènyi entropies,
and arrest the growth of the von-Neumann entropy of $R$ as illustrated in Fig.
17.191919Note in the thermofield double setup the entanglement entropy of the
radiation should not vanish but rather saturate to be consistent with
unitarity since not all radiation escapes to the null infinities in $R$. The
Page curve then involves a transition from linear growth to saturation.

Figure 18: An example of a replica wormhole from Almheiri2020Islands  for the
computation of the $3$rd order Rènyi entropy of the Hawking quanta in $R$ in
the setup described in Fig. 17. Firstly, we consider three copies of the
Euclidean version of the diagrams in Fig. 17. In each copy, the shaded blue
region is the Poincaré disk denoting Euclidean $AdS_{2}$ black hole with an
appropriate time-reparametrization at the boundary of the disk. The white
regions have cuts corresponding to the red intervals in Fig. 17. In (a) we
denote the covering manifold $\widetilde{\mathcal{M}}_{3}$ and in (b) the
actual replica wormhole solution which is
$\mathcal{M}_{3}=\widetilde{\mathcal{M}}_{3}/Z_{3}$ which arises as a solution
to the Euclidean gravity coupled to matter. The fixed points of the $Z_{3}$
action, namely $w_{1}$ and $w_{2}$, appear as conical singularities in the
gravitational solution where the bulk twist operators are inserted. Note their
positions are determined by the equations of motion also.

These replica wormholes are essentially codimension two cosmic branes on the
gravity side (giving rise to conical singularities in two-dimensional gravity)
where the twist operators of the bulk matter theory must be inserted. See Fig.
18 for an illustration. The positions of the these branes should be obtained
by solving the gravitational equations of motion. For the semi-infinite
interval in $R$ containing the Hawking quanta in Fig. 16, it turns out that
there is one such brane, and in the limit $n\rightarrow 1$ with $n$ denoting
the number of replicas, the position of the cosmic brane is exactly that of
the quantum extremal surface which computes the entropy of $B$. In case of the
double interval in Fig. 17, the cosmic branes coincide with the two quantum
extremal surfaces in the limit $n\rightarrow 1$ as well. For a recent
pedagogical review on wormholes in semiclassical gravity including the
explicit construction of these replica wormholes see Arnab_review .

Specifically, the island rule (54) which is now validated by the gravitational
path integral accounting for the replica wormhole saddles, implies that

$S(\rho_{R})=S(\rho_{RI}^{(g)})+\frac{A({\rm QES})}{4G}.$ (81)

Above $\rho_{R}$ denotes the density matrix of the interval(s) in $R$,
$\rho_{RI}^{(g)}$ denotes the density matrix of the quantum fields
(responsible for the Hawking quanta) in $R\cup I$ in the semiclassical
geometry of the black hole glued to Minkowski space(s). Note that $A({\rm
QES})$ above refers simply to the value of the dilaton for the JT theory since
the QES is a point (or two points in the case of the thermofield double). We
should understand the above as a semi-classical statement only – the island is
defined with respect to the semi-classical geometry itself. On the other hand,
if $D$ is the interval in this geometry which is the complement of $R\cup I$
and connects the QES to the boundary of the $AdS_{2}$ region, then the
generalization of the HRT formula given by (53) will imply that

$S(\rho_{B})=S(\rho_{D}^{(g)})+\frac{A({\rm QES})}{4G}$ (82)

Above by $B$ we mean its two copies in the thermofield double case. Since, the
quantum fields in the full Cauchy slice $I\cup D\cup R$ in the semiclassical
description should be in a pure state, it implies that
$S(\rho_{RI}^{(g)})=S(\rho_{D}^{(g)})$, and therefore
$S(\rho_{R})=S(\rho_{B})$. This is then consistent with unitarity of the
evolution of the full (two copies of the) dual $B\cup R$ system.

We can also then claim that the bulk effective field theory operators acting
in the island can be reconstructed in $R$ as an extension of the entanglement
wedge reconstruction described before in section 3. The code subspace is the
span of states in the full (two copies of) $R\cup B$ which can be constructed
in the gravitational description via operator insertions in the same semi-
classical geometry containing the same island $I_{i}$ at leading order in
perturbative EFT corresponding to an interval $R_{i}$ in $R$. The code
subspace is assumed to admit a factorization of the form
$\mathcal{H}_{code}=\mathcal{H}_{R_{i}}\otimes\mathcal{H}_{\overline{R}_{i}}$
where $R_{i}$ here denotes the interval(s) in $R$, and $\overline{R}_{i}$ the
complement of $R_{i}$ that includes (two copies of) $B$. The EFT of bulk
fields in the semi-classical gravity description including the Minkowski space
region(s) imply that
$\mathcal{H}_{code}=\mathcal{H}_{R_{i}}\otimes\mathcal{H}_{D_{i}}\otimes\mathcal{H}_{I_{i}}$
where $D_{i}$ is the complement of the island and $R_{i}$. Then (81) holds for
any $\rho_{R_{i}}^{k}$ obtained from any state
$|k\rangle\in\mathcal{H}_{code}$ by tracing out $\overline{R}_{i}$, and with
corresponding $\rho_{R_{i}I_{i}}^{(g)k}$ on the bulk side obtained by tracing
out $D_{i}$. An extension of the JLMS argument would then imply that the
relative entropies of the semi-classical bulk states and the corresponding
states in the full dual description are identical, i.e.

$S(\rho_{R_{i}}^{m}|\rho_{R_{i}}^{n})=S(\rho_{R_{i}I_{i}}^{(g)m}|\rho_{R_{i}I_{i}}^{(g)n})$
(83)

for two states $|m\rangle$ and $|n\rangle$ belonging to $\mathcal{H}_{code}$.
A repeat of the theorem of Dong, Harlow and Wall described in section 4.2.1
then implies that for any bulk EFT operator $\mathcal{O}_{I_{i}}$ localized in
the island $I_{i}$ and any $|n\rangle\in\mathcal{H}_{code}$, we should have

  1. 1.

For any any $X$ localized in $\overline{R}_{i}$

$\langle n|[X,O_{I_{i}}]|n\rangle=0,$ (84)

and

  2. 2.

there exists an operator $\mathcal{O}_{R_{i}}$ localized in $R_{i}$ such that
its action on the code subspace is identical, i.e.

$\mathcal{O}_{R_{i}}|n\rangle=\mathcal{O}_{I_{i}}|n\rangle,\quad\mathcal{O}_{R_{i}}^{\dagger}|n\rangle=\mathcal{O}_{I_{i}}^{\dagger}|n\rangle.$
(85)

Although the replica wormhole saddles produce results which are compatible
with unitarity, there are crucial subtleties which are explicitly brought out
by the Euclidean (Penington, Shenker, Stanford and Yang) PSSY toy model
Penington:2019kki  in which we can sum over the planar replica wormhole
saddles explicitly (via a Schwinger-Dyson type equation for the resolvent
matrix $1/(\lambda I-\rho_{R})$). It also gives a simple way to see how the
replica wormhole saddles give rise to the island which we will review below.
In this Euclidean model, we represent the state of the $B\cup R$ system in the
maximally entangled form

$|\psi\rangle=\frac{1}{\sqrt{k}}\sum_{i=1}^{k}|\psi_{i}\rangle_{B}|i\rangle_{R}$
(86)

which should be a good approximation to the full state of the Hawking quanta
especially after Page time. The state of the holographic bulk $B$ is captured
by an end of the world (EOW) brane state $i$ as shown in Fig. 19. The
evolution of time is captured by simply dialling $k$. To capture the dynamics
post Page time then we need $k\gg e^{S_{BH}}$ where $S_{BH}$ is the entropy of
the bulk black hole. The bulk theory is pure JT gravity with a dilaton and no
bulk matter, while the EOW brane has vanishing extrinsic curvature and the
Neumann boundary condition for the bulk dilaton is also imposed at its
location. Explicitly, the Euclidean action for the metric $g$ and the dilation
$\phi$ is

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\frac{S_{0}}{4\pi}\left(\int_{\mathcal{M}}\sqrt{g}R+2\int_{\partial\mathcal{M}}\sqrt{h}K\right)$
(87)
$\displaystyle-\frac{1}{2}\left(\int_{\mathcal{M}}\sqrt{g}\phi(R+2)+2\int_{\partial\mathcal{M}}\sqrt{h}\phi
K\right)$ $\displaystyle+\mu\int_{EOW-brane}{\rm d}s,$

with $h$ denoting the induced metric on the boundary and $\mu\geq 0$ the mass
of the EOW brane. The mentioned boundary conditions imply that on the EOW
brane we should impose

$K=0,\quad\partial_{n}\phi=\mu$ (88)

with $\partial_{n}$ denoting the normal derivative. The standard asymptotic
boundary conditions imply that at the boundary

$h=\frac{1}{\epsilon^{2}},\quad\phi=\frac{1}{\epsilon},\quad\epsilon\rightarrow
0.$ (89)

The limit $\epsilon\rightarrow 0$ implies implementing the standard procedure
of holographic renormalization which extracts physical observables of the $B$
system from the regularized action of the classical gravity solution. The
solutions of the gravitational theory are simply the black hole solutions. It
is easy to note that the computations of the replica wormhole saddles simplify
in absence of bulk matter too. The results of the computations are the same.
When $k\gg e^{S_{BH}}$, the replica wormhole saddles produce a dominant
contribution as a result of which the QES in the limit $n\rightarrow 1$ is
close to the bifurcate horizon opening up an island which is the entanglement
wedge of the $R$ system.

Figure 19: The simplified PSSY model with an end of the world brane carrying a
degree of freedom. It has zero extrinsic curvature and the Neumann boundary
condition for the JT gravity dilaton is imposed on it. The left and right
figures correspond to the Euclidean and Lorentzian scenarios. Figure from
Penington:2019kki

The subtlety is that we would naively think that if the EOW brane states
$|i\rangle$ are orthogonal, then the reduced of the $R$ system which follows
from (86) is

$\rho_{R}=\frac{1}{k}\sum_{i=1}^{k}|i\rangle\langle i|_{R}.$ (90)

Furthermore, ${\rm Tr}(\rho_{R}^{2})=1/k$. The latter is however not correct
if we take into account the replica wormhole saddle as illustrated in Fig. 20.
Explicitly we obtain

${\rm
Tr}(\rho_{R}^{2})=\frac{kZ_{(1)}^{2}+k^{2}Z_{(2)}}{(kZ_{(1)})^{2}}=\frac{1}{k}+\frac{Z_{(2)}}{Z_{(1)}^{2}}$
(91)

where $Z_{(2)}$ is the partition function of the two-sided wormhole and
$Z_{(1)}$ is the disk partition function (the denominator comes from the
appropriate normalization of the path integral). This is clearly incompatible
with (90).

