# Information Theoretical Approach to Detecting Quantum Gravitational
Corrections

Behnam Pourhassan1,2,3,4 111email address<EMAIL_ADDRESS>Xiaoping
Shi2, 222email address<EMAIL_ADDRESS>Salman Sajad Wani5 333email
address<EMAIL_ADDRESS>Saif-Al-Khawari5 444email address:
<EMAIL_ADDRESS>, Farideh Kazemian1 555email address:
<EMAIL_ADDRESS>İzzet Sakallı6 666email address:
<EMAIL_ADDRESS>Naveed Ahmad Shah4,7 777email address:
<EMAIL_ADDRESS>Mir Faizal2,4 888email address:
<EMAIL_ADDRESS>1School of Physics, Damghan University, Damghan
3671645667, Iran.  2Irving K. Barber School of Arts and Sciences, University
of British Columbia Okanagan, Kelowna, BC V1V 1V7, Canada.  3Physics
Department, Istanbul Technical University, Istanbul 34469, Turkey.  4Canadian
Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada.  5 Qatar
Center for Quantum Computing, Hamad Bin Khalifa University, Doha, Qatar
6Physics Department, Eastern Mediterranean University, Famagusta 99628, North
Cyprus via Mersin 10, Turkey.  7Department of Physics, Jamia Millia Islamia,
New Delhi- 110025, India.

###### Abstract

One way to test quantum gravitational corrections is through black hole
physics. In this paper, We investigate the scales at which quantum
gravitational corrections can be detected in a black hole using information
theory. This is done by calculating the Kullback-Leibler divergence for the
probability distributions obtained from the Parikh-Wilczek formalism. We
observe that the quantum gravitational corrections increase the Kullback-
Leibler divergence as the mass of the black hole decreases, which is expected
as quantum gravitational corrections can be neglected for larger black holes.
However, we further observe that after a certain critical value, quantum
gravitational corrections tend to decrease again as the mass of the black hole
decreases. To understand the reason behind this behavior, we explicitly obtain
Fisher information about such quantum gravitational corrections and find that
it also increases as the mass decreases, but again, after a critical value, it
decreases. This is because at such a scale, quantum fluctuations dominate the
system and we lose information about the system. We obtain these results for
higher-dimensional black holes and observe this behavior for Kullback-Leibler
divergence and Fisher information depending on the dimensions of the black
hole. These results can quantify the scale dependence and dimension dependence
of the difficulty in detecting quantum gravitational corrections.

###### Contents

  1. I Introduction
  2. II Quantum Gravitational Corrected Geometry
  3. III Kullback-Leibler divergence
  4. IV Fisher Information
  5. V Stability at Quantum Scales
  6. VI Conclusion
  7. VII Acknowledgements

##  I Introduction

Various different proposals for quantizing gravity lead to different
theoretical modifications of low-energy quantum phenomena le1 ; le2 ; le4 ;
le5 ; le6 ; le7 . Thus, it is important to detect effects produced by quantum
gravity, and various tests have been proposed to detect such quantum
gravitational effects qge12 ; qge14 . Among them, it is speculated that black
hole physics can be used to test quantum gravitational effects qge15 ; qge16 .
Although several tests for quantum gravity have been proposed, the scale at
which quantum gravity effects are likely to be observed has not been
rigorously discussed. To properly classify and quantify the dependence of
quantum gravitational effects on the scale, we analyze the probability
distribution of particles emitted from a black hole in the Parikh-Wilczek
formalism pw ; pw1 . As the Parikh-Wilczek formalism considers the back
reaction of particles emitted during black hole evaporation, it is sensitive
to changes in the geometry due to quantum gravitational effects. This feature
of the formalism thus allows the determination of the change in the
probability distribution of the emitting particles due to these effects. To
quantify the scale dependence of such quantum gravitational effects, we use
information-theoretical techniques. We start by using the Kullback-Leibler
divergence kl1 ; kl2 , which measures the difference between two statistical
probability distributions. While not a metric due to its asymmetry, Kullback-
Leibler divergence effectively quantifies the deviation between two
distributions. Thus, a higher Kullback-Leibler divergence would mean that the
corrected probability distribution would have a higher deviation from the
original probability distribution. This, in turn, would make it easier to
experimentally detect the effects of such corrections. Accordingly, the
Kullback-Leibler divergence can clearly quantify and classify the effects of
quantum gravitational corrections. We observe that Kullback-Leibler divergence
first increases as the mass of the black hole decreases, but beyond a certain
critical value, it starts decreasing. We claim that this is due to the loss of
information on very small scales. To quantify this claim, we use Fisher
information, which effectively quantifies information about a parameter that
can be obtained from a given distribution fish1 ; fish2 . We thus directly use
Fisher information to analyze the information we obtain about quantum
gravitational corrections. We demonstrate that Fisher information about
quantum gravity also first increases and then decreases beyond a certain
critical value. Moreover, we find that this critical value for the Kullback-
Leibler divergence and Fisher information depends on dimensions. Our approach
allows us to thoroughly evaluate the influence of quantum gravity on black
holes across various scales and dimensions. This can be directly related to
the possibility of detecting quantum gravitational corrections. We also note
that quantum gravitational effects can occur at different scales in different
dimensions due to the lowering of the Planck scale in higher dimensions k1 ;
k2 ; b1 ; b2 . This observation has motivated the study of higher-dimensional
Schwarzschild black holes ed1 ; ed2 ; ed3 ; ed4 . Thus, we analyze the effects
of quantum gravitational corrections using such higher-dimensional
Schwarzschild black holes. This allows us to explicitly investigate the
effects of dimensions on quantum gravitational corrections, which can be
determined by using an effective quantum corrected metric and by introducing
the concept of a novel quantum mass j6 . In fact, this quantum mass becomes
instrumental in analyzing changes in the probability distribution of emitted
particles in the Parikh-Wilczek formalism pw ; pw1 .

