# Stimulated Emission of Radiation and the Black Hole Information Problem

Christoph Adami<EMAIL_ADDRESS>Department of Physics and Astronomy, Michigan
State University, East Lansing, Michigan 48824, USA

###### Abstract

The quantum theory of black holes has opend up a window to study the
intersection of general relativity and quantum field theory, but perceived
paradoxes concerning the fate of classical information directed at a black
hole horizon, as well as concerning the unitarity of the evaporation process,
have led researchers to question the very foundations of physics. In this
pedagogical review I clarify the ramifications of the fact that black holes
not only emit radiation spontaneously, but also respond to infalling matter
and radiation by emitting approximate clones of those fields in a stimulated
manner. I review early purely statistical arguments based on Einstein’s
treatment of black bodies, and then show that the Holevo capacity of the black
hole (the capacity to transmit classical information through a quantum
channel) is always positive. I then show how stimulated emission turns the
black hole into an almost optimal quantum cloning machine, and furthermore
discuss the capacity of black holes to transmit quantum information. Taking
advantage of an analogy between black hole physics and non-linear optics I
show that a calculation of the evolution of a black hole over time, using a
discretization of the black hole $S$-matrix path integral, yields well-behaved
Page curves suggesting that black hole evaporation is unitary. Finally, I
speculate about possible observable consequences of stimulated emission of
radiation in black holes.

##  I Introduction

Black-hole quantum physics has been an exciting but frustrating area of
research for almost fifty years—ever since Hawking discovered that black holes
described in quantum field theory become less black than their name suggests
Hawking (1975). In classical physics black holes are completely black 111This
holds true only for incident radiation with vanishing impact parameter, as
modes with angular momentum can be scattered. and do not emit any particles
because all classical trajectories must end up in the black hole singularity.
According to a semi-classical calculation in curved-space quantum field
theory, however, black holes emit radiation through the spontaneous emission
of particles near the event-horizon, while the black hole gives up mass in the
process. Ultimately, if no new mass is accreted, black holes may even
evaporate completely, leaving behind only Hawking’s eponymous radiation.

The mere existence of Hawking radiation revealed a daunting problem almost
immediately Hawking (1976). Particles that accrete onto a black hole can be
viewed as carriers of information. For example, a particle’s identity (whether
it is a photon, electron, or proton), its angular momentum, mass, etc., can
encode information that appears to be lost completely and irretrievably behind
the event horizon. While it is expected that the Hawking radiation
theoretically consists out of all kinds of particles (all those that can be
created in pairs via vacuum fluctuations), the thermal nature of the radiation
seems to imply that its quantum numbers are completely uncorrelated to those
that enter the horizon of an already formed black hole, or even of the matter
and radiation that created it. If Hawking radiation was the only energy left
over after the evaporation of the black hole, all information about the state
of matter that initially created the black hole, as well as the information
carried by particles that accreted on to it later, would forever be lost.

This is a more serious problem than is perhaps obvious from the start. Such a
loss of information is not merely inconvenient. It would signal the breakdown
of some of the most fundamental and cherished laws of nature that we have been
able to establish, namely the conservation of probability. Probability
conservation is built into quantum mechanics by describing the time-dependence
of a wavefunction by unitary operators $U=e^{-i/\hbar H}$, where $H$ is a
Hermitian Hamiltonian operator. In quantum field theory, unitarity is ensured
by the unitarity of the $S$-matrix, which is in itself a consequence of a
Hermitian interaction Hamiltonian (see, e.g., Sakurai (1967)). On the one
hand, both quantum mechanics and in particular quantum field theory have been
exceedingly accurate in their prediction of the microscopic properties of
matter and light, to such an extent that it would be shocking if unitarity
would be violated by such macroscopic objects as black holes.

On the other hand, we do not have a consistent theory of quantum gravity, in
particular not one that is expected to accurately describe the late stage of
black-hole evaporation, where gravitational fields are expected to be very
strong. One could therefore not dismiss out of hand that our current theories
simply break down under such extreme circumstances, and that a consistent
theory of quantum gravity would remedy the dilemma. But the following
reasoning suggests that a full-fledged theory of quantum gravity cannot be
necessary to solve the problem of probability conservation in black hole
evaporation. Classical information, as mentioned above, is carried by ordinary
degrees of freedom such as spin, polarization, momentum etc., all of which are
adequately described in a semiclassical theory of gravity. The approximations
involved in the semiclassical theory concern the treatment of the space-time
metric: it is left unquantized, meaning that it is treated as a background
field. While a consistent theory of quantum gravity should treat the space-
time metric as a quantum mechanical variable that can be entangled with other
degrees of freedom, it is not plausible that the uncertainty associated with
the decoherence of the metric will have a significant impact on black hole
dynamics until perhaps the black hole is of Planck mass Wald (1994). Yet,
during the period where quantum effects of the metric are small (large,
massive black holes), predictability would already be lost because the
incident quantum numbers are inaccessible behind the horizon. Indeed, the
principle of microscopic time reversibility relies on predictability at all
times. Thus, it is not necessary to wait for the complete evaporation of the
black hole in order to see a problem with the standard description of black
holes. Moreover, at the point where quantum effects on the metric are expected
to be significant, it is implausible that information can be recovered from
the evaporation of a Planck-size black hole because it is unclear how it could
store that much entropy. Thus, we are encouraged to look for a consistent
treatment of black hole dynamics that allows for predictability at all times,
and explains the apparent lack of coherence in a completely unitary manner.

Before discussing possible solutions to the ”black hole information problem”,
I’d like to emphasize that there are really two such problems. The one that
gets most of the attention is the problem of whether black holes can turn pure
states into mixed states, as Hawking claimed they do Hawking (1976). Even
though this question is often couched in language that suggests that it is a
problem of information conservation (”what happens to the information about
the initial state of the matter and radiation that formed the black hole”?) it
is strictly speaking not a problem of information transmission, but rather a
problem in showing that the quantum state after the evaporation of the black
holes has returned to the pure state it started out as, before the formation
of the black hole. Quite generally, the question should be: ”Is black hole
formation and evaporation described by unitary dynamics?”

The literature usually distinguishes four standard alternatives to deal with
the apparent ”information loss”. I briefly summarize them here, but refer to
Preskill (1993); Fabbri and Navarro-Salas (2005); Unruh and Wald (2017) for a
more thorough exposition. The first alternative is the most conservative: it
claims that information is released with the Hawking radiation after all, or
to put it more concisely, that the state after evaporation of the black hole
somehow returns to purity. This type of scenario has been advocated by Page
and by Bekenstein Page (1993); Bekenstein (1993), and also has more modern
support from theories in which quantum gravity is coupled to dilatons Callan,
Jr and S. Giddings  _et al._ (1992); Almheiri  _et al._ (2020). One of the
most common objections to this scenario is that the thermal nature of the
Hawking radiation precludes it from carrying any information. We will see
below that there is in fact no basis to this objection, as information can be
encrypted in maximum entropy states.

The second alternative posits that, conceivably, the information remains
locked inside of the black hole, but the black hole does not disappear but
rather becomes a stable remnant. This explanation suffers from several
problems, such as the necessity of requiring almost infinitely-degenerate
Planck-sized black holes that should be observable via pair formation.
Moreover, it does not address how those remnants would constitute a pure
state. The third general category of explanations claims that it is possible
that all the accumulated information “comes out at the very end”, that is,
after most (or all) of the mass of the black hole was radiated away. This
suggestion runs into the problem that an arbitrary large amount of information
cannot be radiated away in a finite amount of time. Finally, a fourth
alternative suggested that the information is not lost, but rather is sealed
away into one or more nucleating baby universes. This type of explanation has
now even fewer defenders than it had at its inception.

The second ”information problem” concerns the fate of information that
interacts with an already-formed black hole. While sometimes discussed at the
same time as the problem of the unitarity of the black-hole evaporation
process, these two problems are actually quite separate. While Hawking
radiation (according to the first alternative to information loss discussed
above) might carry the imprint of the formation of the black hole, it is very
unlikely that it carries information about the identity of particles that
interact with the black hole horizon at late times. This problem turns out to
be a problem in quantum channel theory: can information that is absorbed at
the black hole event horizon be reconstituted by an inertial observer that can
only observe radiation emanating from the black hole? Because the black hole
is treated as a static quantity here (the ”communication channel”), statements
about the capacity of this channel do not address the unitarity of the
evaporation process.

In this pedagocical review, I will address both problems using modern methods
of quantum information theory. The problem of communicating via a black hole
is solved by realizing that information is not lost within a black hole
because a perfect copy of the information is always maintained outside of the
event horizon, thanks to a well-known physical mechanism: the stimulated
emission of particles at the event horizon that must accompany the spontaneous
emission (i.e., Hawking radiation) in any consistent theory of black body
radiation. It turns out that particles emitted via stimulated emission provide
the ”quantum hair” Coleman  _et al._ (1992) necessary for a conceptual
understanding of macroscopic black holes.

I will first present a simple statistical argument that appeared in the same
year as Hawking’s paper that points out that the process of stimulated
emission was missing from the latter’s discussion. I then discuss the curved-
space quantum field theory derivation of stimulated emission in response to
early- and late-time modes, and go on to show how to calculate the classical
and quantum information transmission capacity of black holes. We will see that
the capacity to transmit classical information with arbitrary accuracy over a
quantum channel (the so-called ”Holevo capacity”) is always positive, meaning
that information is never lost in black holes. The capacity to transmit
quantum information turns out to vanish in come cases, but we’ll see that this
ensures that the laws of physics are not broken.

I then present an approach that allows us to move beyond the semi-classical
description of black holes to illustrate how black holes might evaporate in a
unitary manner, giving rise to the well-known Page curves. This approach is
speculative in the sense that I have to assume a particular form for the
interaction Hamiltonian of black holes with radiation modes, but is arguably
less speculative than approaches that rely coupling gravity to dilaton fields
or conformal field theories. Moreover, because the coupling between black-hole
and Hawking/partner modes is exactly equivalent to the coupling of modes that
governs optical parametric amplifiers, there is a chance that laboratory
experiments can shed light on black-hole evaporation dynamics in a manner
similar to black-hole analogue experiments. I close with speculations about
observable consequences of stimulated emission in black holes.

##  II Statistical Black Hole Thermodynamics

To set the stage (and my notation), I will repeat a simple ”maximum entropy”
argument due to Bekenstein Bekenstein (1975) to obtain the probability
distribution $p(n)$ of quanta in any of $n$ outgoing modes emitted by a black
hole via spontaneous emission (the distribution of Hawking radiation), using
only Hawking’s result that the mean number of outgoing quanta is Hawking
(1975)

$\displaystyle\langle n\rangle=\frac{\Gamma}{e^{x}-1}\;.$ (II.1)

Here, $\Gamma$ is the absorption coefficient of the black hole (the “gray-body
factor”, so that $1-\Gamma$ is the black hole’s reflectivity), and
$x=\hbar\omega/T_{\rm BH}$222In general, for a charged and rotating black
hole, $x=\hbar\omega/T_{\rm BH}-\hbar m\Omega-\epsilon\Phi$, with $m$ the
azimuthal quantum number, $\epsilon$ the electric charge, $\Omega$ the
rotational frequency, and $\Phi$ the electrical potential. To keep matters
simple, I will only treat uncharged non-rotating black holes here., where
$\omega$ is the mode’s frequency, and $T_{\rm BH}$ the black hole temperature.
Bekenstein derived the result (for a single massless scalar bosonic mode $n$)

$\displaystyle p(n)=(1-e^{-\lambda})e^{-n\lambda}$ (II.2)

simply by demanding that the entropy of the outgoing radiation

$\displaystyle S=-\sum_{n}p(n)\log p(n)$ (II.3)

is maximal, with the constraints $\sum_{n}p(n)=1$ and
$\sum_{n}np(n)=\frac{\Gamma}{e^{x}-1}$ implemented via Lagrange multipliers.
In Eq. (II.2), the Lagrange multiplier $\lambda$ is related to black hole
parameters via

$\displaystyle e^{-\lambda}=\frac{\Gamma}{e^{x}-1+\Gamma}\;.$ (II.4)

It should be noted that distribution (II.2) (which was not given by Hawking)
was also derived independently using full-fledged curved-space quantum field
theory by Wald Wald (1975).

Bekenstein now considered what would happen if a black hole that produces
outgoing radiation with distribution $p(n)$ is immersed in a heat bath with
temperature333This radiation has the distribution
$p_{\star}(n)=(1-e^{-y})e^{-ny}$ where $y=\hbar\omega/T$ and $T$ the
temperature of the heat bath. $T$. Together with his graduate student Amnon
Meisels, Bekenstein found that the distribution for the ensuing outgoing
radiation $p_{o}(n)$ was not compatible with the assumption that this
radiation was composed only of spontaneous emission and radiation scattered
from the surface of the black hole with a reflectivity $1-\Gamma_{0}$ (so that
$\Gamma_{0}$ is the probability that a single quantum is absorbed by the black
hole). Instead, they found that

$\displaystyle p_{o}(n)=\sum_{m=0}^{\infty}p(n|m)p_{\star}(m)\;,$ (II.5)

where $p(n|m)$ is the probability distribution to observe $n$ outgoing
particles given that $m$ such particles were incident on the black hole. This
distribution is independent of the environment that the black hole finds
itself in, and respects the detailed balance condition

$\displaystyle e^{-xm}p(n|m)=e^{-xn}p(m|n)\;.$ (II.6)

This condition implies microscopic reversibility, and allows us to see that
the mean number of outgoing particles $\langle n\rangle$ is given by the sum
of Hawking’s spontaneous emission term (II.1) and an average fraction
$m(1-\Gamma)$ returned outward

$\displaystyle\langle n\rangle=\frac{\Gamma}{e^{x}-1}+m(1-\Gamma)\;.$ (II.7)

By using the result that the fraction that is returned by pure scattering has
to be $m(1-\Gamma_{0})$ (where $1-\Gamma_{0}$ is the pure scattering
reflectivity introduced earlier), Bekenstein and Meisels could show that

$\displaystyle\Gamma=\Gamma_{0}(1-e^{-x})\;.$ (II.8)

This result shows that $1-\Gamma$ is in fact the effective reflectivity of the
black hole, so that the black hole absorptivity $\Gamma$ is the sum of
$\Gamma_{0}$ and a negative contribution due to stimulated emission
$-\Gamma_{0}e^{-x}$, which is present in all modes. Specifically, (II.7) can
be rewritten to read

$\displaystyle\langle n\rangle=(1-\Gamma_{0})m+(m+1)\frac{\Gamma}{e^{x}-1}\;,$
(II.9)

where the $m$ in the second term on the right hand side of Eq. (II.9) refers
to the stimulated response to $m$ incoming particles, and the “1” to the
spontaneously emitted particles (Hawking radiation). These different terms are
indicated in Fig. 2.1, which also shows the number of anti-particles generated
inside of the black hole via spontaneous and stimulated emission (in order to
conserve particle numbers, both spontaneous and stimulated emission occur via
pair formation, as we will see later).

Figure 2.1: Outgoing particles according to Eq. (II.9) (arrows pointing down)
for $m$ incident particles on the horizon. The terms
$\overline{m\frac{\Gamma}{e^{x}-1}}$ and $\overline{\frac{\Gamma}{e^{x}-1}}$
refer to anti-particles generated behind the horizon to conserve particle
number, and will be discussed further below.

The exact form of the distribution $p(n|m)$ is complicated, and had to be
inferred from Eq. (II.5) by a power series expansion. However, it was later
shown to be exactly equal to the result obtained with a full treatment in
curved-space quantum field theory by Panangaden and Wald Panangaden and Wald
(1977). The result was later confirmed by Audretsch and Müller Audretsch and
Müller (1992), who redid the calculation using wave packets, taking into
account the red shift, and studying the effect of both incoming particles and
anti-particles.