Figure 20: The computation of ${\rm Tr}(\rho_{R}^{2})$ involves two saddles.
The first one is the product of two disconnected disks and the second one is a
two-sided replica wormhole. To compute the trace we must join the lines dashed
lines. The disconnected disks then produce a single loop implying a factor of
$k$ while the double sided wormhole produces two loops implying a factor of
$k^{2}$. Thus we obtain the numerator in Eq. (91). Figure from
Penington:2019kki

The reconciliation is achieved if we interpret that the replica wormhole
saddles of JT gravity are essentially computing the path integrals for an
ensemble average of Hamiltonians describing the $B$ system, so that

$\rho_{R}=\frac{1}{k}\sum_{i,j=1}^{k}|i\rangle\langle
j|_{R}\overline{\langle\Psi_{j}|\Psi_{i}\rangle}_{B}$ (92)

and

${\rm
Tr}(\rho_{R}^{2})=\frac{1}{k^{2}}\sum_{i,j=1}^{k}\overline{|\langle\Psi_{j}|\Psi_{i}\rangle|^{2}}_{B}.$
(93)

Supposing that

$\langle\Psi_{j}|\Psi_{i}\rangle_{B}=\delta_{ij}+e^{-S_{0}/2}R_{ij}$ (94)

where $S_{0}$ is the entropy of the JT gravity black hole at zero temperature
and $R_{ij}$ are random phases, we should obtain

$\overline{\langle\Psi_{j}|\Psi_{i}\rangle}_{B}=\delta_{ij},\quad\overline{|\langle\Psi_{j}|\Psi_{i}\rangle|^{2}}_{B}=\delta_{ij}+\mathcal{O}(e^{-S_{0}})$
(95)

where the bar on top denotes averaging over theories. Note that $e^{-S_{0}}$
is exactly the order of magnitude of $Z_{(2)}/Z_{(1)}^{2}$ because of the
respective topologies (two discs vs one with each disk being
$\mathcal{O}(e^{S_{0}})$202020The first term in the action (87) is topological
and accounts for $e^{S_{0}}$ weighting of the disk.). Note it is easy to see
from (91) that when $k\gg e^{S_{0}}$, then the replica wormhole gives the
dominant contribution $Z_{(2)}/Z_{(1)}^{2}$. This in fact generalizes, and we
can readily see that

${\rm
Tr}(\rho_{R}^{n})\approx\frac{k^{n}Z_{(n)}}{(kZ_{(1)})^{n}}=\frac{Z_{(n)}}{Z_{(1)}^{n}}=\mathcal{O}(e^{-(n-1)S_{0}})$
(96)

so that the contribution is entirely from the simply connected replica
wormhole with $n$ asymptotic boundaries (see Fig. 21 for an illustration in
the case of $n=6$). The $Z_{n}$ quotient of these wormholes $Z_{(n)}$ can be
readily analytically continued (the $Z_{n}$ symmetry is simply a rotational
symmetry and quotienting produces a fixed point which is exactly the horizon
in the limit $n\rightarrow 1$ in which the geometry reduces to the original
unreplicated one). It is also then easy to see that in the limit $n\rightarrow
1$, the von Neumann entropy is just the generalized entropy of the QES that is
located at the horizon, i.e.

${\rm Tr}(\rho_{R})=S_{0}+2\pi\phi_{h}=S_{BH},$

where $\phi_{h}$ is the value of the dilaton at the horizon and $S_{BH}$ is
the thermodynamic entropy of the black hole. The latter equality follows from
the action (87)) and note that there is no bulk matter in this model. Thus the
$k\rightarrow\infty$ limit indeed implies that the QES is at the bifurcate
horizon in the Lorentzian picture and the island is the full interior of the
black hole. We explicitly see that the replica wormholes reproduce the correct
QES. In the opposite limit $k\rightarrow\epsilon$ with fixed $S_{0}$, the
fully disconnected geometry dominates (see Fig. 21 for an illustration in the
case of $n=6$) so that

${\rm
Tr}(\rho_{R}^{n})\approx\frac{kZ_{(1)}^{n}}{(kZ_{(1)})^{n}}=\frac{1}{k^{n-1}}.$
(97)

The von-Neumann entropy is then

${\rm Tr}(\rho_{R})=\log k$

in this limit signifying the absence of any island and that we should trace
over the entire black hole spacetime which thus forms the entanglement wedge
of $B$. This toy model then reproduces features of the growth of the islands
in the full Lorentzian computation with the bulk matter fields described
earlier. The island vanishes at early time, and as the black hole evaporates
it forms and eventually encompass the original interior. The flow of time is
indeed represented in the toy model as the growth of $k$ as illustrated in
Fig. 22.

Figure 21: The computation of ${\rm Tr}(\rho_{R}^{6})$ involves several
saddles of which the fully disconnected and the fully connected ones are
explicitly shown above. The first one dominates when $k\ll e^{S_{BH}}$ while
the second one dominates when $k\gg e^{S_{BH}}$. The latter has a $Z_{6}$
rotational symmetry whose action has a fixed point. Figure from
Penington:2019kki .

The toy model also allows us to derive a version of the Petz map via path
integrals with replica wormhole saddles for reconstruction of the operators
acting on the island in the $R$ system. This makes sense if the wormholes
perform an ensemble averaging over theories describing the $B$ system. The
replica wormholes dominate when $k\gg d_{\rm code}e^{S_{0}}$ where $d_{\rm
code}$ is the dimension of the code subspace, and in this limit the mentioned
Petz map produces a perfect reconstruction as expected. Of course when $k\ll
d_{\rm code}e^{S_{0}}$, there is no island and the $R$ system should have no
knowledge of the black hole interior.

Figure 22: As described in the text, the fully disconnected replica wormholes
dominate in the limit $k\ll e^{S_{BH}}$ implying that ${\rm
Tr}(\rho_{R}^{n})=1/k^{n-1}$. This is obtained by tracing over the entire
black hole spacetime which thus forms the entanglement wedge of $B$. In the
limit $k\ll e^{S_{BH}}$, the fully connected replica wormhole geometry (one
single disc with $n$ asymptotic boundaries) dominates. So we obtain ${\rm
Tr}(\rho_{R})=S_{BH}$ when $n\rightarrow 1$. This implies that the
entanglement wedge of $R$ is the interior of the horizon which forms the
island, while the black hole exterior forms the entanglement wedge of $B$.
Figure from Penington:2019kki .

Recently, Bousso and Shahbazi-Moghaddam have revisited the holographic entropy
bounds discussed in the introduction and have examined them in the context of
islands Bousso:2021sji . In particular, they have been able to find some
general conditions for the existence of islands and how holographic entropy
bounds should be revised in the context of island rule. They have argued that
such arguments could guarantee behavior consistent with unitarity in more
general contexts. This also explains absence of islands Manu:2020tty  found in
the context of AdS-Kasner spacetimes coupled to non-gravitating reservoirs.

The Page curve for other entanglement measures such as entanglement negativity
(which is a more suitable measure than the von Neumann entropy in the case of
mixed states) of the Hawking radiation leaking to a bath has also been
recently reproduced in these two-dimensional setups KumarBasak:2020ams ;
KumarBasak:2021rrx  developing on prescriptions obtained in Kudler-
Flam:2018qjo ; Kusuki:2019zsp ; KumarBasak:2020ams  and are consistent with
results obtained from random matrix theory Shapourian:2020mkc . The
entanglement negativity has also been studied in the context of a generalized
version of the PSSY model with a bipartite (non-gravitating) reservoir in
Kudler-Flam:2021efr ; Dong:2021oad . Interestingly, saddles which break
replica symmetry appear. The Page curve in doubly holographic setups with the
(non-gravitating) reservoir subjected to relevant deformations have been
studied in Caceres:2021fuw . In this work, it has been shown that the coarse-
graining of the reservoir generated by the RG flow leads to an increase in the
Page time.

Finally, the emergence of islands should be understood from modular flow which
is the fundamental element of explicit entanglement wedge reconstruction as
discussed in section 3. See Chen:2019iro  for advances made in this direction.

The fact that the replica wormhole saddles of the Euclidean path integrals
including the gravitating regions perform some kind of averaging is a generic
feature which should hold for higher dimensional setups also. In a full
unitary quantum gravity computation in real time in which we should be able to
take into account the microstates of the black hole (such as fuzzballs)
explicitly, the averaging should emerge dynamically via quantum ergodicity.
The late-time self-averaging in many-body dynamics which justifies the
Euclidean computation of Rènyi entropies leading to an universal approximation
has been discussed in PRXQuantum.2.010344 212121See also Vardhan:2021npf  for
a related discussion on how such equilibrium approximation of chaotic many-
body dynamics makes novel predictions for entanglement in Hawking radiation
before Page time.. Nevertheless, it has to be understood from the bulk point
of view.222222In fact JT gravity has a full non-perturbative dual description
in terms of an ensemble of random matrices Saad:2019lba ; Stanford:2019vob .
That such randomness eg random couplings should be described by wormholes has
been already discussed by Giddings:1988wv ; Polchinski:1994zs . In an unitary
description, such averaging should emerge from ergodicity and then the
question is to understand the dual bulk mechanism from explicit microstate
models.

It is already remarkable that the replica wormholes produce results for the
Page curve which are compatible with unitarity. Nevertheless, many crucial
features of the encoding of the black hole interior into the Hawking quanta
and other related phenomena such as quantum information mirroring cannot be
addressed by such an approach. We will examine very soon if we can construct
tractable microstate models to address these fundamental issues.

###  5.2 State-dependence in microstate reconstruction: Alpha bits and
Python’s lunch

The natural question to ask of course is how do we reconstruct (microstate of)
a non-evaporating black hole which is not in contact with an auxiliary
reservoir.232323By non-evaporating we mean with reflecting asymptotic boundary
conditions. Note Hawking radiation is present but the black hole cannot lose
mass. This should be formulated in the general context of entanglement wedge
reconstruction. This question was originally discussed in the context of the
resolution of the AMPS paradox. It was argued by Papadodimas and Raju
Papadodimas:2012aq ; Papadodimas:2013jku ; Raju:2020smc  that the
reconstruction of the black hole interior in the dual boundary theory has to
be necessarily state-dependent and this can resolve the AMPS and related
information paradoxes. The operators needed for the reconstruction of the
interior should themselves be complicated and would be also microstate
dependent. Its consistency with the framework of quantum mechanics has been
debated, see Harlow:2014yoa  for an instance. We will review these issues
pertaining to the AMPS paradox in the next section. In this section, we will
present developments related to state-dependence of black hole interior
reconstruction which has been reformulated in the framework of quantum error
correction with crucial inputs from the extremal surfaces in Hayden:2018khn ;
Akers:2019wxj ; penington2020entanglement  which is manifestly consistent with
the basic postulates of quantum mechanics. We will also discuss how this
formulation leads to the quantification of the complexity of the
reconstruction of the interior Brown:2019rox ; Engelhardt:2021qjs  that is
necessary for the resolution of the AMPS and related paradoxes (more on this
later). Note this discussion relates to the properties of the entanglement
wedge and is also applicable to evaporating black holes.