The phenomenological consequences of quantum gravitational corrections to
black hole thermodynamics have been used to propose tests of quantum
gravitational effects qge17 ; qge18 ; qge19 ; qge10 , noting that the standard
black hole thermodynamics is a semi-classical theory, which is obtained using
quantum field theory in curved spacetime 1a00 ; 1b ; 1c ; 1a0 . In this
approach, the thermodynamic properties of black holes are obtained by
neglecting the quantum gravitational corrections. Using this equilibrium
description, the entropy of black holes scales with the area of their event
horizon, while their temperature scales with the surface gravity 1c ; 1a0 .
This approximation holds true only for sufficiently large black holes. At
these larger scales, the temperature is exceedingly low, allowing us to
disregard thermal fluctuations and use equilibrium thermodynamics. However, as
black holes reduce in size due to Hawking radiation Kanti:2014dxa ;
Pappas:2016ovo ; Sakalli:2022xrb , quantum gravitational corrections can no
longer be neglected. Similarly, at such smaller scales, the temperature rises
and thermal fluctuations cannot be dismissed. These fluctuations can be
explored as perturbative corrections to equilibrium thermodynamics, leading to
logarithmic corrections to the entropy of a black hole 32 ; 32a ; 32b ; 32c ;
32d . Furthermore, it is known that the geometry can be derived from
thermodynamics using the Jacobson formalism gr12 . Therefore, quantum
fluctuations in geometry can be connected to thermal fluctuations in
equilibrium thermodynamics using the Jacobson formalism gr14 . This insight
has spurred research into quantum gravitational corrections for various black
hole scenarios 12P ; 12ap ; 13p ; 19p ; 4p ; hbh . Additionally, the
holographic principle 4a ; 5a  has been used to analyze quantum gravitational
corrections in black hole thermodynamics 6a ; 7a ; 18 ; 18a ; 18b ; 18c ; 18d
. This was achieved by investigating the back reaction of the geometry using
the finite $N$ limit of boundary conformal field theory. These corrections
correspond to $\alpha^{\prime}$ corrections in the geometry, which, in turn,
give rise to higher curvature corrections. These higher curvature corrections
are known to introduce modifications to the standard thermodynamics of black
holes wald1 ; wald2 . Various other approaches have also been used to study
the effects of quantum gravity on black holes 6a ; 7a ; 18 ; 18a ; 18b ; 18c ;
18d . These corrections correspond only to perturbative quantum gravitational
corrections to geometry, which are obtained from perturbative thermal
corrections to equilibrium thermodynamics. These corrections occur at a scale
so small that neglecting the perturbative corrections to equilibrium
thermodynamics is not permissible. However, this scale is still much larger
than the Planck scale, near which the full non-perturbative quantum
gravitational corrections become significant, making it impossible to analyze
the geometry using only perturbative quantum gravitational corrections.

Non-perturbative quantum gravitational corrections to the thermodynamics of
various black holes have been investigated 6ab ; 7ab ; 7ba , leading to the
observation that such corrections can introduce non-trivial modifications to
the thermodynamics of black holes. Non-perturbative quantum gravitational
corrections to the thermodynamics of black holes have also been derived using
string theoretical effects Dabholkar ; ds12 ; ds14 . Additionally, general
arguments based on the properties of conformal field theories have been used
to obtain such corrections main . The influence of full non-perturbative
quantum gravitational corrections on the thermodynamic behavior of small four-
dimensional Schwarzschild black holes bia , Born–Infeld black holes bia1 , AdS
black holes j1 , Myers-Perry black holes j2 , and a system of M2-M5 branes j6
have been investigated. These non-perturbative corrections introduce an
exponential term to the black hole entropy, consequently modifying other
thermodynamic quantities. Furthermore, these corrections play a crucial role
in the short-distance stability analysis of quantum-scale black holes,
affecting their heat capacity, which, in turn, determines their stability.
This has significant implications for black hole evaporation, including the
black hole information paradox Mathur:2009hf ; Almheiri:2019psf ;
Harlow:2014yka ; Preskill:1992tc ; Sakalli:2012zy ; Sakalli:2010yy . Thus, it
becomes important to properly classify the effects of such corrections at
various scales and in various dimensions, which we accomplish in this paper.