##  III Curved-space Semi-classical Quantum Field Theory

The results of Bekenstein and Meisels, Panangaden and Wald, and Audretsch and
Müller did not convince everybody that information in black holes was
preserved even though that work demonstrated microscopic reversibility.
Schiffer, for example, argued that thermal radiation still overpowers
stimulated emission for the vast majority of modes Schiffer (1993).
Furthermore, taking the red shift into account revealed that outgoing modes at
observed frequencies $\sim 1/8\pi M_{\rm BH}$ (where $M_{\rm BH}$ is the mass
of the black hole in units where $G=c=k=\hbar=1$ as usual) ought to be due to
incoming modes that were present just as the black hole was forming, and
therefore must have been enormously blue-shifted with respect to the outgoing
late-time radiation Jacobsen (1991). Thus, it was not clear how those
particular calculations confront the question of what happens to late-time
particles absorbed by an already-formed black hole. I will address this last
point in this section by introducing Sorkin’s treatment of early- and late
incoming modes, and then answer the first question (does stimulated emission
really conserve information?) by explicitly calculating the capacity of the
black hole to transmit information encoded in both early-time and late-time
modes in the section that follows, using the standard methods of quantum
information theory.

To establish notation, I will first treat the simplest case: a perfectly
absorbing black hole ($\Gamma=1$) and only early-time complex massless modes
of energy $\omega_{k}$ ($\omega_{-k}$ for anti-particles). Even though we will
later see that such a choice of absorptivity is inconsistent (and we will
understand why), it is instructive to do this calculation first as it is
simpler than the general case.

To introduce complex fields (which allow us to describe both particles and
anti-particles) might at first glance seem like an unnecessary complication.
Indeed, it is simpler (and still instructive) to study Hawking radiation using
scalar fields only. However, because of the crucial role that negative-
frequency modes play in this discussion, ignoring anti-particles (which are
equivalent to particles traveling backwards in time) obscures some fundamental
aspects of black hole physics, in which time-reversal invariance is key.

After the initial exposition of known (and therefore canonical, if not
classical) results, I will introduce particles to the in-vacuum to understand
how they stimulate the emission of particles in the out-vacuum outside of the
horizon (and the emission of anti-particles beyond the horizon), and then
introduce late-time modes using Sorkin’s trick, allowing me to recover the
gray-body absorptivity (and alleviate any worries about transplanckian
frequencies at past infinity).

Consider the Penrose diagram in Fig. 3.1.

Figure 3.1: Penrose diagram showing the early-time modes $a_{k}$ and $b_{k}$,
created at past infinity ${\mathscr{I}}^{-}$ (just as the black hole formed)
and traveling just outside and just inside of the event horizon towards future
infinity ${\mathscr{I}}^{+}$. The outgoing mode at future infinity is
annihilated by $A_{k}$.

The relation between the operators annihilating the incoming modes $a_{k}$ and
$b_{k}$ and the outgoing mode $A_{k}$ is given by a Bogoliubov transformation
(the annihilation and creation operators satisfy the commutation relations
$[a_{k},a_{k^{\prime}}^{\dagger}]=[b_{k},b_{k^{\prime}}^{\dagger}]=\delta_{k,k^{\prime}}$
and
$[a_{k},a_{k^{\prime}}]=[b_{k},b_{k^{\prime}}]=[a_{k},b^{\dagger}_{k^{\prime}}]=0$)

$\displaystyle
A_{k}=e^{-iH}a_{k}e^{iH}=\alpha_{k}a_{k}-\beta_{k}b^{\dagger}_{-k}\;$ (III.1)

so that the unitary operator $U=e^{-iH}$ (I set $\hbar=1$) maps the in-vacuum
to the out-vacuum:

$\displaystyle|0\rangle_{\rm out}=e^{-iH}|0\rangle_{\rm in}\;.$ (III.2)

It is not an accident that I chose the letter $H$ for the Hermitian operator.
In general, we can write the time-dependent mapping from in-states to out-
states in terms of the time evolution operator

$\displaystyle
U(t_{2},t_{1})=\mathsf{T}e^{-i\int_{t_{1}}^{t_{2}}H(t^{\prime})dt^{\prime}}\;,$
(III.3)

where $\mathsf{T}$ stands for Dyson’s time-ordering operator and $H(t)$ is the
Hamiltonian describing the unitary evolution of the quantum state. This time
evolution operator can be approximated using $N$ small time slices $\Delta t$
so that with $t=N\Delta t$

$U(t)=\mathsf{T}e^{-i\int_{0}^{t}H(t^{\prime})dt^{\prime}}\approx\prod_{i=1}^{N}e^{-i\Delta
tH_{i}},$ (III.4)

where $H_{i}$ is the $i$-th time-slice Hamiltonian. In the static path
approximation (one time slice) and absorbing $\Delta t$ into the interaction
strength, the operator (III.4) becomes the one implementing (III.2) with the
Hamiltonian

$\displaystyle
H=i\sum_{k=-\infty}^{\infty}g_{k}\big{(}a_{k}^{\dagger}b_{-k}^{\dagger}-a_{k}b_{-k}\big{)}\;.$
(III.5)

where $g_{k}$ is an ”interaction strength” that, as we will see, sets the
black hole temperature. Using the Baker-Campbell-Hausdorff theorem we can
relate the coefficients $\alpha_{k}$ and $\beta_{k}$ in (III.1) to $g_{k}$ via

$\displaystyle\alpha_{k}^{2}=\cosh^{2}g_{k}\;,\beta_{k}^{2}=\sinh^{2}g_{k}$
(III.6)

and we have $\alpha_{k}^{2}-\beta_{k}^{2}=1$. The standard arguments of
Hawking Hawking (1975) enforcing analyticity on the solutions to the free
field equations444Despite the appearance of an interaction strength $g_{k}$,
$H$ is a free-field Hamiltonian. allow us to deduce that

$\displaystyle\alpha_{k}^{2}=e^{\omega_{k}/T_{\rm BH}}\beta_{k}^{2}\;,$
(III.7)

and relate $g_{k}$ to $T_{\rm BH}$ since $\frac{\omega_{k}}{T_{\rm
BH}}\approx\log(g_{k})+{\cal O}(g_{k}^{2})$. With these definitions out of the
way, we can write down the out-vacuum state as

$\displaystyle|0\rangle_{\rm
out}=\prod_{k=-\infty}^{\infty}e^{g_{k}\big{(}a_{k}^{\dagger}b_{-k}^{\dagger}-a_{k}b_{-k}\big{)}}|0\rangle_{\rm
in}\;.$ (III.8)

Using the disentangling theorem for SU(1,1), we can evaluate (III.8) to become
(writing the in-vaccum as the product state $|0\rangle_{\rm
in}=|0\rangle_{a}|0\rangle_{b}$)

$\displaystyle|0\rangle_{\rm
out}=\prod_{k=-\infty}^{\infty}\frac{1}{\cosh^{2}g_{k}}\sum_{n_{k},n_{k}^{\prime}}e^{-(n_{k}+n^{\prime}_{-k})\omega_{k}/2T_{\rm
BH}}|n_{k},n^{\prime}_{-k}\rangle_{a}|n_{k}^{\prime},n_{-k}\rangle_{b}\;.$
(III.9)

This is enough for us to recover the probability distribution of outgoing
particles (II.2) (albeit for the case $\Gamma=1$ as we do not treat reflection
here), by calculating the density matrix of outgoing radiation (the radiation
in ”region I”, see Fig 3.1) via tracing out the interior of the black hole
(region II)

$\displaystyle\rho_{I}={\rm Tr}_{\rm II}|0\rangle_{\rm out}\langle
0|=\prod_{k}\rho_{k}\otimes\rho_{-k}\;.$ (III.10)

As expected, the density matrix factorizes into a particle term and an anti-
particle term with

$\displaystyle\rho_{k}=\frac{1}{1+\beta_{k}^{2}}\sum_{n=0}^{\infty}\biggl{(}\frac{\beta_{k}^{2}}{1+\beta_{k}^{2}}\biggr{)}^{n}|n_{k}\rangle\langle
n_{k}|=(1-e^{-\omega_{k}/T_{\rm
BH}})\sum_{n_{k}=0}^{\infty}e^{-n_{k}\omega_{k}/T_{\rm
BH}}|n_{k}\rangle\langle n_{k}|\;.$ (III.11)

This expression implies Bekenstein’s (and therefore Hawking’s) result for the
single mode spontaneous emission probability Eq. (II.2), as $p(n)=\langle
n|\rho_{k}|n\rangle$ (note that $e^{-\lambda}=e^{-x}$ for total absoption).
The mean number of outgoing particles becomes

$\displaystyle\sum_{k=-\infty}^{\infty}{}_{\rm out}\langle
0|a_{k}^{\dagger}a_{k}|0\rangle_{\rm
out}=\sum_{k=-\infty}^{\infty}\beta_{k}^{2}\;,$ (III.12)

which is the celebrated Planck distribution of Hawking radiation since

$\displaystyle\beta_{k}^{2}=\frac{e^{-\omega_{k}/T_{\rm
BH}}}{1-e^{-\omega_{k}/t_{\rm BH}}}\;.$ (III.13)

Before considering the impact of particles entering the in-vacuum, let us take
a closer look at the Hamiltonian (III.5) that we used to map past-infinity
states to future-infinity states. This Hamiltonian is, as a matter of fact,
very common in quantum optics, where it is known as the ”squeezing”
Hamiltonian that describes optical parametric amplification (generally, all
quantum amplification processes can be described by Bogoliubov transformations
Leonhardt (2010)). Quantum amplification is inherently a nonlinear process. In
the simplest system, a pump photon with frequency $\omega_{p}$ is converted
into two photons, called the signal and idler, with frequencies $\omega_{s}$
and $\omega_{i}$, where $\omega_{p}=\omega_{s}+\omega_{i}$. This process,
called parametric downconversion, creates entangled photon pairs via the
Hamiltonian (for a single mode)

$\displaystyle H_{\rm
OPA}=i\eta(a^{\dagger}_{s}b^{\dagger}_{i}-a_{s}b_{i})\;,$ (III.14)

where $\eta$ is the coupling strength (which depends on the pump amplitude)
and $a^{\dagger}_{s}$ and $b^{\dagger}_{i}$ are the creation operators for the
signal and idler modes, respectively (”OPA” stands for ”optical parametric
amplification”). If we compare Hamiltonian (III.14) to Eq. (III.5), we see
that the role of positive and negative frequency modes of Hawking radiation
are here played by the signal and idler modes, and indeed the wavefunction of
the signal-idler pair is simply Nation  _et al._ (2012)

$\displaystyle|\Psi(t)\rangle=\frac{1}{\cosh\eta t}\sum_{n=o}^{\infty}e^{-\eta
tn/2}|n_{i}\rangle|n_{s}\rangle\;,$ (III.15)

which we can compare to Eq. (III.9). In contrast to (III.9) that has both
particles and anti-particles, the wave function for the signal/idler mode is
that of a real (rather than a complex) scalar field and is explicitly time-
dependent. Other quantum amplification processes that can be described by
Bogoliubov transformations are the Unruh effect Unruh (1976) (particle
creation by an accelerated observer) and the dynamical Casimir effect Moore
(1970) (particle creation by oscillating mirrors), but only in the Unruh
effect and Hawking radiation are the ”signal” and ”idler” modes causally
disconnected. So, while the mathematics of mapping ”in” to ”out” states is
similar, the physics of the processes can be quite different.

Bekenstein and Meisels found that (just as Einstein had derived Einstein
(1917)) not only does the vaccum emit radiation spontaneously, it can also be
stimulated to do so. To see this effect in curved-space quantum field theory,
we need to consider $m$ particles in the initial state

$\displaystyle|\psi\rangle_{\rm
out}=e^{-iH}|m\rangle_{a}|0\rangle_{b}=e^{-iH}\frac{1}{\sqrt{m!}}(a_{k}^{\dagger})^{m}|0\rangle_{\rm
in}=\frac{1}{\sqrt{m!}}(A_{k}^{\dagger})^{m}e^{-iH}|0\rangle_{\rm in}\;.$
(III.16)

with $H$ from Eq. (III.5) and using the Bogoliubov transformation (III.1).
Carrying out this calculation for $m_{k}$ particles in a single mode $k$
incident on the black hole leads to an outgoing density matrix in region I
(outside the black hole horizon) Adami and Ver Steeg (2014)

$\displaystyle\rho_{\rm I}={\rm Tr}_{\rm II}|\psi\rangle_{\rm
out}\langle\psi|=\rho_{k|m}\otimes\rho_{-k|0}\;.$ (III.17)

Here, the density matrix $\rho_{k|m}$ of outgoing particles given that $m$
particles were incident in mode $k$ is (the density matrix of outgoing anti-
particles $\rho_{-k|0}$ looks just like the particle matrix with no incoming
particles, shown in (III.11))

$\displaystyle\rho_{k|m}=\frac{1}{(1+\beta_{k}^{2})^{m+1}}\sum_{n=0}^{\infty}\left(\frac{\beta_{k}^{2}}{1+\beta_{k}^{2}}\right)^{n}{{m+n}\choose{n}}|m+n\rangle\langle
m+n|\;,$ (III.18)

which can be rewritten as

$\displaystyle\rho_{k|m}=\sum_{n=0}^{\infty}p(n|m)|n\rangle\langle n|$
(III.19)

with the conditional probability (here $x=\frac{\omega_{k}}{T_{\rm BH}}$ as
before)

$\displaystyle p(n|m)=(1-e^{-x})^{m+1}{{m+n}\choose{n}}e^{-nx}\;.$ (III.20)

If there are no particles entering the black hole ($m=0$), Eq. (III.20)
reduces to $p(n)=(1-e^{-x})e^{-nx}$, which is the $\Gamma\to 1$ limit of
(II.2). However, it is easily checked that the detailed balance condition
(II.6) does not hold for the conditional probability (III.20). We will now see
that this is due to our setting $\Gamma=1$, which will turn out to be
inconsistent: if there is stimulated emission, then the black hole cannot be
fully absorbing: it must return something to the outside world.

In order to correctly describe scattering off of the black hole horizon in
curved-space quantum field theory, I will use a trick due to Sorkin Sorkin
(1987), which will ultimately allow us to recover Bekenstein and Meisels’
$p(n|m)$ that observes detailed balance. Sorkin’s insight comes from the
observation that when we discuss the capacity of a black hole to transmit
information, we are not really interested in the information that was encoded
in the particles that were present during the formation of the black hole (the
modes $a_{k}$ and $b_{-k}$). After all, whether a collapsing star is going to
form a black hole in the future is uncertain, and choosing the timing of
informational modes in such a way that they travel just outside of the black
hole would be rather difficult. Besides, we know that such modes will be
exponentially redshifted. Sorkin instead introduces late-time modes $c_{k}$
that at future infinity are exponentially blue-shifted555Note that in order to
keep with the previous definition of $a_{k}$ and $b_{-k}$ modes, I have
changed the nomenclature of Sorkin (1987). with respect to the early-time
modes $a_{k}$ and $b_{-k}$, and therefore commute with them. Sorkin’s late-
time mode along with the early-time modes are shown in Fig. 3.2.

Figure 3.2: Penrose diagram showing early-time modes ($a_{k}$ and $b_{-k}$)
and late-time modes $c_{k}$. Late-time modes are scattered at the horizon with
probability ${\cal R}$ (the black hole reflectivity). A perfectly absorbing
black hole has ${\cal R}=0$.

The Bogoliubov transformation that connects the outgoing mode $A_{k}$ to
$a_{k}$ and the late-time mode $c_{k}$ can now be written as Sorkin (1987):

$\displaystyle A_{k}$ $\displaystyle=$ $\displaystyle
e^{-iH}a_{k}e^{iH}=\alpha_{k}a_{k}-\beta_{k}b^{\dagger}_{-k}+\gamma_{k}c_{k}\;,$
(III.21)

with $\alpha_{k}^{2}-\beta_{k}^{2}+\gamma_{k}^{2}=1$ to ensure unitarity. What
is the Hamiltonian $H$ that gives rise to this transformation? It turns out
that it is given by the sum of the term we already had (which turned out to be
formally equivalent to the Hamiltonian of an active optical element) and the
Hamiltonian of a passive optical element: a beam splitter Leonhardt (2010):

$\displaystyle
H=\sum_{k=-\infty}^{\infty}ig_{k}(a_{k}^{\dagger}b_{-k}^{\dagger}-a_{k}b_{-k})+ig^{\prime}_{k}(a_{k}^{\dagger}c_{k}-a_{k}c_{k}^{\dagger})\;.\
\ \ \ \ \ $ (III.22)

The beam-splitter term in (III.22) describes the interaction of late-time
modes with the black hole horizon using an interaction strength
$g_{k}^{\prime}$, which we will be able to relate to the black hole’s
reflectivity ${\cal R}$.

