# Navigating the Conjectural Labyrinth of the Black Hole Information Paradox

Galina Weinstein  Reichman University, The Efi Arazi School of Computer
Science, Herzliya; University of Haifa, The Department of Philosophy, Haifa,
Israel.

###### Abstract

This paper explores the enduring black hole information and firewall
paradoxes, challenges that have prompted many proposals, conjectures, and
theories. Noteworthy among these are the ER $=$ EPR conjecture and AdS/CFT
correspondence, which suggest possible avenues toward the yet-to-be-realized
unified theory of quantum gravity. This discourse offers a comprehensive
analysis of the aforementioned paradoxes, drawing upon insights from efforts
to reconcile the schism between general relativity and quantum mechanics.

###### Contents

  1. 1 Introduction
  2. 2 The black hole information paradox and its spinoffs
    1. 2.1 Hawking’s black hole information paradox
    2. 2.2 Hawking reformulates the information paradox
    3. 2.3 The firewall paradox
  3. 3 The stretched horizon and complementarity
    1. 3.1 The stretched horizon and the membrane paradigm
    2. 3.2 The four postulates of complementarity
    3. 3.3 The semi-classical stretched horizon
    4. 3.4 Black hole complementarity
  4. 4 The Firewall and complementarity
    1. 4.1 Contradictions among the four postulates
    2. 4.2 A drama: the firewall as a physical barrier
    3. 4.3 A critique of the firewall
  5. 5 Bits, qubits, and error-correcting codes
    1. 5.1 Hayden and Preskill’s mirror
    2. 5.2 The $A=R_{B}$ conjecture and the Hayden-Harlow protocol
    3. 5.3 Quantum error-correcting codes and the holographic principle
  6. 6 Avoiding a firewall
    1. 6.1 Typical and atypical black holes
    2. 6.2 The ER = EPR conjecture
    3. 6.3 A firewall is unnecessary
    4. 6.4 Causality and chronology protection as a barrier
  7. 7 A zoo of proposals
    1. 7.1 A shift in perspective
    2. 7.2 Fuzzballs, Remnants, and the final burst
    3. 7.3 The Page time paradox
    4. 7.4 Are we in the midst of a crisis?

##  1 Introduction

General relativity has been accepted for over a century, but some anomalies
challenge its validity. One significant anomaly comes from the discrepancy
between quantum mechanics’ principle of unitarity and the no-hair theorem and
the equivalence principle of general relativity, particularly when looking at
Hawking radiation from black holes. This conflict leads to the information and
firewall paradox, discussed in sections 2 and 4.

Resolving the information loss paradox requires finding a consistent framework
that reconciles the unitarity of quantum mechanics with the no-hair theorem of
general relativity. Various proposals and conjectures, such as the stretched
horizon and complementarity, $A=R_{B}$ and the firewall, the ER $=$ EPR
proposal (the connection between Einstein-Rosen bridges and entanglement or
Einstein-Rosen-Podolsky pairs), AdS/CFT (a mathematical equivalence between a
gravity theory in anti-de Sitter space-time and a conformal quantum field
theory on its boundary), and complexity theory, have been explored to address
this paradox and find a satisfactory resolution, as discussed in sections 3,
4, and 6. However, these attempts have not yet led to a paradigm shift, which
would involve the development of a comprehensive theory of quantum gravity
capable of replacing both general relativity and quantum field theory. Such a
theory would encompass and explain the limitations of both theories.

It is conceivable that AdS/CFT could play a role in this future theory,
potentially forming a part of the new framework that replaces the current
ones. Regarding reconciling general relativity and quantum mechanics, AdS/CFT
provides a powerful tool known as holographic duality. It is holographic since
the gravitational theory lives in (at least) one extra dimension [Hart].
According to the AdS/CFT correspondence, a theory of gravity in AdS space-time
is mathematically equivalent to a quantum field theory (with no gravity)
living on the boundary of that space-time. In other words, via the AdS/CFT
correspondence, the CFT is also encodes the dynamics of the bulk (the interior
of the AdS space), where gravity operates.

Another perspective to consider AdS/CFT is looking at this duality through the
following lens: AdS can be likened to ”gravity in a box,” where the boundary
of the box (AdS space) accommodates a CFT, which subsequently encodes the
gravitational dynamics occurring within the box [Kap].

However, we must provide a clear and accurate qualification for these
statements: we either do CFT or have an AdS spacetime, never both
simultaneously. The theories are considered entirely equivalent: any physical
(gauge-invariant) quantity that can be computed in one theory can also be
computed in the dual [Hart]. This duality allows researchers to study certain
aspects of quantum gravity using well-understood quantum field theories.

While AdS/CFT has been successful in providing insights into the nature of
quantum gravity and has been extensively studied in the context of AdS space-
time, its direct application to our universe is more challenging because
AdS/CFT primarily deals with AdS space-time. In contrast, our universe is
believed to be closer to dS space-time. AdS space-time has a negative
cosmological constant, while dS space-time has a positive one. The duality
between gravity in dS space-time and a quantum field theory on its boundary is
not fully understood. However, some researchers have proposed extensions or
modifications of AdS/CFT that could potentially apply to dS space-time.

The black hole information paradox has puzzled physicists and computer
scientists for years. Daniel Harlow and Patrick Hayden’s work shows that
combining computational complexity with quantum mechanics might offer a new
perspective. They suggest that getting information out of a black hole before
it vanishes might be almost impossible, even with quantum computers. While
this doesn’t solve everything, it gives us a fresh perspective on the problem.
As I show in section 5, today, many in the scientific community, especially
computer scientists, see the blend of computation and quantum mechanics as a
promising path forward for understanding black holes and their intricate
information dynamics.

As I discuss in Section 7, Albert Einstein had already acknowledged the
inconsistency between his general theory of relativity and quantum mechanics.
He saw the need for a unified field theory grounded in space-time and gravity
rather than on a quantum basis. Einstein did not focus on black holes,
believing anything beyond the Schwarzschild singularity lacked physical
significance. From his viewpoint, the anomaly lay in what he saw as problems
in quantum mechanics, specifically entanglement, which is now widely accepted
as a fundamental aspect of quantum mechanics. The major conflicts between his
general relativity and quantum mechanics emerged after his death, especially
concerning what he disliked: the Schwarzschild singularity.

As I show in section 7, the consensus shifted towards unifying gravity within
the quantum framework after advances in quantum mechanics, a change evident in
modern theories like the AdS/CFT correspondence. Recent theories revolve
around quantum mechanics as foundational, linking to quantum information and
computation, as demonstrated by the unitary transformation principle. The
debates around the black hole information paradox have been pivotal, with
theories like the fuzzball proposition, remnants, and the final burst offering
explanations, each with unique implications and criticisms, and further
discussions like the Page Time Paradox hinting at unresolved complexities in
our understanding of black hole physics.

##  2 The black hole information paradox and its spinoffs

###  2.1 Hawking’s black hole information paradox

Let us commence our discussion with Stephen Hawking’s definition: ”The no hair
theorem implied that all information about the collapsing body was lost from
the outside region apart from three conserved quantities: the mass, the
angular momentum, and the electric charge” [Hawk2].

Hawking initially suggested in 1975 that the process of black hole evaporation
results in a loss of information about the state of particles that fell into
the black hole, making the final state of the black hole and all the particles
and radiation associated with it unpredictable if the black hole evaporated
completely. If the black hole were to evaporate completely, the system’s final
state would be the state of all the radiation emitted, along with any
remaining particles outside the black hole.

Hawking’s original calculations suggested that the emitted radiation would be
in a mixed state due to the apparent loss of information about the particles
that fell into the black hole. Unlike a pure state, described by a single
vector in the Hilbert space, where we have complete knowledge about the
system, a mixed state represents a lack of complete knowledge. This implied
that a non-unitary S-matrix (scattering matrix) would describe the system’s
evolution (from infalling particles to outgoing radiation), signifying a
departure from standard quantum mechanics where a unitary S-matrix describes
processes. Unitarity ensures the total probability of all possible outcomes of
any event always sums to 1, corresponding to the conservation of information.
A unitary S-matrix ensures that probabilities are conserved in quantum
processes. If the S-matrix is non-unitary, the evolution of the quantum system
(the black hole and all the particles and radiation associated with it) is not
deterministic, leading to unpredictability in the system’s final state after
the complete evaporation of the black hole. [Hawk1].

###  2.2 Hawking reformulates the information paradox

Jacob Bekenstein and Hawking explored the idea that black holes can be
associated with entropy, a measure of a system’s disorder. Bekenstein
calculated the amount of information a black hole could contain and expressed
it in terms of its area (not the volume), which led to the formulation of
black hole entropy. A black hole’s information content can be quantified and
linked to its entropy.

In other words, if we describe mathematically how information is stored in
qubits, the information of all the material that forms the black hole gets
encoded (represented) on the black hole’s event horizon (the boundary).

According to Gerard ’t Hooft, all phenomena within a volume can be described
by degrees of freedom residing on the surface bounding that volume. This
implies the world is a two-dimensional lattice of bits, with the bounding
surface acting as a viewing screen onto which three-dimensional events are
projected. Susskind proposed the holographic principle based on ’t Hooft’s
ideas, positing that all the information contained within a volume of space
can be encoded on a boundary surface surrounding that volume, much like a two-
dimensional hologram can represent a three-dimensional object [Suss95].

The Bekenstein-Hawking formulation suggests this idea by demonstrating that a
black hole’s entropy (and hence information) scales with its surface area
rather than its volume.

The holographic principle was formulated before discovering the AdS/CFT
correspondence. The AdS/CFT correspondence later provided a concrete
realization of the holographic principle within the context of string theory
by establishing a precise duality between a gravity theory in a higher-
dimensional space (AdS space) and a lower-dimensional quantum field theory
(CFT) defined on its boundary.

However, our observable universe appears more accurately described by dS
space, complicating attempts to formulate a holographic description. In dS
space, the boundary is at temporal infinity, making the definition of a
holographic dual more elusive. The formulation and successful realization of
holography in the context of AdS space via the AdS/CFT correspondence was a
significant leap, as it showed how the holographic principle could be made
precise and calculable.