##  II Quantum Gravitational Corrected Geometry

In this section, we will analyze the effects of quantum gravitational
corrections on the geometry of a higher-dimensional Schwarzschild black hole.
We begin by first reviewing the properties of a higher-dimensional black hole.
The metric of $D$-dimensional Schwarzschild black hole can be expressed as
follows SPR :

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r))}+r^{2}d\Omega_{D-2}^{2},$ (1)

where the metric function $f(r)$ is given by,

$\displaystyle f(r)=1-\frac{16\pi G_{D}M}{(D-2)\Omega_{D-2}r^{D-3}}.$ (2)

Here, $G_{D}$ represents the $D$-dimensional Newton’s constant, and $M$
represents the black hole mass. Additionally, $d\Omega_{D-2}^{2}$ represents
the metric of the ${D-2}$-dimensional unit sphere with an area of
$\Omega_{D-2}=2{\pi}^{\frac{D-1}{2}}/\Gamma\left({\frac{D-1}{2}}\right)$.
Using this metric, the radius of the horizon can be determined by the
condition $f(r=r_{0})=0$ and can be explicitly expressed as $r_{0}=\left(16\pi
G_{D}M/(D-2)\Omega_{D-2}\right)^{\frac{1}{D-3}}$. The original equilibrium
entropy for the black hole is given by $S_{0}=\Omega_{D-2}r_{0}^{D-2}/4G_{D}$.
Substituting this expression for the radius of the horizon into $S_{0}$, the
original equilibrium entropy of the black hole can be written as

$\displaystyle S_{0}=\frac{1}{4G_{D}}\Omega_{D-2}\left(\frac{16\pi
G_{D}M}{(D-2)\Omega_{D-2}}\right)^{\frac{D-2}{D-3}}.$ (3)

Hawking temperature $T_{0}$ of the $D$-dimensional Schwarzschild black hole is
given by $T_{0}=(D-3)/4\pi r_{0}$. It has been argued that non-perturbative
quantum gravitational effects produce an exponential correction to the
equilibrium entropy of the black hole, such that main ; Dabholkar ; ds12 ;
ds14 ; exp12 ; exp14

$\displaystyle S_{Q}=S_{0}+\eta e^{-S_{0}},$ (4)

where $S_{Q}$ is the quantum gravitationally corrected entropy of the black
hole and $\eta=[0,1]$ is a parameter.

It has been observed that the thermodynamic behavior of small black holes is
sensitive to the value of $\eta$ bia ; bia1 ; j1 ; j2 . In the limit where
$\eta=0$, we obtain the original equilibrium entropy, and in the limit where
$\eta=1$, we obtain the corrections predicted by string theory Dabholkar ;
ds12 ; ds14 . However, since the parameter governing quantum gravitational
corrections to the black hole depends on the details of the approach 12P ;
12ap ; 13p ; 19p ; 4p ; hbh , we shall treat $\eta$ as a general parameter and
analyze the behavior of the black hole for various values of $\eta$ bia ; bia1
; j1 ; j2 .

We can now utilize this quantum gravitationally corrected entropy to analyze
the effects of quantum gravitational corrections on the geometry of a
$D$-dimensional Schwarzschild black hole. It is well-established that this
geometry can be derived through thermodynamics in the Jacobson formalism gr12
. Consequently, at short distances, quantum fluctuations in the geometry can
be linked to thermal fluctuations in the thermodynamics of the black hole gr14
. Therefore, it becomes possible to use corrections to the equilibrium entropy
of a black hole to derive quantum gravitational corrections to its metric.
This approach is designed to ensure that the quantum-corrected metric directly
yields the quantum gravitational corrected entropy. In pursuit of this
objective, we introduce a modified metric for a Schwarzschild black hole,
which naturally generates the corrected entropy in an exponential form. Thus,
the original metric (1) undergoes modification to a new quantum-corrected
effective metric, with $f(r)_{Q}$ replacing the original $f(r)$, such that

$\displaystyle f(r)_{Q}=1-\frac{16\pi G_{D}\left(1-\eta e^{-\frac{\pi
r_{0}^{D-2}}{G_{D}}}\right)^{D-3}M}{(D-2)\Omega_{D-2}r^{D-3}}.$ (5)

The standard entropy obtained from this quantum-corrected metric is the
quantum gravitational-corrected entropy of the black hole given in Eq. (4).
This quantum-corrected metric can also be used to investigate the effects of
quantum gravitational corrections on various thermodynamic quantities.
Consequently, we observe that the temperature is modified due to quantum
gravitational effects as $T_{Q}=f^{\prime}(r_{0})_{Q}/4\pi=(D-3)/4\pi
r_{0}(1-\eta e^{-S_{0}})$.