The Bogoliubov coefficients $\alpha_{k}$, $\beta_{k}$, and $\gamma_{k}$ can be
written in terms of $g_{k}$ and $g^{\prime}_{k}$ by using (III.22) in
(III.21),

$\displaystyle\alpha_{k}$ $\displaystyle=$
$\displaystyle\cos(g^{\prime}_{k}w)\;,$ (III.23) $\displaystyle\beta_{k}$
$\displaystyle=$
$\displaystyle\frac{g_{k}}{g^{\prime}_{k}}\frac{\sin(g^{\prime}_{k}w)}{w}\;\;,$
(III.24) $\displaystyle\gamma_{k}$ $\displaystyle=$
$\displaystyle-\frac{\sin({g^{\prime}_{k}w})}{w}\;,$ (III.25)

where $w=\sqrt{1-(g_{k}/g^{\prime}_{k})^{2}}$.

We can now proceed as before and construct the out-state using this amended
Hamiltonian

$\displaystyle|0\rangle_{\rm
out}=e^{-iH}|0\rangle_{a}|0\rangle_{b}|0\rangle_{c}\;,$ (III.26)

now acting on a product state of early-and late-time modes (as we have assumed
these modes commute). We can calculate the density matrix of radiation outside
the black hole horizon by tracing the full density matrix over region II (the
inside of the black hole that contains both modes $b_{-k}$ and $c_{k}$)

$\displaystyle\rho_{\rm I}={\rm Tr}_{\rm II}|0\rangle_{\rm out}\langle 0|\;.$
(III.27)

Just as before, the anti-particle density matrix $\rho_{-k|0}$ factorizes, and
we find

$\displaystyle\rho_{k|0}=\frac{1}{1+\beta_{k}^{2}}\sum_{n_{k}=0}^{\infty}\left(\frac{\beta_{k}^{2}}{1+\beta_{k}^{2}}\right)^{n}|n_{k}\rangle\langle
n_{k}|\;.$ (III.28)

This expression is formally identical to expression (III.11) when written in
terms of $\beta_{k}^{2}$, except that due to the term $\gamma_{k}^{2}$ in the
unitarity relation $\alpha_{k}^{2}-\beta_{k}^{2}+\gamma_{k}^{2}=1$ we now have

$\displaystyle\beta_{k}^{2}=\frac{\Gamma}{e^{\omega_{k}/T}-1}\;,$ (III.29)

where $\Gamma$ is the absorptivity of the black hole (for this particular
mode), with $\Gamma=1-\gamma_{k}^{2}$. Thus, (III.28) is just the standard
density matrix of spontaneous emission (a.k.a. Hawking radiation) including
gray-body factors. It reduces to (III.11) in the unphysical limit
$\Gamma\rightarrow 1$.

Now that we have seen how we can use Sorkin’s trick along with a simple beam-
splitter term to recover the gray-body factor in Hawking’s radiation (note
that Hawking obtained this factor in a very different manner, by following
particles from inside the black hole backwards in time into region I), we can
study how the black hole reacts to particles that fall into the black hole at
late times, in mode $c_{k}$. Note that because these modes will not be
significantly red-shifted by the black hole, the corresponding Hawking
radiation will be fully commensurate with the size of the black hole and a
”transplanckian” problem is avoided.

I will now construct the outgoing state $|\psi\rangle_{\rm out}$ when $m_{k}$
late-time particles are directed at the horizon (all in the same mode $k$),

$\displaystyle|\psi\rangle_{\rm out}=e^{-iH}|m_{k}\rangle_{\rm in}\;,$
(III.30)

with $H$ from (III.22) and where $|m_{k}\rangle_{\rm
in}=|0\rangle_{a}|0\rangle_{b}|m_{k}\rangle_{c}$, i.e., the state with $m_{k}$
incoming particles in mode $c_{k}$ on ${\mathscr{I}}^{-}$. Using Eq. (III.21),
we can immediately calculate the number of particles emitted into mode $A_{k}$
if $m_{k}$ were incident in mode $c_{k}$, since

$\displaystyle{{}_{\rm in}}\langle\psi|A_{k}^{\dagger}A_{k}|\psi\rangle_{\rm
in}=\beta_{k}^{2}+m_{k}\gamma_{k}^{2}\;.\ \ \ \ \ \ $ (III.31)

Using $\gamma_{k}^{2}=1-\alpha_{k}^{2}+\beta_{k}^{2}$, we can see that we have
recovered Bekenstein and Meisels’ result (II.7), using (III.29) and
identifying $\gamma_{k}^{2}=1-\Gamma$. Furthermore, we can also recover

$\displaystyle\Gamma=\Gamma_{0}(1-e^{-\omega_{k}/T_{\rm BH}})\;,$ (III.32)

by noting that $\alpha_{k}^{2}=\Gamma_{0}$ actually sets the reflectivity of
the black hole, that is, ${\cal R}$ in Fig. 3.2 is just $1-\alpha_{k}^{2}$. We
can also see that on account of Eqs. (III.23-III.25), we have

$\displaystyle\biggl{(}\frac{g_{k}^{\prime}}{g_{k}}\biggr{)}^{2}=\frac{\gamma_{k}^{2}}{\beta_{k}^{2}}=1+\frac{1-\alpha_{k}^{2}}{\alpha_{k}^{2}}e^{\omega_{k}/T_{\rm
BH}}\;,$ (III.33)

which implies that because the reflectivity ${\cal
R}=1-\alpha_{k}^{2}=\Gamma_{0}\leq 1$, the parameter $g_{k}^{\prime}$ that
sets the scattering rate for modes of energy $\omega_{k}$ is bounded from
below by $g_{k}$, which is itself bounded from above by 1 since $g_{k}\sim
e^{-\omega_{k}/T_{\rm BH}}$.

This explains why setting $\Gamma=1$ could never be consistent. As previously
discovered by Bekenstein and Meisels using maximum entropy arguments only
Bekenstein and Meisels (1977), the total absorptivity of the black hole
$\Gamma$ is the product of the ”bare” absorptivity $\Gamma_{0}$ times the
factor $1-e^{-\omega_{k}/T_{\rm BH}}$ due to stimulated emission, so that
$\Gamma<1$ always. A ”classical” black hole has $\Gamma_{0}=1$ (for incoming
$s$-waves) but a quantum black hole can never be totally black, and as a
consequence information is never lost in black hole dynamics. To make this
statement quantitative, we should proceed to calculate the capacity of quantum
black holes to transmit classical information, using the tools of quantum
information theory.

##  IV Classical Information Capacity of Quantum Black Holes

The primary application of the classical theory of information due to Shannon
Shannon (1948) was to quantify the capacity of channels to transmit
information. One of the surprising results of that theory is that it is
possible to send information through a channel with perfect accuracy even when
there is substantial noise that affects the transmitted message. For example,
we might imagine the problem of communicating by throwing copies of two
different books into a fire repeatedly: books that have exactly the same
number of words, and weigh the same666I choose books that differ only in their
words but not in their mass to connect with a paradox where, throwing these
books into a black hole, the identity of the books would be lost because the
increase in the mass of the black hole is the same for either book.. The two
different books allow us to encode information into a sequence of zeros and
ones (denoting which of the books we choose to incinerate). But how could an
individual that can only observe the flames extract the information encoded in
the series of books? The answer is that the information is not lost in the
flames: the two different books when burning give rise to slightly different
ways in which the flames and smoke behave when turning the pages to ashes (due
to the different ways in which words are arranged on the pages). While we
certainly do not have the technology to detect these differences among the
much more pronounced variation they are embedded in, this information is in
principle accessible. And as a consequence, error correction techniques will
allow us to retrieve this information with perfect accuracy. One way to do
this is to coat the pages of the books with different pyrotechnic colorants
for each, making information retrieval trivial.

Classical black holes, however, are very different from fire. If no marker
carrying the information is available to the observer (as it is in the case of
communication through fire), no amount of error correction can recover the
information. Because Schwarzschild black holes (classical, non-spinning,
neutral black holes) are only characterized by their mass, absorbing two books
with different writing but equal mass would give rise to the same exact final
state. Such dynamics would dictate that two separate phase-space trajectories
merge into one, which is a direct violation of microscopic reversibility: an
abomination.

In 1973, Bekenstein painted the picture of information loss that is still
being discussed today Bekenstein (1973):

> “We imagine a particle goes down a (…) black hole. As it disappears some
> information is lost with it”.

In fact, Bekenstein estimated that the information loss must be at least one
bit, which is the amount of uncertainty created by not knowing whether the
particle still existed behind the horizon or not. It is now clear that this
line of thinking is fundamentally rooted in a misunderstanding of the concept
of information in physics, namely, that information is necessarily attached to
the object that encodes it. Information, rather, is a relative state between
an observer and a system Adami (2016), and more importantly, is not tied to
the encoding body. In the case of the fiery communication described earlier,
the information was first encoded in ones and zeros, then translated to two
kinds of books, and then retrieved into a list of ones and zeros after the
color of the flames was decoded. In communication through black holes,
information is first encoded in particles (for example, the polarization of
photons, or particle/anti-particle identity). The particles are absorbed at
the event horizon, which stimulates the emission of exact copies of those
particles outside (and inside) the horizon (I will discuss how such a process
complies with the no-cloning theorem in section IV.2). The process of
stimulated emission copies the information from the absorbed particle and
transfers it to other carriers outside the black hole, where they are
accessible to an observer so that separate phase-space trajectories remain
separate.

###  IV.1 Classical late-time capacity

We can follow the fate of information interacting with a black hole by using a
preparer to encode information into late-time quantum states that are then
sent into the event horizon. For example, we can imagine a preparer $X$ who
sends packets of $n$ particles with probability $p(n)$. The internal state of
the preparer can be described by the density matrix
$\rho_{X}=\sum_{n}p(n)|n\rangle\langle n|$, with entropy
$S(\rho_{X})=H[p]=-\sum_{n}p(n)\log p(n)$. After the particles interact with
the black hole, our preparer is now correlated with it because the final state
is now the density matrix

$\displaystyle\rho_{\,{\rm
I,II},X}=\sum_{n}p(n)|\psi_{n}\rangle\langle\psi_{n}|\otimes|n\rangle_{X}\langle
n|\;,$ (IV.1)

where $|\psi\rangle_{n}=e^{iH}|0,0,n\rangle_{abc}$. Tracing over the black
hole interior (region II), we obtain the joint density matrix of the radiation
field in region I and the preparer $X$:

$\displaystyle\rho_{\,{\rm I},X}$ $\displaystyle=$
$\displaystyle\sum_{n}p(n)\rho_{k|n}\otimes|n\rangle_{X}\langle n|$ (IV.2)

with entropy

$\displaystyle S(\rho_{\,{\rm I},X})=H[p]+\sum_{n}p(n)S(\rho_{k|n})$ (IV.3)

owing to the block-diagonal form of Eq. (IV.2). The mutual entropy between
radiation field and preparer is then simply given by

$\displaystyle H(X;{\rm I})$ $\displaystyle=$ $\displaystyle S(\rho_{\,\rm
I})+S(\rho_{X})-S(\rho_{\,{\rm I},X})$ (IV.4) $\displaystyle=$ $\displaystyle
S\left(\sum_{n}p(n)\rho_{k|n}\right)-\sum_{n}p(n)S(\rho_{k|n})\;,\ \ $

which turns out to be the Holevo bound Holevo (1973). The latter constitutes
the maximum amount of classical information that can be extracted from a
quantum measurement, and its maximum (over the probability distribution of
signal states) is the capacity of a quantum channel to transmit classical
information Holevo (1998). In other words, the black hole channel capacity is
the Holevo capacity, with Hawking radiation playing the role of channel noise.

As a simple example, consider a binary channel where the preparer either sends
no particle (with probability $1-p$) or one particle (with probability $p$)
into the black hole. As Hawking radiation does not depend on this decision,
the information would be lost if stimulated and scattered particles were not
present in the total radiation field outside the horizon. In order for
information to be recovered, an outside observer must be able to make
measurements on the radiation field that betray the preparer’s decision. If we
send in $n$ particles in mode $k$ (to signal a logical ‘1’) then we obtain
$\rho_{k|n}$ in region I, which is diagonal in the number basis (its general
expression is given in the Appendix). For a single incident particle,

$\displaystyle\rho_{k|1}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{k}^{2}}{(1+\beta_{k}^{2})^{2}}\sum_{n_{k}=0}^{\infty}\left(\frac{\beta_{k}^{2}}{1+\beta_{k}^{2}}\right)^{n_{k}}\\!\\!\\!(1+n_{k}\frac{\gamma_{k}^{2}}{\alpha_{k}^{2}\beta_{k}^{2}})|n_{k}\rangle\langle
n_{k}|\;.$ (IV.5)

This density matrix is clearly non-thermal except in the unphysical limit
$\gamma_{k}=0$, where the probability of absorbing a single quantum would
exceed 1.

Let us calculate the capacity $\chi=\max_{p}H(X;{\rm I})$ for the worst-case
scenario: a perfectly black hole (no reflection, i.e., $\alpha_{k}^{2}=1$).
This is the worst-case scenario because any radiation reflected at the horizon
would allow us to recover the information sent in even if the vast majority of
particles disappear behind the horizon. That observation illustrates the true
magic of Shannon’s theorem: information is only lost if absolutely no trace of
the information is left for us to decode.

For the non-reflecting black hole, the mutual entropy $H(X;{\rm I})$ is
maximized at $p=1/2$, and the capacity can be written in terms of the
parameter $z=e^{-\omega_{k}/T_{\rm BH}}$ as Adami and Ver Steeg (2014)

$\displaystyle\chi=1-\frac{1}{2(1+z)^{3}}\sum_{m=0}^{\infty}\biggl{(}\frac{z}{1+z}\biggr{)}^{m}(m+1)(m-2z)\log(m+1)\;.$
(IV.6)

The capacity $\chi$ is positive for all values $0\leq z\leq 1$, which implies
that the channel’s capacity never vanishes, and information can always be
recovered with perfect accuracy (see Fig. 4.1).

Figure 4.1: Capacity $\chi$ for a binary non-reflecting black hole channel as
a function of the parameter $z=e^{-\omega_{k}/T_{b}h}$. $z=0$ corresponds to a
black-hole with vanishing mass, while $z\to 1$ as the mass of the black hole
tends to infinity. The solid line represents the late-time capacity (IV.6).
The dashed line is the capacity for early-time modes, which is obtained from
the late-time capacity by rescaling $z\to\frac{z}{z-1}$ Adami and Ver Steeg
(2014). Note that since $g_{k}\approx z$, we can see this plot also as
depicting the dependence of the information transmission capacity on the mode
coupling strength in the black-hole Hamiltonian (III.5).

As the black hole shrinks (and the Hawking temperature increases), the
capacity to transmit information (using particles with a given energy
$\omega_{k}$) decreases. But a decreased capacity does not indicate
information loss. Instead, it tells us what the maximum rate of perfectly
accurate information transmission is. Thus, a capacity of 0.8 bits, for
example, indicates that if we are sending in information at 1 bit per second,
we can only retrieve perfectly accurate information at the rate of at most 0.8
bits per second, on account of the error correction necessary to protect the
information. Non-optimal schemes of error correction will yield smaller rates
of error-free information transmission.

Incidentally, using the conditional probability for perfect absorption of
early-time modes in Eq. (III.20) still leads to a non-vanishing capacity,
though significantly reduced (see Fig. 4.1). It turns out that this early-time
capacity was previously (and independently) derived for the Unruh channel (the
case of accelerated observers, which is another case of quantum
amplification). When replacing the black hole interaction strength $g_{k}$
with the acceleration $r$ (and keeping $g_{k}^{\prime}=g_{k}$ to ensure zero
reflectivity), the black hole and the Unruh capacities are identical Brádler
(2011). In both cases, using the probability distribution $p(n|0)$ for
spontaneous emission (pure Hawking radiation) yields a vanishing capacity.
This quantifies what has been conjectured many times before: Hawking radiation
carries no signal, no information. It is only noise.

