# Quantumness near a Schwarzschild black hole

S. Haddadi  0000-0002-1596-0763<EMAIL_ADDRESS>Faculty of Physics,
Semnan University, P.O. Box 35195-363, Semnan, Iran  M. A. Yurischev
0000-0003-1719-3884 Federal Research Center of Problems of Chemical Physics
and Medicinal Chemistry, Russian Academy of Sciences, Chernogolovka 142432,
Moscow Region, Russia  M. Y. Abd-Rabbou  0000-0003-3197-4724 Mathematics
Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo,
Egypt  M. Azizi  0000-0002-1935-5923 Faculty of Physics, Semnan University,
P.O. Box 35195-363, Semnan, Iran  M. R. Pourkarimi  0000-0002-8554-1396
Department of Physics, Salman Farsi University of Kazerun, Kazerun, Iran  M.
Ghominejad  0000-0002-0136-7838<EMAIL_ADDRESS>Faculty of Physics,
Semnan University, P.O. Box 35195-363, Semnan, Iran

###### Abstract

The merging of quantum information science with the relativity theory presents
novel opportunities for understanding the enigmas surrounding the transmission
of information in relation to black holes. For this purpose, we study the
quantumness near a Schwarzschild black hole in a practical model under
decoherence. The scenario we consider in this paper is that a stationary
particle in the flat region interacts with its surroundings while another
particle experiences free fall in the vicinity of a Schwarzschild black hole’s
event horizon. We explore the impacts of Hawking radiation and decoherence on
the system under investigation and find that these effects can limit the
survival of quantum characteristics, but cannot destroy them completely.
Hence, the results of this study possess the potential to yield valuable
insights into the comprehension of the quantum properties of a real system
operating within a curved space-time framework.

Quantumness; Schwarzschild black hole; Hawking radiation; Decoherence

##  I INTRODUCTION

In 2000, Zurek introduced the concept of information-theoretic quantum discord
between two classically identical definitions of mutual information [1].
Quantum discord can be regarded as a measure of a violation of the
classicality of a joint state of two quantum subsystems. In the next year, he,
alongside Ollivier, incorporated the optimization procedure into his original
definition of quantum discord [2]. Therefore, for the first time in their
paper, we encountered the term “quantumness” of correlations. Quantum
correlations [3], a fundamental aspect of quantum mechanics, are crucial in
numerous quantum information processing tasks, including quantum cryptography,
quantum communication, and quantum computation [4]. Nonetheless, these
correlations are susceptible to fragility owing to the effect of a variety of
decoherence [5]. Literally, decoherence is the phenomenon wherein a quantum
system interacts with its environment, resulting in the loss of quantum
coherence and quantum correlations [6]. Generally, we know that quantum
systems interact with their environment in the real world [7]. As a result,
quantum correlations present in the initial state may experience a decay over
time. In the field of quantum computing, the computation process is
susceptible to accumulating errors caused by various decoherence mechanisms.
These errors can disrupt the accuracy of quantum gates and ultimately
influence the precision of the final results. Unfortunately, such errors have
the potential to affect the correlations necessary for quantum algorithms to
function properly.

On the one hand, environmental noise, which includes various external factors
such as electromagnetic radiation and temperature fluctuations, has the
potential to cause stochastic fluctuations in the parameters of a considered
quantum system [8]. These fluctuations possess the potential to induce
uncontrolled phase shifts and energy exchanges, thereby disturbing the quantum
correlations. On the other hand, dephasing occurs when different parts of a
quantum system’s wave function acquire different phases owing to interactions
with surrounding [9]. The phenomenon of dephasing can lead to the dissipation
of quantum correlation and coherence between quantum states [10, 11]. The
correlations that are most vulnerable to dephasing are those that pertain to
superpositions and phase differences, such as those that exist in Bell states
utilized for quantum entanglement. Moreover, the spontaneous emission may lead
to a loss of coherence in quantum states and result in reduced correlations,
particularly in cases where photon emission affects the entangled states [12].

In general, quantum measurements inherently perturb the measured quantum state
[13, 14, 15]. This disturbance can result in the collapse of the state into an
eigenstate of the measured observable, which can affect the correlations [16].
Repeated measurements can also perturb the correlations over time, especially
where entanglement is involved. However, in order to reduce the destructive
effects of decoherence on quantum correlations, various techniques such as
error correction codes, quantum error correction protocols, and decoherence-
free subspaces [17] are employed. These approaches help protect and extend the
lifetime of quantum correlations in the presence of environmental influences
[18, 19].

The dynamics of quantum correlations under decoherence effects and Hawking
radiation can be quite complex but intriguing, involving an intricate
interplay between gravity, quantum information, and environmental interactions
[20, 21, 22, 23, 24, 25, 26]. Hawking radiation appears near the event horizon
of a black hole due to quantum effects in the vicinity of the event horizon
boundary [27]. Particles are created in pairs near the event horizon, where
one particle falls into the black hole and the other escapes as radiation.
This process causes black holes to gradually lose their mass over time.
Besides, Hawking radiation can be witnessed as a form of decoherence because
of the interaction of quantum fields near the black hole. This process can
introduce disturbances and thermal noise that can affect quantum correlations,
just like environmental decoherence.

Interestingly, the process of Hawking radiation may generate entangled pairs
of particles near the black hole’s event horizon [28, 29, 30]. These produced
entangled pairs can carry away some of the black hole’s energy, contributing
to the mass loss of the black holes. Potentially, Hawking radiation can also
carry away entanglement, affecting the correlations between the infalling and
escaping particles. Notably, the interplay between quantum correlations and
Hawking radiation is linked to the black hole information paradox. Although
the information is conserved according to quantum mechanics, the Hawking
process seems to present that information can be lost when particles fall into
a black hole. This has led to debates regarding whether and how quantum
correlations can be reconstructed or preserved in the presence of Hawking
radiation.