However, to analyze the importance of a quantum gravitationally corrected
black hole, it is useful to define a novel quantum mass j6 . This will also
become important for analyzing the impact of quantum gravitational corrections
on the system. Such a quantum mass would reduce to the original mass when
quantum gravitational corrections are neglected. However, in the limit these
quantum gravitational corrections are considered, the quantum mass replaces
the original mass in the metric. Using the expression for the quantum
corrected metric given by Eq. 5, we can define a novel quantum mass for the
$D$-dimensional Schwarzschild black hole as

$M_{Q}=\left[1-\eta\exp\left(-\frac{\pi
r_{0}^{D-2}}{G_{D}}\right)\right]^{D-3}M.$ (6)

For large black holes, the effects of quantum gravitational corrections can be
neglected, and the analysis can be performed using the quantum mass, which
will coincide with the analysis conducted using the original mass of the black
hole; deviation will occur only at quantum scales.

##  III Kullback-Leibler divergence

In the previous sections, we analyzed the modifications to a higher-
dimensional black hole from quantum gravitational corrections. Now, we
quantify the deviations from the standard behavior due to such corrections.
This will be done using Kullback-Leibler divergence kl1 ; kl2 , which measures
how much two probability distributions differ from each other. Even though
Kullback-Leibler divergence is not symmetric, and hence not a metric, it does
give a realistic estimation of how far a given probability distribution is
from another probability distribution. It is possible to obtain the
probability distribution of the particles emitted from a black hole during its
evaporation using the Parikh-Wilczek formalism pw ; pw1 , which considers the
back reaction of the emitted particles on the black hole geometry and proposes
that the entropy of the black hole can be expressed in terms of the
probability distribution of the emitted particles. As the entropy of the black
hole is modified by quantum gravitational corrections, this probability
distribution in the Parikh-Wilczek formalism will also be modified. We will
explicitly analyze such modifications to the probability distribution of the
emitted particles, and use them to estimate the deviation of the behavior of a
quantum-corrected black hole from the original black hole. For a higher
dimensional Schwarzschild black hole, we assume that $p_{n}$ is the
probability that the black hole evaporates by radiating $n$ particle. We can
write this probability as

$p_{n}=\frac{\Omega_{n}}{\Omega_{total}},$ (7)

where $\Omega_{n}$ is the number of microstates when a particle evaporates
into $n$ particle, and $\Omega_{total}=\sum_{1}^{\infty}\Omega_{n}$ is the
total number of the possible microstates of the system. It has been
demonstrated that for a Schwarzschild black hole of mass $M$, $p_{n}$ can be
expressed in terms of $M$ pw1 . Although the expression was explicitly
analyzed for a four-dimensional black hole, the arguments used to obtain this
distribution pw1  are very general and hold in any dimension. Thus, we can
write this probability distribution for a higher-dimensional Schwarzschild
black hole as

$p_{n}=\frac{(4\pi G_{D}M^{2})^{n-1}}{(n-1)!}e^{-4\pi G_{D}M^{2}}.$ (8)

The probability distribution $p_{n}$ is normalized. As the black hole
evaporates by emitting $n$ particles, the probability that one among
$\Omega_{n}$ microstates occurs is $q_{\alpha}$. Since it is assumed that all
of the $\Omega_{total}$ microstates occur with the same probability, we can
write $q_{\alpha}=1/\Omega_{n}$. The total probability distribution for both
the probabilities of the number of particles that a four-dimensional
Schwarzschild black hole emits during its evaporation and the possible
microstates has been obtained pw1 . This can be easily generalized to higher
dimensions, where this probability distribution can be written as

$P_{n}(0)=q_{\alpha}\times
p_{n}=\frac{1}{\Omega_{total}}=\frac{1}{\sum_{n=1}^{\infty}\frac{(4\pi
G_{D}M^{2})^{n-1}}{(n-1)!}e^{-4\pi G_{D}M^{2}}}=e^{-4\pi G_{D}M^{2}},$ (9)

where the zero in $P_{n}(0)$ denotes the absence of any quantum gravitational
corrections. Here, we observe that this probability distribution depends on
the mass of the black hole.