Within the AdS/CFT correspondence, a black hole in an AdS space can be
described as a CFT living on the boundary of that space. This correspondence
suggests that the evolution of black holes can be described by the evolution
of a certain quantum system (described by the CFT) that doesn’t have gravity.
In quantum field theories, the evolution of systems is unitary, meaning that
information is conserved over time. Through this lens, the process of black
hole evaporation and the subsequent dynamics could be described in a unitary
fashion, suggesting that all the information about the state of particles
(matter) thrown into the black hole isn’t lost but is encoded in the quantum
state of a CFT living on the boundary of the AdS space in which the black hole
resides.

While Hawking initially proposed that information gets lost in black hole
processes, the AdS/CFT correspondence suggests otherwise. Even though the
details about the particles that fell into the black hole seem lost due to the
black hole evaporation, this original form of the information is not truly
lost, but instead encoded and transferred into the Hawking radiation.
Extracting the information from the radiation would involve a process of
decoding. The decoding process would hypothetically allow one to retrieve
information about the initial state of the particles that fell into the black
hole by analyzing the characteristics of the outgoing Hawking radiation.

In 2005, Hawking noted a shift in understanding regarding the black hole
information paradox. He conceded that the AdS/CFT correspondence has shown no
information loss. In other words, the evolution and evaporation of black holes
can be described in a unitary, information-conserving manner through the dual
quantum field theory description. This represents a reversal from Hawking’s
earlier stance, which suggested that information could be lost in black holes,
thereby contributing to the black hole information paradox[Hawk1].

The shift in understanding, primarily brought about by the AdS/CFT
correspondence, moved the discourse from information loss to the conservation
of information and the notions of encoding and decoding the information in the
context of black holes. However, despite this shift, resolving the information
paradox remains difficult. The paradox now hinges on understanding how the
information is encoded in the Hawking radiation and how, in theory, it could
be decoded or retrieved. However, the practicality and exact mechanism of the
decoding are among the complex issues intertwined with the black hole
information paradox.

###  2.3 The firewall paradox

An observer falling into a black hole would be in free fall. From the
perspective of the falling observer, there would be no dramatic indication at
the event horizon that they have crossed any special boundary. This is the ”no
drama” scenario. The transition would appear smooth, and they would continue
freely falling inward. The observer would not feel any sudden forces and
disruptions when crossing the event horizon, assuming the black hole is
sufficiently massive to have a weak tidal gradient at the horizon. The
equivalence principle states that locally (in small enough regions of
spacetime), the effects of gravity are indistinguishable from acceleration in
a non-gravitational field. So, as the falling observer approaches and crosses
the event horizon of a large black hole, the equivalence principle would still
be valid locally. The observer falling into the black hole would feel like
they were freely floating; even if they were to drop a small object next to
them, it would fall at the same rate.

When considering Hawking radiation over the lifespan of a black hole, early
and late radiation would be entangled due to the continuous process of
particle pair creation near the event horizon. Suppose the Hawking radiation
is in a pure state. In that case, the total system, including the early and
late radiation, must maintain an entanglement reflecting the quantum
correlations established at their emission. Thus, the purity of Hawking
radiation suggests that late radiation is fully entangled with early
radiation. The lack of a dramatic event for an infalling observer indicates
that the late radiation is fully entangled with modes behind the black hole
horizon.

This scenario hints at a type of cloning that contradicts quantum mechanics
principles, specifically the strong subadditivity of entropy. Strong
subadditivity is a fundamental inequality related to quantum entropies which
states that for any three quantum systems $A$, $B$, and $C$, the following
inequality holds: $S_{AB}+S_{BC}\geq S_{B}+S_{ABC}$.

The _AMPS paradox_ , or _Firewall Paradox_ , named after the authors Ahmed
Almheiri, Donald Marolf, Joseph Polchinski, and James Sully, arises from
analyzing the behavior of an old black hole (a black hole in the late stages
of its evaporation). A contradiction arises if one assumes that the late
Hawking radiation is entangled with the early Hawking radiation (to preserve
quantum purity) and that the infalling observer encounters no drama. According
to quantum mechanics, a quantum system (like a qubit) can be entangled with
only one other system at a time. However, in this scenario, it would imply
that the late Hawking radiation is entangled with both the early radiation and
the modes inside the horizon, hinting at a sort of cloning of entanglement,
which is not allowed.

AMPS demonstrate an entropy violation. The entropy is decreasing for an old
black hole, which violates the strong subadditivity condition and other
derived relations. If entropy is decreasing, and given the scenario of the
AMPS paradox described above, the strong subadditivity of entropy is violated.
This violation hints at a deeper inconsistency in understanding the nature of
black holes, Hawking radiation, and quantum entanglement [AMPS].

##  3 The stretched horizon and complementarity

###  3.1 The stretched horizon and the membrane paradigm

In 1986, Douglas A. Macdonald, Richard H. Price, and Kip S. Thorne introduced
the notion of a ”stretched horizon” and the classical _membrane paradigm_. The
stretched horizon is a surface that is modeled to curve slightly outside the
event horizon of a black hole [Pr-Th], [Pr-Th-Mac] (see discussion in [Wall-
mem.18], [Wall-mem.19]).

According to the classical membrane paradigm, black holes are conceptualized
as a two-dimensional surface or membrane. When describing a black hole,
talking about a surface or a membrane is useful, even though black holes are
not compact objects with a surface. The stretched horizon is a conceptual and
mathematical construct rather than a real physical surface. By imagining a
surface with membrane-like properties slightly outside the event horizon of a
black hole, physicists can simplify certain complex problems and handle them
more manageable.

The stretched horizon idea facilitates the analysis of black hole dynamics and
interactions, making the complex physics of black holes more tractable and
comprehensible [Par-Wil]:

1) In Schwarzschild coordinates, the metric becomes singular (i.e., undefined)
at the event horizon, leading to a coordinate singularity. Alternative
coordinates like Kruskal-Szekeres and Eddington-Finkelstein coordinates are
smooth through the event horizon and do not have this problem. While these
coordinate systems can eliminate the coordinate singularity at the event
horizon, considering a surface slightly outside the true horizon, one can
avoid the coordinate singularities.

2) The event horizon is also the location of an infinite redshift: light
emitted from just outside the horizon will be infinitely redshifted, i.e., its
energy will be reduced to zero as seen by an observer at infinity.

3) As one approaches the event horizon, tidal gravitational forces become
infinitely large. However, for a sufficiently massive black hole, these forces
can be small at the event horizon and diverge as one approaches the
singularity at $r=0$. The curvature of space and the tidal gravitational
forces are less severe near a massive black hole’s event horizon than a small
black hole. An observer falling into a massive black hole might not experience
significant tidal forces until they reach the singularity at $r=0$.

4) Within the context of Schwarzschild coordinates, at the event horizon, the
Jacobian matrix becomes singular (it loses its invertibility), and the
transformation between different coordinate systems degenerates at that point.
If the Jacobian is singular at the event horizon, it leads to a degenerate
metric in the transformed coordinate system. When one switches to a coordinate
system better suited for analyzing the geometry near the event horizon, such
as Kruskal-Szekeres coordinates, the coordinate singularity at the event
horizon is eliminated, and the metric tensor remains non-degenerate. The
Jacobians do not become singular at the event horizon in these coordinates,
and a well-behaved metric tensor can be obtained.

The stretched horizon is chosen to be timelike. It has a well-defined geometry
and doesn’t suffer from the singularities associated with a null event
horizon. On a timelike surface, it is possible to have a family of time-like
observers who hover just outside the true horizon. These observers are called
”fiducial observers” and have a well-defined time experience. Unlike an
observer at the true event horizon who would experience infinite gravitational
time dilation, these fiducial observers experience a large but finite redshift
relative to observers far from the black hole. Because it is timelike, the
stretched horizon allows for a clear definition of boundary conditions and a
non-degenerate, well-behaved, induced metric.

In short, the stretched horizon is not only about overcoming coordinate
singularities. It behaves more like ordinary space, so standard techniques
from classical physics can be applied.

###  3.2 The four postulates of complementarity

In 1993, Leonard Susskind, Lárus Thorlacius, and John Uglum introduced the
concept of black hole complementarity [STU]. The stretched horizon becomes
invisible to an observer who falls through it, demonstrating a complementarity
between observations made by infalling and distant observers. By considering a
2D model of gravity, Susskind et al. outline the behavior, kinematics, and
statistical fluctuations of the stretched horizon, emphasizing its importance
in understanding black hole formation and evaporation while considering
quantum mechanical principles.

Susskind et al. initially develop four postulates to establish the foundation
of black hole complementarity [STU], [AMPSS]:

_Postulate 1_. _Unitarity_ : The formation and evaporation of a black hole, as
observed by a distant observer, can be entirely described within the framework
of standard quantum theory. A unitary S-matrix describes the evolution from
the infalling matter to the outgoing Hawking-like radiation.

_Postulate 2_. _Effective field theory_ : Outside the stretched horizon of a
massive black hole, physics can be described as a good approximation by a set
of semi-classical field equations. Although a consistent formulation of four-
dimensional gravity is lacking, a simplified two-dimensional gravity model
effectively illustrates a stretched horizon. These semi-classical equations,
part field-theoretic and part thermodynamic describe the average energy flow
and the evolution of the horizon.

_Postulate 3_ : To a distant observer, a black hole appears to be a quantum
system with discrete energy levels. The dimension of the subspace of states
describing a black hole of mass $M$ is the exponential of the Bekenstein
entropy $S_{BH}(M)$ without any infinite additive constant in the entropy.

While Susskind et al. outline the above three postulates focusing on the
perspective of an outside observer, they also bring up an additional
assumption concerning the experience of an infalling observer as per the
equivalence principle. This assumption is a form of a fourth postulate,
although not explicitly labeled as such in the text. It contrasts with
Postulate 1, as it hints at a divergence in the observed phenomena between an
external observer and an infalling observer, particularly around the event
horizon of a black hole:

_Postulate 4_. _No drama_ : A freely falling observer experiences nothing
unusual when crossing the event horizon. The low-energy dynamics probed by
this observer near their worldline are well-described by the Lorentz-invariant
effective field theory (this theory ensures that the effective descriptions
are consistent with the fundamental symmetries of the underlying quantum field
theory).