Studying the combined effects of Hawking radiation and decoherence requires a
deep understanding of both quantum field theory and quantum information
theory, as well as a grasp of the underlying principles of black holes and
gravity. While some progress has been made in exploring these interactions
theoretically [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49], many aspects are still the subject of ongoing research and
debate. With this in mind, we are motivated to explore the quantumness
(quantum correlations, quantum coherence, and non-locality) of a physical
system consisting of two qubits in the background of the Schwarzschild black
hole under the influence of both the decoherence effect and Hawking radiation.

##  II Measures of quantum information correlation

###  II.1 Local quantum uncertainty

The local quantum uncertainty (LQU) is used when dealing with measurements and
interactions at a specific location or in a limited region of space. It is a
foundational characteristic of quantum mechanics that challenges our classical
intuition and also underlies many of the unique behaviors and phenomena
observed in the quantum world. Indeed, it is a measure of quantum correlations
based on the Wigner-Yanase skew information $\mathcal{I}$ [50, 51]. The LQU
pertaining to subspace of party $A$ with the measurement operator $H_{A}$,
upon optimization over all local observables on $A$, is explicitly given as
[52]

$\mathcal{U}(\varrho):=\min_{H_{A}}\mathcal{I}\left(\varrho,H_{A}\right).$ (1)

Therefore, the concept of LQU is precisely characterized as the minimum level
of quantum uncertainty that is inherently linked to a singular measurement
pertaining to one of the subsystems of the bipartite system, $AB$. It is
noteworthy to mention that LQU is a genuine measure of quantum correlations,
and it has been demonstrated that LQU satisfies all of the requisite physical
conditions that are necessary to qualify as a criterion of quantum
correlations. Gilorami et al. [52] were able to execute optimization for
qubit-qudit systems successfully and subsequently introduced the measure (1)
in a specific form as

$\mathcal{U}=1-\lambda_{\max}^{\mathcal{W}},$ (2)

where $\lambda_{\max}^{\mathcal{W}}$ represents the highest eigenvalue of the
three-by-three symmetric matrix $\mathcal{W}$, which comprises entries

$\mathcal{W}_{\nu\mu}=\textmd{tr}\\{\sqrt{\varrho}(\sigma_{\nu}\otimes
I)\sqrt{\varrho}(\sigma_{\mu}\otimes I)\\},$ (3)

where $\sigma_{\nu}$ and $\sigma_{\mu}$ represent the set of Pauli matrices
with $\nu,\mu=x,y,z$.

###  II.2 Local quantum Fisher information

Quantum Fisher information (QFI) is a concept from quantum metrology, which
deals with the precision of measurements in the quantum realm [53, 54, 55,
56]. In quantum mechanics, the Fisher information quantifies how much
information about a parameter of interest is contained in the outcomes of
measurements. It provides a measure of how well a quantum state can be
distinguished from nearby states in terms of the parameter being measured
[57]. The local quantum Fisher information (LQFI) measure $\mathcal{F}$ is
based on the QFI (indicated by $F$) and it is equivalent to the optimal LQFI
when the measurement operator $H_{A}$ is applied to the subsystem $A$ within
the bipartite system $AB$. Thus, the LQFI is as follows [57]

$\mathcal{F}(\varrho):=\min_{H_{A}}F\left(\varrho,H_{A}\right).$ (4)

Specifically, one can write the following formula for LQFI if the subsystem
$A$ is a qubit

$\mathcal{F}=1-\lambda_{\max}^{\mathcal{M}},$ (5)

where $\lambda_{\max}^{\mathcal{M}}$ denotes the highest eigenvalue of the
3-by-3 symmetric matrix $\mathcal{M}$, which its elements are

$\mathcal{M}_{\nu\mu}=\sum_{i,j;q_{i}+q_{j}\neq
0}\frac{2q_{i}q_{j}}{q_{i}+q_{j}}\langle\psi_{i}|\sigma_{\nu}\otimes
I|\psi_{j}\rangle\langle\psi_{j}|\sigma_{\mu}\otimes I|\psi_{i}\rangle,$ (6)

where $q_{i}$ and $|\psi_{i}\rangle$ are respectively the eigenvalues and
eigenstates of $\varrho=\Sigma_{i}q_{i}|\psi_{i}\rangle\langle\psi_{i}|$ with
$q_{i}\geq 0$ and $\Sigma_{i}q_{i}=1$.

##  III Theoretical framework

Let us commence by revisiting the precise definition of metric Hawking
radiation as it pertains to the Dirac field model in Schwarzschild black hole.
Typically, the mathematical model known as the Dirac equation is utilized to
explain the behavior of particles in relation to a space-time that is curved
in a general manner. This phenomenon is commonly expressed through the
following formula [58, 59]

$\left[\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Gamma_{\mu}\right)\right]\Psi=0,$
(7)

where $\gamma^{a}$ is Dirac matrix, $\Gamma_{\mu}$ is spin connection,
$e_{a}^{\mu}$ is the inverse of tetrad $e_{\mu}^{a}$, and $\mu$ is mass of the
Dirac field. In the background of Schwarzschild, the metric may be delineated
as a mathematical construct that characterizes the spatial and temporal
properties of space-time. This metric can be defined as

$ds^{2}=-fdt^{2}+\frac{1}{f}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (8)

with $f=1-2Mr^{-1}$, where $M$ denotes the mass of the black hole and $r$ is
the radial coordinates. In this paper, the gravitational constant, reduced
Planck constant, speed of light, and Boltzmann constant are assumed to be
equal to one for simplicity.