As the arguments are general, these also hold for a quantum-sized black hole,
with the mass being replaced by the novel quantum mass of the black hole. This
is because the quantum mass naturally reproduces the quantum-corrected entropy
of the quantum-sized black hole. As the entropy obtained from the Parikh-
Wilczek formalism pw ; pw1  has to be considered with the entropy of the black
hole obtained from standard methods, the quantum gravitationally corrected
entropy produced by the Parikh-Wilczek formalism pw ; pw1  should also
coincide with the quantum gravitationally corrected entropy produced using
standard methods. Thus, we can also write the probability distribution for a
quantum-corrected higher-dimensional Schwarzschild black hole, using the novel
quantum mass $M_{Q}=[1-\eta\exp{-(\pi r_{0}^{D-2}/G_{D}})]^{D-3}M$

${P}_{n}(\eta)=\frac{1}{\sum_{n=1}^{\infty}\frac{(4\pi
GM_{Q}^{2})^{n-1}}{(n-1)!}e^{-4\pi GM_{Q}^{2}}}=e^{-4\pi GM_{Q}^{2}}.$ (10)

Since $\eta$ is a very small constant, the equation for the modified mass can
be approximated as $M_{Q}=M(1+(3-D)\eta e^{-\frac{\pi r^{D-2}}{g}})$. It may
be noted that at ${P}_{n}(\eta)|_{\eta=0}=P_{n}(0)$. We can use the Kullback-
Leibler divergence kl1 ; kl2  to measure deviations from quantum gravitational
corrections. The Kullback-Leibler divergence is given by

$\displaystyle D_{KL}({P}_{n}(\eta)||P_{n}(0))$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\Omega_{total}}{P}_{n}(\eta)\log\frac{{P}_{n}(0)}{P_{n}(\eta)}=\sum_{n=1}^{\Omega_{total}}e^{-4\pi
GM_{Q}^{2}}\log\frac{e^{-4\pi GM_{Q}^{2}}}{e^{-4\pi GM^{2}}}$ (11)
$\displaystyle=$ $\displaystyle e^{-4\pi M^{2}G\eta\alpha}\log\frac{e^{-4\pi
GM_{Q}^{2}}}{e^{-4\pi GM^{2}}}=e^{(4\pi GM^{2}\eta\alpha)}(4\pi
GM^{2}\eta\alpha).$

where $\alpha=(D-3)e^{{-\pi r^{D-2}}/{G}}$.

It is interesting to note that for this case, although
$D_{KL}({P}_{n}(\eta)||P_{n}(0))$ is well defined,
$D_{KL}(P_{n}(0)||{P}_{n}(\eta))$ is not. The reason is that for two
distributions $P(X)$ and $Q(Y)$ to have a valid $D_{KL}(P(X)||Q(Y))$, $\forall
P(X)=0$, we must have $Q(Y)=0$ and $\Re_{Y}\subseteq\Re_{X}$. The first
condition is easily satisfied by both divergences when the distributions are
taken to be ${P}_{n}(\eta)$ and $P_{n}(0)$. We see the range of the variable
for ${P}_{n^{\prime}}(\eta)\in[1,\Omega^{\prime}_{total}=e^{4\pi GM_{Q}^{2}}]$
and $P_{n}(0)\in[1,\Omega_{total}=e^{4\pi GM^{2}}]$. It is easy to see that
$M\geq M_{Q}$ for all positive values of $\eta$, hence,
$\Re_{n^{\prime}}\subseteq\Re_{n}$. Thus, only
$D_{KL}({P}_{n}(\eta)||P_{n}(0))$ is well defined.

We observe from Fig. 1, that the Kullback-Leibler divergence depends not only
on the size of the black hole but also on its dimensions. As the dimensions
increase, the Kullback-Leibler divergence becomes significant even for
relatively larger masses. This implies that in higher dimensions, quantum
gravitational effects can occur even at larger scales. It has already been
pointed out that the Planck scale decreases as dimensions increase, and,
consequently, quantum gravitational effects can become significant k1 ; k2 .
Here, we have been able to rigorously demonstrate it using Kullback-Leibler
divergence. However, what is rather unexpected is that the Kullback-Leibler
divergence again decreases, as the mass becomes lower than a certain critical
value. Thus, it seems that to measure quantum gravitational corrections, a
certain critical scale exists, and this scale depends on the dimensions. Above
this scale, the effect of these corrections becomes negligible, which is
expected. Also, going below this critical scale, the effect of the corrections
diminishes. The reason for such a reduction seems to be that below this scale,
the quantum fluctuations dominate the systems, and we seem to lose information
about the system. The loss of information corresponds to a loss of making
predictions, and hence the Kullback-Leibler divergence between the original
and corrected probability distributions vanishes. This also implies that there
is a certain critical value at which we can test quantum gravity. We observe
that as we increase the dimensions, the range of this value reduces, and this
makes it harder to detect quantum gravitational corrections in extra
dimensions.

Figure 1: $D_{KL}$ as a function of $M$ and $\eta$ for $D=4$,$D=5$,$D=6$, and
$D=11$.