Figure 23: In presence of a horizon, there are multiple extremal surfaces
attached to the boundary $\partial A$ of a boundary subregion. As shown above,
there are two with areas $\mathcal{A}_{2}$ and $\mathcal{A}_{1}$. If the
boundary subregion inlcudes more than half of the full boundary, then
$\mathcal{A}_{2}>\mathcal{A}_{1}$. However, $\mathcal{A}_{1}$ is homologous to
$\overline{A}$ and $\mathcal{A}_{2}$ is homologous to $A$. The homology
constraint implies that the entanglement wedge of $A$ is $a$, the bulk region
bounded by $A$ and $\mathcal{A}_{2}$ (and that of $\overline{A}$ is similarly
$\overline{a}$) in the thermal state dual to a black hole. The bulk region
$a^{\prime}$ bounded by the two minimal surfaces, which includes portion of
the exterior of the black hole, cannot be reconstructed at the boundary. As
discussed in text, the situation is different in case of a microstate. Figure
from Hayden:2018khn .

The crucial point in Hayden:2018khn  is that state-dependence in the
(approximate) reconstruction of the interior occurs when we consider a
boundary sub-region instead of the whole dual system. Furthermore, state
dependence actually implies dependence on the code subspace and not dependence
on the specific bulk density matrix with support in this code subspace.
Geometrically it originates from the dependence of the entanglement wedge on
the choice of the code subspace and it is simply determined by the maximally
mixed state in this code subspace. To illustrate this form of state-dependence
in the reconstruction of the interior, consider a region $A$ at the boundary
as shown in Fig. 23. There are two extremal RT surfaces with areas
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ respectively. Here $\mathcal{A}_{2}$
is homologous to $A$ and $\mathcal{A}_{1}$ to its complement $\overline{A}$.
If $A$ is larger than half the boundary, then
$\mathcal{A}_{2}>\mathcal{A}_{1}$. The bulk is separated into three regions
$a$, $a^{\prime}$ and $\overline{a}$ where $a(\overline{a})$ is the region
bounded by $\mathcal{A}_{2}(\mathcal{A}_{1})$ and $A(\overline{A})$, while
$a^{\prime}$ is the region between the two minimal surfaces containing the
black hole horizon with area $\mathcal{A}_{0}$ and portions of the black hole
exterior also. Furthermore, $\mathcal{A}_{2}-\mathcal{A}_{1}<\mathcal{A}_{0}$
and so we can define

$\alpha=\frac{\mathcal{A}_{2}-\mathcal{A}_{1}}{\mathcal{A}_{0}}$

so that $0\leq\alpha\leq 1$. The code subspace is

$\mathcal{H}_{\rm
code}=\mathcal{H}_{a}\otimes\mathcal{H}_{a^{\prime}}\otimes\mathcal{H}_{\overline{a}}.$

In the semi-classical limit $G\rightarrow 0$, the dimension of the full code
subspace is

$e^{S_{\rm BH}}=e^{\frac{\mathcal{A}_{0}}{4G}}.$

Consider the thermal state dual to the actual black hole geometry. Then the
homology constraint implies that the entanglement wedge of $A$ is $a$ and that
of $\overline{A}$ is $\overline{a}$. Any bulk operator
$\mathcal{O}_{a^{\prime}}$ localized in $\mathcal{H}_{a^{\prime}}$ whether it
is in the interior of the black hole or in the exterior cannot be
reconstructed either in $\mathcal{H}_{A}$ or in $\mathcal{H}_{\overline{A}}$.
This is not surprising at all. Firstly note that only
$\mathcal{H}_{a^{\prime}}$ which contains the horizon region will have
dimension $\mathcal{O}(e^{1/G})$ whereas $\mathcal{H}_{a}$ and
$\mathcal{H}_{\overline{a}}$ will have dimension $\mathcal{O}(1)$ in the limit
$G\rightarrow 0$. This implies that none of the microstates can be resolved as
should be the case in the thermal ensemble. Only in the case of a microstate,
we should be able to access (reconstruct) states in $\mathcal{H}_{a^{\prime}}$
at the boundary.

To be precise, we assume that for any typical microstate we can consider the
same semi-classical geometry up to where the surfaces $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ are located (indeed valid for fuzzballs which differ from
the black hole significantly only at the scale of the horizon Mathur:2005zp ).
This assumption is however not crucial for what follows – it is enough if the
location of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ have sub-leading state
dependence. Crucially, in a microstate (with a smooth geometry) there will be
no homology constraint. For reasons which will be clear later, we should
consider a mixed state in the bulk. In order to do this, let us entangle the
black hole with a reference system $R$ (note this does not imply that they are
in physical contact or $R$ is acting as a reservoir of Hawking quanta). Let
$|\psi\rangle$ be a pure state in $\mathcal{H}_{\rm
code}\otimes\mathcal{H}_{R}$. The question is then can we decode an operator
$\mathcal{O}_{a^{\prime}}$, localized in $\mathcal{H}_{a^{\prime}}$, in $A$.
It will be possible if the entanglement wedge of $A\cup R$ is $a\cup
a^{\prime}\cup R$. This would be so if
$S_{gen}(\mathcal{A}_{2})>S_{gen}(\mathcal{A}_{1})$ implying that the quantum
extremal surface corresponding to $A$ should be $\mathcal{A}_{1}$ instead of
$\mathcal{A}_{2}$ for all states $|\psi\rangle\in\mathcal{H}_{\rm
code}\otimes\mathcal{H}_{R}$. Therefore, we need (using
$S(aa^{\prime})_{\psi}=S(\overline{a}R)_{\psi}$ for a pure state
$|\psi\rangle$):

$S(a)_{\psi}+\frac{\mathcal{A}_{2}}{4G}>S(aa^{\prime})_{\psi}+\frac{\mathcal{A}_{1}}{4G}=S(\overline{a}R)_{\psi}+\frac{\mathcal{A}_{1}}{4G}.$
(98)

As discussed above, in the limit $G\rightarrow 0$, $S(a)_{\psi}$ is
$\mathcal{O}(1)$. Also the triangle inequality implies

$|S(R)_{\psi}-S(\overline{a}R)_{\psi}|\leq
S(\overline{a})_{\psi}=\mathcal{O}(1).$

So (98) reduces to the condition that if

$4GS(R)_{\psi}<\mathcal{A}_{2}-\mathcal{A}_{1},$ (99)

then the entanglement wedge of $A$ is $a\cup a^{\prime}$. This requires that
$|\psi\rangle$ belongs to a code subspace
$\mathcal{H}_{\mathcal{S}}\subseteq\mathcal{H}_{\rm code}$ of dimension
$d_{S}=d_{R}$ such that the above inequality is satisfied for any state in
this subspace. The reconstruction criterion then arises from the maximally
mixed state in the subspace for which $S(R)_{\psi}=\log d_{S}$ implying that

$d_{S}<e^{\frac{\mathcal{A}_{2}-\mathcal{A}_{1}}{4G}}=e^{\alpha S_{\rm BH}}.$
(100)

Irrespective of whichever state we choose in this subspace
$\mathcal{H}_{\mathcal{S}}$, any bulk operator $\mathcal{O}_{a^{\prime}}$
localized in $\mathcal{H}_{a^{\prime}}$ can be reconstructed via the same
operator $\mathcal{O}_{A}$ in $\mathcal{H}_{A}$. Nevertheless, this operator
$\mathcal{O}_{A}$ will depend on the choice of the code subspace – outside
this subspace there exists states (eg. the fully entangled state of the black
hole interior and $R$) the action of $\mathcal{O}_{a^{\prime}}$ on which
cannot be simulated by $\mathcal{O}_{A}$ since the entanglement wedge of $A$
does not contain $a^{\prime}$ for such states. The state-dependence of
$\mathcal{O}_{A}$ is via the choice of this appropriate code subspace alone.
As the size of $A$ grows, clearly $\alpha\rightarrow 1$. Therefore, the code
subspace involves all typical states in the full Hilbert space and there is no
state dependence in the reconstructed operators $\mathcal{O}_{A}$ which have
the same action as $\mathcal{O}_{a^{\prime}}$ acting in the interior of the
black hole.

The quantum error correcting protocol which reproduces the above desired
behavior is the universal subsystem recovery channel first considered in
Hayden:2017xed . Suppose there is a quantum channel that applies a Haar-random
unitary $U$ to $n$ qubits, throws away a fraction of them and transmits the
rest. Let the input Hilbert space be $\mathcal{H}_{\rm in}$. If one retains a
fraction $(1+\alpha)/2$ of the qubits, then one will be able to decode
(recover) any subspace in the input Hilbert space which has $\alpha n$ qubits
(this subspace therefore has dimension $2^{\alpha n}$.) The universal subspace
recovery map however can only approximately reverse the channel. Sometimes,
there has to be an error $\epsilon>e^{-\eta n}$ with $\eta>0$. In the context
of reconstruction of microstates this implies error

$\epsilon>e^{-\eta/G}$

which should be typically non-perturbatively small in $G$. The explicit
universal recovery map is closely related to the twirled Petz map
Hayden:2018khn  which has tantalizing connections with the modular Hamiltonian
as discussed before. The bulk region $a^{\prime}$ between the minimal surfaces
contains the $\alpha$-bits in the language of universal subspace error
correction.

Such issues in bulk reconstruction arise not only in the case of black holes
but also when such competing extremal surfaces enclose bulk matter with high
entropy – we refer the reader to Akers:2019wxj ; Akers:2020pmf  for extensive
discussions. In particular Akers:2020pmf  discusses how the quantum extremal
surface prescriptions should be refined in such generic circumstances with
information-theoretic interpretations.

Intuitively it should be hard to reconstruct operators lying in $a^{\prime}$
between the two minimal surfaces in Fig 23 from $A$ even within the code
subspace essentially due to state-dependence. This is similar to the Hayden-
Harlow conjecture of exponential complexity of reconstruction of operators in
the interior of the black hole restricted to Hawking radiation to be discussed
in the following subsection. A remarkable geometric way of capturing this
complexity was proposed in Brown:2019rox  based on the generic existence of
non-minimal extremal surfaces forming a Python-lunch geometry (sandwiched
between minimal extremal surfaces or behind one of them) and interpreting this
geometry in terms of tensor network constructions. This proposal further
develops earlier proposals for the holographic dictionary for complexity
Susskind:2014rva ; Brown:2015bva ; Brown:2015lvg . For a recent review on
quantum complexity and its holographic description see Shira_review .