Within the framework of Schwarzschild space-time, it is now possible to derive
the Dirac equation as

$\displaystyle-\gamma_{0}f^{-1/2}\frac{\partial\Psi}{\partial
t}+\gamma_{1}f^{1/2}\left[\frac{\partial}{\partial
t}+r^{-1}+\frac{f^{-1}M}{2r^{2}}\right]\Psi$ (9)
$\displaystyle+\frac{\gamma_{2}}{r}\frac{\partial\Psi}{\partial\theta}+\frac{\gamma_{3}}{r\sin\theta}\frac{\partial\Psi}{\partial\varphi}=0.$

By solving equation (LABEL:Dirac_equation), it is feasible to derive the
outgoing solutions of positive (fermions) frequency for both the outside and
inside domains surrounding the event horizon as [59]

$\Psi_{k,out}^{\mathrm{I}+}=\mathcal{S}e^{-i\omega u},$ (10)

and

$\Psi_{k,in}^{\mathrm{II}+}=\mathcal{S}e^{i\omega u},$ (11)

where $\mathcal{S}$ is a 4-component Dirac spinor, $k$ is a wave vector,
$\omega$ is a monochromatic frequency of Dirac filed, and $u$ is the retarded
time expressed as $u=t-r^{*}$ with $r^{*}=r+2M\ln[rf/2M]$ which is the
tortoise coordinated.

By employing Damour and Ruffini’s suggestion [60], an analytic extension can
be made for the aforementioned equation, thereby providing a comprehensive
foundation for the positive energy modes. As a result, it becomes possible to
obtain the Bogoliubov transformations [61, 62] that pertain to the creation
and annihilation operators in both the Schwarzschild and Kruskal coordinates
by quantizing the Dirac fields in the Schwarzschild and Kruskal modes,
respectively. Upon suitably normalizing the state vector, one can express the
expressions of the Kruskal vacuum and excited states with mode $k$ as

$\displaystyle|0\rangle_{k}^{+}$
$\displaystyle\rightarrow\alpha\left|0_{k}\right\rangle_{\mathrm{I}}^{+}\left|0_{-k}\right\rangle_{\mathrm{II}}^{-}+\beta\left|1_{k}\right\rangle_{\mathrm{I}}^{+}\left|1_{-k}\right\rangle_{\mathrm{II}}^{-},$
$\displaystyle|1\rangle_{k}^{+}$
$\displaystyle\rightarrow\left|1_{k}\right\rangle_{\mathrm{I}}^{+}\left|0_{-k}\right\rangle_{\mathrm{II}}^{-},$
(12)

where $\alpha=(e^{-\omega_{k}/T}+1)^{-1/2}$ and
$\beta=(e^{\omega_{k}/T}+1)^{-1/2}$ with $T=1/8\pi M$ which is the Hawking
temperature [63]. Besides, $\left|n_{k}\right\rangle_{\mathrm{I}}^{+}$ and
$\left|n_{-k}\right\rangle_{\mathrm{II}}^{-}$ are respectively represented as
the orthonormal bases for outside and inside domains of the event horizon. For
the sake of simplicity, it is imposed that $\omega_{k}=\omega=1$,
$\left|n_{k}\right\rangle_{\mathrm{I}}^{+}=\left|n\right\rangle_{\mathrm{I}}$,
and
$\left|n_{-k}\right\rangle_{\mathrm{II}}^{-}=\left|n\right\rangle_{\mathrm{II}}$.

Figure 1: A schematic diagram for the physical model with Alice’s particle–$A$
in a flat region and Bob’s particle–$B$ near the event horizon of the
Schwarzschild black hole. The dashed lines indicate the quantum correlation
between two particles $A$ and $B$. Input state is given in Eq. (13) and output
states are provided in Eqs. (IV.0.1) and (IV.0.2).

Finally, in order to investigate the collective effects of Hawking and
decohering effects on quantum correlation, we shall consider a system
comprising two particles (Alice’s particle–$A$ and Bob’s particle–$B$) that
begin in a state characterized by generic Bell-diagonal state as follows

$\varrho_{AB}=\frac{1}{4}\left(I_{A}\otimes
I_{B}+\sum_{i=1}^{3}c_{i}\sigma_{i}^{A}\otimes\sigma_{i}^{B}\right),$ (13)

where $\sigma_{1}=\sigma_{x}$, $\sigma_{2}=\sigma_{y}$ and
$\sigma_{3}=\sigma_{z}$, and
$c_{i}=\textmd{tr}_{AB}(\varrho_{AB}\sigma_{i}^{A}\otimes\sigma_{i}^{B})$
satisfies $0\leq|c_{i}|\leq 1$.

In the present study, we suppose that the particle belonging to Bob is
situated within a Schwarzschild space-time, which is situated proximal to the
event horizon, whilst the particle belonging to Alice is kept within a static
and flat space-time. Hence, the state of our considered system (13) after
interaction of Bob’s particle with the Hawking radiation (12) is given by [37]

$\displaystyle\varrho_{AB_{\mathrm{I}}B_{\mathrm{II}}}=$ (14)
$\displaystyle\frac{1}{4}\Big{[}(1+c_{3})\\{\alpha^{2}|000\rangle\langle
000|+\alpha\beta(|000\rangle\langle 011|+|011\rangle\langle 000|)$
$\displaystyle+\beta^{2}|011\rangle\langle 011|+|110\rangle\langle
110|\\}+(1-c_{3})\\{\alpha^{2}|100\rangle\langle 100|$
$\displaystyle+\alpha\beta(|100\rangle\langle 111|+|111\rangle\langle
100|)+\beta^{2}|111\rangle\langle 111|+|010\rangle\langle 010|\\}$
$\displaystyle+(c_{1}-c_{2})\\{\alpha(|000\rangle\langle
110|+|110\rangle\langle 000|)+\beta(|011\rangle\langle 110|$
$\displaystyle+|110\rangle\langle
011|)\\}+(c_{1}+c_{2})\\{\alpha(|010\rangle\langle 100|+|100\rangle\langle
010|)$ $\displaystyle+\beta(|010\rangle\langle 111|+|111\rangle\langle
010|)\\}\Big{]}.$

##  IV LQU and LQFI under decoherence

Quantum correlation and coherence are of paramount importance in both quantum
information processing and quantum communication protocols. Nonetheless, as
stated in the introduction, the interaction of quantum systems with their
surroundings triggers a phenomenon commonly referred to as decoherence, which
results in the loss of quantum coherence and the disruption of intricate
quantum correlations. Consequently, the system may exhibit more classical
behavior and forfeit some of its distinctive quantum characteristics.