The capacity of the binary channel using an encoding of $n$ particles as the
logical ‘1’, and either 0 particles or $n$ anti-particles as the logical ’0’
exceeds the one shown in Fig. 4.1 (as the adventurous reader can confirm using
the expressions in the Appendix) because these methods of encoding are
significantly more robust to noise, that is, to the interference of the
stimulated emission signal with the spontaneously emitted Hawking radiation.
We will use such an encoding in the following section when discussing in which
way a black hole can be understood in terms of quantum cloning machines.

###  IV.2 Black holes are quantum cloning machines

I wrote earlier that the stimulated emission process essentially ”clones” the
incoming particles so that these copies are available outside of the event
horizon and information is not lost. Perfect cloning is, of course, forbidden
in quantum mechanics: the ”quantum no-cloning theorem” is a direct consequence
of the linearity of quantum mechanics (see, for example, Dieks (1982);
Wootters and Zurek (1982)). The proof of this theorem is simple, and worth
repeating here. Suppose we define a cloning operator $U_{C}$ in quantum
physics so that it will copy arbitrary quantum states $|\phi\rangle$

$\displaystyle U_{C}|\phi\rangle|0\rangle=|\phi\rangle|\phi\rangle\;$ (IV.7)

onto a prepared ancilla state $|0\rangle$. After the copying operation, the
original quantum state is in a product state with its copy, as desired. The
action of this operator on a quantum superposition
$\sigma|\phi\rangle+\tau|\psi\rangle$ (with complex $\sigma$ and $\tau$ that
satisfy $|\sigma|^{2}+|\tau|^{2}=1$) does not produce a product state of that
superposition, however, as

$\displaystyle
U_{C}(\sigma|\phi\rangle+\tau|\psi\rangle)|0\rangle=\sigma|\phi\rangle|\phi\rangle\
+\tau|\psi\rangle|\psi\rangle\neq(\sigma|\phi\rangle+\tau|\psi\rangle)(\sigma|\phi\rangle+\tau|\psi\rangle)$
(IV.8)

instead. In fact, this hypothetical ”cloning operator” $U_{C}$ in (IV.7) turns
out to be an ”entanglement operator”, and is typically (in the space of
qubits) the ”controlled NOT” (CNOT) operator
$U_{C}=P_{0}\otimes\mathbbm{1}+P_{1}\otimes\sigma_{x}$, where $P_{0}$ and
$P_{1}$ project on the respective states, and the Pauli matrix $\sigma_{x}$
flips a bit. In the following I discuss the cloning of binary states (quantum
bits, or qubits), but the formalism can be extended to quantum states of
arbitrary dimension Cerf and Fiurasek (2006).

While perfect cloning of arbitrary states is not possible, it is of course
possible to clone ”known” (that is, prepared) states, with the help of an
operator as described above that uses projectors onto those known basis
states. It is, however, also possible to make ”approximate” copies of
arbitrary quantum states, using unitary operators that are referred to as
”quantum cloning machines”. A quantum cloning machine is designed to maximize
the probability that a quantum state $|\psi\rangle$ (or, in general, $N$
identically prepared states $|\psi\rangle^{\otimes N}$) is cloned into a state
$|\psi_{\rm out}\rangle=U|\psi_{\rm in}\rangle|0\rangle|R\rangle$ (or $M$
copies of that state, with $M>N$) with a unitary transformation $U$ acting on
the input state $|\psi_{\rm in}\rangle$, a ”blank” state $|0\rangle$ that will
ultimately hold those multiple copies of the cloned state, and an ancillar
state $|R\rangle$. The formalism of quantum cloning machines was introduced by
Buzek and Hillery Buzek and Hillery (1996), and has given rise to a large body
of work (see, e.g., the reviews Cerf and Fiurasek (2006); Scarani  _et al._
(2005)).

Figure 4.2: Schematics of an $N\to M$ quantum cloning machine ($M>N$). The
machine receives as input the product of $N$ identical copies of the quantum
state, supplemented by an internal state $R$ and $M-N$ states in a prepared
blank state $|0\rangle$. The $M$ approximate clones are mixed states
$|\tilde{\psi}\rangle\langle\tilde{\psi}|$, while the $M-N$ anti-clones are
the output of an approximate logical NOT transformation of the clones,
indicated by $\overline{|\tilde{\psi}\rangle\langle\tilde{\psi}|}$.

For the general case of an $N\to M$ cloning machine (see Fig. 4.2), the
accuracy of the cloning process is determined by a fidelity measure, defined
as the expectation value of one copy of the $M$ cloned states $\rho_{\rm
out}^{j}$ evaluated in the basis of the input state

$\displaystyle F_{j}=\mbox{}_{\rm in}\langle\psi|\rho^{j}_{\rm
out}|\psi\rangle_{\rm in}\;.$ (IV.9)

The largest possible $N\to M$ cloning fidelity of universal quantum cloning
machines (devices that copy any input state $|\psi\rangle$ with the same
fidelity) is Gisin and Massar (1997); Bruß  _et al._ (1998)

$\displaystyle F_{\rm opt}=\frac{M(N+1)+N}{M(N+2)}\;.$ (IV.10)

For example, the optimal (and therefore maximal) fidelity of a $1\to 2$
approximate cloning machine is $F_{1\to 2}=5/6$. In the limit $M\to\infty$,
the cloning fidelity approaches $\frac{N+1}{N+2}$, which happens to be the
fidelity of the best possible state preparation via state estimation (using
classical information only) from a finite quantum ensemble Massar and Popescu
(1995). Quantum cloning machines that reach the cloning limit are called
”optimal cloners”, and can be constructed using simple quantum optical
elements (parametric amplifiers and beam splitters) for discrete states such
as polarization Simon  _et al._ (2000), but also for continuous variables
Fiurášek (2001); Braunstein  _et al._ (2001).

Because black holes stimulate the emission of copies of accreting particles,
we can seek to apply the formalism of quantum cloning machines to black holes
to answer the question: How well do black holes clone quantum states? We will
see that the answer is ”almost perfectly”.

Since the cloning transformation that takes $|\psi\rangle_{\rm in}$ into
$|\psi\rangle_{\rm out}$ is a unitary transformation, our candidate for the
cloning operator is $U=e^{-iH}$, which transforms late-time incoming particles
into outgoing radiation with Eq. (III.22) as the Hamiltonian. But before we do
this, let us first study a much simpler case: the fully absorbing ($\Gamma=1$)
black hole with only early-time particles, described by Eq. (III.5).
Understanding this case will set important limits to allow us to better
understand the general cloner based on (III.22).

Let us apply $U=e^{-iH}$ to an arbitrary incoming quantum state $|\psi_{\rm
in}\rangle$ (a state that includes the blank and ancillar states), where
$|\psi\rangle$ encodes information in the “particle-antiparticle” basis using
a so-called ”dual rail” encoding. In a dual rail encoding, the logical one
$|1\rangle_{L}$ is encoded with $N$ particles and zero anti-particles in mode
$a$ (in region I outside the horizon), while for the logical zero particles
and anti-particles are interchanged, i.e.,

$\displaystyle|1\rangle_{L}$ $\displaystyle=$
$\displaystyle|N,0\rangle_{a}|0,0\rangle_{b}$ (IV.11)
$\displaystyle|0\rangle_{L}$ $\displaystyle=$
$\displaystyle|0,N\rangle_{a}|0,0\rangle_{b}\;.$ (IV.12)

Naturally, sending information to future infinity using particles or anti-
particles that are just outside of the black hole horizon after the formation
of the black hole is not feasible as we remarked above before introducing
late-time modes: this is purely an exercise to set some limits. We will even
consider the limit where information is encoded in modes just inside the black
hole horizon (the $b$ modes) for the same reason.

We can construct an arbitrary quantum state from the logical states
(IV.11-IV.12) by writing

$\displaystyle|\psi\rangle_{\rm in}=\sigma|1\rangle_{L}+\tau|0\rangle_{L}\;.$
(IV.13)

However, because the quantum cloner defined by the unitary mapping $U=e^{-iH}$
using Hamiltonian (III.5) is ”rotationally invariant” (the action of the
cloning machine does not depend on the state), we can simply choose $\sigma=1$
and $\tau=0$, that is, we will attempt to clone $N$ particles.

Writing $|\psi_{\rm
in}\rangle=|N,0\rangle_{a}|0,0\rangle_{b}=|1\rangle_{L}^{\otimes N}$ we obtain
using (III.5)

$\displaystyle|\psi_{\rm out}\rangle=e^{-iH}|\psi_{\rm
in}\rangle=\frac{1}{\alpha_{k}^{2+N}}\sum_{jj^{\prime}}^{\infty}e^{-(j+j^{\prime})\frac{\omega}{2T}}\sqrt{\binom{j+N}{N}}|j+N,j^{\prime}\rangle_{a}|j^{\prime},j\rangle_{b}\;.$
(IV.14)

In order to describe an $N\to M$ cloning machine and calculate its fidelity,
we need to fix the number of particles and antiparticles in region I to $M$.
In quantum optics, this ”postselection” to a fixed number of clones is
achieved by entangling the quantum state with a trigger signal and then
measuring the trigger. If $M$ particles are detected in the trigger, then we
know that only the component of (IV.14) with $M$ clones survives. For black
holes, we can perform the same operation, but must send the trigger towards
future infinity (but away from the black hole horizon). Even though the
quantum state reconstruction is conditioned on the trigger, it only uses the
quantum states coming from the black hole.

Fixing the number of output clones to $M$ reduces the state (IV.14) to

$\displaystyle|\psi\rangle_{M}\sim\sum_{j=0}^{M-N}\sqrt{\binom{M-j}{N}}|M-j,j\rangle_{a}|j,M-N-j\rangle_{b}\;,\
\ $ (IV.15)

which, it turns out, is (up to normalization) identical to the wavefunction
emanating from the optical quantum cloner of Simon et al. Simon  _et al._
(2000) that achieves the optimal fidelity (IV.10). Thus, for quantum states
sent into a black hole at early times so that they remain just outside the
horizon, the black hole behaves as a universal, optimal, quantum cloning
machine.

Let us now study what happens to quantum states behind the horizon (modes
$b$). Just as in the optical realization of the optimal unversal cloner, the
quantum state behind the horizon contains $M-N$ anticlones of the initial
state $|N,0\rangle_{a}$, that is, ”inverted” states obtained from the initial
states by the application of the optimal universal NOT gate Buzek  _et al._
(1999); Gisin and Popescu (1999). The fidelity of these anticlones is

$\displaystyle F_{\rm anti}=\frac{N+1}{N+2}$ (IV.16)

for each anticlone, which as we already saw is the fidelity of the best
possible state preparation via state estimation Massar and Popescu (1995).
This implies that the black hole has induced the maximal disturbance on the
states behind the horizon.

Consider now instead the fidelity of cloning when $N$ anti-particles are sent
into mode $b$. We are now following anti-particles that are traveling just
inside the horizon: $|\psi\rangle_{\rm in}=|0,0\rangle_{a}|0,N\rangle_{b}$.
Thinking of these modes as impinging on the black hole horizon from inside the
black hole (a horizon that looks to those modes as a mirror, that is, a white
hole with $\Gamma=0$), we expect those modes to stimulate clones of these
anti-particles inside the black hole, but also anticlones on the outside.

Let us calculate the fidelity of those anticlones generated on the outside
region. The wavefunction is

$\displaystyle|\psi\rangle_{M}\sim\sum_{j=0}^{M}\sqrt{\binom{j+N}{N}}|j,M-j\rangle_{a}|M-j,j+N\rangle_{b}\;,$
(IV.17)

which implies the probability to observe $j$ particles and $M-j$ antiparticles
outside the horizon

$\displaystyle
p(j,M-j|N)=\frac{\binom{j+N}{N}}{\sum_{j=0}^{M}\binom{j+N}{N}}\;.$ (IV.18)

As can easily be checked using (IV.9), the fidelity of these anticlones is
also $N+1/N+2$, that is, it is equal to the fidelity of the anticlones when
sending in $|N,0\rangle_{a}$. But the anticlones of the antiparticle states
are, of course, clones of the particle states!

In summary, sending in $N$ particles in mode $a$ creates particle clones in
region I with optimal fidelity. These clones could in principle be entangled
with particles that the $N$ incoming particles were entangled with before they
were sent into the black hole, very much like in the quantum optical case,
where the initial particles are entangled with a trigger Simon  _et al._
(2000) (we will exploit this later when calculating the quantum capacity of
black holes in section V). Sending in $N$ antiparticles in mode $b$ towards
the horizon (which, to those modes, looks like a perfectly reflecting mirror)
gives rise to classical particle clones in region I instead. They are
classical because they cannot share entanglement with any particles that the
$N$ antiparticles inside the black hole could have been entangled with777The
classical roots of the probability $\frac{N+1}{N+2}$ are also apparent by
noting that this is Laplace’s rule of succession: the likelihood that an event
takes place given that we have $N$ successive observations of it.. We note
that those anti-particles traveling towards a white hole horizon can be seen
as particles moving away from the horizon (backward in time) after absorption
on a non-reflecting horizon. Thus, it is plausible that white holes are just
time-reversed black holes. Box 1 makes this analogy more clear, and shows how
stimulated emission saves microscopic time-reversal invariance.

Box 1: Microscopic Time-Reversal Invariance in the Presence of Black Holes

Newtonian gravity is invariant under time-reversal at the micro-level: every
particle trajectory in a gravitational field, when time-reversed, gives rise
to another plausible particle trajectory. This is illustrated schematically in
Fig. 4.2(a) (left panel), where for the sake of simplicity the effect of the
gravitational field on a particle is shown as if a perfect mirror reflected
the particle. Under time reversal, the trajectory simply reverses (middle
panel). If we use charge-conjugation symmetry, we see that this trajectory is
equivalent to that of an anti-particle moving backwards in time (anti-
particles are depicted as dashed lines). Comparing the first and the last
panel, we note that time reversal is equivalent to CP symmetry, thus embodying
the celebrated CPT theorem.

Figure 4.2: Schematic representation of particle and anti-particle
trajectories in the presence of white- and black hole horizons. Stimulated
particles/anti-particles are rendered in red, solid lined denote particle
trajectories, while dashed lines illustrate anti-particles. (a) Classical
trajectory reflected at a mirror (or falling back under the influence of
gravity), its time-reversed trajectory, and the equivalent trajectory in which
particles moving forward in time are replaced by anti-particles moving
backwards in time. (b) Classical trajectory of a particle absorved by a black
hole horizon, its time-reversed trajectory, and the equivalent process. (c):
Quantum trajectories including stimulated emission effects.

However, classical black holes violate this theorem. If instead of a perfect
mirror the particle encounters a perfectly absorbing black hole horizon (Fig.
4.2(b), left panel) reversing the arrow of time (middle panel) does not
produce a time-reversed picture of the left panel, since a particle from
inside of the black hole cannot escape to the outside. Instead, it is
”reflected” at the horizon, that is, from inside the black hole must act like
the perfect mirror in Fig. 4.2(a). Replacing particles by anti-particles
moving backwards in time produces the right panel, which clearly is not the
anti-particle version of the left panel, thus breaking CPT invariance. Adding
Hawking radiation (spontaneous emission of particles) to this picture does not
restore CPT invariance. However, the stimulated emission of radiation does.
Fig. 4.2(c) shows the stimulated particle/anti-particle pair in red (for
illustrative purposes I disregard factors of $\beta^{2}$ here) that must
accompany the absorption process (the case $\Gamma=1$ is shown here). Time-
reversing this trajectory produces the middle panel, where the particle from
inside of the black hole indeed reflects at the horizon, but it also
stimulates a particle/anti-particle pair outside of the horizon (shown in
red). In fact, this is the white-hole stimulated emission process described in
the main text ($\Gamma=0$). Re-interpreting particles moving backwards in time
as anti-particles moving forwards in time produces the right panel in Fig.
4.2(c), which indeed is the same process as in the left panel, only with
particles replaced with anti-particles. Thus, stimulated emission of pairs
restores CPT invariance. The spontaneous emission of pairs (Hawking radiation)
is not shown in these diagrams as it has no influence on CPT invariance: it
only serves to safeguard the no-cloning theorem.