##  IV Fisher Information

In the previous section, we analyzed the Kullback-Leibler divergence between
the original and corrected probability distributions and observed that it
reduces with masses lower than a certain critical value. We speculated that
this was due to a loss of information. In this section, we will quantify this
argument by using the concept of Fisher information. Fisher information
measures the information about a parameter that can be obtained from a
probability distribution fish1 ; fish2 . To analyze how much information can
be obtained about quantum gravitational corrections from the probability
distribution of particles emitted from a black hole during its evaporation, we
obtain Fisher information of the parameter $\eta$. As Fisher information is
related to Kullback-Leibler divergence fish4 ; fish5 , we measure Fisher
information by first defining Kullback-Leibler divergence between two
corrected probability distributions over different values of $\eta$; namely:
$P_{n}(\eta)$ and $P_{n}(\eta+\delta\eta)$. We measure the Fisher information
associated with $\eta$, as we move in the $\eta$ space between $\eta$ and
$\eta+\delta\eta$. In the limiting case, we will get our previous results
back, when we set $\eta=0$. Now we can use the Taylor expansion to expand
$P_{n}(\eta+\delta\eta)$, and find its relation to $P_{n}(\eta)$

$P_{n}(\eta+\delta\eta)\approx P_{n}(\eta)+\delta\eta\frac{\partial
P_{n}(\eta)}{\partial\eta}=P_{n}(\eta)+\delta P_{n}(\eta).$ (12)

Here, we define the correction to the probability distribution from $\eta$ to
$\eta+\delta\eta$ as $\delta P_{n}(\eta)$. We can write the Kullback-Leibler
divergence between the probability distributions at $\eta$ and
$\eta+\delta\eta$ as

$\displaystyle D_{KL}(P_{n}(\eta+\delta\eta)||P_{n}(\eta))$ $\displaystyle=$
$\displaystyle\sum^{\Omega}_{n=1}(P_{n}(\eta+\delta\eta)\log\frac{P_{n}(\eta+\delta\eta)}{P_{n}(\eta)}).$
(13)

We write the argument of the logarithm as $(1+\delta
P_{n}(\eta)/P_{n}(\eta))$, and thus obtain

$\displaystyle D_{KL}(P_{n}(\eta+\delta\eta)||P_{n}(\eta))$ $\displaystyle=$
$\displaystyle\sum^{\Omega}_{n=1}\left(P_{n}(\eta)+\delta
P_{n}(\eta)\log\frac{P_{n}(\eta)+\delta P_{n}(\eta)}{P_{n}(\eta)}\right)$ (14)
$\displaystyle=$
$\displaystyle\sum^{\Omega}_{n=1}\left(P_{n}(\eta)\log\left(1+\frac{\delta
P_{n}(\eta)}{P_{n}(\eta)}\right)\right)+\sum^{\Omega}_{n=1}\log\left(1+\frac{\delta
P_{n}(\eta)}{P_{n}(\eta)})\delta P_{n}(\eta)\right).$

We expand the logarithm term up to the second order in $\delta P_{n}$, and
neglect other higher order terms. Thus, using $\log(1+{\delta
P_{n}(\eta)}/{P_{n}(\eta)})\approx({\delta
P_{n}(\eta)}/{P_{n}(\eta)})-({(\delta P_{n}(\eta))^{2}}/{2P_{n}(\eta)^{2}})$,
we write

$\displaystyle
D_{KL}(P_{n}(\eta+\delta\eta)||P_{n}(\eta))\approx\sum^{\Omega}_{n=1}\delta
P_{n}(\eta)+\frac{1}{2}\sum^{\Omega}_{n=1}\left(P_{n}(\eta)\frac{(\delta
P_{n}(\eta))^{2}}{P_{n}(\eta)^{2}}\right).$ (15)

We can use the expression for the probability distributions, and obtain an
explicit expression for the Kullback-Leibler divergence in terms of the mass
of the black hole. Moreover, we express $\delta
P_{n}(\eta)=\delta\eta{\partial P_{n}(\eta)}/{\partial\eta}$ , and $(\delta
P_{n}(\eta))^{2}=(\delta\eta)^{2}({\partial P_{n}(\eta)}/{\partial\eta})^{2}$.
Putting these back in the expression above we can express the Kullback-Leibler
divergence as

$\displaystyle D_{KL}(P_{n}(\eta+\delta\eta)||P_{n}(\eta))$
$\displaystyle\approx$
$\displaystyle\sum^{\Omega}_{n=1}\delta\eta\frac{\partial
P_{n}(\eta)}{\partial\eta}+\sum\frac{1}{2}(\delta\eta)^{2}P_{n}(\eta)\Bigg{(}\frac{1}{P_{n}(\eta)}\frac{\partial
P_{n}(\eta)}{\partial\eta}\Bigg{)}^{2}$ (16) $\displaystyle\approx$
$\displaystyle\delta\eta\frac{\partial\sum^{\Omega}_{n=1}\exp{4\pi
GM^{2}(1+\eta\alpha)}}{\partial\eta}$
$\displaystyle+\frac{1}{2}\delta\eta^{2}\sum^{\Omega}_{n=1}\exp{4\pi
GM^{2}(1+\eta\alpha)}\Big{(}\frac{\partial}{\partial\eta}(\log(\exp{4\pi
GM^{2}(1+\eta\alpha)})\Big{)}^{2}.$