The physical processes that give rise to decoherence in quantum systems are
known as decohering channels. These channels are often depicted as noise or
perturbations that impact the quantum state and cause it to become mixed or
probabilistic, thereby deviating from its pure state [4, 64]. In broad terms,
the behavior of particles that independently interact with various
environments can be elucidated through the solutions derived from the relevant
Born-Markov-Lindblad equations.

In the case of any quantum state that is initially established as $\varrho$,
the final quantum state that emerges due to the effects of decohering
channels, with the assistance of the Kraus operator approach, can be
ascertained by

$\varepsilon\left(\varrho\right):=\sum_{i}K_{i}\varrho(K_{i})^{\dagger}.$ (15)

In the above equation, the Kraus operators $K_{i}$ are denoted as the single-
qubit quantum channels. Note that these operators are required to conform to
the closure condition, which mandates that the summation of
$(K_{i})^{\dagger}K_{i}$ must always be equal to the identity matrix $I$.

In the following, we study the quantum correlation attributes that are
associated with our considered system, which is subject to decoherence
effects. In a realistic setting, we consider that the system is independently
coupled to two different decohering channels, such as phase damping (PD) and
depolarizing (De).

####  IV.0.1 PD channel

A PD channel is a quantum communication concept that models a specific type of
noise or decoherence that can affect a quantum system during transmission or
storage. It represents the loss of information about the phase of a quantum
state while preserving its energy. In other words, the amplitude of the
quantum state remains unchanged, but its phase information becomes uncertain
or randomized due to interactions with the environment. Mathematically, the PD
channel can be represented using Kraus operators. For a single-qubit, the
Kraus operators of PD channel are

$K_{0}=|0\rangle\langle 0|+\sqrt{1-p}|1\rangle\langle 1|,\quad
K_{1}=\sqrt{p}|1\rangle\langle 1|.$ (16)

The channel applies one of these Kraus operators with probabilities determined
by $p$ to the input quantum state.

By employing Eqs. (14), (15) and (16), we can obtain the decohered state after
passing Alice’s particle through the PD decohering channel. Finally, after
tracing over all degrees of freedom in the physically inaccessible region
(inside the region of the event horizon–$\mathrm{II}$, see Fig. 1), the
system’s state takes the form

$\displaystyle\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}}$
$\displaystyle=v^{+}|00\rangle\langle 00|+\mu^{+}|01\rangle\langle
01|+v^{-}|10\rangle\langle 10|$ $\displaystyle+\mu^{-}|11\rangle\langle
11|+u^{-}|11\rangle\langle 00|+u^{+}|10\rangle\langle 01|$
$\displaystyle+u^{+}|01\rangle\langle 10|+u^{-}|00\rangle\langle 11|,$ (17)

where $v^{\pm}=\alpha^{2}(1\pm c_{3})/4$, $\mu^{\pm}=[2-\alpha^{2}(1\pm
c_{3})]/4$, and $u^{\pm}=\alpha\sqrt{1-p}(c_{1}\pm c_{2})/4$.

Using now the general formulas presented in [65], we derive the analytical
expression of LQU (2) for the above state as follows,

$\mathcal{U}(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})=1-\max\\{\mathcal{W}_{11}^{\textmd{PD}},\mathcal{W}_{33}^{\textmd{PD}}\\},$
(18)

where

$\displaystyle\mathcal{W}_{11}^{\textmd{PD}}=$
$\displaystyle\left(\sqrt{\gamma_{1}}+\sqrt{\gamma_{2}}\right)\left(\sqrt{\gamma_{3}}+\sqrt{\gamma_{4}}\right)$
$\displaystyle+\frac{(\mu^{+}-v^{-})(\mu^{-}-v^{+})+4|u^{-}u^{+}|}{\left(\sqrt{\gamma_{1}}+\sqrt{\gamma_{2}}\right)\left(\sqrt{\gamma_{3}}+\sqrt{\gamma_{4}}\right)},$
$\displaystyle\mathcal{W}_{33}^{\textmd{PD}}=$
$\displaystyle\frac{1}{2}\bigg{[}\left(\sqrt{\gamma_{1}}+\sqrt{\gamma_{2}}\right)^{2}+\left(\sqrt{\gamma_{3}}+\sqrt{\gamma_{4}}\right)^{2}$
$\displaystyle+\frac{(\mu^{-}-v^{+})^{2}-4|u^{-}|^{2}}{\left(\sqrt{\gamma_{1}}+\sqrt{\gamma_{2}}\right)^{2}}-\frac{(\mu^{+}-v^{-})^{2}-4|u^{+}|^{2}}{\left(\sqrt{\gamma_{3}}+\sqrt{\gamma_{4}}\right)^{2}}\bigg{]},$
(19)

with
$\gamma_{1,2}=(v^{+}+\mu^{-}\pm\sqrt{(v^{+}-\mu^{-})^{2}+4|u^{-}|^{2}})/2$ and
$\gamma_{3,4}=(\mu^{+}+v^{-}\pm\sqrt{(\mu^{+}-v^{-})^{2}+4|u^{+}|^{2}})/2$.

Besides, utilizing the general equations from [65], the analytical expression
of LQFI (5) for the state (IV.0.1) can be obtained as

$\mathcal{F}(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})=1-\max\\{\mathcal{M}_{11}^{\textmd{PD}},\mathcal{M}_{33}^{\textmd{PD}}\\},$
(20)

where

$\mathcal{M}_{11}^{\textmd{PD}}=\frac{k_{1}k_{2}}{k_{3}},\quad\mathcal{M}_{33}^{\textmd{PD}}=1-4\left(\frac{|u^{-}|^{2}}{v^{+}+\mu^{-}}+\frac{|u^{+}|^{2}}{\mu^{+}+v^{-}}\right),$
(21)

with

$k_{1}=64\left(v^{+}v^{-}+\mu^{+}\mu^{-}+\gamma_{1}\gamma_{2}+\gamma_{3}\gamma_{4}+2|u^{+}u^{-}|\right),$
$k_{2}=(\mu^{+}+v^{-})\gamma_{1}\gamma_{2}+(v^{+}+\mu^{-})\gamma_{3}\gamma_{4},$

and

$k_{3}=\left[1-\left(\gamma_{1}-\gamma_{2}\right)^{2}-\left(\gamma_{3}-\gamma_{4}\right)^{2}\right]^{2}-4\left(\gamma_{1}-\gamma_{2}\right)^{2}\left(\gamma_{3}-\gamma_{4}\right)^{2}.$