Let us now analyze the more realistic cloning scenario where we send quantum
states into the already-formed black hole at late times, using modes $c$ (see
Fig. 3.2) that are strongly blue-shifted with respect to the early-time modes
$a$ and $b$. Because we also use the beam-splitter term with strength
$g_{k}^{\prime}\geq g_{k}$, we are studying ”gray holes” with arbitrary
reflectivity.

We first discuss $1\rightarrow M$ cloning. Because the full Hamiltonian
(III.22) is also rotationally invariant, we can again restrict ourselves to
study cloning of one particular state. To clone the state
$|1\rangle_{L}=|0,0\rangle_{a}|0,0\rangle_{b}|1,0\rangle_{c}$, for example, we
obtain (I omit the subscript $k$ in the particle numbers and coefficients in
the following, as we send in modes of only one particular frequency):

$\displaystyle\rho_{a}={\rm Tr}_{bc}\left(U|1\rangle_{L}\langle
1|U^{\dagger}\right)=\rho_{k|1}\otimes\rho_{-k|0}\;,$ (IV.19)

where $\rho_{k|1}$ is given by Eq. (IV.5) derived earlier and $\rho_{k|0}$
given by the standard result (III.28).

With (III.28) and (IV.5), the $1\rightarrow M$ cloning fidelity can be
calculated as before (with $\xi=\frac{\gamma^{2}}{\alpha^{2}\beta^{2}}$)

$\displaystyle F_{1\rightarrow
M}=\frac{\sum_{j=0}^{M}\frac{M-j}{M}p(M-j|1)p(j|0)}{\sum_{j=0}^{M}p(M-j|1)p(j|0)}=\frac{3+\xi+2\xi
M}{3(2+\xi M)}\;.\ \ $ (IV.20)

Let us investigate this result in a number of physical limits. As the black
hole becomes more and more reflective, $\Gamma_{0}=\alpha^{2}\rightarrow 0$,
which implies $\xi=\frac{e^{\omega/T_{\rm
BH}}(1-\Gamma)}{\Gamma_{0}^{2}}\rightarrow\infty$. In this case, the fidelity
(IV.20) approaches the optimal value

$\displaystyle\lim_{\Gamma_{0}\rightarrow 0}F_{1\rightarrow
M}=\frac{2}{3}+\frac{1}{3M}\;,$ (IV.21)

as is seen in Fig. 4.3. For arbitrary $N$, the limit is exactly equal to the
Gisin-Massar optimal fidelity (IV.10) of an $N\to M$ cloning machine. We can
recognize this result as the special case we treated earlier: If the black
hole perfectly reflects incoming states, the black hole behaves just as if
early-time modes ($a$-modes) were traveling just outside the horizon (except
for the redshift). Because those modes (by definition) never enter the black
hole, this is akin to a perfectly reflecting black hole.

Figure 4.3: Cloning fidelity $F_{1\to M}$ of the quantum black hole as a
function of the number of copies $M$, for different values of the quantum
absorption probability $\Gamma_{0}$ and a fixed $z=e^{-\omega/T_{\rm
BH}}=0.01$. Adapted from Adami and Ver Steeg (2015).

Another limit of note is that of full absorption: $\Gamma_{0}\rightarrow 1$.
In that case $\xi\to 1$ and $F_{1\to M}\to 2/3$, the fidelity of a classical
cloning machine. It can be shown in general that for full absorption, the
$N\rightarrow M$ cloning fidelity is equal to $N+1/N+2$ independently of
$\omega/T_{\rm BH}$, which is the result we obtained earlier when sending $N$
antiparticles in mode $b$ directly behind the horizon. This is again not
surprising, as the absorption of $c$-modes stimulates the emission of $b$
anti-modes behind the horizon, who in turn give rise to anticlones of the
antiparticles, that is, clones. But they must be ”classical” clones, so the
fidelity is that of state estimation by classical means.

While in the limit $\Gamma_{0}\rightarrow 1$ the best we can do to reconstruct
the quantum state is to make classical measurements that allow us to optimally
estimate the quantum state, note that in the limit $N\to\infty$ the
probability to do this correctly tends to one, implying that the quantum state
information can be reconstructed with arbitrary accuracy. In a sense, this
result mirrors a result from calculating the classical capacity of the black
hole channel, where you can show Adami and Ver Steeg (2014) that in the limit
$N\to\infty$ the capacity of the quantum black hole channel to transmit
classical information becomes equal to the noiseless channel capacity, even
for full absorption.

Figure 4.4: Cloning fidelity $F_{1\to M}$ as a function of the number of
copies $M$, for different $z=e^{-\omega/T_{\rm BH}}$ and a fixed quantum
absorption probability $\Gamma_{0}=0.95$. Adapted from Adami and Ver Steeg
(2015).

In the previous discussion we had kept the ratio between the mode energy
$\omega$ and the mass of the black hole constant (by keeping
$z=e^{-\omega/T_{\rm BH}}$ constant). In the limit of large black holes (where
the Hawking temperature approaches zero), the limit $\omega/T_{\rm
BH}\to\infty$ implies $\xi\to\infty$ (as long as $\Gamma_{0}<1$), and we again
recover the optimal universal quantum cloning fidelity (IV.21), as seen in
Fig. 4.4. Given that even modest-sized black holes have $\omega/T_{\rm BH}\geq
10$, most black holes are therefore nearly-optimal universal quantum cloners,
unless the absorption probability is exactly equal to 1. These results also
hold for $N\to M$ cloning machines. Just as in the case $N=1$, the black hole
cloner approaches the optimal cloner in the limit $T_{\rm BH}\to 0$ or
$\Gamma_{0}\to 0$, and turns into a classical cloning machine in the limit
$\Gamma_{0}\to 1$ and for $M\to\infty$.

Now that we have seen in which way the black hole is a nearly optimal quantum
cloning machine, we can turn our attention to the quantum channel aspects of
the black hole. In the previous section we focused our attention on how
classical information fares when sent into the black hole horizon. But in our
discussion of quantum cloning, we clearly had the opportunity to study how
entangled quantum states are affected by black holes. For example, in order to
make sure that we measure exactly $M$ copies of the initial quantum state at
future infinity, it was necessary to entangle a late-time particle with
another whose state we would measure. It turns out that the quantum channel
defined by the mapping (IV.14) is an example of so-called ”cloning channels”
Brádler (2011). More precisely, the black hole acts a weighted ensemble of
cloning machines. In the next section, we will study the fate of quantum
information (more precisely, quantum entanglement) that interacts with the
black hole horizon. We should be mindful that there is no law of physics that
prescribes that quantum entanglement must be preserved when interacting with a
black hole. We will find that for the most part it is, but if the black hole
is perfectly absorbing, it is not.

##  V Quantum Information Capacity

Quantum information is a relatively new concept within the canon of
theoretical physics. While classical information (defined in 1948) was a
concept that was available to workers in the field of quantum gravity who
worried about information conservation, quantum information theory rose to
prominence in the mid 1990s spurred on by Peter Shor’s discovery Shor (1994)
that a quantum algorithm can factor numbers faster than any classical
algorithm. One of the central results of classical information theory is the
calculation of the capacity of various channels to transmit information, and
the results in the previous section borrow heavily from that theory.
Understanding the transmission of quantum information through a quantum
channel requires an entirely different formalism, however, mainly because
quantum information is something altogether different from classical
information. As a consequence, we will see that for most channels it is not
even possible to write down a closed expression for the capacity.

Classical information characterizes the relative state of two systems,
specifically where one system can make predictions about the state of another.
Quantum information is entanglement, that is, it is given by a quantum state
relative to another. However, knowing this relative state does not make it
possible for one system to make predictions about the other because entangled
states are not separable: when two states are entangled they become one. For
example, recall Eq. (III.9): the quantum state at future infinity (for the
simplified situation without a beam splitter that would give rise to gray-body
factors) with no initial particles present at past infinity,

$\displaystyle|0\rangle_{\rm
out}=e^{-iH}|0,0\rangle_{a}|0,0\rangle_{b}=\prod_{k=-\infty}^{\infty}\frac{1}{\cosh^{2}g_{k}}\sum_{n_{k},n_{k}^{\prime}}e^{-(n_{k}+n^{\prime}_{-k})\omega_{k}/2T_{\rm
BH}}|n_{k},n^{\prime}_{-k}\rangle_{a}|n_{k}^{\prime},n_{-k}\rangle_{b}\;.$
(V.1)

This is a highly entangled state with both particles and anti-particles
present behind and in front of the horizon, because the Bogoliubov
transformation is, at heart, an entangling operation. Now let us study what
happens to entangled states interacting with black holes.

When we say that we want to ”send quantum entanglement through a quantum
channel”, what we mean is that the entanglement that one party has with a
quantum system is transferred to another party. Say the two parties are called
”Alice” and ”Bob”, and Alice is entangled with a reference system called ”R”.
To simplify things further, suppose Alice’s state is a qubit. In that case we
can without loss of generality write the entangled state between Alice’s qubit
and R as

$\displaystyle|\psi\rangle_{\rm
in}=\sigma|0\rangle_{A}|0\rangle_{R}+\tau|1\rangle_{A}|1\rangle_{R}\;,$ (V.2)

with complex coefficients $\sigma$ and $\tau$ that satisfy the condition
$|\sigma|^{2}+|\tau|^{2}=1$. Now imagine that Bob is an inertial observer at
future infinity, and Alice (for the present discussion) delivers her qubit
into the forming black hole at past infinity. Is it possible for Bob to be
entangled with R in the same way that Alice was? We will see in a moment that
for the situation described here (namely a channel descibed by the unitary
$U=e^{-iH}$ with the Hamiltonian given by Eq. (III.5)), the answer is
definitely ”No”: the quantum capacity of this channel vanishes. This, however,
does not signal a breakdown of any law of physics: there are plenty of quantum
channels with vanishing capacity. But understanding how black holes affect
quantum entanglement is still an interesting question, which we now delve
into.

Before we define and then calculate the quantum capacity of this channel, we
need to discuss quantum channels more generally. It turns out that the
mathematics of quantum channels is far more complex than that of classical
channels, with many cases where it is not just impossible to calculate the
capacity, we cannot even write down an expression for it! The reason for this
is that the capacity of channels is defined asymptotically. Classically, the
capacity is the maximum rate at which information can be sent through the
channel with vanishing error rate, in the limit where the number $n$ of uses
of the channel goes to infinity. Because classically each use of the channel
is independent of the previous use, such a limit is easily taken. In quantum
physics, however, one message sent earlier can be entangled with a message
sent later. Because these uses of the channel are now not independent anymore,
the limit $n\to\infty$ becomes highly nontrivial. Let us first define the
channel.

We begin with the entangled state between Alice and R written as in (V.2).
This state becomes the input to the unitary dynamics of the Bogoliubov
transformation $U=e^{-iH}$ with a suitable $H$ (see Fig. 5.1).

Figure 5.1: Quantum channel with entangled input state $|\psi\rangle_{RA}$
(the entanglement between R and A is indicated by the dashed vertical line),
and outputs B and E. Generally, the ”environment” E is an unobserved variable
that provides the noise in the channel. For the black hole quantum channel, A
is at past infinity (or when discussing late-time signaling, A sends in her
quantum state at future infinity) while B is an inertial observer at future
infinity. The complementary channel output E is the inside of the black hole,
region II in Fig. 3.1. The ouput of the joint channel is the pure state
$|\Psi\rangle_{RBE}$. The density matrices $\rho_{RB}$ and $\rho_{BE}$ are
obtained from $|\Psi\rangle_{RBE}$ by tracing over the unobserved output.

After the action of $U$, we ask whether the ouput of the channel, namely Bob’s
quantum state at future infinity, is now entangled with R in the same manner
as Alice’s qubit was. Note that the channel has two outputs: the recipient A,
as well as a secondary ouput denoted by E. This secondary output is called the
”complementary” channel: it is where everything that does not make its way to
B must go. In classical information theory, what does not make its way to the
receiver is lost to the environment, and in principle someone who has access
to it could reconstitute the information from it, that is, such an agent could
”eavesdrop” on the channel (hence sometimes this observer is called ”Eve”).
One of the most important differences between classical and quantum channels
is that it is strictly impossible to both have quantum information perfectly
reconstructed by B and by E: it is forbidden by the no-cloning theorem.

Let us now return to the asymptotic property of channels. This is the second
important distinction between classical and quantum channels. As mentioned
earlier, the classical capacity is an asymptotic quantity: it is the rate at
which information can be sent through the channel with arbitrary accuracy in
the limit of $n\to\infty$ uses of the channel. The quantum capacity for Alice
to send a single qubit (her share of the entangled state $|\psi_{RA}\rangle$)
to Bob is known as the ”coherent information”, and is defined as Barnum  _et
al._ (1998)

$\displaystyle C_{1}(|\psi\rangle_{RA})=\underset{|\psi\rangle_{RA}}{\rm
max}\bigl{[}S(B)-S(E)\bigr{]}\;,$ (V.3)

where $S(B)$ is the von Neumann entropy of Bob’s density matrix $\rho_{B}={\rm
Tr}_{RE}(|\Psi\rangle_{RBE}\langle\Psi|)$ given by von Neumann (1927)

$\displaystyle S(B)=-{\rm Tr}_{B}\,\rho_{B}\log\rho_{B}\;,$ (V.4)

and $S(E)$ is the entropy of the complementary channel, defined in a similar
manner. Note that Shannon entropy [for example, Eq. (II.3)] is simply a von
Neumann entropy evaluated in the basis in which the density matrix is
diagonal, defined by von Neumann 21 years before Shannon introduced his
”uncertainty function”888It is interesting in this context to note that it was
von Neumann who suggested to Shannon to call his measure entropy because (as
recounted in Tribus and McIrvine (1971)) “your uncertainty function has been
used in statistical mechanics under that name”..

That quantum channels are not ”asymptotically” additive becomes quite obvious
when you can show that two channels that each have zero $C_{1}$ could in fact
transmit quantum information when used in parallel Smith and Yard (2008),
something that is completely impossible for classical channels. The true
quantum capacity of a channel is

$\displaystyle
C_{Q}=\underset{n\to\infty}{\lim}\frac{1}{n}C_{1}(|\psi\rangle_{RA}^{\otimes
n})\;,$ (V.5)

where $|\psi\rangle_{RA}^{\otimes n}$ represents $n$ copies of the state
$|\psi\rangle_{RA}$. However, the calculation of this limit $n\to\infty$ is in
most cases intractable (see, for example Wilde (2013), which should be
consulted for a more comprehensive introduction to quantum channels). There
are some exceptions: channels where the ”regularization” [the limit
$n\to\infty$ in (V.5)] is unnecessary. One such example is the ”symmetric”
quantum channel: a channel where the outputs B and E in the channel are
interchangeable. This is a very peculiar channel, because if B and E are
interchangeable, then we can say that both B and E receive the same amount of
quantum information. But this is impossible according to the no-cloning
theorem, and as a consequence the quantum channel capacity of a symmetric
channel vanishes. This situation is, perhaps, analogous to the classical
binary channel where a bit is equally likely to be flipped or not. That
channel also has zero capacity.

It turns out that the channel defined by the mapping (V.1) is a symmetric
channel, as can be seen by the symmetry between the $a$ modes in front of the
horizon and the $b$ modes behind it. As a consequence we know this capacity,
and it is zero. This, in hindsight, is not surprising. The input state to this
channel is zero-dimensional: the vacuum. We would need at least an entangled
qubit as input to have a nonzero capacity.

We will now study a channel with input. We use a dual-rail encoding of the
logical bit like before, but rather than using particles and anti-particles,
we instead encode the qubit in two particle modes that are available to Alice,
for example the basis states $|10\rangle_{a}$ and $|01\rangle_{a}$. This way,
we can simplify the calculation by ignoring the anti-particle component in
(III.5). Doing this will also allows us to drop the subscript $k$, which we
used mostly to indicate particle/anti-particle status. For simplicity, we will
first show the calculation without the beam-splitter term [that is,
Hamiltonian (III.5)], then later perform the full calculation with Hamiltonian
(III.22).