###  3.3 The semi-classical stretched horizon

Susskind and colleagues describe their ”semiclassical stretched horizon” as
being just one Planck unit larger in area than the global event horizon
itself. They suggest that for an external observer to not reference events
inside a black hole, a theory based on Postulates 1-3 is needed. To address
this, they introduce the concept of a stretched horizon, a visible timelike
curve positioned just ahead of the black hole’s event horizon. Susskind et al.
state that Postulates 1-3 are best implemented by viewing the stretched
horizon through a thermodynamic lens, which is assumed to have a microphysical
foundation. Despite the coarse-grained thermodynamic description (which
overlooks the minute details and focuses on broader, averaged properties),
Susskind et. al. believe there is an underlying microphysical basis, i.e., a
finer-grained level of description of the stretched horizon based on
microscopic physical properties. This finer description would be more detailed
and might explain the thermodynamic behaviors observed at the coarse-grained
level regarding underlying quantum mechanical processes [STU].

In the finer-grained level of description of the stretched horizon, when
information hits the stretched horizon, it gets scrambled and mixed up with
other information, but it is not lost. The scrambling is so effective that it
appears to an outside observer that the information has been destroyed,
although, in principle, it has been conserved. Black holes evaporate over
time, and as they do so, they re-emit the scrambled information. This re-
emission of Hawking radiation is considered a quantum mechanical process,
where pairs of particles are created near the horizon, one falling in and the
other escaping, carrying away a bit of the black hole’s mass and potentially
some scrambled information. The stretched horizon paradigm addresses the
dynamics of black holes at the quantum level by providing a framework to
analyze the quantum gravitational effects near the black hole horizon.

The properties and behaviors of the semi-classical stretched horizon are
similar to a physical membrane with mechanical, electrical, and thermal
characteristics, as seen by an outside observer. Due to the gravitational
dynamics around a black hole, the stretched horizon is portrayed as extremely
complex and chaotic, with features akin to a viscous fluid and thermal
properties governed by its area. When something approaches the black hole, the
membrane absorbs and scrambles the information before it crosses the event
horizon and re-emits it. The idea that information is scrambled and re-emitted
before reaching the event horizon contradicts the no-drama scenario. The
stretched horizon surface could show dramatic interactions contradicting
general relativity’s no-drama scenario. The membrane paradigm posits that
physics outside the black hole should not be affected by dynamics inside it.
The formulation helps separate the regions inside and outside the black hole
to focus on observable phenomena.111I refer to David Wallace’s discussion to
gain further insights into the membrane paradigm and its implications on black
hole information loss [Wall18], [Wall-mem.18], [Wall-mem.19].

###  3.4 Black hole complementarity

In 1993, Susskind et al. demonstrated that black hole complementarity
reconciles two seemingly contradictory observations: the infalling observer
and the outside observer. They posited that the experience of an observer
falling into a black hole (no drama) and that of an observer remaining outside
(who perceives the horizon as a dynamical membrane or stretched horizon and
sees information scrambled and re-emitted) are both valid. However, these
experiences cannot be reconciled due to the fundamental limitations imposed by
the event horizon. In other words, the nature of the event horizon creates a
fundamental division between the experiences of inside and outside observers.

Niels Bohr introduced the idea of complementarity to reconcile the wave-
particle duality of quantum mechanics. Susskind et al. invoke the concept of
complementarity in the context of understanding the nature of the stretched
horizon and the quantum behaviors surrounding black holes.

Susskind et al. suggest that different experimental setups reveal different
aspects of quantum systems, and these aspects, though seemingly contradictory,
are complementary and collectively contribute to the full understanding of the
black hole information paradox. Like the wave-particle duality, they suggest
that depending on the experiment conducted, one may detect a quantum membrane
(stretched horizon) around a black hole. Both experiments provide valid but
complementary information about the black hole’s nature. They hint at a
scenario where different experiments, akin to the different setups used to
observe particle or wave behavior, will reveal either the quantum membrane
aspect or some other aspect of black holes: ”An experiment of one kind will
detect a quantum membrane, while an experiment of another kind will not.”
However, no observer can ascertain the results of both types of experiments
simultaneously. The membrane is not an invariant that all observers can agree
upon, indicating a level of complementary understanding required to reconcile
different observational outcomes [STU]. [LPSTU].

Susskind et al. also touch on the interaction of infalling matter with the
atoms of the stretched horizon, leading to a thermal state and subsequent
evaporation, hinting at the quantum mechanical interactions at play. The
observed states and behaviors can be viewed as complementary descriptions of
the black hole’s behavior from different observational and theoretical
frameworks [STU], [Pol].

##  4 The Firewall and complementarity

###  4.1 Contradictions among the four postulates

In 2013, AMPS evaluated the consistency of Susskind et al.’s four postulates
regarding black hole complementarity in an ”old” black hole scenario. They
considered a black hole formed from a pure state and decaying over time; they
analyzed its Hawking radiation and divided it into an early part and a late
part, especially focusing on a scenario after the black hole has emitted half
of its initial Bekenstein-Hawking entropy. AMPS show that their derivation
contradicts Susskind et al.’s Postulates 1, 2, and 4 for an old black hole.
According to Postulate 1, the state of Hawking radiation is pure. According to
Postulate 2, the physics outside the stretched horizon can be described semi-
classically, implying a predictable and ”free” evolution of modes outside the
horizon. By applying Postulates 1 and 2, AMPS conclude that an infalling
observer would encounter high-energy modes. This contradicts Postulate 4,
which posits that an infalling observer should experience nothing of the
ordinary while crossing the horizon [AMPS].

Let us explore the intricacies of AMPS’ argument. Postulate 2 means low-energy
dynamics near an observer’s worldline still obey the familiar Lorentz-
invariant field theory. Also, according to Postulate 4, an observer falling
into a black hole is unlikely to come across extremely high-energy quanta.
When considering black holes that form and decay, Postulate 1 indicates that
the Hawking radiation from the black hole remains in a pure state. AMPS divide
the Hawking radiation into an early part and a late part. After the Page time,
when a black hole emitted half of its initial Bekenstein-Hawking entropy, the
number of states in the early radiation vastly outnumbers the states in the
late radiation. Postulate 1 implies that the state of Hawking radiation
remains pure, even as it is divided between early and late radiation.

AMPS write equations highlighting the interaction between outgoing Hawking
modes of radiation and infalling modes, which lead to the creation of Hawking
radiation. They show how these modes, especially the ones from late radiation,
are experienced by an infalling observer. This is central to the firewall
paradox because this derivation suggests that the late radiation is deeply
entangled with the early radiation for an old black hole (one after the Page
time). Because of this deep entanglement between early and late radiations, an
observer falling into the black hole would encounter high-energy modes (a
”firewall”) as they approach and cross the event horizon. This contradicts
Postulate 4, which asserts that an infalling observer would experience nothing
unusual upon crossing the event horizon of the black hole.

AMPS use Eddington-Finkelstein coordinates to visually portray how the
postulates, when applied to an old black hole scenario, lead to a
contradiction concerning the experience of an observer falling into a black
hole and the encounter with high-energy modes contrary to the postulates of
black hole complementarity.

The specific mode (mode $b$) that is part of the Hawking radiation emitted by
the black hole starts as a well-defined Hawking photon at a large distance
from the black hole (as per postulate 1). Still, as the observer encounters it
nearer to the black hole, its wavelength is much shorter (much smaller than
the Schwarzschild radius). In other words, the mode’s frequency is much higher
(or its wavelength much shorter) than one would expect near the event horizon
of the black hole, contradicting Postulate 4.

AMPS define $a^{\dagger}a$, representing a quantum measurement of the
infalling modes, which are modes that an observer falling into the black hole
would naturally measure. Suppose the observer’s measurements show that the
field is in an eigenstate of this number operator. In that case, it implies
they detect a definite number of particles in the infalling mode, and it would
contradict Postulate 1, which insists that the state of Hawking radiation is
pure. $b^{\dagger}b$ is the number operator for the outgoing modes associated
with the Hawking radiation emitted from the black hole. Suppose the observer’s
measurements align with an eigenstate of this operator. In that case, it
contradicts Postulate 4, which states that a freely falling observer should
experience nothing of the ordinary when crossing the horizon. If the results
the observer obtains are contingent on when they fall into the black hole,
then it would violate Postulate 2. This postulate posits that the results
should be universal, regardless of when the observer falls in [AMPS].

###  4.2 A drama: the firewall as a physical barrier

The main conclusion of AMPS’ paper is that if one assumes that the process of
black hole evaporation is unitary and follows the postulates of black hole
complementarity, a contradiction arises, leading to the unexpected result of a
”firewall,” a barrier to information retrieval from black holes, at the event
horizon. This firewall would be a highly energetic boundary that would
incinerate anything that attempts to cross it, contradicting the idea of a
smooth horizon expected from classical general relativity. The firewall would,
therefore, very much be a ”drama” for an infalling observer, Alice. According
to AMPS, she would encounter the firewall as she approached the event horizon
and would be immediately destroyed. On the other hand, the processes near the
event horizon look very different for an outside observer, Bob. Bob, who stays
safely outside the black hole and never falls in, will never see Alice cross
the event horizon. Due to the extreme gravitational time dilation effects, as
Alice gets closer and closer to the horizon, she appears to move more slowly.
From Bob’s viewpoint, Alice will asymptotically approach the horizon but never
quite cross it. Alice would appear thermalized to Bob in the stretched-horizon
hot Planckian layer just above the event horizon. She seems absorbed and
radiated away as part of the black hole’s Hawking radiation. From Bob’s
perspective, Alice appears to be destroyed by the firewall as she
asymptotically approaches the horizon. The firewall, therefore, behaves much
like the stretched horizon in terms of its position relative to the black
hole. However, while the stretched horizon is a benign region that an observer
can pass through without noticing anything unusual, the firewall is a highly
energetic and destructive barrier that an observer cannot pass through
unscathed.

In their argument, AMPS examine the behavior of black holes past the Page time
and the scrambling time. The scrambling time gives an insight into how quickly
information can become hidden in the black hole’s microstates. When
information enters a black hole, it gets mixed up with the chaotic state of
information within it. The scrambling time is when this newly introduced
information gets dispersed and mixed up to become virtually irretrievable from
the previous chaos within the black hole. The quicker the scrambling time, the
quicker the information gets dispersed within the black hole, making it harder
or impossible to retrieve later. Scrambling time is described concerning black
holes being the fastest scramblers in nature. Fast scramblers are systems that
achieve a minimal bound on the scrambling time, indicating that they scramble
information exceptionally quickly. Black holes are examples of fast scramblers
due to their purported ability to scramble information at a rate that
approaches the theoretically lower limit. This scrambling makes it exceedingly
difficult to retrieve any information about the state of particles that have
fallen into the black hole.