####  IV.0.2 De channel

The De channel refers to a type of quantum channel that introduces noise into
a quantum system by causing a loss of information about the initial quantum
state. This noise tends to depolarize the quantum state, meaning it reduces
the coherence and fidelity of the quantum information. A De channel is often
represented using a Kraus operator sum representation. For a single-qubit, the
Kraus operators of De channel can be written as

$\displaystyle K_{0}=\sqrt{1-p}(|0\rangle\langle 0|+|1\rangle\langle
1|),\,K_{1}=\sqrt{\frac{p}{3}}(|0\rangle\langle 1|+|1\rangle\langle 0|),$
$\displaystyle K_{2}=i\sqrt{\frac{p}{3}}(|0\rangle\langle 1|-|1\rangle\langle
0|),\,K_{3}=\sqrt{\frac{p}{3}}(|0\rangle\langle 0|-|1\rangle\langle 1|).$ (22)

According to the previous method, by exploiting the Eqs. (14), (15) and (22),
and after tracing over all degrees of freedom in the region $\mathrm{II}$ (see
Fig. 1), the final decohered state takes the following form

$\displaystyle\varrho_{AB_{\mathrm{I}}}^{\textmd{De}}$
$\displaystyle=\vartheta^{+}|00\rangle\langle 00|+\eta^{+}|01\rangle\langle
01|+\vartheta^{-}|10\rangle\langle 10|$
$\displaystyle+\eta^{-}|11\rangle\langle 11|+\kappa^{-}|11\rangle\langle
00|+\kappa^{+}|10\rangle\langle 01|$
$\displaystyle+\kappa^{+}|01\rangle\langle 10|+\kappa^{-}|00\rangle\langle
11|,$ (23)

where

$\vartheta^{\pm}=\frac{1}{12}[2p\mp\alpha^{2}(c_{3}\pm 1)(-3+2p)],$
$\eta^{\pm}=\frac{1}{12}[6-2p\pm\alpha^{2}(c_{3}\pm 1)(-3+2p)],$
$\kappa^{\pm}=\frac{1}{12}\alpha[3(c_{1}\pm c_{2})-2p(c_{1}\pm 2c_{2})],$

The analytical expression of LQU (2) for our state (IV.0.2) can be obtained as

$\mathcal{U}(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})=1-\max\\{\mathcal{W}_{11}^{\textmd{De}},\mathcal{W}_{33}^{\textmd{De}}\\},$
(24)

where

$\displaystyle\mathcal{W}_{11}^{\textmd{De}}=$
$\displaystyle\left(\sqrt{\theta_{1}}+\sqrt{\theta_{2}}\right)\left(\sqrt{\theta_{3}}+\sqrt{\theta_{4}}\right)$
$\displaystyle+\frac{(\eta^{+}-\vartheta^{-})(\eta^{-}-\vartheta^{+})+4|\kappa^{-}\kappa^{+}|}{\left(\sqrt{\theta_{1}}+\sqrt{\theta_{2}}\right)\left(\sqrt{\theta_{3}}+\sqrt{\theta_{4}}\right)},$
$\displaystyle\mathcal{W}_{33}^{\textmd{De}}=$
$\displaystyle\frac{1}{2}\bigg{[}\left(\sqrt{\theta_{1}}+\sqrt{\theta_{2}}\right)^{2}+\left(\sqrt{\theta_{3}}+\sqrt{\theta_{4}}\right)^{2}$
$\displaystyle+\frac{(\eta^{-}-\vartheta^{+})^{2}-4|\kappa^{-}|^{2}}{\left(\sqrt{\theta_{1}}+\sqrt{\theta_{2}}\right)^{2}}-\frac{(\eta^{+}-\vartheta^{-})^{2}-4|\kappa^{+}|^{2}}{\left(\sqrt{\theta_{3}}+\sqrt{\theta_{4}}\right)^{2}}\bigg{]},$
(25)

with
$\theta_{1,2}=(\vartheta^{+}+\eta^{-}\pm\sqrt{(\vartheta^{+}-\eta^{-})^{2}+4|\kappa^{-}|^{2}})/2$
and
$\theta_{3,4}=(\eta^{+}+\vartheta^{-}\pm\sqrt{(\eta^{+}-\vartheta^{-})^{2}+4|\kappa^{+}|^{2}})/2$.

Moreover, we obtain the analytical expression of LQFI (5) for the mentioned
state as

$\mathcal{F}(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})=1-\max\\{\mathcal{M}_{11}^{\textmd{De}},\mathcal{M}_{33}^{\textmd{De}}\\},$
(26)

where

$\mathcal{M}_{11}^{\textmd{De}}=\frac{l_{1}l_{2}}{l_{3}},\quad\mathcal{M}_{33}^{\textmd{De}}=1-4\left(\frac{|\kappa^{-}|^{2}}{\vartheta^{+}+\eta^{-}}+\frac{|\kappa^{+}|^{2}}{\eta^{+}+\vartheta^{-}}\right),$
(27)

with

$l_{1}=64\left(\vartheta^{+}\vartheta^{-}+\eta^{+}\eta^{-}+\theta_{1}\theta_{2}+\theta_{3}\theta_{4}+2|\kappa^{+}\kappa^{-}|\right),$
$l_{2}=(\eta^{+}+\vartheta^{-})\theta_{1}\theta_{2}+(\vartheta^{+}+\eta^{-})\theta_{3}\theta_{4},$

and

$l_{3}=\left[1-\left(\theta_{1}-\theta_{2}\right)^{2}-\left(\theta_{3}-\theta_{4}\right)^{2}\right]^{2}-4\left(\theta_{1}-\theta_{2}\right)^{2}\left(\theta_{3}-\theta_{4}\right)^{2}.$

Notice that the analytical expressions of $\mathcal{W}_{11}$,
$\mathcal{W}_{33}$, $\mathcal{M}_{11}$, and $\mathcal{M}_{33}$ in the cases of
high- and low-temperatures for PD and De channels are reported in appendix A.