Using $U=e^{-iH}$ with Hamiltonian (III.5) on the input state
$(\sigma|01\rangle_{a}+\tau|10\rangle_{a})|0\rangle_{b}$, it is not difficult
to see that the channel we have constructed is in fact the same as the quantum
cloner discussed in (IV.14). We did not calculate the quantum capacity of the
quantum cloning channel, so we will do this now. First, consider the $N\to M$
single use of the channel, which we term ${\cal N}_{N\to M}$. The $U$ in Fig.
5.1 is then the quantum cloning machine QCM in Fig. 4.2. Using the single-shot
formula for the quantum capacity (V.3), Bradler et al. were able to show that
Brádler  _et al._ (2010)

$\displaystyle C_{1}({\cal N}_{N\to
M})=\log_{2}(M+1)-\log_{2}(M-N+1)=\log_{2}\left(\frac{M+1}{M-N+1}\right)\;.$
(V.6)

While the calculation in Brádler  _et al._ (2010) was not performed in the
context of black holes, once we realize that the quantum black hole channel is
simply a cloning transformation, their results carry through. Of course,
taking into account the redshift will modify these results, but using late-
time particles as the signal (as we will do later) should recover this
expression as late-time particles do not suffer a redshift.

Now, the full quantum channel is not an $N\to M$ cloning machine, as the
number of output particles is not fixed. Instead, the general channel is a
superposition of cloning channels Brádler (2011). Let us focus on the case
with a single input qubit. The channel ${\cal N}$ is then the superposition

$\displaystyle{\cal N}=\sum_{M=1}^{\infty}p_{M}{\cal N}_{1\to M}\;,$ (V.7)

where Brádler  _et al._ (2010); Brádler (2011); Brádler and Adami (2014)

$\displaystyle p_{M}=\frac{1}{2}(1-z)^{3}M(M+1)z^{M-1}\;.$ (V.8)

Here (as before), $z=e^{-\omega/T_{\rm BH}}$, and
$\sum_{M=1}^{\infty}p_{M}=1$. Note that the complex phases $\sigma$ and $\tau$
do not appear anywhere, which is due to the earlier observation that cloning
channels are invariant under unitary rotations of the input state, so we can
simply set $\sigma=1$, for example.

We would like to calculate the one-shot capacity of the channel ${\cal
N}=\sum_{M=1}^{\infty}p_{M}{\cal N}_{1\to M}$, but up to this point we only
know the capacity of a cloning channel with fixed $N$ and $M$, Eq. (V.6).
Fortunately, it is possible to show that the capacity of a convex mixture of
channels is equal to the mixture of capacities (see Appendix A in Brádler and
Adami (2014)):

$\displaystyle C_{1}({\cal N})=\sum_{M=1}^{\infty}p_{M}C_{1}({\cal N}_{1\to
M})\;.$ (V.9)

However, as I remarked before, in general the one-shot capacity is not equal
to the quantum capacity of the channel because in quantum physics calculating
a capacity requires the regularization (V.5), which is intractable in the
general case. On the other hand, we also saw that there are exceptions where
regularization is not required. We will now see that the cloning channels are
another such exception, but to understand this exception we have to first
become familiar with another property of channels, termed degradability
Devetak and Shor (2005).

Roughly speaking, a channel is said to be degradable if the output of the
channel can be ”degraded” in such a manner that it looks like noise. Recall
that in our construction Fig. 5.1 the channel ${\cal N}$ maps the input state
$\rho_{A}={\rm Tr}_{R}|\psi\rangle_{RA}\langle\psi|$ to the output, that is
$\rho_{B}={\cal N}(\rho_{A})$. The noise of the channel is represented by the
channel from Alice to the environment, which is called the ”complementary
channel” ${\cal N}_{c}$, so that $\rho_{E}={\cal N}_{c}(\rho_{A})$.

A degradable channel is one where there exists a mapping ${\cal D}$ so that

$\displaystyle{\cal D}(\rho_{B})=\rho_{E}\;,$ (V.10)

that is, where the map ${\cal N}$ followed by the map ${\cal D}$ equals the
map ${\cal N}_{c}$: ${\cal N}_{c}={\cal D}\circ{\cal N}$. It is not difficult
to prove (see, for example, Appendix A in Cubitt  _et al._ (2008)) that all
degradable channels have additive capacities, meaning that the regularization
(V.5) is unnecessary and that therefore $C_{Q}=C_{1}$.

Because a degradation always adds noise, a channel that is not degradable is
one where Bob’s output is noisier than Eve’s. There are also channels where
the environment ”simulates” the channel itself (as opposed to the converse
where the output simulates the noise). Such channels are called ”anti-
degradable”, and naturally they require the existence of an anti-degrading map
${\cal D}_{c}$ so that ${\cal D}_{c}(\rho_{E})=\rho_{B}$. The relationship
between these maps is sketched in Fig. 5.2.

Figure 5.2: Relationship between the channel ${\cal N}$, the complementary
channel ${\cal N}_{c}$, and the degrading and anti-degrading maps ${\cal D}$
and ${\cal D}_{c}$.

Let us check if we can find a degradable map for the cloning channel. To keep
things simple, let us focus on the simplest $1\to 2$ cloning machine. One way
to describe this channel is via its action on an arbitrary qubit, written in
the Bloch-state representation as

$\displaystyle\rho_{A}=\frac{1}{2}(\mathbbm{1}+\hat{n}\cdot\vec{\sigma})\;,$
(V.11)

where $\hat{n}\in{\mathbbm{R}}^{3}$ is the unit vector in the Bloch sphere,
and $\vec{\sigma}$ are the Pauli matrices. The vector $\hat{n}$ is determined
by the complex coefficients $\sigma$ and $\tau$ of an arbitrary qubit, the
exact form of which is not important for this argument. The optimal cloner
returns Brádler (2011)

$\displaystyle\rho_{B_{i}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathbbm{1}+\frac{2}{3}\hat{n}\cdot\vec{\sigma})\;,$
(V.12) $\displaystyle\rho_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mathbbm{1}+\frac{1}{3}(n_{c}\sigma_{x}-n_{y}\sigma_{y}+n_{z}\sigma_{z})\right)\;,$
(V.13)

for $i=1,2$. One thing we can do to turn $\rho_{B_{i}}$ into $\rho_{E}$ is to
apply a depolarizing map that shrinks the Bloch vector by a factor of two. But
this is not enough, as there is a minus sign multiplying $\sigma_{y}$ in
(V.13). However, applying a complex conjugation after the depolarization will
indeed turn $\rho_{B}$ into $\rho_{E}$. Such combined maps are called
”conjugate degrading” maps, and the cloning channel is therefore conjugate
degradable. It is possible to prove that not only is the optimal $1\to 2$
cloner conjugate degradable, but so are all $N\to M$ cloners, and by extension
the general cloning channel (V.7) Brádler (2011). Fortunately, conjugate
degradable channels also are additive, so that the capacity of the quantum
cloning channel, and therefore the quantum capacity of the black hole channel
(as they are one and the same thing) is given by (V.9). Evaluating (V.9) using
(V.6) (with $N=1$) and (V.8) we arrive at the expression Brádler and Adami
(2014)

$\displaystyle
C_{Q}=\frac{1}{2}(1-z)^{3}\sum_{M=1}^{\infty}M(M+1)z^{M-1}\log\left(\frac{M+1}{M}\right)\;,$
(V.14)

which is shown in Fig. 5.3 as a function of $z=e^{-\omega/T_{\rm BH}}$. Note
that unlike the classical capacity of the black hole, which was still positive
even as the temperature of the black hole diverges ($M_{\rm BH}\to 0$), the
quantum capacity vanishes in this limit.

Figure 5.3: Quantum capacity of the channel (V.7) with a single incoming qubit
($N=1$) as a function of $z=e^{-\omega/T_{\rm BH}}$.

The previous discussion of the capacity of black holes to transmit quantum
information was intentionally naive (we sent in information early during the
formation of the black hole but neglected the red shift, and we did not treat
scattering off the black hole horizon). However, we will see that we can reuse
some of that calculation in the full case that we discuss now.

To correctly treat scattering off of the black hole horizon with particles
sent towards the horizon long after the formation of the black hole, we define
(as before, in Eq. (III.21)) the outgoing annihilation operator

$\displaystyle A_{k}$ $\displaystyle=$ $\displaystyle
e^{-iH}a_{k}e^{iH}=\alpha_{k}a_{k}-\beta_{k}b^{\dagger}_{k}+\gamma_{k}c_{k}\;,$
(V.15)

except for simplicity we will use a scalar as opposed to a complex field since
we will not need to describe anti-particles. In the following, I will only
treat the case $\alpha_{k}=0$ (a perfectly reflecting black hole, which we can
call a ”white hole”), and a perfectly absorbing black hole ($\alpha_{k}=1$).
The reason I only treat these extreme cases is that, in this formalism, they
are the only ones that are tractable. I will discuss the ”gray hole channel”
using the formalism of Gaussian states later.

###  V.1 Quantum capacity of perfectly white holes

If we set $\alpha_{k}=0$, the Bogoliubov transformation (V.15) becomes

$\displaystyle A_{k}=\gamma_{k}c_{k}-\beta_{k}b^{\dagger}_{k}\;,$ (V.16)

This transformation is formally identical to the transformation (III.1), which
was the naive description of black holes without scattering. Using
$\gamma_{k}^{2}=1+\beta_{k}^{2}$, the outgoing density matrix given $m$
incident late-time particles is precisely (III.19). I previously pointed out
(right after that equation) that it was not consistent to claim that the case
treated there should be seen as a perfectly absorbing black hole ($\Gamma=1$),
as detailed balance would be violated. In fact, we now see that the case
described there (and therefore the case studied by Hawking in his initial
publication Hawking (1975)) only consistently describes a perfectly reflecting
black hole: a white hole. This makes sense in hindsight: if you take a look at
the Penrose diagram Fig. 3.1 that describes the time evolution of the
operators $a_{k}$ and $b_{k}$, it is clear that the particles that travel
towards future infinity just outside of the horizon (the modes $a_{k}$) never
enter the horizon999This is also consistent with how Hawking introduced his
gray body factor, by following particles from region II backwards in time into
region I. If particles can be transmitted with $\Gamma=1$ from the inside to
the outside (a black hole from the inside), then its time reversal must be a
white hole with $\Gamma=0$ from the outside).. Thus, if viewed as the
signaling particles, they are never absorbed and for them the black hole is a
perfect mirror.

We can now be confident that the capacity shown in Fig. 5.3, when treating
late-time particles incident on the horizon, is in fact the quantum capacity
of a white hole. It is finite for all but vanishing-mass black holes, which
implies that (given suitable quantum error correction methods), Alice’s
quantum state can be perfectly reconstructed by Bob. Now let us take a closer
look at the complementary channel (the one to Eve, that is, beyond the
horizon). The quantum no-cloning theorem tells us that this capacity better
vanish, as otherwise quantum information could be perfectly reconstructed by
both Eve and Bob.

It turns out that the complementary channel ${\cal N}_{c}$ gives rise to a
quantum state straddling the horizon that is separable, meaning that the
regions inside and outside of the horizon are not entangled Brádler (2011):
the horizon has ”broken” the potential entanglement between Eve and Alice.
Channels that do this are known as ”entanglement-breaking channels” Brádler
_et al._ (2010), and have zero capacity. In particular, it can be shown that
all anti-degradable channels (see Fig. 5.2) are entanglement-breaking Brádler
_et al._ (2010). Since the cloning channel is degradable (along with being
conjugate degradable), the no-cloning theorem ensures that the complementary
channel is anti-degradable and therefore entanglement-breaking.

###  V.2 Quantum capacity of perfectly black holes

We now study the case $\alpha_{k}=1$, that is, black holes that do not reflect
any of the incoming radiation. In this case, the Bogoliubov transformation
reads

$\displaystyle A_{k}=a_{k}-\beta_{k}b^{\dagger}_{k}+\gamma_{k}c_{k}\;,$ (V.17)

but because $\alpha_{k}^{2}-\beta_{k}^{2}+\gamma_{k}^{2}=1$, setting
$\alpha_{k}=1$ implies $\beta_{k}=\gamma_{k}$ [we previously identified this
case with $g_{k}=g_{k}^{\prime}$ in (III.5)]. Acting with $U=e^{-iH}$ on an
initial state without any incoming particles $|000\rangle_{abc}$ using
Hamiltonian (III.5) but with $\alpha_{k}=1$ and
$\beta_{k}^{2}=\gamma_{k}^{2}\equiv g_{k}^{2}$ results in

$\displaystyle
U|000\rangle_{abc}=\frac{1}{1+\frac{1}{2}\beta_{k}^{2}}\sum_{n=0}^{\infty}\sum_{m=0}^{n}\left(\frac{2\beta_{k}}{2+\beta_{k}}\right)^{n}(-\beta_{k})^{m}\sqrt{n\choose
m}|n-m\rangle_{a}|m\rangle_{b}|m\rangle_{c}\;,$ (V.18)

In order to encode a qubit in a dual-rail fashion, we also need to study how a
single-late time particle (in mode $c_{k}$) fares under the transformation. We
find Brádler and Adami (2014)

$\displaystyle U|001\rangle_{abc}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{1+\frac{1}{2}\beta_{k}^{2}}\right)^{2}\sum_{n=0}^{\infty}\sum_{m=0}^{n+1}\left(\frac{2\beta_{k}}{2+\beta_{k}}\right)^{n}(-\beta_{k})^{m}\sqrt{n\choose
m}\times$ (V.19)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\sqrt{m+1}|n-m\rangle_{a}|n\rangle_{b}|m+1\rangle_{c}+g_{k}\sqrt{n-m+1}|n-m+1\rangle_{a}|n\rangle_{n}|m\rangle_{c}\right)\;.$

If Alice’s qubit is described by the density matrix $\rho_{\rm in}$, we can
show that the output matrix is a superposition of channels101010We should not
confuse the depolarizing map ${\cal D}_{M}$ with the degrading map ${\cal D}$
introduced in Eq. (V.10), as the depolarizing map is neither degrading nor
anti-degrading. ${\cal D}_{M}$ (where $M$ is again the number of clones)
Brádler and Adami (2014)

$\displaystyle\rho=\sum_{M=1}^{\infty}p_{M}{\cal D}_{M}(\rho_{\rm in}).$
(V.20)

Let us look at the first term, $M=1$. The output under this channel can be
calculated to be

$\displaystyle{\cal D}_{1}(\rho_{\rm in})=\frac{1}{3}\rho_{\rm
in}+\frac{1}{3}\mathbbm{1}\;,$ (V.21)

where $\mathbbm{1}$ is the unit matrix. It turns out that this is the output
of a quantum depolarizing channel Calderbank and Shor (1996); Adami and Cerf
(1997); King (2003), whose action on an input $\rho_{\rm in}$ is given by
$\rho_{\rm depol}=(1-p)\rho_{\rm in}+\frac{p}{2}\mathbbm{1}$ ($p$ is the
depolarizing parameter of the channel). Thus, the quantum channel for full
absorption ($\alpha_{k}=1$) is a quantum depolarizing channel where the
depolarizing parameter is given by the ”classical” cloning fidelity $F=2/3$,
which we recall from section IV.2 is the worst cloning fidelity we can
achieve. More importantly, it was shown in Brádler (2011) that a depolarizing
channel with $F=2/3$ is in fact entanglement-breaking, and therefore has zero
capacity. Indeed, all channels ${\cal D}_{M}$ in (V.20) have this property,
and therefore the quantum capacity for a perfectly black hole vanishes.

In hindsight, we could have guessed this result. After all, the quantum
channel with $\alpha_{k}=1$ is the complementary channel for the
$\alpha_{k}=0$ channel, viewed from behind the horizon. And because that
channel has positive capacity, the capacity of its complementary channel must
vanish so that we cannot reconstruct quantum information in two different
places.

Clearly, the channels with $\alpha_{k}=0$ and those with $\alpha_{k}=1$ are
extreme cases. Because the reflectivity of a black hole depends primarily on
the impact parameter of scattering, it is important to understand the black
hole quantum channel for all values between $\alpha_{k}=0$ and $\alpha_{k}=1$.
To study the capacity for black holes with arbitrary $0\leq\alpha_{k}^{2}\leq
1$, we are going to have to deploy more sophisticated artillery.