AMPS conclude by stating that following the scrambling time, ”we have to
expect either drama for the infalling observer or novel physics outside the
black hole [AMPS].

Recall that all black holes can be completely described by just three
externally observable classical parameters: mass, electric charge, and angular
momentum. Any other information about the matter that formed a black hole and
that fell into it (”hair”) is not accessible to outside observers. The no-hair
theorem suggests that black holes reach a simple, stable configuration once
they have settled down or aged. How long does it take for a black hole to lose
its ”hair,” i.e., for information about in-falling matter to become scrambled
and indiscernible across its horizon? One can liken this process to a
localized perturbation that is introduced at a specific location on the black
hole’s stretched horizon, thereby temporarily disrupting the uniformity of the
horizon. Over time, this localized disturbance diffuses and spreads out over
the entire stretched horizon, returning the system to a state of thermal
equilibrium. The time it takes for the perturbation to spread out and cover
the entire stretched horizon uniformly is the scrambling time. This is the
time it takes for a black hole to ”become bald”, i.e., lose any distinguishing
features or ”hair” and reach a state where it can only be described by its
mass, charge, and angular momentum. But there isn’t a mathematical proof for
this identification [Sek-Suss].

The firewall would form around when the black hole emitted about half of its
initial entropy in Hawking radiation, corresponding to the Page time. The
logic behind this timing is based on the idea that after the Page time, the
outgoing radiation begins to carry away information about the black hole’s
interior, leading to a potential violation of the no-cloning theorem of
quantum mechanics if there’s no dramatic change at the horizon like a
firewall.

###  4.3 A critique of the firewall

In 2017, William Unruh and Robert Wald criticized the firewall that suggests
violating the principle of equivalence, as it would entail a dramatic and
locally detectable change for infalling observers at the horizon. The firewall
proposition doesn’t provide a clear mechanism for why or how a high-energy
barrier would suddenly emerge at the horizon of an old black hole. Such a
mechanism must be consistent with known physics, and any _ad hoc_ addition
raises concerns about its physical legitimacy.

Unruh and Wald suggest that the firewall proposal might be an overreaction to
the information paradox. They assert that while the paradox is indeed
puzzling, it doesn’t necessarily mandate such a drastic revision of our
understanding of black hole horizons, especially one that conflicts with well-
established principles. They believe there are other potential resolutions to
the information paradox that do not require the introduction of firewalls.
They argue for the validity of quantum field theory in regions away from the
Planck scale. They suggest that the semiclassical analysis of black hole
evaporation doesn’t truly violate quantum theory.

Unruh and Wald find such suggestions neither plausible nor palatable. They
critique those adjusting and modifying quantum mechanics to get around the
information loss problem. They find it ironic that some researchers, in their
quest to preserve the principles of quantum mechanics, propose modifications
or radical alternatives that might undermine quantum mechanics. Such drastic
measures could violate quantum theory in situations where it should reliably
work [Un-Wal].

##  5 Bits, qubits, and error-correcting codes

###  5.1 Hayden and Preskill’s mirror

Consider again black holes that have passed their Page time. Recall that these
would be black holes that have radiated away more than half their initial
entropy and are on their way to evaporating eventually. As black holes near
the end of their Hawking radiation process, they would be considered old. In
2007, Patrick Hayden and John Preskill suggested that such old black holes
behave more like an _information mirror_ rather than a black abyss that
swallows and destroys information irretrievably. Hayden and Preskill outlined
a scenario in which Alice throws her diary into the black hole; instead of
being lost, the information is reflected in Hawking radiation. This is akin to
how a mirror reflects light [Hay-Pres].

Hayden and Preskill imagined Alice, who decided to throw a diary containing
sensitive information into an old black hole to ensure its secrecy. They
speculated that Bob, an expert with advanced recovery skills, might later be
able to retrieve the information contained in the diary from the Hawking
radiation emitted by the black hole as it slowly evaporates over a long
period.

They first described a deterministic model where the black hole’s internal
dynamics operate predictably. In this model, when Alice’s diary enters the
black hole, it is transformed into a series of bits combined with the black
hole’s internal bit string. This combined string is then subjected to a known
permutation, which is deterministic and known to Bob. With his advanced
knowledge of black hole dynamics and the known permutation, Bob observes the
bits emitted from the black hole via Hawking radiation. Once Bob receives a
small number of bits more than the original size of the diary, he can decode
the entire diary. The extra bits Bob needs to receive provide enough
information, alongside his understanding of the permutation and the black
hole’s internal state, to deduce the diary’s content. Initially, the black
hole was considered a perfect medium to conceal and destroy the information in
Alice’s diary. However, Hayden and Preskill suggest that with enough
understanding of the black hole’s internal dynamics and collecting a slightly
larger amount of data, Alice’s diary can be decoded and recovered, challenging
the idea of secure destruction of information [Hay-Pres].

Suppose Alice has quantum information encoded in a $k$-qubit quantum memory,
$M$, with the quantum state represented in a Hilbert space of dimension
$2^{k}$. Hayden and Preskill then introduce a third party, Charlie, with a
reference system $N$, initially maximally entangled with Alice’s memory $M$.
Alice, aiming to destroy the information in $M$, throws it into an old black
hole. The black hole’s complex and destructive dynamics would render the
information in $M$ irretrievable. Bob tries to extract a subsystem from the
Hawking radiation emitted by the black hole, post-Alice’s disposal act.

Suppose the subsystem Bob extracts is maximally entangled with Charlie’s
reference system $N$. In that case, this entanglement implies that the quantum
information initially in $M$ has somehow made its way out of the black hole
via Hawking radiation. In other words, the above entanglement is a channel
that allows the transfer of quantum information from $M$ out of the black
hole. By utilizing this entanglement, Bob can recover the quantum information
initially in $M$.

Suppose the initial state of $M$ was in a pure state (i.e., not entangled with
any other reference system), and the subsystem Bob extracts is maximally
entangled with $N$. In that case, it again suggests that the information from
$M$ has made its way out of the black hole. The maximal entanglement with $N$
allows Bob to recover the exact state in $M$ in his chosen subsystem and
retrieve the original quantum information Alice tried to destroy. In other
words, in principle, in both cases, the result is that Bob can recover the
quantum information Alice attempted to destroy by extracting a subsystem from
the Hawking radiation, which is maximally entangled with $N$. Bob, wanting to
recover Alice’s memory $M$, collects the Hawking radiation emitted by the
black hole. Hayden and Preskill propose that by collecting the Hawking
radiation and waiting for just a little more than $k$ qubits to be emitted,
Bob would then have enough information to begin decoding the state of Alice’s
qubits.

Information of infalling qubits (i.e., Alice’s memory $M$) gets scrambled with
the black hole’s state and is not permanently lost inside black holes. Hence,
information that falls into a black hole is later emitted via Hawking
radiation and can, in principle, be recovered, thereby aligning with the
quantum mechanical principle of information conservation. This goes against
the traditional belief stemming from the no-hair theorem that, from Bob’s
point of view, information is permanently lost in black holes [Hay-Pres].

By suggesting a mechanism for information retrieval in 2007, Hayden and
Preskill provided a potential resolution to the black hole information loss
paradox.

Once Alice’s information from $M$ enters the black hole, it undergoes
scrambling, wherein the information it carries becomes thoroughly mixed with
the state of the black hole in a highly chaotic but deterministic, unitary
evolution. This scrambling is hypothesized to happen extremely fast, making
the information ostensibly lost, though it’s conserved in a quantum mechanical
sense. If the black hole is a perfect scrambler, then in principle, the
information about Alice’s memory $M$ is not destroyed but is distributed non-
locally within the emitted Hawking radiation.

In the Hayden-Preskill model, two different scenarios are considered to
explore the dynamics of information retrieval from a black hole [Hay-Pres]:

1\. In the first scenario discussed above, Alice encodes quantum information
in a $k$-qubit quantum memory $M$ and throws it into the black hole while she
stays outside. The argument proposes that if the black hole has evaporated
past its halfway point, then Bob can retrieve the information from the emitted
Hawking radiation shortly after Alice’s quantum memory enters the black hole.

2\. Alice carries the quantum memory $M$ in the second scenario as she falls
into the old black hole. Here, the principle of black hole complementarity is
invoked to reconcile the difference in experiences between Alice and Bob.
According to black hole complementarity, Alice and Bob will have fundamentally
different, non-overlapping experiences. From Alice’s perspective, she falls
into the black hole with $M$. From Bob’s perspective, he may retrieve the
information from the Hawking radiation outside the black hole.

Let us now explore the variation of the first scenario, where Alice herself
falls into the black hole carrying the quantum information $M$.

Consider Alice once again. This time, she falls into the black hole with $M$
while Bob stays outside and tries to decode $M$ from the Hawking radiation
emitted by the black hole. After the Page time, the black hole could swiftly
return the quantum information it receives via Hawking radiation to Bob. This
timely return means that from Bob’s perspective, the information is retrieved
and not lost to the black hole, thus not violating the principle of unitarity.
While Alice sees the information fall into the black hole, Bob retrieves the
information from the Hawking radiation, and neither can verify the other’s
experience. Recall that black hole complementarity posits no single global
description of events but that the descriptions from Alice and Bob’s
perspectives are valid.

According to Hayden and Preskill, black hole complementarity suggests that the
information is not cloned. The idea of cloning in their scenario refers to the
possibility of having two copies of the same quantum information $M$: one
inside the black hole (with Alice) and one outside (recovered by Bob from the
Hawking radiation). If the information could be recovered from the outside
while still being on the inside, it would seemingly violate the no-cloning
theorem.

Hayden and Preskill show that once Alice falls into the black hole, she cannot
communicate with Bob to compare her information with what Bob recovers from
the Hawking radiation. Similarly, suppose Bob decides to jump into the black
hole after recovering the information from the Hawking radiation. In that
case, he won’t be able to compare notes with Alice or communicate his findings
to anyone outside the black hole. The conditions inside the black hole prevent
such verification. This unverifiability is crucial to avoiding a direct
violation of the no-cloning theorem [Hay-Pres].