The mechanism for nucleation for such Python lunches was proposed in
Engelhardt:2021qjs . Essentially the argument in the case of the black hole
setup of Fig 23 is that we should first consider the appropriate code subspace
where the interior can be decoded in $A$. In the maximally mixed state in this
code-subspace the Hawking pairs will experience disentanglement and the bulk
entropy gradients will be larger than the Hartle-Hawking state. Using results
from Wall:2012uf ; Marolf:2019bgj ; Akers:2019lzs , then it has been argued
that due to blueshift of the entropy gradients when extrapolated in the past
there would be nucleations of non-minimal (highly non-classical) extremal
surfaces (since it can compete with the leading order $1/G$ term) behind the
horizon ensuring that they do exist also at late time (this is somewhat
similar to the existence of the transition in the extremal surfaces we have
seen before in the context of reproduction of the Page curve). Even if one
considers the full boundary, one needs to consider a code subspace for
discussing the interior outgoing modes and then there will be such a bulge
surface $\gamma_{\rm bulge}$ behind the outermost extremal surface
$\gamma_{\rm aptz}$ (the appetizer of the Python’s lunch) – see Fig. 24. The
complexity of the decoding (defined below) within this code subspace will then
be half the difference in the generalized entropies of $\gamma_{\rm bulge}$
and $\gamma_{\rm aptz}$.

Figure 24: In order to consider excited modes in the black hole interior, one
first needs to restrict to a code subspace where the geometry will have a
Python’s lunch – a non-minimal extremal surface $\gamma_{\rm bulge}$ behind
the outermost minimal extremal surface $\gamma_{\rm aptz}$. The throat refers
to the asymptotic region. There can be other locally minimal surfaces behind
$\gamma_{\rm bulge}$ (not shown in figure). Figure from Engelhardt:2021qjs .

For the moment, let us assume that such a Python lunch geometry is generic
when one considers appropriate code subspaces where interior modes are excited
and can be defined in terms of the macroscopic geometry corresponding to the
maximally mixed state in this code subspace. In this case, one can justify the
quantification of the complexity of decoding the interior from tensor network
models. Such models were also used to derive the holographic complexity
conjectures Susskind:2014rva ; Brown:2015bva ; Brown:2015lvg  which need to be
modified in the presence of the Python lunches. For an illustration consider
such a wormhole with a Python lunch in the two-sided thermofield double
geometry represented in the form of a tensor network in Fig. 25. We discuss
the circuit complexity of decoding following Engelhardt:2021qjs .

Figure 25: A tensor network dual to a Python’s lunch geometry such that the
map from the input state in the left Hilbert space and the black dots
representing bulk legs to the right Hilbert space is an approximate isometry.
It has two locally minimal cuts $\gamma_{\rm aptz}$ and $\gamma_{\rm min}$
with $\gamma_{\rm aptz}>\gamma_{\rm min}$. Furthermore, there is a maximal cut
$\gamma_{\rm bulge}$ between these two cuts. The triangles denote isometries
whereas the squares have one or more of their legs projected to $|0\rangle$
state. Figure from Engelhardt:2021qjs .

The tensor network represents the map from the bulk represented as black dots
in Fig 25 to the tensor product of the left and right CFT Hilbert spaces.
Triangles represent isometries and the squares involve postselection – an
isometry where one (or more) of the legs is projected to $|0\rangle$. In a
generic tensor network of such type, there will be two locally minimal cuts
$\gamma_{\rm min}$ (global minimum) and $\gamma_{\rm aptz}$ (representing the
locally minimal extremal surfaces) and a $\gamma_{\rm bulge}$ locally maximal
cut in the middle. When we view the figure from the left to right it
represents an approximate isometry from the tensor product of bulk legs and
left CFT Hilbert space to the right CFT Hilbert space. For this to be the
case, we need to assume that the bonds cut in $\gamma_{\rm aptz}$ have larger
dimension than those in the $\gamma_{\rm min}$ cut plus the dimensions of the
bulk legs (denoted as black dots) between these two cuts. However, the problem
is in the middle because the map from the $\gamma_{\rm bulge}$ cut plus the
bulk legs in between the $\gamma_{\rm bulge}$ and the $\gamma_{\rm aptz}$ cut
to the $\gamma_{\rm aptz}$ cut is not an isometry as the Hilbert space
dimension of the latter is smaller. To make it work in practice we need to
somehow implement the postselection (which is not unitary as it involves
projection) in a different way via unitary circuits. This can be done via
Grover search algorithm in the manner discussed first in Yoshida:2017non  – we
need to do sequential unitary transformations such that we bring the qubits
that are supposed to be postselected already in $|0\rangle$ state. The
complexity of the decoding of the Python lunch is essentially that of this
Grover search algorithm. By general arguments this complexity is

$C=O(\tilde{C}2^{m/2})$

where $m$ is the number of qubits which should be postselected (equal to the
difference between the bulge and appetizer cuts) and $\tilde{C}$ is related to
the overall size of the network.242424For a discussion on complexity utilizing
the Petz map in the context of a subregion see Zhao:2020wgp .

Now in the gravitational analogue $\gamma_{\rm aptz}$ and $\gamma_{\rm min}$
are surfaces with locally minimal generalized entropies. By our earlier
discussion the code subspace that can be reconstructed in the right Hilbert
space has dimension

$S_{gen}(\gamma_{\rm aptz})-S_{gen}(\gamma_{\rm min})$

with generalized entropies defined by considering the maximally mixed state in
the code subspace. The condition $S_{gen}(\gamma_{\rm
aptz})>S_{gen}(\gamma_{\rm min})$ is the reconstructibility criterion
analogous to that for the tensor network to be an isometry to the right
Hilbert space. Furthermore, $\gamma_{\rm bulge}$ has locally maximal
generalized entropy and for the Python lunch geometry $S_{gen}(\gamma_{\rm
bulge})>S_{gen}(\gamma_{\rm aptz})$. Then the tensor network analogy suggests
complexity of decoding should be

$C=\mathcal{O}(\tilde{C}\exp\frac{1}{2}\left(S_{gen}(\gamma_{\rm
bulge})-S_{gen}(\gamma_{\rm aptz}))\right).$

The exponential dependence in $G^{-1}$ comes crucially only from the
difference of the generalized entropies while the dependence on $G^{-1}$ in
$\tilde{C}$ (which is of same order in $G^{-1}$ as the generalized entropies)
is only linear. So the former accounts for exponential complexity. A similar
discussion can be repeated for a single sided microstate geometry shown in Fig
24 also.

The understanding of the mechanism for generation of exponential complexity
already alerts us towards the need for more explicit microstate geometries
that can describe appropriate code subspaces. We will further motivate
microstate models in the following subsection.

Interestingly a converse of the Python’s lunch conjecture has been discussed
in Engelhardt:2021mue . It has been claimed that bulk operators between the
boundary and the outermost extremal surface should have a simple
reconstruction at the boundary in the sense that they can be recovered
efficiently from a dual coarse-grained state with an effective local modular
Hamiltonian. See also Levine:2020upy  for a related discussion. Recent
discussions of the reconstruction of the experience of a bulk observer in the
dual conformal field theory can be found in Jafferis:2020ora  via the use of
modular flow and through emergent properties of von-Neumann algebras in
Leutheusser:2021frk  .

###  5.3 Decoding the interior in real time

####  5.3.1 Quantum information mirroring and the resolution of the AMPS
paradox via complexity

Quantum information theory has been applied to understand how the black hole
acts as a quantum channel with a motivation to resolve the AMPS paradox. Such
analyses predict very non-trivial features of how the interior gets encoded
into the outgoing Hawking quanta in real time.

An extension of Page’s argument implies that old black holes (past its Page
time) would act as quantum information mirrors following a thought experiment
and its analysis due to Hayden and Preskill Hayden_2007  as illustrated in
Fig. 26. A typical state of an old black hole would have Hawking quanta in the
exterior ($E$) maximally entangled with the interior modes ($B$). If the black
hole is a fast scrambler, then qubits ($D$) thrown into it would be maximally
scrambled with $B$ (one could understand this as an action of a random unitary
operator $U$252525Hayden and Preskill showed that this unitary operator need
not have exponentially large number of gates for a Page-like argument to work.
It suffices to pick the unitary randomly from a unitary two-design, which can
be achieved by a quantum circuit of depth $\mathcal{O}(\log n)$ where
$n=S_{BH}$ is the total number of qubits at the horizon with $S$ being the
black hole entropy. If each step in the circuit takes Plank time, then it
should be redshifted to $r_{s}$ in the time of the asymptotic observer. It
then reproduces the scrambling time as the circuit time which is $r_{s}\log
S_{BH}$. This argument was refined in Sekino:2008he ) resulting in a remnant
$B^{\prime}$ and newly radiated Hawking quanta $R$. For a formal statement, we
consider that $D$ is maximally entangled with a reference system $S$, so that
post-scrambling an extension of Page’s arguments would imply that $S$ and
$B^{\prime}$ have no mutual information (entanglement). However, the
information in $D$ would be transferred to the combined $R\cup E$ system of
Hawking quanta implying information mirroring which would essentially happen
at scrambling time. The latter has been computed based on growth of thermal
commutators by Shenker and Stanford Shenker:2013pqa . In the context of black
holes, it is essentially the time (from the point of view of an asymptotic
observer) it takes a light ray to reach Planck distance close to the stretched
horizon, which is $\approx r_{s}\log S_{BH}$, with $S_{BH}$ being the entropy
of the black hole. For a detailed review of the Hayden-Preskill protocol and
progress on experimental realization of quantum simulators which achieves the
scrambling needed to realize it see Lata_review .

Figure 26: An illustration of the Hayden-Preskill thought experiment which
argues for quantum information mirroring in old black holes – figure taken
from Harlow:2014yka . A quantum diary $D$ entangled with a reference state $S$
is thrown into an old black hole $B$ which is entangled with its radiation
$E$. The diary $D$ and $B$ then gets scrambled by the unitary evolution
operator $U$ after which we obtain the remaining black hole $B^{\prime}$ and
some more radiation $R$. If $R$ has only a few more qubits than $D$, then we
can show that mutual information between $S$ and $B^{\prime}$ should be
negligible given that $U$ is random. The information in $D$ is then in the
$E\cup R$ system after the scrambling time.