Since the argument say $F((P_{n}(\eta))$ of the summations in the expressions
above does not have an explicit $\eta$ dependence, we can write
$\sum_{n=1}^{\Omega}F(P_{n}(\eta))=\Omega F(P_{n}(\eta))$. Thus, we can write

$\displaystyle D_{KL}(P_{n}(\eta+\delta\eta)||P_{n}(\eta))$
$\displaystyle\approx$
$\displaystyle(\delta\eta)^{2}\sum^{\Omega}(\frac{1}{2}{\exp{4\pi
GM^{2}(1+\eta\alpha)}\frac{\partial}{\partial\eta}(log(\exp{-4\pi
GM^{2}(1+\eta\alpha)})})^{2}.$ (17)

The first time is zero as $\Omega=\exp{4\pi GM^{2}(1+\eta\alpha)}$ and so
$\sum^{\Omega}_{n=1}\exp{-4\pi GM^{2}(1+\eta\alpha)}=1$. Moreover, since
Kullback-Leibler divergence has a minima at $\eta=0$, the first derivative of
Kullback-Leibler divergence is zero at $\eta=0$,

$D_{KL}(P_{n}(\eta+\delta\eta)||P_{n}(\eta))=\frac{1}{2}(\delta\eta)^{2}\textbf{E}\left(\Bigg{(}\frac{\partial\log
P(\eta)}{\partial\eta}\Bigg{)}^{2}\right).$ (18)

Here E denotes the expectation value. Thus, the Fisher Information $FI$ can be
obtained from Kullback-Leibler divergence as

$FI(\eta)\approx\frac{\partial^{2}}{\partial^{2}\eta}D_{KL}(P_{n}(\eta+\delta\eta)||P_{n}(\eta))|_{\delta\eta=0}=\textbf{E}\Bigg{(}\frac{\partial\log
P(\eta)}{\partial\eta}\Bigg{)}^{2}\Bigg{|}_{\delta\eta=0}.$ (19)

Now using the explicit expression for the Kullback-Leibler divergence between
a corrected value and an original value at $\eta=0$, we can also write an
explicit expression for the Fisher information as (with $\alpha=(D-3)e^{{-\pi
r^{D-2}}/{G}}$ )

$FI\approx(8\pi GM^{2}\alpha)^{2}.$ (20)

We observe from 2, that the Fisher information about quantum gravity decreases
for larger black holes; this is expected, as the quantum gravitational
corrections only have negligible effects at larger scales. We also observe
that Fisher information decreases with mass after a certain critical value.
This explains the decrease in the Kullback-Leibler divergence, as the quantum
fluctuations dominate at such a scale, and we lose all information about the
system. Therefore, there seems to be a critical scale, at which we have
maximum Fisher information about the quantum gravitational corrections, and it
is around such a scale that the Kullback-Leibler divergence also maximizes
between the original and quantum gravitationally corrected probability
distributions. Using Fisher information, we could identify the reason for the
decreases in Kullback-Leibler divergence below this scale. We also observe
that, like the Kullback-Leibler divergence, the dependence of Fisher
information with scale also depends on the dimensions.

Figure 2: $FI$ as a function of $M$ i.e size of the blackhole for $D=4$,$D=5$,
$D=6$, and $D=11$.

##  V Stability at Quantum Scales

We will observe that quantum gravitational corrections can change the
stability at quantum scales. This can be done by first using the quantum
corrected metric to explicitly calculate the corrections to other
thermodynamic quantities. Therefore, we can express the Helmholtz free energy
of the black hole corrected for quantum gravitational effects as

$\displaystyle F_{Q}$ $\displaystyle=$ $\displaystyle-\int
S_{Q}dT_{Q}=\frac{\Omega_{D-2}}{16\pi
G_{D}}r_{0}^{D-3}+\frac{\eta(D-3)}{4\pi(D-2)}A\ \Gamma\left(\frac{-1}{D-2},A\
r_{0}^{D-2}\right),$ (21)

where $A\equiv\left(\Omega_{D-2}/4G_{D}\right)^{\frac{1}{D-2}}$. We can use
this expression for quantum gravitationally corrected Helmholtz free energy to
obtain the expression for quantum gravitationally corrected internal energy

$\displaystyle E_{Q}=F_{Q}+T_{Q}S_{Q}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{D-2}}{16\pi
G_{D}}r_{0}^{D-3}+\eta\frac{(D-3)}{4\pi(D-2)}A\Gamma(\frac{-1}{D-2},Ar_{0}^{D-2})$
(22) $\displaystyle+\frac{(D-3)}{r_{0}(1-\eta e^{-S_{0}})}(S_{0}+\eta
e^{-S_{0}}).$