Hayden and Preskill’s thought experiment proposed that information thrown into
a black hole can be recovered quickly from the Hawking radiation, given that
the black hole has already emitted more than half of its initial entropy. From
a quantum mechanical point of view, the catch here is the potential violation
of the no-cloning theorem. If Bob can reconstruct the state outside the black
hole while Alice still possesses it inside it, it seems the state has been
cloned. Black hole complementarity suggests that no single observer can see
both the inside and outside of a black hole, and thus, for any observer, the
information is either inside or outside, but never both. This prevents the
apparent cloning in the Hayden-Preskill scenario. However, AMPS challenged
black hole complementarity by pointing out a contradiction. Suppose one
accepts the postulates used in their argument. In that case, it leads to the
conclusion that there must be a firewall, a highly energetic barrier just
outside the event horizon, which would burn anything falling into the black
hole. This firewall would be at odds with the no-drama scenario, which
predicts that free-falling observers should experience no special effects when
crossing the event horizon of a large black hole. So, if one accepts the
conclusions of the AMPS paper and still wants to maintain the consistency of
quantum mechanics, then a firewall would seem to be required in the Hayden-
Preskill scenario.

###  5.2 The $A=R_{B}$ conjecture and the Hayden-Harlow protocol

AMPS then posited that a sufficiently old black hole could have a firewall at
its event horizon instead of having a smooth spacetime interior. The firewall
forms at the Page time (when a black hole has emitted about half its particles
and half the information). Post the Page time, the modes (or quantum states)
just behind the black hole horizon (labeled as $A$) can only be entangled with
modes in front of the horizon (labeled as $B$) if they are identified with the
radiation that has been emitted (labeled as $R$). This is expressed as
$A=R_{B}$ [AMPS], [Suss12].

AMPS argue that qubit $B$, obtained from late radiation from the black hole,
can’t be entangled with both qubit $A$ (behind the horizon) and qubit $R$
(from early radiation) due to the monogamy of entanglement, leading to the
conclusion that there must be a firewall at the horizon where Alice would
encounter high-energy particles instead of smoothly crossing the event
horizon. The idea of the firewall contradicts the traditional understanding
that a freely falling observer should experience nothing unusual as she
crosses the horizon (due to the equivalence principle).

Daniel Harlow and Hayden showed that the procedure Alice would have to
undertake to verify the entanglement between qubits $B$ and $R$ or $R_{B}$
(and thereby concluding about the entanglement between qubits $B$ and $A$)
would be computationally infeasible within the time frame before she falls
into the black hole.

Harlow and Hayden use a computational complexity argument to suggest that
decoding the information contained in the Hawking radiation is computationally
infeasible for Alice until the black hole has almost completely evaporated.
This argument posits a form of a decoding task that would be essential for
Alice to perform to extract the information about Charlie’s state from the
Hawking radiation. Harlow and Hayden’s decoding process is set within the
AdS/CFT framework, exploring the practicality and computational feasibility of
extracting information from Hawking radiation as envisioned by the AdS/CFT
correspondence [Har-Hay].

Harlow and Hayden’s approach attempts to address the practical feasibility of
an infalling observer (Alice) being able to measure the Hawking radiation to
distinguish between these entanglements before the black hole completely
evaporates. They model an old black hole as a system of qubits and employ
Page’s theorem to represent the state of the black hole as old, which would be
maximally mixed on a subsystem. Alice performs various quantum measurements
before and after crossing the event horizon of a black hole.

Consider the quantum state $\ket{\psi}$ of the black hole and its Hawking
radiation. Utilizing Schmidt decomposition, the quantum state of the black
hole and its radiation is represented with specific bases for two different
subsystems, $B$ and $H$. The subsystems $B$, $H$, and $R$ represent different
parts of $\ket{\psi}$. Subsystem $B$ is related to a specific mode in the
near-horizon area of the black hole, $H$ represents the black hole horizon,
and $R$ represents the radiation emitted from the black hole. A unitary
transformation ($U_{R}$) relates the chosen bases (chosen for subsystems $B$
and $H$ within the quantum state $\ket{\psi}$) to the natural bases sets of
orthogonal vectors that arise naturally when the quantum state is decomposed
using the Schmidt procedure, demonstrating the entanglement between the
subsystems $B$, $H$, and $R$. $U_{R}$ re-expresses $\ket{\psi}$ in terms of
the natural bases arising from the Schmidt decomposition. $U_{R}$ transitions
the state representation from the chosen bases to the natural bases, where the
entanglement structure of $\ket{\psi}$ becomes more apparent.

Alice needs to perform the transformation $U^{\dagger}_{R}$ using a quantum
computer to distill the purification of subsystem $B$ from the radiation
before she can observe a violation of entanglement monogamy as per the AMPS
experiment.222Distillation in the context of quantum entanglement is akin to
purification. Both processes aim to obtain a set of purer or more highly
entangled states from a set of less entangled or mixed states. The analogy
with distillation or purification in chemistry is often made to help
intuitively understand the process. Just like in a chemical process where you
might distill a liquid to remove impurities and obtain a purer substance, in
quantum information theory, you would distill (or purify) a mixed quantum
state to obtain a state of higher entanglement, which is often more useful for
various quantum information processing tasks. The terms ”distillation” and
”purification” convey the idea of extracting and isolating the most useful
part of something from a less pure or mixed state.

This distillation process aims to obtain a clearer and more usable form of
subsystem $B$ from the radiation, which is essential for further analysis.
Once the purification is distilled, Alice can enter the black hole to make the
necessary observations. However, due to the extreme computational complexity
of implementing the unitary transformation $U^{\dagger}_{R}$, doing so would
take longer than the lifetime of the black hole itself. While the black hole
can produce certain states without requiring exponential time, distilling $B$
from the radiation poses a significant computational challenge. Alice won’t
have enough time to complete the necessary computations and observations to
detect a violation of entanglement monogamy before she crosses the event
horizon, which would result in her inability to obtain the desired information
or validate/invalidate the AMPS paradox (a firewall).

A circuit ($U_{dyn}$) can produce $\ket{\psi}$ in polynomial time. Even with
an efficient $U_{dyn}$, the inverse of this circuit cannot be used to decode
Hawking radiation due to the challenges in acting on degrees of freedom ($H$)
inside the black hole. This highlights a boundary in our understanding and
manipulation of black hole information. This inability is a prominent issue in
black hole physics because it implies a lack of a complete quantum mechanical
description of its interior. The distilling transformation $U^{\dagger}_{R}$
is established, which could act solely on radiation. However, this is
nonconstructive because it only proves there is such a transformation without
providing a method to achieve it. This leaves a gap in understanding the
complexity of the distillation process, which is crucial for further analyses.
The absence of a constructive method to achieve the transformation
$U^{\dagger}_{R}$ leaves a crucial aspect of the black hole quantum mechanics
— the complexity of the distillation process — unexplored. The complexity of
this process is pivotal for understanding the practicality and feasibility of
distilling and analyzing quantum information from black hole radiation [Har-
Hay], [Har] (see explanation in [Win]).

###  5.3 Quantum error-correcting codes and the holographic principle

In quantum computing, quantum error correction is a set of techniques to
maintain the integrity of quantum information against errors due to
decoherence and other quantum noise. Drawing inspiration from these
techniques, computer scientists and physicists have explored the idea that the
AdS/CFT correspondence might be understood through the lens of quantum error
correction. The essence is to map the bulk to the boundary like a quantum
error-correcting code. The Ryu-Takayanagi (RT) formula and quantum error
correction are a crucial bridge between these two areas. This formula links
the entanglement entropy of a sub-region on the boundary with the entropy of
the bulk, specifically tying it to the minimal surface in the bulk.

This perspective hints at a deep connection between quantum computation,
quantum error correction, and holography that behaves much like a quantum
error-correcting code. Specifically, the low-energy Hilbert space
$\mathcal{H}_{bulk}$ of bulk effective field theory has a boundary described
by a Hilbert space, $\mathcal{H}_{boundary}$. When discussing the low-energy
Hilbert space $\mathcal{H}_{bulk}$ of the bulk effective field theory, we
refer to the AdS/CFT correspondence used to explore aspects of effective field
theories. In other words, at low energies, the bulk behaves in a way that a
boundary theory can effectively capture without gravity. The emphasis on low
energy means that this correspondence breaks down at extremely high energies
or distances of order the Planck length [Kap].

Chris Akers, Netta Engelhardt, Daniel Harlow, Geo Penington, and Shreya
Vardhan believe that the exterior of the black hole, i.e., Hawking radiation,
seems to carry information about the interior, challenging the classical
notion of black holes as information sinks. The Hawking radiation that Bob,
the outside observer, observes (which carries information) seems to conflict
with Alice’s uninterrupted fall into the black hole. Drawing on the AdS/CFT
correspondence, which posits that the information about the inside of a black
hole can be encoded on its boundary, Akers, Engelhardt, Harlow, Penington, and
Vardhan proposed that the black hole interior can be represented as a kind of
quantum error-correcting code on the boundary.

A correspondence called the dictionary $V$ maps states in the low-energy
Hilbert space $\mathcal{H}_{bulk}$ of bulk effective field theory into the
boundary Hilbert space $\mathcal{H}_{boundary}$. The quantum error-correcting
code is a process (isometric map) represented by $V$, which takes logical
states in the logical Hilbert space $\mathcal{H}_{bulk}$. It encodes them in
the physical Hilbert space $\mathcal{H}_{boundary}$ (an isometry is a linear
map which obeys $V^{\dagger}V=I$). This encoding is redundant, i.e., some of
the quantum information about the bulk can be lost and inaccessible without
affecting our ability to reconstruct the bulk’s physics. $V$ has certain
holographic properties that determine how information can be reconstructed.

Alice’s fall into the black hole represents information being thrown into the
bulk (inside the black hole). Bob, staying outside, only has access to the
boundary (the event horizon and surroundings). From Alice’s perspective, she
continues to experience the interior when she falls into a black hole. But
from Bob’s perspective, this information is scrambled and appears as part of
the Hawking radiation outside the black hole. Over time, more of Alice’s
reality seems to be externalized.

Through this framework, Bob might recover and decode some information about
Alice and the black hole’s interior without ever entering it. This information
would be encoded in the Hawking radiation he observes. However, the complexity
remains exponential, suggesting that the decoding would be computationally
infeasible and take an incredibly long time. While the information isn’t truly
lost, extracting it would be exceptionally complex. The computational
requirements for decoding this scrambled information are so enormous that it
would seem virtually impossible.