###  V.3 Quantum capacity of black holes with arbitrary transmissivity

Previously, I pointed out the relationship between the Bogoliubov
transformation engendered by (III.5) and those that we encounter in quantum
optics in order to motivate a discussion of quantum dynamics in terms of
optical elements: the two-mode squeezer and the beam splitter. In this section
we are going to use this analogy in order to marshal the considerable quantum
optics literature of so-called Gaussian states, and study the quantum capacity
of Gaussian channels. This will allow us to make some statements about the
quantum capacity of black holes with arbitrary transmissivity, but we will
also see that, because of the problem of regularization of quantum capacities,
we cannot as yet answer all questions about the capacity of those channels.

Earlier, we discussed an encoding of information using states with defined
particle number. For example, a logical zero would be encoded using $m$ anti-
particles, and a logical one would correspond to sending $m$ particles
instead. However, creating quantum states with defined particle number is
exceedingly difficult. In standard quantum optics applications, it is more
convenient to construct states with a defined mean number of particles
$\langle m\rangle$ instead. A typical Gaussian quantum state with fixed mean
number of particles is a thermal state, defined by the density matrix

$\displaystyle\rho_{\rm therm}=\frac{1}{\langle
m\rangle+1}\sum_{m=0}^{\infty}\left(\frac{\langle m\rangle}{\langle
m\rangle+1}\right)^{m}|m\rangle\langle m|\;.$ (V.22)

This is a thermal state because the mean number of particles $\langle
m\rangle={\rm Tr}(\rho_{\rm therm}a^{\dagger}a)$ is

$\displaystyle\langle m\rangle=\frac{e^{-\omega/T}}{1-e^{-\omega/T}}\;,$
(V.23)

and we immediately recognize that the output of a black hole channel without
any incident particles (III.11) is, in fact, a thermal state with mean
particle number $\langle m\rangle=\beta_{k}^{2}$ in each mode $k$.

Thermal states are a particular example of the more general Gaussian states,
which are defined in terms of correlation matrices acting on quadratures,
rather than the creation and annihilation operators that we have used
throughout. The relation between quadrature operators $q,p$ and
creation/annihilation operators $a^{\dagger},a$ for a single mode $k$ is
simply (I have reinstated $\hbar$ here)

$\displaystyle
q_{k}=\sqrt{\frac{\hbar}{2}}(a_{k}+a_{k}^{\dagger})\;,\;p_{k}=i\sqrt{\frac{\hbar}{2}}(a_{k}-a_{k}^{\dagger})$
(V.24)

so that $q_{k}$ and $p_{k}$ observe the standard uncertainty relation

$\displaystyle[q_{k},p_{k^{\prime}}]=i\hbar\delta_{kk^{\prime}}\;.$ (V.25)

To express an arbitrary density matrix $\rho$ in the $n$-mode quadrature basis
Weedbrook  _et al._ (2012), first we define the column vector of $2n$
operators

$\displaystyle{\bf x}=[q_{1},p_{1},\cdots,p_{n},q_{n}]^{\top}$ (V.26)

so that $x_{1}=q_{1}$, $x_{2}=p_{1}$, $x_{3}=q_{2}$ and so forth until
$x_{2n}=p_{n}$. Then we can write the commutation relation for all $2n$
operators as

$\displaystyle[x_{i},x_{j}]=i\hbar\Omega_{ij}\;,$ (V.27)

where the matrix $\bm{\Omega}$ is the direct sum of matrices $\bm{\omega}$ for
each mode:

$\displaystyle\bm{\Omega}=\bigoplus_{k=1}^{n}\bm{\omega}=\begin{pmatrix}\bm{\omega}&&\\\
&\ddots&\\\ &&\bm{\omega}\end{pmatrix}\;,\;\bm{\omega}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\;.$ (V.28)

The first moment of a density matrix $\rho$ can then be expressed in this
basis as

$\displaystyle\bm{\bar{x}}={\rm Tr}{(\rho\bm{x})}.$ (V.29)

The all-important second moment of $\rho$ is given by the matrix $\bm{V}$ with
elements

$\displaystyle V_{ij}=\frac{1}{2}{\rm Tr}(\rho\\{\Delta x_{i},\Delta
x_{j}\\})$ (V.30)

where $\Delta x_{i}=x_{i}-\bar{x}_{i}$ and $\\{\ ,\ \\}$ is the anti-
commutator. Gaussian quantum states are then defined as those states for which
higher-order moments beyond the second moment vanish, and the covariance
matrix $\bm{V}$ is a real symmetric matrix that satisfies the uncertainty
principle Weedbrook  _et al._ (2012)

$\displaystyle\bm{V}+i\frac{\hbar}{2}\bm{\Omega}\geq 0\;.$ (V.31)

The ”positivity” requirement for a matrix such as that written in (V.31)
stipulates that all the eigenvalues of the matrix need to be positive, which
puts constraints on the real-valued elements.

With these preliminaries out of the way, we can study how Gaussian states
behave under the transformations of the black hole channel. As before, we need
to look at how the Bogoliubov transformation affects the two vacuum modes
$a_{k}$ and $b_{k}$, as well as the signal mode $c_{k}$. We can write this
transformation in matrix form [see Eqs. (III.23-III.25) for the definitions of
$\alpha_{k}$, $\beta_{k}$, and $\gamma_{k}$]

$\displaystyle\begin{pmatrix}A_{k}\\\ B^{\dagger}_{k}\\\ C_{k}\\\
\end{pmatrix}=\begin{pmatrix}\alpha_{k}&-\beta_{k}&\gamma_{k}\\\
\beta_{k}&1+\frac{\beta_{k}^{2}}{1+\alpha_{k}}&-\frac{\beta_{k}\gamma_{k}}{1+\alpha_{k}}\\\
-\gamma_{k}&\frac{\beta_{k}\gamma_{k}}{1+\alpha_{k}}&1+\frac{\gamma_{k}^{2}}{1+\alpha_{k}}\end{pmatrix}\begin{pmatrix}a_{k}\\\
b^{\dagger}_{k}\\\ c_{k}\\\ \end{pmatrix}\;.$ (V.32)

We now have to write this transformation in terms of operators acting on
quadratures instead. When we do this (see Brádler and Adami (2015) for the
details), we can write the action of the channel in terms of its effect on the
covariance matrix of an input Gaussian state (the first moments of the
Gaussian state can always be set to zero). In particular, an input Gaussian
one-mode state (sent in at late time) $\bm{V}_{\rm in}$ transforms with a
transmission matrix $\bm{T}$ and a noise matrix $\bm{N}$ as

$\displaystyle{\bm{V}}_{\rm out}={\bm{T}}{\bm{V}}_{\rm
in}{\bm{T}}^{\top}+{\bm{N}}\;.$ (V.33)

Here, $\bm{T}$ and $\bm{N}$ are $2\times 2$ matrices (they act on the single
late-time mode) that take the very simple (and diagonal) form

$\displaystyle{\bm{T}}=\begin{pmatrix}\sqrt{\gamma_{k}^{2}}&0\\\
0&\sqrt{\gamma_{k}^{2}}\end{pmatrix}\;\ \
{\bm{N}}=\begin{pmatrix}\alpha_{k}^{2}+\beta_{k}^{2}&0\\\
0&\alpha_{k}^{2}+\beta_{k}^{2}\end{pmatrix}\;.$ (V.34)

This form is particularly pleasing because it allows us to compare this
channel directly to a complete characterization of all possible one-mode
Gaussian (OMG) channels that previously appeared in the literature Weedbrook
_et al._ (2012); Schäfer  _et al._ (2013). Of the eight channels listed there,
three make an appearance in the black hole Gaussian channel. First, let us
examine the parameter space of this channel, which is characterized by its
transmission potential (parameterized by $\gamma_{k}^{2}$), and the noise
level (described by $\alpha_{k}^{2}+\beta_{k}^{2}$). This much is not
surprising: $\gamma_{k}$, after all, is the amplitude of the Bogoliubov
transformation affecting our signal state, while the black hole’s modes
$a_{k}$and $b_{k}^{\dagger}$ are in a vacuum state and provide the noise to
the channel.

Fig. 5.4 depicts the OMG channel parameterized by the noise level and the
transmissivity of the channel.

Figure 5.4: Allowed regions of the parameter space (white) of the black hole
OMG channel in terms of transmissivity $\gamma_{k}^{2}$ and noise level
$\alpha_{k}^{2}+\beta_{k}^{2}$. Unphysical regions are in gray.

While the regions in gray are not physical channels, we would like to
calculate the capacity of the channel to transmit quantum information for the
permissive region. Different conditions lead to the disallowed regions: two of
them stem from the fact that the black hole’s diffraction parameter
$\alpha_{k}$ is bounded from below and above by zero and 1, respectively. In
fact, these bounds also correspond to the positivity condition for the output
Gaussian state, which is equivalent to the condition $g^{\prime}_{k}\geq
g_{k}$. The second condition that rules out the lower left triangle in Fig.
5.4 is the condition that the mean particle number $\langle n_{k}\rangle\geq
0$, as $\alpha_{k}^{2}+\beta_{k}^{2}=(2\langle
n_{k}\rangle+1)(1-\gamma_{k}^{2})\geq 1-\gamma_{k}^{2}$, using $\langle
n_{k}\rangle=\frac{\beta_{k}^{2}}{\alpha_{k}^{2}-\beta_{k}^{2}}$, consistent
with (V.23).

Within the family of OMG channels, the channel with $\gamma_{k}^{2}<1$ is
known as the “lossy channel” ${\cal C}_{\rm loss}$ (area to the left of the
vertical dashed line in Fig. 5.4). For black holes, this corresponds to the
cases where the effective transmissivity $\Gamma=1-\gamma_{k}^{2}$ lies
between zero and one (recall that the effective transmissivity is strictly
smaller than the black hole “bare” transmissivity $\alpha_{k}^{2}$, the
parameter that characterizes the beam-splitter, since
$\Gamma=\alpha_{k}^{2}(1-e^{-\omega_{k}^{2}/T_{\rm BH}})$. OMG channels exist
for which $\gamma_{k}^{2}>1$: these are the so-called “amplifying channels”,
which, as the name implies, amplify the incoming signal. While it is unclear
whether black holes exist with such a property, we’ll discuss these
hypothetical channels here for completeness.

The allowable region of the channel is bounded by the two lines
$\alpha_{k}^{2}+\beta_{k}^{2}=1+\gamma_{k}^{2}$ and
$\alpha_{k}^{2}+\beta_{k}^{2}=-1+\gamma_{k}^{2}$, which correspond to the two
cases for which we have been able to calculate the quantum capacity of the
black hole channel in section V.1 ($\alpha_{k}^{2}=0$) and section V.2
($\alpha_{k}^{2}=1$). For the latter case we determined that the quantum
capacity vanishes, and indeed the analysis of Caruso  _et al._ (2006) revealed
that the entire region between $\alpha_{k}^{2}=0.5$ (the balanced beam
splitter) and $\alpha_{k}^{2}=1$ corresponds to a channel that can be written
as the composition of an arbitrary channel and an anti-degradable channel, and
therefore must have zero capacity111111We noted earlier that anti-degradable
channels are additive and must have zero capacity as they are entanglement-
breaking.. This area is shaded in yellow in Fig. 5.5.

Figure 5.5: Quantum capacity for the black hole OMG channel. The region in
yellow has vanishing capacity, while for the purple region we have a non-zero
lower bound for the capacity. For the region in white (bounded from below by
${\cal I}_{\rm coh}=0$), the capacity is currently not known because it could
be super-additive.

We can now attempt to calculate the quantum capacity for the lossy channel. To
do this, we must calculate the coherent information of the channel, and
optimize it. As we discussed previously, this can only be achieved if the
quantum capacity is additive, but for the channels with $\alpha_{k}^{2}<1/2$
we do not know this. However, if we can derive a positive lower bound using
the additive capacity (V.14), then we can be assured that the capacity is
positive. If the lower bound is zero, then we can only say that it is possible
that the capacity is non-zero, but we simply do not know.

Let us then find out under what circumstances the limit $n\to\infty$ of the
”$n$-shot” coherent information $\lim_{n\to\infty}{\cal I}_{\rm coh}(n)$ is
positive. The coherent information ${\cal I}_{\rm coh}(n)$ has previously been
calculated for both the lossy and the amplifying channel. In particular, it is
possible to show that Brádler and Adami (2015)

$\displaystyle{\cal I}_{\rm coh}(n)=g(n)-2g(\xi)\;,$ (V.35)

where $\xi=\sqrt{1+4n\alpha^{2}}$ and

$\displaystyle g(x)=(1+x)\log(1+x)-x\log x\;.$ (V.36)

It turns out that in the limit $n\to\infty$ $g(\xi)$ vanishes Brádler (2015),
while

$\displaystyle{\cal I}_{\rm
coh}=\lim_{n\to\infty}g(n)=\frac{\beta_{k}^{2}}{\alpha_{k}^{2}-\beta_{k}^{2}}\log\frac{\beta_{k}^{2}}{\alpha_{k}^{2}}+\log\frac{\gamma_{k}^{2}}{\alpha_{k}^{2}}\;.$
(V.37)

The dashed curve in Fig. 5.5 corresponds to the boundary where ${\cal I}_{\rm
coh}=0$. We thus see that for parameters where ${\cal I}_{\rm coh}>0$ (purple
region in Fig. 5.5) the quantum capacity must be positive (as it is lower-
bounded by (V.37)), while the quantum capacity in the white region still
cannot be determined.

For $\gamma_{k}^{2}>1$, the expression (V.37) also turns out to be a lower
bound Brádler and Adami (2015). The case $\gamma_{k}=1$, however, has to be
treated separately. In this limit ${\cal I}_{\rm coh}$ diverges as
$n\to\infty$, however, this channel is trivial: it represents the so-called
”zero-added classical noise” channel Holevo (2007), termed ${\cal B}_{2}$ in
Schäfer  _et al._ (2013). For this channel, the transmission matrix ${\bm{T}}$
in (V.34) is the identity, and the noise vanishes. Such channels have infinite
capacity also in classical physics Cover and Thomas (1991).

To summarize this section, we have seen that it is possible to make statements
about the quantum capacity of black holes for values of the beam-splitter
variable $\alpha_{k}^{2}$ other than the extreme cases $\alpha_{k}=0$ (for
which we saw that the capacity is positive), and $\alpha_{k}=1$, where the
capacity vanishes. We saw that when the beam-splitter absorbs more than it
reflects ($\alpha_{k}^{2}\geq 1/2$) then the quantum capacity must vanish (so
as to conform to the no-cloning theorem). When, in turn, the beam-splitter
reflects more than it transmits ($0\leq\alpha_{k}^{2}<1/2$) the capacity is
positive for some parameters (those for which ${\cal I}_{\rm coh}>0$), but
since for the remaining parameter region ${\cal I}_{\rm coh}\leq 0$, we cannot
establish whether the quantum capacity is positive since ${\cal I}_{\rm coh}$
is only a lower bound to the capacity. Needless to say at this point: a
vanishing quantum capacity does not point to a flaw in the laws of physics.
Rather, when it vanishes it does so because we must conform to the laws of
physics, which stipulate that quanta cannot be cloned.

##  VI Unitary Evaporation of Black Holes

In everything we have been discussing up to this point, the black hole was
treated as a static quantity: it had already formed, and its mass was fixed at
$M_{\rm BH}$. This approximation, which essentially treats the gravitational
force as a background field, was necessitated by taking the static-path
approximation to the time-dependent operator (III.26). In this approximation,
the back-reaction of the radiation on the metric field is neglected: that is
the essence of the semi-classical approach.

However, this assumption also precludes us from studying the evaporation of
the black hole microscopically. Hawking noticed early on that the energy of
the outgoing Hawking radiation must be provided by the black hole, and that
therefore the black hole must ultimately disappear. But this seemed to open up
another fundamental problem: if we were to assume that a black hole can form
from a quantum mechanical pure state (a state $\rho$ with vanishing von
Neumann entropy) that in the future produces Hawking radiation with entropy
$S_{H}$, then (since Hawking radiation is thermal) the final state after black
hole evaporation would be a mixed state with positive entropy Hawking (1976).
However, in a closed system such a transition from a pure state to a mixed
state is forbidden: it is tantamount to the non-conservation of probability, a
state of affairs I have previously referred to as an abomination.