So, now everything is ready to start a comparative analysis between the LQU
and LQFI that were discussed above. The aim is to explore the qualitative as
well as quantitative aspects of the quantum correlations in the vicinity of a
Schwarzschild black hole when subjected to the effects of decoherence. Without
loss of generality, we are motivated to consider an initial maximally non-
classically correlated state (pure Bell state) throughout this paper under the
condition $c_{1}=-c_{2}=c_{3}=1$ because we are interested in knowing how much
quantumness between the non-classically correlated particles of Alice and Bob
is lost under the influences of Hawking radiation and decoherence.

(a)
(b)

Figure 2: LQU (a) and LQFI (b) versus $p$ when particle $A$ is exposed to PD
channel and particle $B$ is positioned near the event horizon in a
Schwarzschild black hole.

Figure 2 delineates the quantum correlations of a non-classically correlated
two-qubit state, which are quantified through the LQU and LQFI measures under
the influence of Hawking radiation for particle $B$ while the particle $A$
passes through the PD channel. As evidenced in Fig. 2(a), the quantum
correlation indicated by the function $\mathcal{U}$ (18) diminishes
progressively as the severity of the decoherence parameter of the PD channel
intensifies, ultimately arriving at an absence of quantum correlation.
Additionally, the maximum values of quantum correlation are observed to
decrease with increasing Hawking temperature. More specifically, at $T=0.1$,
the LQU initiates at its maximum value ($\mathcal{U}=1$) for $p=0$, after
which the quantum correlations decline with escalations in $p$. Moreover, for
$T=0.5,1,3$ and at the onset of the interaction $(p\simeq 0)$, the values of
$\mathcal{U}$ are almost $0.75,0.60$, and $0.49$, respectively. There also
exists a locus in these instances where an abrupt alteration in the behavior
of the function transpires, wherein the decay rate intensifies. Figure 2(b)
elucidates that the function $\mathcal{F}$ (20) under analogous conditions
exhibits comparable behavior to the function $\mathcal{U}$. The quantum
correlation, as quantified by $\mathcal{F}$, decreases as $p$ increases.
However, the decay rate of $\mathcal{F}$ is inferior to that of $\mathcal{U}$.
Moreover, at discrete $T$ values, the function $\mathcal{F}$ initiates at
$1,0.88,0.73,$ and $0.58$ for $T=0.1,0.5,1,$ and $3$, respectively. Notice
that the abrupt transition point for LQFI is less pronounced and shifts toward
escalating $p$. More precisely, at $T=0.1$, both functions $\mathcal{U}$ and
$\mathcal{F}$ are smooth. However, quantum correlations experience sudden
transitions at first LQU for $p=0.05$, $p=0.11$ and $p=0.18$, and then LQFI
for $p=0.09$, $p=0.20$ and $p=0.31$ at temperatures $T=0.5$, $T=1$, and $T=3$,
respectively. Overall, at $p=1$, the quantum correlations estimated by both
functions are dissipated.

(a)
(b)

Figure 3: LQU (a) and LQFI (b) versus $T$ when particle $A$ is exposed to PD
channel and particle $B$ is positioned near the event horizon in a
Schwarzschild black hole.

In Fig 3, we examine the quantum correlations of the two functions
$\mathcal{U}$ and $\mathcal{F}$ across continuous values of the Hawking
temperature $T$ and discrete values of the PD channel decoherence parameter
$p$. The first qubit, $A$, passes through the PD channel, whilst the second
qubit, $B$, is impacted by Hawking radiation. The outcomes depicted in Figs.
3(a) and 3(b) demonstrate that LQU and LQFI do not attain their maximal values
for any value of $p$. Additionally, the behaviors of the functions are not
considerably contingent on elevated temperatures. Intensifying $p$ increases
the decoherence of the system, as the amounts of $\mathcal{U}$ and
$\mathcal{F}$ at $p=0.1$ exceed that at $p=0.7$. Moreover, in agreement with
Fig. 2, incrementing $p$ attenuates the temperature dependence. The extent of
quantum correlations changes with $p$ values, as the maximum bounds of quantum
correlations diminish, and discrepancies between the curves become evident as
decoherence strength intensifies.

(a)
(b)

Figure 4: LQU (a) and LQFI (b) versus $p$ when particle $A$ is undergoing De
channel and particle $B$ is positioned near the event horizon in a
Schwarzschild black hole.

(a)
(b)

Figure 5: LQU (a) and LQFI (b) versus $T$ when particle $A$ is undergoing De
channel and particle $B$ is positioned near the event horizon in a
Schwarzschild black hole.

The numerical outcomes of the functions LQU (24) and LQFI (26) as a function
of the De decoherence parameter are delineated in Fig. 4, accounting for
discrete Hawking temperature values. It is discernible that the two functions
$\mathcal{U}$ and $\mathcal{F}$ decay monotonically toward a non-zero steady-
state value, diminishing as the decoherence parameter $p$ escalates, before
re-surging to augment toward a residual value at superior $p$. Fig. 4(a)
demonstrates the non-zero steady-state value is contingent on temperature; for
$T=0.1,0.3,1,$ and $3$ at $p=1$, the residual $\mathcal{U}$ values are almost
$0.33,0.18,0.12,$ and $0.08$, respectively. Besides, Fig. 4(b) elucidates the
residual value of LQFI under analogous conditions as $\mathcal{F}$
$=0.33,0.27,0.20,$ and $0.14$, respectively.