Hawking initially advocated black hole information loss; see section 2. His
prediction was based on an effective description (i.e., effective field theory
or semiclassical calculations). Crucially, Hawking radiation appeared to carry
no information about the state of the matter that had fallen into the black
hole. Black holes emitted Hawking radiation (which appeared to be thermal and
thus information-free). Eventually, they evaporated, seemingly erasing any
information about the matter that collapsed to form them in the first place.

The problem with Hawking’s calculation was the apparent non-unitarity. On the
other hand, although the CFT side of AdS/CFT involves effective field theory,
it is a fully unitary description. Thus, if information gets lost in the bulk
(like in a black hole), it should still be recoverable on the boundary. The
challenge is understanding how.

The effective description appeared to show no information, and it was coarse-
graining. The semiclassical calculation is an effective and coarse-grained
description, i.e., it doesn’t consider all possible fine details or quantum
effects. The fact is that information was not lost but was in such a scrambled
form that Hawking’s semiclassical calculations couldn’t capture it. Hence,
while the information might be recoverable in principle (due to the error-
correcting properties of holography), doing so would be computationally
prohibitive because of the immense complexity involved [AEHPS].

##  6 Avoiding a firewall

###  6.1 Typical and atypical black holes

According to Juan Maldacena and Susskind, an AdS eternal black hole is
equivalent to two entangled black holes connected by an ER bridge [Suss20].

Eternal black holes in AdS space are typically considered to be in
equilibrium. But in 2013, Donald Marolf and Joseph Polchinski discussed black
holes formed by collapse in AdS/CFT, writing: “While our black holes have been
explicitly formed by collapse, to high accuracy they resemble the eternal
black holes that dominate any corresponding microcanonical ensemble.

In other words, under certain conditions, the black holes formed by collapse
resemble the eternal black holes to a high degree of accuracy in a certain
microcanonical ensemble comprising all possible configurations of a black hole
at a specified energy. Marolf and Polchinski clarify that the term ”collapse”
here is used in a broader sense than usual. Typically, collapse might refer to
a rapid process where a massive celestial object collapses to form a black
hole in a short time relative to the AdS time scale associated with AdS space.
However, in this context, the term also encompasses much slower processes of
collapse that may occur over exponentially long times. Marolf and Polchinski
broaden the collapse concept to include slower processes. In the usual rapid
collapse, the resultant black holes occupy a subspace of small entropy, which
limits the resemblance to eternal black holes. By considering a broader range
of collapse processes, including much slower ones, the resultant black holes
exhibit a closer resemblance to eternal black holes. Thus, more varieties of
black hole states may be included within the microcanonical ensemble of all
possible configurations a black hole could have at a particular energy level.
This would provide a richer set of black hole states for theoretical
examination and analysis [Pol-Mar].

However, the above analogy is limited. While it may provide a useful
mathematical and theoretical framework for certain analyses, it doesn’t fully
capture the complexities and specifics of _conventional_ collapsing black hole
formation and behavior. Conventional black holes formed by collapse, i.e.,
Schwarzschild and Kerr black holes, are considered ”typical.” The typical
nature of the black hole is understood in terms of how these black holes are
formed through known processes (e.g., stellar collapse) in our universe. On
the other hand, AdS eternal black holes are theoretical constructs that live
in a self-consistent but adS universe. Therefore, eternal black holes could be
considered ”atypical” due to their specific mathematical and physical
properties that make them less representative of the black holes in our
universe,

Marolf and Polchinski claim that, under certain high-energy scenarios, black
holes formed by collapse can closely resemble eternal black holes when they
reach a stationary state. This resemblance might occur within a microcanonical
ensemble that specifies a fixed energy range. However, the insights gained
from studying AdS eternal black holes might not straightforwardly transfer to
the physics of black holes in a dS space, which is a closer representation of
our universe. Specifically, eternal black holes may not serve as direct models
for understanding black holes in our universe, which is more accurately
described by dS space. AdS and dS spaces have fundamentally different
geometries, which lead to differing gravitational physics. The behaviors,
interactions, and even the formation of black holes can significantly differ
between these two settings. Our universe has a positive cosmological constant,
characteristic of dS space. This plays a significant role in the large-scale
dynamics of the universe, including the behavior of black holes. AdS space, on
the other hand, has a negative cosmological constant. Moreover, while AdS/CFT
correspondence is a powerful tool within AdS space, no analogous precise
duality is known in dS space, making the transition of insights from AdS to dS
non-trivial. Given these fundamental differences, insights derived from
eternal black holes in AdS space may not hold in dS, and the lack of precise
dS/CFT correspondence makes such translations even more challenging.

A Kerr/CFT correspondence was developed to provide a holographic description
of extremal rotating (Kerr) black holes by relating the microstates of a black
hole to a two-dimensional CFT [HMB]. But extremal black holes have a zero
Hawking temperature, so they do not emit Hawking radiation, unlike their non-
extremal counterparts. Extremal black holes, with their zero Hawking
temperature, do not pose the same informational challenges; the firewall
paradox does not arise similarly. In other words, the information and firewall
paradoxes primarily concern non-extremal black holes that emit Hawking
radiation, leading to the potential loss of information. The Kerr/CFT
correspondence focused on extremal (or near-extremal) black holes does not
directly address and resolve the firewall and information paradoxes associated
with Hawking radiation. Its primary aim is to provide a holographic
description to understand the microstate counting and the entropy of such
black holes, aiding in the broader understanding of quantum gravitational
phenomena in these extreme conditions.

###  6.2 The ER = EPR conjecture

In the AdS/CFT correspondence context, a two-sided black hole is an eternal
black hole in a maximally extended spacetime, with two exterior regions
connected by a wormhole. On the other hand, a one-sided black hole is more
like an astrophysical black hole formed from collapse and doesn’t have this
sort of maximally extended spacetime. A one-sided black hole initially lacks
Hawking radiation but emits Hawking quanta over time. These quanta are
entangled with the black hole, which, according to ER $=$ EPR, could imply a
kind of wormhole connection. The entanglement entropy increases until it
reaches a maximum at the Page time, after which it decreases as more quanta
are emitted, signifying a change in the entanglement structure. After the Page
time, the Hawking radiation begins to carry away information about the
interior of the black hole, as the emitted radiation becomes less entangled
with the black hole and more entangled with the earlier emitted radiation. The
change in entanglement entropy and the shrinking of the black hole have
implications for the smoothness of the horizon.

In 2013, Maldacena and Susskind suggested that the two-sided black hole is
initially similar to the setup where there are two black holes, Alice’s and
Bob’s, connected by a non-traversable ER bridge due to entanglement. However,
as Hawking radiation comes into play, Alice’s black hole is replaced by this
radiation. Their description suggests that the entanglement between the
remaining black hole (Bob’s) and the Hawking radiation could be depicted as a
modified ER bridge with many exits corresponding to the many particles in the
Hawking radiation. The transformation of a two-sided black hole into a one-
sided black hole plus Hawking radiation, as described, is a way to illustrate
the complex and evolving entanglement relationships as Hawking radiation is
emitted and interacts with the black hole system [Mald-Suss].

In the Penrose diagram of an eternal black hole, the two sides represent two
asymptotically flat or AdS regions of spacetime connected by an ER bridge.
Maldacena and Susskind use this diagram to create an analogy: one side (right
side) represents Bob’s black hole while the other (left side) metaphorically
represents Hawking radiation. This analogy helps to provide a geometric
picture of the entanglement between the black hole and its Hawking radiation.
Maldacena and Susskind argue that this geometric representation of
entanglement does not necessitate a firewall at the horizon. The entanglement
between Bob’s black hole and the Hawking radiation (interpreted geometrically
as a wormhole) maintains the event horizon’s smoothness, adhering to general
relativity predictions while accounting for the quantum entanglement. The
firewall argument arises primarily in one-sided black holes, like those
described by the Schwarzschild solution or other astrophysical black holes,
especially when considering the implications of Hawking radiation and the
information paradox over long periods. It challenges the classical notion of
the no-drama scenario at the event horizon, as predicted by general
relativity. However, if the ER $=$ EPR conjecture holds, it implies that the
entangled particles created by Hawking radiation are geometrically connected
to the one-sided black hole’s interior via a wormhole. This setup might offer
a way to resolve the monogamy of entanglement problem without invoking a
firewall, potentially allowing for the preservation of the equivalence
principle [Mald-Suss].

###  6.3 A firewall is unnecessary

If there were a firewall, it would be a highly energetic, disruptive boundary
at the event horizon of a black hole, which contradicts the predictions of
general relativity. General relativity suggests that the event horizon should
be smooth and unremarkable from a local perspective. The entanglement between
the black hole and the Hawking radiation could lead to a buildup of
entanglement entropy, which was feared to result in a firewall. The ER $=$ EPR
conjecture potentially resolves this conflict. By interpreting the
entanglement between the black hole and the Hawking radiation as a wormhole,
Maldacena and Susskind provide a geometric, spacetime-based framework to
understand the quantum entanglement without disrupting the smoothness of the
event horizon. The wormhole acts as a conduit for entanglement, potentially
allowing for information reconciliation without needing a firewall. The
geometry of spacetime via wormholes could provide a way to preserve
information about particles falling into the black hole, aligning with quantum
mechanical principles while maintaining a smooth horizon as per general
relativity.

Maldacena and Susskind mention two possible interpretations of the AMPS
argument [Mald-Suss]:

1\. In the first scenario, Alice can create a high-energy particle (in mode
$A$) when she distills information from radiation ($R_{B}$) far from the black
hole. If this is the case, this action alone doesn’t disrupt the smoothness of
the black hole’s horizon—no firewall is formed. In other words, a firewall is
unnecessary since the high-energy particle was not present before Alice’s
action.

2\. In the second scenario, the high-energy particle is present regardless of
Alice’s actions, implying there is a firewall.

In Maldacena and Susskind’s view, the excitation or the high-energy particle
Alice encounters at the horizon results from her interaction with qubit
$R_{B}$ before she falls into the black hole, mediated through a wormhole. The
ER $=$ EPR conjecture proposes a connection between quantum entanglement and
spacetime geometry, potentially providing a new way to understand the black
hole interior. In this picture, the early radiation from the black hole
(entangled with the black hole) could provide a complementary description of
the black hole’s interior. In the case of a black hole, the early radiation
emitted by the black hole is entangled with the interior of the black hole.
According to ER $=$ EPR, a wormhole might mediate this entanglement. When
Alice interacts with a qubit from the early radiation (qubit $R_{B}$), this
interaction might affect the state at the horizon, potentially creating the
excitation (or firewall) she encounters as she falls into the black hole. This
happens through the wormhole connecting qubit $R_{B}$ and the black hole’s
interior.