The natural question that arises is how easy it would be to decode the qubits
in $D$ from $E\cup R$. Before considering this question we need to encounter
first the most pragmatic operational way to resolve the AMPS paradox that
leads to an additional feature of the encoding of the interior into the
Hawking quanta. In order to present this, it is useful to restate the AMPS
paradox. Once again consider the old black hole for which the interior modes
$B$ should be maximally entangled with $E$, the already radiated quanta.
Unitarity of the time-evolution would imply that the newly emitted quanta $R$
that decouples from the black hole can be purified by a factor $E_{R}$ of $E$
so that $\rho_{RE_{R}}$ is a pure state. However, $R$ must also be maximally
entangled with (a factor of) $B$ if semiclassical EFT holds at the horizon. A
simple way to resolve this paradox is to invoke the philosophy of black hole
complementarity PhysRevD.48.3743 ; PhysRevD.50.2700  which postulates that the
violation cannot be observed operationally via a quantum complexity conjecture
due to Harlow and Hayden Harlow:2013tf  (see Aaronson:2016vto  for a nice
discussion). Essentially this argument states that in order to distill the
factor $E_{R}$ from $E$ needed to purify $R$, it would take time which is
exponential in the entropy of the black hole (at the time when $R$ was
emitted). This conjecture can be made more precise by evoking pseudorandom
encoding of the interior into the outgoing Hawking quanta Kim:2020cds  (in
microstate models we will be able to relate inherent pseudorandom/chaotic
dynamics with Python lunches that can macroscopically amplify small
excitations). Since the black hole evaporates in time that is polynomial in
its entropy, the violation of monogamy of entanglement cannot be demonstrated
operationally.

This discussion raises a few fundamental questions. Firstly, what exactly does
an operational resolution mean? Does it mean we need to modify the framework
of quantum mechanics actually to describe black holes although we cannot
operationally test violation of its postulates? Or does it mean that somehow
these Hilbert spaces, especially $B$ (the interior) and $E$ (the pre-Page time
Hawking quanta) are actually not separable but only in an operational sense?
Then in this operational framework would the AMPS paradox be resolved by
complexity? The Euclidean replica wormhole saddles which connect $E$ and $B$
would support the latter point of view. However, could we actually understand
how to make sense of such an operational framework in real time? The latter is
challenging as we would also need to validate the usual semi-classical picture
of the black hole (at least from the point of view of measurements of the EFT
observables). In fact, the Page curve can be computed using only the semi-
classical geometry as discussed before and therefore it should be valid for a
large class of measurements.

The second class of issues are related with the encoding of the information in
Hawking radiation. If rapid mirroring of the information thrown in after Page
time could happen together with the complex encoding of the Hawking interior,
then which of these two possibilities could be true: (a) one can decode the
qubits $D$ thrown into the black hole from the Hawking quanta immediately
after scrambling time without explicit knowledge of the interior encoded in
$E$ (decoding the latter would take time exponential in the entropy of the
black hole), or (b) we can only decode it after we actually know the interior
after an enormously long time. The discussion on islands in the form of
entanglement wedge of $R$ would actually prefer the first possibility. In the
setups discussed in the previous subsection, the newly emitted Hawking quanta
would be in the entanglement wedge of the holographic system $B$ and not yet
in the bath region. The physical separation between these regions would
indicate that the mirrored information could be readily decoded soon after the
new Hawking quanta emerges in the entanglement wedge of $B$. However, one
still needs to demonstrate that the decoding is possible without the knowledge
of the island (the black hole interior) that is encoded in this new radiation
since the knowledge of how the island has been modified by the infalling bits
should eventually leak into the bath. Therefore, one needs to identify such
features of the encoding of the infalling qubits which co-exist with the
complex encoding of the interior, and explain their physical origins. One
could formulate these questions also more generally without the setups of the
previous subsection by advocating an emergent infrared holographic theory that
describes the near-horizon geometry of the black hole and which could play the
role of $B$.

In the next subsection, we will review a microstate model that could be
promising for finding answers to such questions.

####  5.3.2 Microstate dynamics: Towards understanding quantum black holes in
real time

It is useful to study tractable models of black hole microstate dynamics to
gain insights into how the features of quantum information mirroring and
complex encoding of interior appear in the Hawking radiation, and also for
understanding how the principle of black hole complementarity and the
averaging implied in replica wormholes could emerge operationally. The
fuzzball program Mathur:2005zp , if developed to its full potential, would be
able to reveal these mysteries. However, at present, it will be very difficult
to study the quantum dynamics of fuzzballs in a sufficiently detailed way. The
same could be said about large $N$ BFSS matrix models Banks:1996vh , etc. We
will argue that certain simplified models could be promising for understanding
many (if not all) aspects of these issues. This class of models described in
PhysRevD.102.086008  essentially simplify some of the otherwise untractable
aspects of quantum gravity in a suitable way, and can be studied in the same
spirit as the setups described earlier in this review. One key aspect of this
class of models is to give prominence to hair degrees of freedom on the
horizon which are essentially (approximately) conserved (non-)gravitational
charges whose role in the information paradox have been emphasised in
PhysRevLett.116.231301 ; Strominger:2017aeh . The crucial and novel aspect of
the hair that will be of interest to us is how it can enable an operational
definition of separability of the interior and exterior Hilbert spaces while
resolving AMPS type paradoxes that threaten black hole complementarity, and
enable mechanisms for the complex encoding of interior, and quantum
information mirroring where decoding could be possible without the knowledge
of the encoding of the interior.

A class of models: The setup of these microstate models can be motivated from
the fragmentation instability Maldacena:1998uz  of the near-horizon geometry
of the near-extremal black hole mediated by the Brill instantons of semi-
classical gravity. These imply the fragmentation of the near-horizon geometry
into several two-dimensional throats of the type $AdS_{2}\times X$, where $X$
is a compact space that has the topology of the horizon and $AdS_{2}$ is the
two-dimensional anti-de Sitter space. At the boundaries of the instanton
moduli space where the centers of some of the throats come within Planckian
distance proximity to each other, there is a proliferation of soft modes.
However, quantum gravity effects also become large, so the semi-classical
computation becomes intractable. Heuristically, we can assume that these
fragmented throats crystallise into a stable configuration forming a lattice
of $AdS_{2}$ spaces as shown in Fig. 27. These throats should interact with
each other via mobile hair which are the (non-)gravitational charges of the
original unfragmented part of the geometry and other soft modes that live
here. The lattice is simpy a discretization of the compact $X$ space that has
the same topology of the horizon. Additionally, we can allow the $AdS_{2}$
spaces to join in their interiors and form complex networks. Here, we will
restrict to the simplest version where the $AdS_{2}$ throats do not join with
each other and are infinitely extended in the ingoing Eddinton-Finkelstein
radial coordinate.

Figure 27: An illustration of the microstate models comprising of a lattice of
two-dimensional $AdS_{2}$ throats coupling to each other via mobile hair
carrying gravitational charges. The lattice is a discretization of the
horizon, a compact space. The $AdS_{2}$ throats may join in the interior
forming complex networks. Hawking evaporation occurs due to appropriate
asymptotic boundary conditions at the throats. This figure is from
PhysRevD.102.086008 .

For further tractabilibity and simplification, we consider that each $AdS_{2}$
throat in the lattice is described a by semiclassical JT-gravity theory with
conformal bulk matter. The semi-classical picture is thus valid for the
observer infalling at any throat, however non-local measurements will see
struture at the horizon. We will soon discuss how the energy absorbing and
relaxation dynamics of the semi-classical black hole emerges from this
microstate model. We begin our discussion by first arresting Hawking radiation
by enforcing the usual reflecting boundary conditions in each $AdS_{2}$ throat
so that we can study the intrinsic properties of the microstates first. Note
that we retain only a coarse-grained (effective infrared) description of each
throat which is dual to a quantum dot in the Planckian lattice. Regardless, we
should enforce total energy conservation of this combined lattice of quantum
dots and mobile hair system in absence of Hawking radiation. Although, we do
modify the effective description of horizon physics, usual statistical
arguments would imply that the results of typical measurements would be almost
the same as in a microcanonical ensemble which we will describe below. For
simplicity, we will consider the horizon to be $S^{1}$, so the lattice would
be a ring (chain) with periodic boundary conditions.

The state of the $i$-th throat (dual to the $i$-th quantum dot) can then be
described by $t_{i}(u)$ where $u$ is the time of the asymptotic observer.
Essentially $t_{i}(u)$ is the time-reparametrization mode of the $i$-th throat
which determines the correlation functions in the localized state in terms of
the vacuum correlation functions as in the Sachdev-Ye-Kitaev model
Sachdev_1993 ; Kitaev_2015  which could be the (UV complete) theory governing
the quantum dots. In other words, $t_{i}(u)$ is the time determining the state
of the $i$-th throat as a function of a common vacuum state time which is
identified with the time of the vacuum observer. For ease of simulations, it
is useful to define $\tau_{i}(u)$ via

$t_{i}(u)=\tanh\left(\frac{\tau_{i}(u)}{2}\right)$ (101)

where $\tau_{i}(u)$ maps to the common time of a thermal state (instead of the
vacuum state) with $\beta=2\pi$. In absence of coupling between throats and
sources for the bulk matter, each individual throat should have conserved
$SL(2,R)$ charges which are:

$\displaystyle\mathcal{Q}_{i}^{0}$ $\displaystyle=$
$\displaystyle\frac{\tau_{i}^{\prime\prime\prime}}{\tau_{i}^{\prime
2}}-\frac{{\tau_{i}^{\prime\prime}}^{2}}{{\tau_{i}^{\prime}}^{3}}-\tau_{i}^{\prime},$
$\displaystyle\mathcal{Q}_{i}^{+}$ $\displaystyle=$ $\displaystyle
e^{\tau_{i}}\left(\frac{\tau_{i}^{\prime\prime\prime}}{\tau_{i}^{\prime
2}}-\frac{{\tau_{i}^{\prime\prime}}^{2}}{{\tau_{i}^{\prime}}^{3}}-\frac{\tau_{i}^{\prime\prime}}{\tau_{i}^{\prime}}\right),$
$\displaystyle\mathcal{Q}_{i}^{-}$ $\displaystyle=$ $\displaystyle
e^{-\tau_{i}}\left(\frac{\tau_{i}^{\prime\prime\prime}}{\tau_{i}^{\prime
2}}-\frac{{\tau_{i}^{\prime\prime}}^{2}}{{\tau_{i}^{\prime}}^{3}}+\frac{\tau_{i}^{\prime\prime}}{\tau_{i}^{\prime}}\right).$
(102)

We denote these collectively as $\vec{\mathcal{Q}}_{i}$. The Casimir of these
$SL(2,R)$ charges is the Arnowitt-Deser-Misner (ADM) mass $M_{i}$ of the
$AdS_{2}$ throat, i.e.