In Fig. 5, the universal behaviors of LQU and LQFI as a function of the
Hawking temperature $T$ across discrete values of the De decoherence parameter
$p$ are displayed. Fig. 5(a) illustrates the behavior of LQU, which initially
decreases from its initial values and then stabilizes at a specific value
corresponding to the given decoherence parameter. For high temperatures, the
function $\mathcal{U}$ becomes a constant with a fixed value that varies
according to the value of the decoherence parameter $p$. Similarly, Fig. 5(b)
demonstrates that LQFI exhibits analogous behavior to LQU, but with higher
upper bounds. In general, although the quantitative behavior of LQU and LQFI
is different as expected, their qualitative behavior is the same.

(a)
(b)

Figure 6: LQU (a) and LQFI (b) versus $p$ when particle $A$ is exposed to PD
and De channels independently and particle $B$ is positioned near the event
horizon in a Schwarzschild black hole with $T=0.1$.

Figure 6 presents a comparative study of LQU and LQFI functions by considering
the effects of Hawking radiation and decohering channels as a function of $p$
at the fixed value of Hawking temperature $T=0.1$. Figure 6(a) illustrates how
the variation of $p$, from $0$ to $1$, causes differences in the behaviors of
LQU related to the impacts of PD and De channels. At low values of $p$,
especially when $p$ tends to zero, the behavior of mentioned curves are
coincident, but they are not totally similar. At low values of $p$, the LQU
subjected to both PD and De channels is decreasing and the value of LQU under
the PD channel is slightly more than that of under the De channel even when
$p$ is close to zero. For a better illustration, a magnified frame of that
region is presented in the inset plot. However, after their intersection and
especially at high values of $p$, their behaviors are extremely different and
the LQU values related to the PD channel are much lesser than the values of
the De channel. In other words, after the intersection, the LQU curve
extracted by considering the De channel keeps its value, approximately, but
the LQU curve subjected to the PD channel tends to zero. The second diagram of
Fig. 6, plot (b), provides a comparison of LQFI by considering the influences
of PD and De channels. Similar to the first plot, there are resemblances in
their behavior. They behave similarly to each other in the lower and middle
regions of the $p$ interval. In Fig. 6(b) also there is an intersection. Just
like previously described in Fig. 6(a), after the intersection the LQFI curve
related to the PD channel tends to zero harshly and the curve related to the
De channel maintains its value, nearly. It is worth mentioning that the curves
in Figs. 4 and 6 have local minima around $p=0.7$ to $p=0.9$.

(a)
(b)

Figure 7: LQU (a) and LQFI (b) versus $T$ when particle $A$ is exposed to PD
and De channels independently and particle $B$ is positioned near the event
horizon in a Schwarzschild black hole with $p=0.1$.

Figure 7 just like Fig. 6 presents a comparison of quantum correlations (LQU
and LQFI) gained by considering the impacts of PD and De channels under the
background of a Schwarzschild black hole, but this time they are functions of
Hawking temperature $T$ that it varies from 0 to 5 and the value of $p$ is
fixed to $0.1$. Both the first and second plots of Fig. 7 during the whole
interval of $T$ have equal gradients, approximately, in other words, there is
no intersection between them. The values of quantum correlations subjected to
the PD channel are higher than those of the De channel when $T$ varies in its
interval. Note that according to Figs. 6 and 7, it seems more appropriate to
plot LQU and LQFI as a function of $p$ with a fixed value of $T$ to find their
significant differences when exposed to PD and De channels.

##  V Quantum coherence under decohering channels

Quantum coherence is the property of quantum systems in which the different
states of the system are in a superposition, meaning they exist simultaneously
[66]. It is essential for many quantum technologies, such as quantum computing
and quantum cryptography. When a system is in a state of coherence, it can
exhibit interference effects, which allow for precise control and manipulation
of the system’s behavior. However, coherence is fragile and can be easily
disrupted by interactions with the environment, leading to a phenomenon known
as decoherence. Therefore, maintaining coherence is a significant challenge in
the development of practical quantum technologies.

Various techniques are at disposal for the evaluation of quantum coherence
[67]. Among the various measures of coherence, the $l_{1}$-norm of coherence
has been widely employed in the realm of quantum physics. It is expressed in
the following manner

$C(\varrho)=\sum_{i\neq j}\mid\varrho_{i,j}\mid.$ (28)

From Eq. (28), the $l_{1}$-norm of coherence value can be expressed as the
summation of the absolute values of the off-diagonal elements pertaining to
the selected basis [7].

Using, as for LQU and LQFI, the system’s states are given in Eqs. (IV.0.1) and
(IV.0.2), we arrive at formulas for the $l_{1}$-norm of coherence under
decohering channels

$C(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})=2(|u^{+}|+|u^{-}|),$ (29)

and

$C(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})=2(|\kappa^{+}|+|\kappa^{-}|).$
(30)

It is interesting to mention that based on Refs. [68, 69], the analytical
expressions of quantum coherence given in Eqs. (29) and (30) present the
quantum consonance. The analytical expressions of
$C(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})$ and
$C(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})$ at high and low temperatures are
given in appendix B.

Figure 8 shows the evolution of quantum coherence between two qubits when
qubit $A$ is affected by PD and De channels and qubit $B$ is in the vicinity
of the event horizon in a Schwarzschild black hole. As shown in Fig. 8(a),
quantum coherence decays with increasing parameter $p$ for both PD and De
channels as expected. Because the superposition of quantum states, or
interference patterns, is destroyed when those quantum systems interact with
their environments. From this figure, quantum coherence has bigger values when
qubit $A$ is affected by the PD channel than the De channel for $0<p<0.74$,
However, quantum coherence tends to zero more quickly when qubit $A$ is
subjected to PD channel after $p=0.74$.

In Fig. 8(b), quantum coherence has been plotted versus Hawking temperature
$T$ under the effect of PD and De channels. As depicted in this figure,
quantum coherence remains unchanged for weak values of $T$ and then it
decreases to get an approximately fixed value for both channels like the
previous cases (see LQU and LQFI in Fig. 7). However, despite the previous
case, quantum coherence in PD and De channels nearly has the same value of
$0.67$ when $T$ tends to infinity.