We contemplated this abomination when it appeared that classical information
was lost inside of the black hole, but were able to recognize in the previous
sections that all these problems arise simply from ignoring the stimulated
emission process. However, there is no ”incoming” signaling particle when
discussing black hole evaporation, so stimulated emission will not help us
understand how this process unfolds. To understand how black hole evaporation
returns space-time to the pure state it started out as, we need a description
of the interaction of black holes with radiation that goes beyond the semi-
classical approach. The dynamics that such a treatment should reveal is that
of the celebrated ”Page curves”. Page first discussed how the quantum entropy
of one system might depend on the ”size” of a subsystem that it is entangled
with, while both together are in a pure state. Specifically, Page asks us to
imagine a pure initial state formed using $n$ particles, $|\psi\rangle_{\rm
in}=|n\rangle$. After this state is entangled with another system, the density
matrix of the outgoing system becomes $\rho_{\rm
out}=\sum_{i=0}^{n}p_{i}|i\rangle_{\rm out}\langle i|$, with entanglement
entropy

$\displaystyle S_{e}(\rho_{\rm out})=-\sum_{i=0}^{n}p_{i}\log p_{i}\;.$ (VI.1)

The maximal entropy is reached when all the states $|i\rangle\langle i|$ are
equiprobable, so that $S_{\rm max}=\log(n+1)$. Page imagined that as the black
hole pure state decoheres, the entanglement entropy of the outgoing radiation
must also increase until it reaches its maximal value Page (1993). As the
black hole continues to evaporate, Page argued (using a toy quantum mechanical
model) that the entanglement entropy must start to decrease (after a time now
dubbed the ”Page time”), as in Fig. 6.1.

Figure 6.1: A typical Page curve showing the entanglement entropy of Hawking
radiation as a function of the ”size” of the subsystem (determined by the
number of particles in the subsystem). If we assume that the subsystem sizes
increases as the black hole evaporates, we can take the subsystem size as a
measure of elapsed time since formation of the black hole from a pure state.

If a black hole evaporated via a unitary process (that is, if black hole
evaporation can be described by an $S$-matrix), Page argued that ultimately
the entanglement entropy of the black hole must disappear, leaving only a
vacuum with zero entropy, in contradiction to Hawking’s assessment that black
holes must turn pure states into mixed states. Unfortunately, the semi-
classical treatment of black holes prevents us from testing this prediction
directly: we do not know what the interaction Hamiltonian $H_{\rm int}$ in
(III.3) is (whose matrix elements would form the black hole $S$-matrix).
Indeed, to proceed we had used the free-field Hamiltonian consisting of the
two-mode squeezer and the beam-splitter analogues, which, along with using the
single-time-slice approximation of the path integral, gave rise to a
consistent picture of black hole dynamics when interacting with classical or
quantum information.

It is important to realize at this juncture that the two-mode squeezing (or
”optical parametric amplifier”, OPA) Hamiltonian (III.14) is itself an
approximation that assumes that the number of pump quanta is so large that the
down conversion process does not change the ”store” of pump quanta. In other
words, in this approximation it is assumed that the down-conversion process
does not ”react back” on the pump, which is therefore ”undepletable”, much
like the black hole mass is held constant in the semi-classical approximation.
But unlike in quantum gravity where we do not know how to move beyond this
approximation, in quantum optics it is possible to write down the interaction
between the pump modes and the signal and idler modes that represents the
canonical extension of the OPA to depletable pumps: it is a tri-linear
Hamiltonian

$\displaystyle H_{\rm
tri}=ir(d_{p}a^{\dagger}_{s}b^{\dagger}_{i}-d_{p}^{\dagger}a_{s}b_{i})\;.$
(VI.2)

Here, the annihilation operators $a_{s}$ and $b_{i}$ refer to the signal and
idler modes as before, but $d_{p}$ and $d^{\dagger}_{p}$ create and annihilate
pump modes instead. The coupling constant $r$ is related to the gain $\eta$ of
the OPA in Eq. (III.14) and the expected number of pump modes, and is in
principle time-dependent.

Given that the quantum optics analogy has been so successful when transferred
to black hole dynamics, what if we used the interaction Hamiltonian (VI.2) to
calculate the black hole $S$-matrix, where the pump modes play the role of
black hole modes, and the signal and idler modes are identified with the
Hawking and partner modes (just as before)? This was in fact attempted by
Nation and Blencowe Nation and Blencowe (2010), and later by Alsing Alsing
(2015). Both found that the entanglement entropy of the Hawking modes
decreases after reaching a maximum, but they could not reproduce Page curves
because, using effectively a one-time-slice or ”static path” approximation
(SPA) of the path integral as in (III.3), the calculation quickly became
unreliable as the time step $\Delta t$ is taken to be large. A good
introduction to the quantum optics/black hole physics analogy using trilinear
Hamiltonians can be found in Flórez Gutiérrez (2022).

We will now see what happens if we use the tri-linear Hamiltonian (VI.2) to
calculate the $S$-matrix of black hole evaporation, by going beyond the SPA
and approximating the black hole $S$ matrix using enough time slices that
$\Delta t$ can be kept small. In this way, we can follow the evaporation of
the black hole (or, in the words of quantum optics, the depletion of the pump)
accurately as long as the number of initial quanta $n$ is not too high. While
for black holes the number $n$ surely must be astronomical, we will have to
keep this number comparatively small since the evaluation of the path integral
can only be done numerically.

We begin by writing the initial state at $t=0$ as

$\displaystyle|\Psi(0)\rangle=|n\rangle_{d}|0\rangle_{a}|0\rangle_{b}\equiv|n\rangle_{d}|0\rangle_{ab}\;.$
(VI.3)

Here, the Hawking modes (annihilated by operators $a_{k}$ in region I, as in
Fig. 3.1) and the partner modes (annihilated by $b_{k}$ in region II) are
interacting with black hole modes created and annihilated by $d^{\dagger}_{k}$
and $d_{k}$, see Fig. 6.2.

Figure 6.2: Schematics of a black hole (approximate Schwarzschild radius in
bold) where black hole modes $d_{k}$ are transformed into Hawking modes
$a_{k}$ and partner modes $b_{k}$ at the horizon.

We write the time evolution of the joint state
$\mathop{|\Psi(t)\rangle}\nolimits$ in terms of black hole $S$-matrix acting
on $|\Psi(0)\rangle$ as

$\mathop{|\Psi(t)\rangle}\nolimits=S(t,0)\mathop{|\Psi(0)\rangle}\nolimits=\mathsf{T}e^{-i\int_{0}^{t}H_{\mathrm{tri}}(t^{\prime})dt^{\prime}}\mathop{|n\rangle}\nolimits_{d}\mathop{|0\rangle}\nolimits_{ab}\;,$
(VI.4)

using the trilinear Hamiltonian

$H_{\mathrm{tri}}=\sum_{k=-\infty}^{\infty}ir_{\omega_{k}}(t)\big{(}d_{k}a_{k}^{\dagger}b_{k}^{\dagger}-d_{k}^{\dagger}a_{k}b_{k}\big{)}\;.$
(VI.5)

Here, $r_{\omega_{k}}(t)$ is the time-dependent coupling strength that sets
the Hawking temperature $T_{\rm BH}(t)$ and the black hole mass $M_{\rm
BH}(t)=\frac{1}{8\pi T_{\rm BH}(t)}$, via the standard relation Nation and
Blencowe (2010)

$\displaystyle T_{\rm BH}(t)=\frac{\omega_{k}}{2\ln{\rm cotanh}\
r_{\omega_{k}}(t)}\;.$ (VI.6)

In the following, I will again focus on a single mode $k$ with energy
$\omega_{k}$, and omit the index $k$ for convenience.

In order to evaluate (VI.4), we need to introduce small time slices $\Delta t$
that allow us to discretize the path integral so that with $t=N\Delta t$

$U(t)=\mathsf{T}e^{-i\int_{0}^{t}H_{\mathrm{tri}}(t^{\prime})}dt^{\prime}\approx\prod_{i=1}^{N}e^{-i\Delta
tH_{i}}\;.$ (VI.7)

In (VI.7), the $i$-th time-slice Hamiltonian
$H_{i}=ir_{0}\big{(}da_{i}^{\dagger}b_{i}^{\dagger}-d^{\dagger}a_{i}b_{i}\big{)}$
acts on the black hole state and the $i$-th slice of the $ab$ Hilbert space
$\mathop{|0\rangle}\nolimits_{a_{i}b_{i}}$. The initial value of the coupling
strength $r_{0}$ simply sets the energy scale, and we can set $r_{0}=1$ in the
following without loss of generality.

Let us now apply the discretized (VI.7) to the initial state, so that Hillery
and Zubairy (1982)

$\mathop{|\Psi(t)\rangle}\nolimits_{dab}=W\mathop{|n\rangle}\nolimits_{d}\mathop{|0\rangle}\nolimits_{ab}=\prod_{i=1}^{N}e^{-i\Delta
tH_{i}}\mathop{|n\rangle}\nolimits_{d}\mathop{|0\rangle}\nolimits_{ab}\;,$
(VI.8)

where I defined he time-sliced basis

$\displaystyle\mathop{|0\rangle}\nolimits_{ab}\overset{\rm
df}{=}\mathop{|0\rangle}\nolimits_{a_{N}b_{N}}\otimes\ldots\otimes\mathop{|0\rangle}\nolimits_{a_{1}b_{1}}\;$
(VI.9)

as well as the unitary operator acting on time slice $i$

$\displaystyle W^{(i)}=e^{-i\Delta tH_{i}}$ (VI.10)

so that

$\displaystyle W=W^{(N)}\otimes\ldots\otimes W^{(1)}\;.$ (VI.11)

Assuming that the basis states for each time slice appear as product states in
Eq. (VI.8) implies that after a black hole mode has been converted to Hawking
and partner modes, those modes will never interact with the black hole again,
as depicted in Fig. 6.3(a).

While this is certainly reasonable for the outgoing Hawking modes, this is
questionable for the partner modes $b$ behind the black hole horizon. In fact,
this is an approximation that is also often made in quantum optics, where the
non-linear crystal is assumed to be so thin that the two modes (the signal and
idler modes) do not interact with the crystal degrees of freedom after they
have been produced. We will test this assumption later by allowing the $b$
modes to interact with the black hole again, as depicted in Fig. 6.3(b).

Figure 6.3: Action of the discretized $S$-matrix $W^{(i)}$ on the discretized
basis states. (a) By assuming that once a black hole mode is converted, the
Hawking and partner mode never interact with the black hole again allows us to
write the sliced basis as a product (VI.8). (b) Relaxing this assumption for
the $b$ modes creates a far more complex set of interactions.

Let’s evaluate the first time slice:

$\mathop{|\Psi(1)\rangle}\nolimits=W^{(1)}|n\rangle_{d}\mathop{|0\rangle}\nolimits_{a_{1}b_{1}}=\sum_{j=0}^{n}U_{nj}^{(1)}\mathop{|j\rangle_{d}|n-j\rangle}\nolimits_{a_{1}b_{1}}\;,$
(VI.12)

with amplitudes $U_{nj}^{(1)}$ determined below. The probability
$p(j|n)=|U_{nj}^{(1)}|^{2}$ reflects the probability to convert $n$ black hole
modes into $j$ Hawking and partner modes in one interaction, and I will
outline its calculation (and others like it) below. The full quantum state
after time $t$ in this approximation becomes

$\displaystyle\mathop{|\Psi(t)\rangle}\nolimits_{dab}=\sum_{j_{1}...j_{N}}U_{nj}^{(1)}\ldots
U_{j_{N-1}j_{N}}^{(N)}|j_{N}\rangle_{d}|j_{N-1}-j_{N}\rangle_{a_{N}b_{N}}\ldots|n-j_{1}\rangle_{a_{1}b_{1}}\;.$
(VI.13)

As I pointed out earlier, it is possible to approximate the path integral
using a single time-slice in the static path approximation (SPA), see e.g.
Arve  _et al._ (1988); Lang  _et al._ (1993). Such an approximation can yield
good results at very low temperatures, when self-consistent temporal
fluctuations can safely be ignored. However, SPA calculations of the black
hole entropy using the trilinear Hamiltonian lead to an oscillating behavior
of the black hole entropy Nation and Blencowe (2010); Alsing (2015),
suggesting that self-consistency of fluctuations are an important element of
Page curves.

Using the time-dependent out-density matrix

$\displaystyle\rho_{\rm
out}(t)=\mathop{|\Psi(t)\rangle}\nolimits_{dab}\langle\Psi(t)|$ (VI.14)

we can define the black hole density matrix $\rho_{\rm BH}$ by tracing over
the Hawking and partner modes:

$\displaystyle\rho_{\rm BH}(t)={\rm Tr}_{ab}\,\rho_{\rm out}(t)\;,$ (VI.15)

so that the black hole entropy is

$\displaystyle S_{\rm BH}(t)=-{\rm Tr}_{d}\,\rho_{\rm BH}(t)\log\rho_{\rm
BH}(t)\;.$ (VI.16)

The density matrix $\rho_{\rm BH}$ can be written entirely in terms of the
probabilities $p(j|i)=|U_{ij}|^{2}$ introduced earlier, which stand for the
probability to turn $i$ black hole modes into $j$ Hawking/partner modes (there
are always as many partner modes as there are Hawking modes since they are
always created in pairs). We find

$\displaystyle\rho_{\rm BH}(t)=\sum_{j_{N}=0}^{n}\limits
p_{j_{N}}|j_{N}\rangle\langle j_{N}|$ (VI.17)

where

$p_{j_{N}}=\sum_{j_{1}>j_{2}>\ldots>j_{N-1}}^{n}|U_{nj_{1}}^{(1)}|^{2}\ldots|U^{(N-1)}_{j_{N-2}j_{N-1}}|^{2}|U_{j_{N-1}j_{N}}^{(N)}|^{2}.$
(VI.18)

The probabilities $p(j|i)=|U_{ij}|^{2}$ are difficult to evaluate. Unlike in
the case when we were dealing with the ”free-field” Hamiltonian (III.5) that
allowed the associated $U=e^{iH}$ to be factorized using the SU(2) and SU(1,1)
disentangling theorems, the unitary operator $W^{(i)}$ does not appear to be
factorizable in a simple way. The usual formal factorization formulas Magnus
(1954); Suzuki (1976); Trotter (1959) are not suitable for practical
calculations.

In the absence of a disentanglement decomposition of $W^{(i)}$, we might
entertain the idea to simply perform a Taylor expansion of the exponential in
(VI.10) in terms of $r_{0}t$. However, even for moderate $r_{0}t$, the Taylor
expansion is prohibitively inefficient, requiring of the order of about
$2^{500}$ terms for $n=50$ and $\Delta t=1/15$. Fortunately, a method
developed by Brádler Brádler (2015) makes it possible to evaluate matrix
elements of $W^{(i)}$ in terms of an integer lattice known as a generalized
Dyck path Stanley (1999) as long as $W^{(i)}$ acts on any state generated by
the repeated action of $da^{\dagger}b^{\dagger}$ on a ground state
$|0\rangle$, defined by $d^{\dagger}ab|0\rangle=0$. It so happens that the
basis elements $\\{|j\rangle_{d}|00\rangle_{ab}\\}_{j=0}^{n}$ spanning the
input Hilbert space of $W^{(i)}$ are all ground states of $H_{i}$.

Using Brádler’s nearly miraculous Dyck-path representation of $W^{(i)}$ (which
generates a polynomial rather than exponential number of terms) we can
evaluate $\rho_{\rm BH}$ for black holes with initial quanta up to $n=50$,
using the discretized path integral (VI.13). Fig. 6.4 shows the black hole
entropy $S_{\rm BH}$ as a function of the number of time slices used, for a
small $\Delta t=0.15$, for black holes with $n=5$, $n=20$, and $n=50$. As the