We observe that various thermodynamic quantities for a higher-dimensional
black hole can also be directly obtained from the quantum gravitationally
corrected metric. However, to analyze the stability of a quantum
gravitationally corrected black hole, it is useful to define a novel quantum
mass j6 . We can explicitly express the quantum gravitationally corrected
specific heat in terms of the novel quantum mass as follows:

$\displaystyle C_{Q}$ $\displaystyle=$
$\displaystyle\frac{-[{S}^{\prime}(M_{Q})_{Q}]^{2}}{{{S}^{\prime\prime}}(M_{Q})_{Q}},=-\frac{[S_{0}^{\prime}(1-\eta
e^{-S_{0}})]^{2}}{S_{0}^{\prime\prime}(1-\eta
e^{-S_{0}})+\eta(S_{0}^{\prime})^{2}e^{-S_{0}}}$ (23) $\displaystyle=$
$\displaystyle-\frac{(D-2)S_{0}(1-\eta e^{-S_{0}})^{2}}{1+[(D-2)S_{0}-1]\eta
e^{-S_{0}}}$

In equation 23, the numerator is clearly positive, which means that in order
to have a positive specific heat, the denominator should be negative. Using
the condition for thermodynamics stability, $C_{Q}\geq 0$, we obtain
$1+[(D-2)S_{0}-1]\eta e^{-S_{0}}\leq 0.$ Now for example, for $D=4$ with
$G_{4}=G$, we have, $1+(8\pi GM^{2}-1)\eta e^{-4\pi GM^{2}}\leq 0.$

Figure 3: The lower limit of $\eta$ versus $r_{0}$ to satisfy the stability
condition (positive specific heat).

For a fixed mass of a black hole, we can find the lower limit for $\eta$ for
which the black hole is stable. This is illustrated in Fig. 3, where the lower
limit of $\eta$ is plotted with respect to $r_{0}$. In Fig. 3, we observe
three different regions. On the right, we plot the case with $(D-2)S_{0}>1$.
The corrections are negative at large radii, and the correction terms vanish
at large radii. In the center, we plot the case with $(D-2)S_{0}\approx 1$.
Here the correction coefficient diverges. n the left, we plot the case with
$(D-2)S_{0}<1$. The correction coefficient $\approx 1$ as the horizon radius
decreases. The third case agrees with the earlier observations about quantum
gravitational corrections main , so physically $\eta\approx 1$. We know that
the Schwarzschild black hole is unstable (at $\eta=0$), but in the presence of
the exponential correction with the correction coefficient $\eta\geq 1$, it
may become stable at small radii. In plots of Fig. 4, we can see the behavior
of the specific heat in terms of $r_{0}$ and see that in the presence of the
exponential correction, the Schwarzschild black hole is stable at small radii.
Although, we have only analyzed two cases, i.e., $D=4$ (Fig. 4 (a)) and $D=5$
(Fig. 4 (b)), we expect to find similar behavior in other dimensions.

##  VI Conclusion

In this paper, we analyze the effects produced by quantum gravitational
corrections on a black hole. This was done by calculating the Kullback-Leibler
divergence between the probability distribution of the original particles
emitted during the evaporation of a black hole, and the probability
distribution corrected by quantum gravitational corrections. We observe that
Kullback-Leibler divergence depends both on the scale and dimensions of the
system, and increases as the mass decreases, as expected. However, after a
critical value, it starts to decrease as the mass decreases. This is because
after this critical point, the quantum fluctuations dominate, and we lose
information about the system. This is explicitly demonstrated to be the case
using Fisher information. We also observe that Fisher information about
quantum gravitational corrections exhibits similar behavior; first increasing
and then decreasing as the mass decreases. Fisher information first increases
as quantum gravitational corrections become significant as the size of the
black hole decreases. However, after a certain critical value, the quantum
fluctuations dominate the system, and we again start losing information about
quantum gravitational corrections. This explains why Fisher information first
increases and then decreases as the the size of the black hole decreases. This
observation has direct implications for detecting quantum gravitational
corrections. It easiest to measure the effects produced by quantum gravity at
the scale at which Kullback-Leibler divergence is maximum. Similarly, as the
range of this value for which the Kullback-Leibler divergence is high also
decreases with increasing dimensions, it will be very hard to detect quantum
gravitational effects in extra dimensions. This is further quantified by the
amount of Fisher information about quantum gravitational corrections. Thus, we
have been able to quantify the scale dependence and dimension dependence of
the difficulty of detecting quantum gravity. We have also commented on the
stability of these black holes at quantum scales.

Figure 4: Specific heat versus $r_{0}$ with $G_{D}=1$ for Schwarzschild black
hole.

##  VII Acknowledgements

İ.S. would like to acknowledge networking support of COST Action CA18108 -
Quantum gravity phenomenology in the multi-messenger approach. He also thanks
to TÜBİTAK and SCOAP3 for their support.

## Data Availability

There is no associated data in this paper.

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