$M_{i}={\mathcal{Q}_{i}^{0}}^{2}-\mathcal{Q}_{i}^{+}\mathcal{Q}_{i}^{-}=-2\,{\rm
Sch}(\tau_{i}(u),u)+{\tau_{i}^{\prime}}^{2},$ (103)

where $\rm{Sch}$ denotes the Schwarzian derivative

${\rm
Sch}(f(u),u)=\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3}{2}\frac{f^{\prime\prime
2}}{f^{\prime 2}}.$ (104)

On top of these lattice $SL(2,R)$ charges, we need to consider additionally
mobile hair charges which we can take to be $SL(2,R)$ charges too representing
the gravitational charges of the unfragmented geometry. These we denote as
$\vec{\mathsf{Q}}_{i}$ which follow discretized Klein-Gordon type equation
when decoupled from the lattice charges. We denote $\vec{A}\cdot\vec{B}$ as
the $SL(2,R)$ invariant dot product of two $SL(2,R)$ vectors $\vec{A}$ and
$\vec{B}$.262626Explicitly,
$\vec{A}\cdot\vec{B}=A^{0}B^{0}-\frac{1}{2}(A^{+}B^{-}+A^{-}B^{+}).$ The
simplest equations of motion of this system of gravitational lattice charges
$\vec{\mathcal{Q}}_{i}(u)$ and mobile hair $\vec{\mathsf{Q}}_{i}(u)$, which
reproduce desired phenomenological properties of a classical black hole, take
the form:

$\displaystyle M_{i}^{\prime}$ $\displaystyle=$
$\displaystyle-\lambda(\vec{\mathcal{Q}}_{i-1}+\vec{\mathcal{Q}}_{i+1}-2\vec{\mathcal{Q}}_{i})\cdot\vec{\mathsf{Q}}_{i}^{\prime},$
$\displaystyle\vec{\mathsf{Q}}_{i}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma^{2}}(\vec{\mathsf{Q}}_{i-1}+\vec{\mathsf{Q}}_{i+1}-2\vec{\mathsf{Q}}_{i})$
(105)
$\displaystyle+\frac{1}{\lambda^{2}}(\vec{\mathcal{Q}}_{i-1}+\vec{\mathcal{Q}}_{i+1}-2\vec{\mathcal{Q}}_{i}).$

where $\lambda>0$ is the coupling between lattice charges and hair, and
$\sigma$ determines the velocity of propagation of hair in the continuum
limit. The first equation above gives the equations for evolution of
$t_{i}(u)$ and the second equation determines the evolution of the hair. These
equations have unique solutions provided we specify the initial lattice
charges, and the hair charges and their time-derivatives (we assume initial
synchronicity, i.e. $t_{i}(u=u_{0})=u_{0}$ at initial time $u_{0}$ but this is
not necessary) and can be solved numerically following PhysRevD.101.066001 .
Generalizations of the above equations with higher derivative corrections and
desired phenomenological features are possible but we do not discuss them
here. These equations imply a specific form of null matter in the $AdS_{2}$
throats, but alternatively we can simply think of these as the equations
determining the quantum dots and hair. Crucially, in the full interacting
system we have only one global $SL(2,R)$ symmetry, namely that of the original
unfragmented geometry.

Note that we should treat the lattice charges $\vec{\mathcal{Q}}_{i}$ semi-
classically invoking a large-$N$ limit in each throat, but the hair
$\vec{\mathsf{Q}}_{i}$ should be understood as an open quantum system
interacting with the lattice. However, for present purposes it will be
sufficient to consider coherent states of the hair and treat it classically.
The equations (5.3.2) imply that the full system has a conserved energy of the
form:

$\mathcal{E}=\mathcal{E}_{\mathcal{Q}}+\mathcal{E}_{\mathsf{Q}}$ (106)

which is simply a sum of the energy in the lattice charges and that in the
hair with

$\displaystyle\mathcal{E}_{\mathcal{Q}}$ $\displaystyle=$
$\displaystyle\sum_{i}M_{i}=\sum_{i}\mathcal{Q}_{i}\cdot\mathcal{Q}_{i},$
$\displaystyle\mathcal{E}_{\mathsf{Q}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{3}}{2}\sum_{i}\vec{\mathsf{Q}}_{i}^{\prime}\cdot\vec{\mathsf{Q}}_{i}^{\prime}$
(107)
$\displaystyle+\frac{\lambda^{3}}{2\sigma^{2}}\sum_{i}(\vec{\mathsf{Q}}_{i}-\vec{\mathsf{Q}}_{i-1})\cdot(\vec{\mathsf{Q}}_{i}-\vec{\mathsf{Q}}_{i-1}).$

Thus $\mathcal{E}_{\mathcal{Q}}$ is simply the sum of the ADM masses of the
throats and $\mathcal{E}_{\mathsf{Q}}$ is the (discretised) kinetic energy of
the hair. Note that $\lambda>0$ is necessary for the positivity of the average
energy in the microcanonical ensemble as we will see below.

The microstates: The microstates of the black hole can be identified with
stationary solutions of (5.3.2). One can readily prove that in such
microstates, we should have

$\vec{\mathcal{Q}}_{i}=Q\vec{\xi}+\vec{\mathcal{Q}}_{i}^{\perp},\quad{\rm
with}\quad\vec{\mathcal{Q}}_{i}^{\perp}\cdot\xi=0.$ (108)

The $SL(2,R)$ charge vector $\vec{\xi}$ thus spontaneously breaks the global
$SL(2,R)$ symmetry. We can set this global frame $\vec{\xi}$ in the
$0$-direction without loss of generality so that
$\vec{\mathcal{Q}}_{i}^{\perp}$ have only $+$ and $-$ components. We also
normalize $\vec{\xi}$ such that $\vec{\xi}\cdot\vec{\xi}=1$. Furthermore, for
stationarity we need the hair charges to have the following configuration:

$\vec{\mathsf{Q}}_{i}(u)=\vec{\mathsf{Q}}_{i}^{\rm
loc}+\vec{\mathsf{Q}}_{i}^{\rm mon}(u)+q_{i}^{\rm rad}(u)\vec{\xi}$ (109)

where

$\vec{\mathsf{Q}}_{i}^{\rm
loc}=-\frac{\sigma^{2}}{\lambda^{2}}\vec{\mathcal{Q}}_{i}+\vec{\mathcal{K}},$
(110)

are locked to the lattice charges,

$\vec{\mathsf{Q}}_{i}^{\rm mon}=\alpha\vec{\xi}u$ (111)

is a monopole component with a homogeneous
$\vec{\mathsf{Q}}_{i}^{\prime}=\alpha\xi$ in the direction of the global
frame, and $q_{i}^{\rm rad}(u)$ are oscillating hair components decoupled from
the lattice and satisfying the normal mode equations

${q_{i}^{\rm rad}}^{\prime\prime}=\frac{1}{\sigma^{2}}(q_{i-1}^{\rm
rad}+q_{i+1}^{\rm rad}-2q_{i}^{\rm rad}).$ (112)

To avoid redundancy, we can set

$\sum_{i}q_{i}^{\rm rad}=\sum_{i}{q_{i}^{\rm rad}}^{\prime}=0.$ (113)

Crucially the Fourier transform of $q_{i}^{\rm rad}(u)$ will have support only
on the discrete normal modes with non-vanishing frequencies.

It follows from (5.3.2) that $\sum_{i}\vec{\mathsf{Q}_{i}^{\prime}}$ is
conserved, and therefore $\alpha$, the monopole charge, should not change when
the microstate is perturbed.

Finally, we can define the microcanonical ensemble as the collection of
microstate solutions subject to the constraints, that (i) the total $M_{i}$
should add up to the ADM mass of the black hole, i.e.
$\mathcal{E}_{\mathcal{Q}}=M$ , and (ii) $t_{i}(u)$ and hence $\tau_{i}(u)$
should be real, continuous and have continuous first and second derivatives at
all lattice sites (necessary to define the mass $M_{i}$ that is proportional
to the Schwarzian derivative). This implies two set of possibilities. Firstly
let’s set $\xi^{0}=1$ and $\xi^{\pm}=0$ without loss of generality as
mentioned before. Then the first set of possibilities which satisfy both
constraints are those which satisfy $0\leq M_{i}\leq Q^{2}$, $Q>0$,
$\sum_{i}M_{i}=M$ and

$\displaystyle\mathcal{Q}_{i}^{0}=-Q,\quad\mathcal{Q}_{i}^{+}=-\rho_{i}\sqrt{Q^{2}-M},$
$\displaystyle\mathcal{Q}_{i}^{-}=-\frac{1}{\rho_{i}}\sqrt{Q^{2}-M}$ (114)

with

$\displaystyle\sqrt{\frac{Q-\sqrt{M_{i}}}{Q+\sqrt{M_{i}}}}\leq\rho_{i}\leq\sqrt{\frac{Q+\sqrt{M_{i}}}{Q-\sqrt{M_{i}}}}.$
(115)

Remarkably, these imply that $t_{i}^{\prime}\geq 0$ (and hence
$\tau_{i}^{\prime}\geq 0$), i.e. the arrows of time of all the lattice sites
should be aligned towards the future. Thus a global arrow of time is a
consequence of the equations of motion! The second set of possibilities lead
to similar inequalities which align the global arrow of time towards the past.
We discard this set and define the microcanonical ensemble with the mentioned
set of (in)equalities.

In the microstate solutions, the total energy of the hair
$\mathcal{E}_{\mathsf{Q}}$ further splits into three parts, i.e.

$\mathcal{E}_{\mathsf{Q}}=\mathcal{E}_{\mathsf{Q}}^{\rm
pot}+\mathcal{E}_{\mathsf{Q}}^{\rm mon}+\mathcal{E}_{\mathsf{Q}}^{\rm rad}.$
(116)

with (setting $\xi^{0}=1$ and $\xi^{\pm}=0$ without loss of generality as
mentioned before )

$\displaystyle\mathcal{E}_{\mathsf{Q}}^{\rm pot}$ $\displaystyle=$
$\displaystyle-\frac{\sigma^{2}}{2\lambda}\sum_{i}(\mathcal{Q}_{i}^{+}-\mathcal{Q}_{i-1}^{+})(\mathcal{Q}_{i}^{-}-\mathcal{Q}_{i}^{-}),$
$\displaystyle\mathcal{E}_{\mathsf{Q}}^{\rm mon}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda^{3}\alpha^{2},$
$\displaystyle\mathcal{E}_{\mathsf{Q}}^{\rm rad}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{3}}{2}\sum_{i}{{q_{i}^{\rm
rad}}^{\prime}}^{2}+\frac{\lambda^{3}}{2\sigma^{2}}\sum_{i}(q_{i}-q_{i-1})^{2}.$
(117)

Clearly, if $\lambda>0$, then both $\mathcal{E}_{\mathsf{Q}}^{\rm mon}$ and
$\mathcal{E}_{\mathsf{Q}}^{\rm red}$ are positive. Although
$\mathcal{E}_{\mathsf{Q}}^{\rm pot}$ need not be positive, its ensemble
average is zero. Thus the average energy is positive if $\lambda>0$.