ER $=$ EPR posits a potentially smooth interior linked to the exterior by
entanglement-mediated wormholes (interpretation 1), while AMPS suggests a
disruptive firewall at the horizon (interpretation 2). ER $=$ EPR suggests
that interaction with early radiation could change the state at the horizon
due to entanglement. In contrast, AMPS argue that the entanglement between
early and late radiation creates a contradiction leading to the firewall.

While ER $=$ EPR may provide a way to reconcile the seemingly smooth nature of
the event horizon with quantum entanglement, it’s a conjecture that has not
been proven. If correct, ER $=$ EPR might eliminate the need for firewalls,
providing a smooth spacetime structure inside black holes. However, the exact
nature of the black hole interior, the reality of firewalls, and the validity
of the ER $=$ EPR conjecture are all open questions requiring further
investigation.

The difference between the AMPS claim and Maldacena and Susskind’s claim
mainly lies in their interpretation of the entanglement scenario and its
consequences for the structure of the black hole’s event horizon and interior.
While AMPS suggest a firewall at the horizon due to a violation of
entanglement monogamy, Maldacena and Susskind propose a more harmonious
resolution through the ER $=$ EPR conjecture, indicating that the black hole’s
interior and the early radiation could be two complementary descriptions of
the same reality, mediated by quantum entanglement and possibly connected by
wormholes. AMPS argue against the first scenario, stating that Alice couldn’t
have created the particle due to the distance of the distillation process from
the black hole and the particle’s observed motion. Contrary to AMPS, Maldacena
and Susskind argue that Alice created the high-energy particle when she
distilled $R_{B}$. They suggest that a corresponding quantum in mode A is
created if Alice distills one such mode, but the horizon remains smooth if no
other disturbance occurs, i.e., if Alice does not distill any $R_{B}$.

The big question is how Alice could create a particle in mode A from a
distance. Maldacena and Susskind propose that the particle reaches mode A not
outside the black hole but via an ER bridge. In scenarios where Hawking
radiation is replaced by a second entangled black hole, fitting with the ER
$=$ EPR conjecture, they feel more confident in this explanation. Even though
Maldacena and Susskind believe the firewall proposal is incorrect, the
challenging concept pushed them to thoroughly explore the nexus between
quantum information science and quantum gravity.

While such ideas’ mathematical elegance and potential explanatory power are
enticing, they remain speculative frameworks within which physicists can
explore and resolve the AMPS paradox without empirical verification. But this
speculative nature is a known aspect of quantum gravity, especially when
delving into realms that are currently beyond the reach of experimental
testing and observation.

Susskind mentions that if there are firewalls at the horizons of black holes,
they could straightforwardly resolve the paradox without necessitating a
rethinking of quantum mechanics. However, the mechanism through which these
firewalls would form is yet to be identified or understood. This proposition
preserves the standard interpretations of quantum mechanics but at the cost of
introducing a radical new phenomenon at the event horizons of black holes. On
the other hand, the ER $=$ EPR conjecture leads to a more profound and
potentially revolutionary understanding of quantum mechanics, tying together
quantum entanglement and geometric connectivity [Suss16]. Yet, this idea also
hinges on wormholes, theoretical constructs that haven’t been observed.

As discussed in section 4, Harlow and Hayden approach the firewall problem
using computational considerations. The Harlow-Hayden conjecture provides a
computational complexity-based resolution to the AMPS paradox. They propose
that recovering information from a black hole by manipulating the Hawking
radiation is computationally infeasible within the time frame before the black
hole evaporates. This approach ties more closely to information theory and
computational complexity, providing a framework that may feel more grounded to
some.

Maldacena and Susskind, on the other hand, delve into the realm of wormholes
and ER bridges to propose a geometric interpretation of entanglement, aiming
to resolve the paradox by connecting quantum entanglement with spacetime
geometry. While the Harlow-Hayden approach might seem more accessible or
realistic due to its grounding in computational theory, it’s still within a
theoretical realm yet to be empirically validated. Furthermore, the acceptance
of one approach over the other could be influenced by advancements in
theoretical understanding or experimental techniques that could provide
evidence supporting one conjecture.

###  6.4 Causality and chronology protection as a barrier

According to Harlow and Hayden, a key obstruction to the AMPS experiment is
the high computational complexity required to distill purification from
Hawking radiation, which is likely to take an exponential amount of time
relative to the black hole’s entropy, far exceeding the black hole’s
evaporation time.

Susskind suggests that there could be a principle other than computational
complexity that prevents the experiment from being conducted, specifically,
causality. Hawking’s chronology protection conjecture argues against the
physical plausibility of time loops (or closed timelike curves) in nature,
which would allow for causality violations akin to time travel. The narrative
wherein Alice extracts $R_{B}$ (a representation of future quantum state $A$)
from early radiation and reintroduces it to the black hole to meet $A$ is
analogized to time travel. It’s as if $A$ has traveled back in time as $R_{B}$
met its future self, creating a closed causal loop. In the AMPS scenario,
introducing a firewall—representing a dramatic deviation from the expected
smooth spacetime near the horizon—breaks this loop and enforces chronology
protection, similar to the idea behind Hawking’s conjecture. The firewall
ensures that the entanglement arrangement leading to a causality violation is
interrupted, thus maintaining the causal structure of spacetime.

Harlow and Hayden proposed that performing this distillation task with a
quantum computer would require time exponential in $N$, where $N$ is a
parameter related to the system size or the number of qubits involved. This
exponential time complexity presents a significant computational hurdle as it
means that the time required to complete the distillation grows exponentially
with the size of the system. If the conjecture holds, Alice could not complete
the distillation of $R_{B}$ until after the black hole has evaporated. This
timeline implies that by the time she could verify the entanglement between
$R_{B}$ and $B$, the black hole itself, and thus state A, would no longer
exist.

This scenario effectively acts as a form of chronology protection, ensuring
that the causality-violating scenario envisaged in the AMPS paradox cannot
occur. The chronology protection mechanism here is not a physical barrier like
a firewall but rather a computational limitation that prevents Alice from
extracting and utilizing the information in $R_{B}$ in a way that would lead
to a causality contradiction.

In the case of black holes within AdS space, the evaporation process is
arrested, giving Alice unlimited time to perform her distillation task. Yet,
the AMPS paradox still hints at a necessity for a barrier (a computational
barrier) to prevent causality contradictions, rendering the barrier as a
mechanism for chronology protection even in such non-evaporating scenarios:
The mechanism of chronology protection here is not a physical barrier like a
firewall, but rather a computational limitation that prevents Alice from
extracting and utilizing the information in $R_{B}$ in a way that would lead
to a causality contradiction. The alignment of the evaporation timeline of the
black hole with the exponential time complexity of the distillation task
serves as a synchronicity that preserves the causal order of events and avoids
the potential paradoxes that would arise if Alice could complete the
distillation before the black hole evaporates.

Susskind argues that the Harlow-Hayden conjecture introduces a computational
barrier to the hypothetical sequence of events posed in the AMPS paradox,
serving as chronology protection through the inherent limitations of quantum
computational processes [Suss12],[Suss13], [Suss20].

##  7 A zoo of proposals

###  7.1 A shift in perspective

Einstein spent a significant portion of his later life attempting to formulate
a Unified Field Theory where he aimed to unify gravity with electromagnetism
within the framework of General Relativity. His approach was largely
deterministic and geometrical, attempting to encompass quantum mechanics
within a general relativistic framework [Sau].

Post advancements in understanding quantum mechanics, modern efforts have
largely aimed at incorporating gravity into the quantum mechanical framework.
The prevailing belief among many physicists is that a quantum theory of
gravity is the necessary next step. This is a stark shift from Einstein’s
geometrical, classical approach towards a probabilistic, quantum approach. One
promising pathway is the AdS/CFT correspondence, a part of the larger
holographic principles and string theory framework.

AdS/CFT suggests an emergent nature of gravity and spacetime geometry from a
lower-dimensional quantum theory, contrasting with Einstein’s view of gravity
as a fundamental force described geometrically. This reverses Einstein’s
approach, focusing on deriving gravitational phenomena from quantum mechanics
rather than vice versa. Modern approaches open the door to describing physical
phenomena in terms of quantum information and computation by rooting the
foundations of physics in quantum mechanics. This is not easily accomplished
if starting from a classical general relativistic foundation. Unitary
transformation is aligned with the quantum mechanical view of the universe’s
evolution. In quantum mechanics, the evolution of systems is described by
unitary transformations, which preserve the total probability (or information)
within the system, further tying into the information theoretical description
of the universe.

The change in approach over the decades reflects the accumulation of
experimental evidence supporting quantum mechanics and the difficulties
encountered in attempting to unify physics starting from a classical,
geometrical foundation as Einstein had attempted. This shift towards quantum
foundations and the exploration of emergent gravity through frameworks like
AdS/CFT represents a profound change in our conceptual understanding of the
fundamental forces of nature and how they might be unified. In the AdS/CFT
correspondence context, the boundary CFT is a quantum field theory that obeys
the unitary evolution principles. So, if the duality holds, then unitarity
should also be preserved in the bulk gravitational theory, and thus, there
should be no information loss in black holes. While the advent of the AdS/CFT
correspondence brought a new perspective, and despite its allure, it confronts
challenges in its direct application to our universe’s dS space-time.

###  7.2 Fuzzballs, Remnants, and the final burst

In 2017, Unruh and Wald addressed a long-standing debate in physics about
whether information can truly be lost inside a black hole [Un-Wal]. Over the
years, many alternatives to this idea of information loss have been proposed
[Un-Wal]:

1\. _Fuzzballs_ : the fuzzball concept suggests that during the gravitational
collapse of a star, instead of forming a traditional black hole with an event
horizon, some other structure, termed a fuzzball, might form [Math]. The
Fuzzball proposal implies that since there’s no event horizon, there’s no
associated information loss problem. Therefore, the challenge posed by the
black hole information paradox would not arise.