Both classically and quantum-mechanically, the hair cannot be separated into
interior and exterior components. However, in microstate solutions such a
split operationally emerges since $\vec{\mathsf{Q}}_{i}^{\rm loc}$ is locked
with the lattice $SL(2,R)$ charges which describe the configuration of the
black hole interior, while both $\vec{\mathsf{Q}}_{i}^{\rm mon}$ and
$\vec{\mathsf{Q}}_{i}^{\rm rad}$ do not affect the interior and are thus
decoupled from it. The potential energy term $\mathcal{E}_{\mathsf{Q}}^{\rm
pot}$ is determined solely by $\vec{\mathsf{Q}}_{i}^{\rm loc}$, while
$\mathcal{E}_{\mathsf{Q}}^{\rm mon}$ and $\mathcal{E}_{\mathsf{Q}}^{\rm rad}$
are determined by $\vec{\mathsf{Q}}_{i}^{\rm mon}$ and
$\vec{\mathsf{Q}}_{i}^{\rm rad}$ respectively. Therefore, the total energy
also splits into an interior component which is the sum of
$\mathcal{E}_{\mathcal{Q}}=M$ and $\mathcal{E}_{\mathsf{Q}}^{\rm pot}$, and
the exterior component which is the sum of $\vec{\mathsf{Q}}_{i}^{\rm mon}$
and $\vec{\mathsf{Q}}_{i}^{\rm rad}$. Operationally, therefore, the Hilbert
space of the hair has the structure

$\mathcal{H}_{\mathsf{Q}}=\bigoplus_{\alpha}\mathcal{H}_{\mathsf{Q}_{\alpha}}^{\rm
int}\otimes\mathcal{H}_{\mathsf{Q}_{\alpha}}^{\rm ext}$

with $\alpha$ denoting microstates assuming that the evolution is adiabatic
(we can employ an adiabatically evolving basis) with the off-diagonal terms
being suppressed due to decoherence. Note that this would imply that the
global frame $\vec{\xi}$ which is crucial to make the distinction between the
interior and exterior would be evolving adiabatically too. We would come back
to this in the context of the Hawking evaporation with asymptotic boundary
conditions at the throats that allow the Hawking quanta to escape.

Phenomenological viability: The immediate question is that whether the
ensemble of microstates in the model behave like a semi-classical black hole
with relaxing and energy-absorbing properties. To see this, we once again
return to the semi-classical limit in which Hawking radiation is absent, and
perturb an arbitrary microstate solution in the ensemble by a sequence of
shocks (injections of energies) $e_{iA}$ with $i$ denoting the $i$-th throat
and $A$ referring to the instant $u_{A}$. The equations in (5.3.2) are then
modified to

$\displaystyle M_{i}^{\prime}$ $\displaystyle=$
$\displaystyle-\lambda(\vec{\mathcal{Q}}_{i-1}+\vec{\mathcal{Q}}_{i+1}-2\vec{\mathcal{Q}}_{i})\cdot\vec{\mathsf{Q}}_{i}^{\prime},$
$\displaystyle+\sum_{A}e_{iA}\delta(u-u_{A}),$
$\displaystyle\vec{\mathsf{Q}}_{i}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma^{2}}(\vec{\mathsf{Q}}_{i-1}+\vec{\mathsf{Q}}_{i+1}-2\vec{\mathsf{Q}}_{i})$
(118)
$\displaystyle+\frac{1}{\lambda^{2}}(\vec{\mathcal{Q}}_{i-1}+\vec{\mathcal{Q}}_{i+1}-2\vec{\mathcal{Q}}_{i}).$

Note that the hair is not directly coupling to the shocks (this would imply
the results such as mirroring to be more non-trivial). The shocks are
localized on the lattice and cannot directly affect the delocalized
gravitational charges comprising the hair. The shocks add a total energy of

$\Delta\mathcal{E}=\sum_{i,A}e_{i,A}$

to the system. Starting from a random microstate with the (conserved) monopole
charge $\alpha>0$ and simulating the system via methods of PhysRevD.101.066001
, one finds that

  1. 1.

Any microstate with or without decoupled hair oscillations rapidly settles
down to another microstate with decoupled hair oscillations after the sequence
of shocks.

  2. 2.

The final microstate is determined by the initial microstate in a rather
complex manner. Even for a single shock, one needs to take into account the
initial states of all lattice sites and that of the hair indicating
pseudorandom dynamics.

  3. 3.

Almost all the energy in the shock is absorbed by the change in the total
black hole mass, i.e. $\mathcal{E}_{\mathcal{Q}}=M$. This statement becomes
better (for a typical initial microstate) if we increase the number of sites
keeping the total initial black hole mass and total initial energy fixed.
However, even for five lattice sites, less than $1$ percent of the energy in
shocks is transferred to the hair.

See Fig. 28 for an illustration for the case of a five site lattice suffering
a single shock. The above features imply that a typical microstate with a
positive monopole charge behaves like a semi-classical black hole
qualitatively as far its relaxation dynamics and energy absorption properties
are concerned, and furthermore shows features of pseudorandom dynamics.

(a) Evolution of lattice charges and masses after a single shock

(b) Evolution of the energies after a single shock

Figure 28: The evolution of a randomly chosen microstate after a single shock
in a five site model is shown above. The first site is shocked with energy
injection $e=0.4$ at $u=0$. We set $\lambda=1$, $\alpha=1$ and $\sigma=0.01$.
The system relaxes to another microstate with a different distribution of
masses and lattice charges. Almost the entire energy of the shock is absorbed
by the total mass of the black hole $\mathcal{E}_{\mathcal{Q}}$. The energy in
the hair $\mathcal{E}_{\mathsf{Q}}$ remains approximately constant. These
figures are from PhysRevD.102.086008 .

As a consequence of these phenomenological features, we can argue that the
effective split into interior and exterior emerges dynamically after the
relaxation time (which goes to a finite value in the continuum limit as
observed numerically).272727 $\mathcal{H}_{\mathsf{Q}}^{\rm int}$ is supported
at the inhomogeneous static fixed point configurations (or analogous adiabatic
versions of these in the presence of Hawking radiation) while
$\mathcal{H}_{\mathsf{Q}}^{\rm ext}$ is supported at the normal mode
frequencies only (or analogous adiabatic versions of these in the presence of
Hawking radiation).  In very general situations, we expect such a split to be
operationally valid when the dynamics is coarse-grained over the relaxation
time-scale, and can be demonstrated by employing a suitable coherent state
basis.

Classical information mirroring in hair: The remarkable aspect of these
microstate models is that it has the feature of information mirroring built
into them intrinsically even though the Hawking radiation is arrested via
boundary conditions at the throats. This can be demonstrated readily by
studying the dynamics of the system in response to a sequence of shocks. As
described above, a typical microstate relaxes to another microstate
demonstrating pseudorandom dynamics. A part of the hair
$\vec{\mathsf{Q}}_{i}^{\rm rad}$ decouples from the interior while another
component gets locked to the interior lattice charges. The information of the
shocks is mirrored in the $\vec{\mathsf{Q}}_{i}^{\rm rad}$ or equivalently in
$q_{i}^{\rm rad}$ defined in (109). The crucial point is that there are
features in $q_{i}^{\rm rad}$ which allows us to decode the information of the
shocks without the knowledge of the interior of the initial or final
microstate. Many aspects of the information encoded in the infalling shocks
can thus be decoded by assuming that the interior is in the microcanocial
ensemble. The only necessary information for the decoding is the global frame
$\xi$ in (109) which one can simply obtain from the (decoupled) monopole
component, the analogue of the early radiation in the Hayden-Preskill thought
experiment.

The Fourier transform of $q_{i}^{\rm rad}(u)$ will be supported only at the
non-vanishing normal mode frequencies of (112). It is sufficient to look into
the phases and amplitudes of the positive normal mode frequencies since
$q_{i}^{\rm rad}(u)$ is real. It turns out that only the differences of the
phases of the two positive normal mode frequencies at the various sites are
necessary to decode the information of the shocks. If we encode information
into the time sequence of the location of sites which are shocked and the
differences of energy injections into these shocks are not large, then the
information of the sequence can be decoded from the ordering of the phase
differences of the positive frequency normal modes of $q_{i}^{\rm rad}(u)$ at
various lattice sites. All other features of $q_{i}^{\rm rad}(u)$ will be
contaminated with the details of the initial microstate but this ordering of
phase differences will be determined only by the specific time sequence of the
shocks.

For a non-trivial example, consider the case of five sites and a sequence of
two shocks into two sites. The decoding protocol for which sites have been
shocked and in which sequence is illustrated in Fig. 29. This protocol is
independent of the initial microstate.

(a) A sequence of two shocks into two non-neighboring sites. The dark blue
site is shocked first and the light blue one later. The one shocked earlier
has the highest and the one shocked later has the lowest phase differences
between the positive normal modes.

(b) A sequence of two shocks into two neighboring sites. The dark blue site is
shocked first and the light blue one later. The one shocked earlier has the
lowest and the one shocked later has the highest phase differences between the
positive normal modes.

(c) A sequence of two shocks into the same site. The shocked site has the
highest phase differences between the positive normal modes.

Figure 29: The decoding protocol for information mirroring when two sites are
shocked in the five site microstate model via measurements of the phase
differences between the two positive normal modes in the hair oscillations
which decouple from the interior. Note that this protocol for decoding which
sites have been shocked and in which sequence is the same for any initial
microstate. Although the absolute values of the phase differences are somewhat
sensitive to the initial microstate, the ordering is not.

The information mirroring should carry over to the quantum regime if we
quantize the hair. It would be interesting to find out if the information of
the initial microstate prior to the shock is encoded in observables which are
complementary, i.e. have large non-vanishing commutators with the phase
differences in the normal modes. This should have consequences for
understanding the black hole complementarity as we comment below.

Encoding into the Hawking radiation: Hawking radiation can be readily
incorporated into the microstate model described above by simply coupling each
throat to a bath and implementing transparent boundary conditions following
AEMM  as for instance. These boundary conditions choose a specific global null
frame $\vec{\xi}$ (as for instance in AEMM  there is a null shock when the
boundary conditions are altered). The first question to be asked is whether
the information of the interior leaks out to the Hawking radiation completely.
It is a non-trivial question because although the two dimensional black holes
evaporate away at all the lattice sites, the $SL(2,R)$ directions of the
lattice charges $\vec{\mathcal{Q}}_{i}$ may not homogenize and thus there
would be a remnant information in the interior even after complete
evaporation. One may need this remnant to decode the complete information.
Firstly, we see that the $SL(2,R)$ directions of the lattice charges
$\vec{\mathcal{Q}}_{i}$ align themselves with the null direction $\vec{\xi}$
set by the boundary conditions asymptotically at late time. Nevertheless,
asymptotically $\vec{\mathcal{Q}}_{i}\rightarrow\mathcal{Q}_{i}\vec{\xi}$, so
that the overall magnitude $\mathcal{Q}_{i}$ is non-trivial although the
Casimir (the mass of the black hole) vanishes at each site. However, the hair
(being a smaller subsystem) will also dissipate its energy to the Hawking
radiation that escapes away. The evolution equations (5.3.2) would imply that
the hair charges should lock themselves with the lattice charges
$\vec{\mathcal{Q}}_{i}\approx\mathcal{Q}_{i}\vec{\xi}$ following (110) so that
the potential energy vanishes (since $\vec{\xi}$ is null) and thus the hair
system reaches a fixed point. Furthermore, as discussed below the hair should
be entangled with the Hawking radiation so the full decoding of the interior
should be possible.