Unruh and Wald challenge the fuzzball conjecture based on inconsistencies.
They see the fuzzball idea as a significant departure from classical
understandings of physics. In classical general relativity, they highlight
that sufficiently massive black holes can form even at very low energy
densities and curvatures. For a fuzzball to form instead, there would need to
be a severe breakdown of established physics in areas where these models
should work well. They argue that if a massive shell collapses inwards, seeing
how it could stop its inward momentum just in time to form a fuzzball rather
than a traditional black hole is challenging. This would seem to go against
principles like momentum conservation; for a fuzzball to form, extreme and
sudden stresses would need to appear, which would cause significant
alterations to the effective spacetime metric. These sudden stresses would not
align with the energy densities, creating a scenario that is hard to reconcile
with current physical understanding. Classical black holes and their horizons
are global structures. If the collapsing matter were to behave differently to
form a fuzzball and avoid forming a horizon suddenly, it would imply that the
matter has some form of non-local knowledge – which again challenges
conventional physics. Unruh and Wald find it difficult to reconcile such ideas
with known physics [Un-Wal].

2\. _Remnants_ : Hawking’s calculation suggested that black holes evaporate
completely. However, this calculation is based on assumptions that may not
hold as the black hole approaches the Planck scale. The question remains
whether a black hole can fully evaporate or if it would leave a _remnant_
behind. Some calculations hint at such remnants, but a definitive answer
remains elusive due to the lack of a complete quantum gravity theory. Since
there is no apparent rule preventing a black hole from fully evaporating, it
should do so, behaving like any unstable quantum system. However, the
advocates of the _remnants_ conjecture argue against complete evaporation,
drawing a parallel with the hydrogen atom. The atom does not collapse because
of the uncertainty principle. Similarly, a generalized uncertainty principle
may prevent black holes from evaporating completely. This principle extends
the standard uncertainty principle, incorporating quantum gravitational
effects. In this model, the black hole ceases to radiate once it approaches
the Planck mass, leaving a stable remnant behind. It is suggested that such
remnants might be a candidate for dark matter [Ad-Ch-Sa].

3\. Finally, according to another suggestion, the _final burst_ idea, as a
black hole evaporates and reduces to the Planck scale, all the information
stored within the black hole is suddenly and explosively released, leading to
a pure final state. Although it seems counterintuitive for a Planck-sized
object to release massive amounts of information, Unruh and Wald suggest
looking into models where such behavior could be plausible.

Unruh and Wald raise a key question concerning the remnants conjecture: Can
these Planck scale remnants interact with the external universe? If remnants
do not interact, they contend that the information within the remnants would
be trapped and inaccessible. This would imply that, for all practical
purposes, the system’s final state would still be mixed, rendering the remnant
somewhat moot in the context of the information paradox.

At first glance, the idea of the final burst might seem implausible because it
implies that an enormous amount of information is released from an object that
has reduced to the Planck size, which is extremely small. Normally, one would
expect that releasing a large amount of information would require the release
of a correspondingly large amount of energy. However, just like the remnant
proposal, in the final burst conjecture, the final burst is not considered
radical because it suggests a deviation from our current understanding of
black hole physics only when the black hole reaches the Planck scale. Our
semiclassical understanding of gravity is expected to break down at this
scale, so postulating new behaviors here is not considered extreme. Unruh and
Wald conclude that the final burst proposal remains a viable alternative to
explain the black hole information paradox [Un-Wal].

Unruh and Wald point out that the evolution of a black hole from a pure state
to a mixed state (i.e., information loss) violates the unitarity of quantum
mechanics. However, they also argue that the evolution from a pure state to a
mixed state, as suggested by semiclassical analyses of black hole evaporation,
is not a violation of quantum theory but can be a prediction in certain
contexts. While highlighting the tension between information loss and certain
principles of quantum mechanics, they see scenarios where quantum theory
allows for such evolution (from pure to mixed states) without being violated.
Unruh and Wald advocate for a nuanced view, emphasizing the robustness of
quantum mechanics, even if it means accepting challenging results.

###  7.3 The Page time paradox

In 2018, David Wallace introduced the _Page Time Paradox_ , a variation of the
information paradox [Wall18]. The Page time paradox arises from the contrast
between the predictions of quantum field theory (QFT) and the predictions of
black hole statistical mechanics in the scenario of a black hole radiating
away its energy, as described by Hawking radiation.

Wallace considers a typical thermodynamic system, i.e., a black hole, with a
defined Hilbert space. The black hole is initially in a pure state at a
certain energy level, $E_{0}$, and it cools down by emitting thermal
radiation. The radiation is without memory of the specific state of the black
hole. While the radiation is thermal, each emitted quantum will be in a mixed
state. To preserve the overall purity of the black hole-plus-radiation state,
each quantum of radiation must be entangled with the black hole. According to
unitarity, the total von Neumann entropy of the emitted radiation as the black
hole cools down should be balanced by an equal von Neumann entropy of the
black hole to keep the total state of black hole-plus-radiation pure. This
suggests that the black hole becomes mixed to balance the entropy as radiation
with mixed states is emitted. As the black hole cools down and transitions
through different energy levels, the microcanonical entropy at each level
provides a bound for the von Neumann entropy the black hole can have at that
level. This is due to the definition of the microcanonical ensemble, which
assumes a fixed energy level and an equal probability of being in any
microstate within that energy level.

Over time, as more radiation is emitted, a conflict arises. The unitarity
principle suggests the black hole’s von Neumann entropy should increase to
balance the entropy of the emitted radiation. However, microcanonical entropy
decreases as the black hole energy decreases. Microcanonical entropy bound at
lower energy levels limits how much von Neumann entropy the black hole can
have. The Page Time is the point at which these two effects balance each
other. Importantly, the Page time signals when half of the black hole’s
entropy has been radiated away. After this time, for unitarity to be
preserved, the radiation must carry away more information, i.e., it can’t be
purely thermal. This transition from purely thermal to more informative
radiation is central.

After this point, the emitted radiation can no longer be purely thermal, as
that would require the black hole’s von Neumann entropy to exceed the
microcanonical entropy bound. To resolve this, the later emitted radiation
must be entangled with the earlier emitted radiation, reducing the overall von
Neumann entropy and adhering to the microcanonical entropy bound. However, it
is crucial to note that the QFT calculations giving rise to Hawking radiation
as purely thermal are done in a semi-classical regime, which combines
classical general relativity with QFT rather than a full quantum gravity
calculation. This semi-classical calculation shows the radiation as exactly
thermal, with no such entanglement, thus contradicting the predictions from
black hole statistical mechanics. Wallace called this discrepancy the _Page-
Time Paradox_ [Wall18].

In an interview, Wallace said: ”The Page-time paradox seems to point to a
breakdown of low-energy physics in a place where it has no business breaking
down, because the energies are still low” [Mus]. Wallace’s statement
underscores the unexpected nature of the paradox. When we think of breakdowns
in physics, we imagine extreme conditions, e.g., very high energies. However,
the Page-time paradox arises even when the energies involved are still
relatively low. It is puzzling because, under such low-energy conditions, QFT,
a standard theory, should be perfectly valid. The appearance of this paradox
in such conditions hints at a deeper problem or a missing piece in our
understanding of black hole physics.

###  7.4 Are we in the midst of a crisis?

There is a certain historical cyclicality in the foundations of physics, where
different theoretical frameworks vie for dominance and a unified
understanding. The historical context of the $19^{\textbf{th}}$ century among
the mechanistic, energetic, and electromagnetic worldviews mirrors the current
struggle between general relativity and quantum mechanics frameworks in some
ways. The competing frameworks in the $19^{\textbf{th}}$ century tried to
encapsulate all known physical phenomena within a coherent theoretical
structure. Each had its base entity, ether, energy, or mechanical matter, from
which other phenomena were to be derived. In the late $19^{\textbf{th}}$ and
early $20^{\textbf{th}}$ centuries, paradoxes and anomalies were resolved and
clarified through the advent of the theories of relativity and quantum
mechanics. Similarly, general relativity and quantum mechanics, each
extraordinarily successful in its realm, face challenges regarding new
paradoxes and anomalies.

Thomas Kuhn discerns a cyclical pattern in the history of science. Kuhn’s
cyclical picture of scientific revolutions consists of the following key
elements [Bir]:

1\. _Normal Science_ : This phase is characterized by the widespread
acceptance and practice of a particular paradigm and scientific ideas and
methods. Scientists work on puzzle-solving within this paradigm, further
refining and expanding it.

Certain experimental results and phenomena emerge that don’t fit well within
the current paradigm over time. Initially, these are treated as anomalies and
exceptions. Kuhn distinguishes between a _severe anomaly_ and an _ordinary
anomaly_. Kuhn stipulates that a _severe anomaly_ is recognized when ”all
members of a scientific group will reach the same decision” [Kuhn]. Based on
this criterion, the contentious debate surrounding the black hole information
paradox and its manifestation in the firewall paradox can be aptly categorized
as a severe anomaly.

2\. _Crisis_ : As more anomalies accumulate and the existing paradigm
struggles to account for them, there is growing doubt and a sense of crisis in
the scientific community.

Just as political conflicts can be resolved within a constitution, normal
science resolves anomalies within a paradigm. However, when the system becomes
a problem, a revolution becomes necessary.

3\. _Revolution_ : The crisis eventually leads to searching for a new paradigm
to explain the anomalies better. Kuhn calls this shift from one paradigm to
another a scientific revolution. It is a dramatic, paradigm-shifting change in
the field. The shift between paradigms is not always smooth. Just as political
revolutions require force and propaganda, scientific revolutions involve
external factors and individual biases.

4\. _New Paradigm_ : Accepting a new paradigm ushers in another period of
normal science, and the cycle begins anew: normal science, crisis, revolution,
new paradigm, normal science. According to Kuhn, there isn’t always a clear
translation between the language and concepts of different paradigms. This
leads to the idea that paradigms are incommensurable and fundamentally
incomparable. A paradigm change is likened to a gestalt shift. Scientists
operating within different paradigms might see the world in fundamentally
different ways.

The meta-irony in Kuhn’s ideas is that if scientific paradigms are indeed
subject to cycles of acceptance, crisis, and revolution, then Kuhn’s paradigm
about the nature of scientific revolutions might be subject to the same cycle.
Applying Kuhn’s model to his theory underscores the potential universality of
his insights. This suggests that no theory, including Kuhn’s, is immune from
being revised, replaced, or overthrown as new insights and anomalies emerge.
It’s a humbling reminder of the evolving nature of knowledge and
understanding.

## Acknowledgement

This work is supported by ERC advanced grant number 834735.

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