The AMPS paradox, of course, is not about the final state after complete
evaporation, but the system after Page time. Here, each individual throat
would have its own Page time and therefore the Page time here refers to the
average Page time of these throats (which will have small fluctuation about
the average for a typical microstate of a large black hole). As argued above,
the full system should have an effective split into interior and exterior
because the hair can be split into
$\mathcal{H}_{\mathsf{Q}}=\bigoplus_{\alpha}\mathcal{H}_{\mathsf{Q}_{\alpha}}^{\rm
int}\otimes\mathcal{H}_{\mathsf{Q_{\alpha}}}^{\rm ext}$ when averaged over the
(slowly evolving) relaxation time with $\alpha$ denoting different
microstates. Also the outgoing Hawking radiation essentially forms a sequence
of outgoing shocks. Therefore the information of coarse grained features of
the outgoing Hawking radiation such as the overall energy outflow from the
throats over a period commensurate with the relaxation time (which are
insufficient to fully reconstruct the interior, especially the $SL(2,R)$
directions of the lattice charges) will be encoded into the radiation
component of the hair which essentially forms $\mathcal{H}_{\mathsf{Q}}^{\rm
ext}$ as happened in the case of mirroring of the information of the shocks.
Crucially, these low energy/ coarse-grained features will be encoded in
special observables such as the phase differences of the normal modes in
$\mathcal{H}_{\mathsf{Q}}^{\rm ext}$ from which decoding will be possible
without the knowledge of the finer structure of the microstates. The detailed
knowledge of the interior would be encoded in the complementary observables of
$\mathcal{H}_{\mathsf{Q}}^{\rm ext}$ and also $\mathcal{H}_{\mathsf{Q}}^{\rm
int}$282828In Verlinde:2012cy , one can find a general discussion on how
effective field theory observables in a semiclassical but not necessarily
smooth horizon geometry can decode the black hole interior. This discussion,
cast in the formalism of quantum error correction, can be applied in the
context of our microstate model. The crucial part of the construction of the
recovery map from the local operators acting on the hair and the Hawking
quanta at the horizon, is the knowledge of conditional transition matrices of
the black hole interior. The resolution of the AMPS paradox in this context
has been also discussed in Verlinde:2012cy ..

Finally to understand encoding in the Hawking radiation, we should decipher
which observables acting on the Hawking radiation will be strongly correlated
with features of $\mathcal{H}_{\mathsf{Q}}^{\rm ext}$ that are responsible for
mirroring the infalling qubits and encoding the coarse-grained information
carried out by the outgoing Hawking radiation (these should have simple
encoding in the sense that decoding can be possible by assuming that the
interior is a microcanonical ensemble), and similarly which features of the
Hawking radiation correlate strongly with $\mathcal{H}_{\mathsf{Q}}^{\rm int}$
and the fine grained features of the black hole interior (eg. the lattice
$SL(2,R)$ charges). The encoding of the latter should be complex due to the
underlying pseudorandom dynamics of the microstates (more on the generation of
complexity via the Python lunch mechanism below). Since the split between the
interior and the exterior is only an operational concept that emerges after
averaging over an evolving relaxation timescale, there is no real
contradiction that the Hawking radiation is in a way entangled maximally with
both $\mathcal{H}_{\mathsf{Q}}^{\rm ext}$ and $\mathcal{H}_{\mathsf{Q}}^{\rm
int}$. Furthermore, the coarse-grained and finer information could be encoded
with the best fidelity in observables that have large mutual commutators. This
could be fundamental to understanding why the semiclassical geometry of the
black hole emerges. Additionally, one possible way to realize the expected
near saturation (but not violation) of the strong subadditivity property which
is key to the AMPS paradox is that the tripartite interior, hair and outgoing
Hawking radiation system has a quantum Markov chain like structure (see
Section 4.2.2 for the definition of quantum Markov chain states).

The key to the detailed understanding would be to follow the evolution of the
full tripartite system (after quantizing the hair) and analyze it using
various tools of quantum information theory. This should reveal how black hole
complementarity emerges in an operational sense free of paradoxes along with
the self-averaging properties represented by the semi-classical black hole
geometry and the Euclidean replica wormhole saddles.

Pseudorandom dynamics crucial for Python lunches: The appearance of Python
lunch geometries gives a natural quantification of encoding of the interior
modes into outgoing Hawking radiation as shown in Sec. 5.2. As discussed
there, instead of the vacuum of the bulk matter in each throat, we should
consider first a small subspace of its Hilbert space which we can identify
with the code subspace, and then study the microstate dynamics with the bulk
matter in the throat in the maximally mixed state in this (small) code
subspace (the importance of the maximally mixed state in the code subspace has
been emphasized also). Via the arguments presented in Sec. 5.2 we expect
nucleation of a Python lunch or equivalently a (non-classical) locally maximal
extremal surface in the throat. However, the inherent pseudorandom dynamics
would be needed to ensure a necessary amplification so that the Python’s lunch
manifests macroscopically across a significant fraction of throats (this would
be needed in the higher dimensional setup) before the operational split of the
hair Hilbert space into interior and exterior emerges, i.e. before black hole
complementarity becomes operational. One can also similarly consider a code
subspace of the interior modes of the hair (modes with frequencies smaller
than the first non-trivial normal mode) and study the Python lunch phenomenon.
It will be also interesting to consider code subspaces of interior modes of
bulk matter involving entanglement across several throats since it will give
insights into understanding of the complexity of encoding of the entanglement
of the interior. In all cases, the inherent chaos in microstate dynamics
should connect the Python’s lunch geometric mechanism of generating
exponential complexity Brown:2019rox  in encoding with the pseudorandomness
mechanism discussed in Kim:2020cds .

Outlook: These microstate models hold many promises for partial answers to how
the encoding in Hawking radiation happens with desired features along with the
validation of black hole complementarity, but further improvements would be
necessary. Although we find information mirroring occurring at relaxation time
in the microstates, it is not clear if they have shortest possible scrambling
time for typical microstates. Furthermore, the bath region which collects the
Hawking radiation is itself fragmented into two dimensional spaces. It would
be necessary to glue these baths in a way we can form a connected flat
asymptotic spacetime region. One way to construct more realistic models would
be to figure out how simplified scenarios with characteristics of those
discussed above can emerge from fuzzballs in string theory. The notion of
fuzzball complementarity Mathur:2010kx ; Avery:2012tf  which advocate emergent
holographic descriptions for local measurements by an infalling observer could
be useful.

##  6 Discussion and Outlook

One of the topics we have not been been able to discuss in this review is the
fundamental relationship between tensor networks and gravity. Although tensor
networks do model many aspects of holography, it is unclear if they reproduce
all information theoretic aspects beyond the Ryu-Takanayagi type extremal
surface which computes entanglement entropy. The latter was established first
Swingle:2009bg  in the context of multiscale-entanglement-renormalization
(MERA) tensor networks.292929The correspondence between tensor networks and
holography has also been explored via continuum versions of MERA
Miyaji:2015yva . See also Caputa:2017urj  for a path integral formulation in
which the continuum version of the tensor network is reformulated as an
optimization problem. In Akers:2018fow  it has been argued that existing
quantum error correcting tensor networks in the toy models discussed in
section 4.2.3 are area eigenstates of the bulk gravity theory rather than
representatives of smooth bulk geometries. A better understanding of these
issues could also come from constructing tensor networks which mimic the
highly efficient RG flow described in section 4.4 as it is designed to
reproduce dynamical gravity. These issues are important to understand how we
can simulate aspects of quantum gravity and also strongly interacting quantum
field theories in an efficient qubit regularized way and in real time.

Another interesting direction of research would be related to further
elucidating the models of evaporating black holes in terms of quantum
thermodynamics. These models described in this review involve coupling of a
bath to a holographic system which has quantum matter in the bulk, and
therefore usual ubiquitous outcomes like monotonic growth of entropy in
classical gravity coupled to matter which satisfies the null energy condition,
can be avoided. It could be possible to construct quantum engines in similar
setups. Quantum thermodynamics is a rapidly emerging field which uses quantum
information to generalize the laws of thermodynamics Goold_2016 ;
PhysRevE.93.022126 ; Yunger_Halpern_2016 ; Guryanova_2016 ;
RevModPhys.91.025001 . As for instance, bounds have been established for the
one-shot work cost of creating a state and also for extractable work from the
state in terms of the hypothesis testing relative entropy with respect to the
Gibbs ensemble PhysRevE.93.022126 . Recently, even within the classical
approximation in the bulk (with classical bulk matter) and assuming an
infinite memoryless bath, it has been shown in Kibe:2021qjy  that for an
instantaneous transition between thermal rotating states in holography, the
quantum null energy condition Bousso_2016  of the boundary theory bounds the
growth of thermodynamic entropy (temperature) for a fixed increase in
temperature (entropy) from both above and below. Similarly, the rates of
growth of entanglement can also be bounded from both above and below.
Furthermore, one can recover the Landauer erasure principle and also
understand how to construct erasure tolerant quantum memory Banerjee:2022dgv .
It would be interesting to pursue how one can construct various protocols by
exploiting suitable quantum bulk matter in a semi-classical black hole
geometry and interacting with a dynamical reservoir at the boundary. Also how
the efficiency and work cost of such processes can be bounded via tools of
quantum information theory especially the quantum null energy condition and
its possible generalizations.

It is quite likely that the fundamental understanding bulk emergence and
quantum black holes in holography will lead us to more fundamental and novel
connections between quantum information and many-body dynamics.

Furthermore, we have restricted ourselves here to asymptotically anti-de
Sitter space. For interesting discussions especially on how islands
(entanglement wedges) may or may not generalize consistently for other
asymptotic boundary conditions see Geng:2020fxl ; Geng:2021hlu ;
Shaghoulian:2021cef  (an explicit discussion on de-Sitter space is in
Shaghoulian:2021cef ). In the future, it would be of deep interest to
understand bulk emergence in the context of both asymptotically flat and
cosmological spacetimes. In the latter context, the approach of Bzowski:2012ih
, as for instance, may lead to promising results by extrapolating our existing
understanding of anti-de Sitter spaces after incorporating extremal surfaces
and consistent entanglement wedges.

###### Acknowledgements.

The research of TK is supported by the Prime Minister’s Research Fellowship
(PMRF). PM and AM ackowledge support from the Institute of Eminence scheme of
IIT Madras funded by the Ministry of Education of India. AM acknowledges the
support of the Ramanujan Fellowship of the Science and Engineering Board of
the Department of Science and Technology of India, the new faculty seed grant
of IIT Madras and the CEFIPRA/IFCPAR grant no 6304-3.

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