(a)
(b)

Figure 8: Quantum coherence versus $p$ and $T$ when particle $A$ is subjected
to PD and De channels independently and particle $B$ is located near the event
horizon in a Schwarzschild black hole with (a) $T=0.1$ and (b) $p=0.1$.

(a)
(b)

Figure 9: Bell non-locality versus $p$ and $T$ when particle $A$ is subjected
to PD and De channels independently and particle $B$ is located near the event
horizon in a Schwarzschild black hole with (a) $T=0.1$ and (b) $p=0.1$.

##  VI Bell non-locality in the Schwarzschild space-time

Bell non-locality is a quantum phenomenon characterized by the presence of
correlations between measurements performed on entangled particles that defy
the predictions of classical physics [70, 71]. This intriguing phenomenon
strongly indicates that the behavior of the particles in question is
inherently non-local and thus cannot be explained by invoking local hidden
variables [72].

For systems of dimension $2\times 2$, the detection of the non-locality of a
quantum state $\varrho$ can be achieved through the violation of the Bell-CHSH
inequality, as proposed by Clauser, Horne, Shimony and Holt [72]. This
inequality is defined as follows

$|\langle B_{\textmd{CHSH}}\rangle_{\varrho}|\leq 2,$ (31)

with $\langle B_{\textmd{CHSH}}\rangle_{\varrho}=\textmd{tr}[\varrho
B_{\textmd{CHSH}}]$ where $B_{\textmd{CHSH}}$ is the Bell operator related to
the quantum CHSH inequality. The maximum value of the inequality (31), namely
$B_{\max}(\varrho)=\max|\langle B_{\textmd{CHSH}}\rangle_{\varrho}|$, is
related to the quantity $M(\varrho)=\max_{i<j}(\omega_{i}+\omega_{j})$ with
$B_{\max}(\varrho)=2\sqrt{M(\varrho)}$ in which $\omega_{i}$ ($i=1,2,3$) being
the eigenvalues of a $3\times 3$ matrix $X^{\dagger}X$, where $X$ here is a
positive matrix with elements
$x_{nm}=\textmd{tr}(\varrho\sigma_{n}\otimes\sigma_{m})$ [73]. The violation
of inequality (31) is manifestly evident if and only if $M(\varrho)$ exceeds
the value of 1. Furthermore, $M(\varrho)$ can be utilized as an effective
measure to evaluate the degree of Bell non-locality violation for a bipartite
state [74].

For the quantum state expressed in Eq. (IV.0.1), we can obtain

$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})=2\sqrt{M(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})},$
(32)

where

$M(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})=\max\\{M_{1}^{\textmd{PD}},M_{2}^{\textmd{PD}}\\},$
(33)

with

$M_{1}^{\textmd{PD}}=8(|u^{+}|^{2}+|u^{-}|^{2}),$

and

$M_{2}^{\textmd{PD}}=4(|u^{+}|+|u^{-}|)^{2}+(v^{+}+\mu^{-}-v^{-}-\mu^{+})^{2}.$

Also, we obtain the maximum value of the inequality (31) for the state given
in Eq. (IV.0.2) as follows

$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})=2\sqrt{M(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})},$
(34)

where

$M(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})=\max\\{M_{1}^{\textmd{De}},M_{2}^{\textmd{De}}\\},$
(35)

with

$M_{1}^{\textmd{De}}=8(|\kappa^{+}|^{2}+|\kappa^{-}|^{2}),$

and

$M_{2}^{\textmd{De}}=4(|\kappa^{+}|+|\kappa^{-}|)^{2}+(\vartheta^{+}+\eta^{-}-\vartheta^{-}-\eta^{+})^{2}.$

Note that the analytical formulas of
$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})$ and
$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})$ for high- and low-
temperature limits are reported in appendix B.

Figure 9 is devoted to showing the behavior of Bell non-locality based on Eqs.
(32) and (34) when particle $A$ is exposed to PD and De channels independently
and qubit $B$ is close to the event horizon in a Schwarzschild black hole. In
Fig. 9(a), Bell non-locality has been drawn versus $p$ in which the maximum
value of non-locality starts at 3 when there is no interaction between qubit
$A$ and the environment, i.e., $p=0$. Bell non-locality decreases for both
channels when particle $A$ begins to interact with the environment, but, it
decreases for the De channel more rapidly. As shown in this figure by an
arrow, Bell non-locality based on
$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})$ in Eq. (34) becomes less
than 2 when $p>0.44$, however,
$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})$ in Eq. (32) goes never
under 2. Hence, Bell inequality is violated for all values of $p$ in the PD
channel.

By looking at Figs. 6, 8(a) and 9(a), one can see that LQU, LQFI, and quantum
coherence in our considered system when qubit $A$ is exposed to PD channel
have greater values than De channel for weak values of $p$. But, Bell non-
locality in the PD channel is more robust than the De channel for all values
of $p$.

We have plotted Bell non-locality versus $T$ for De and PD channels in Fig.
9(b). It is clear that Bell non-locality has a fixed value for low Hawking
temperature like previous cases in Figs. 7 and 8. Then, it goes down smoothly
to get values less than 2 for which non-locality effects disappear in the De
channel when $T>2.1$, and we have
$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{De}})\simeq 1.8$ for
$T\rightarrow\infty$. Nonetheless, Bell non-locality value for the PD channel
becomes less than 2, i.e.
$B_{\max}(\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}})\simeq 1.9$, when
$T\rightarrow\infty$. Thus, one may infer that PD and De channels have more
serious effects on quantum correlations or other characteristics of
quantumness (quantum coherence and non-locality) than the effect of Hawking
temperature.

##  VII Conclusion and outlook

The research conducted in the field of convergence of quantum information
theory and black hole physics is promising in revealing new perspectives
regarding the fundamental characteristics of gravity in the realm of quantum
scales. Such research has the potential to have significant implications for
our understanding of black holes as well as the complex fabric of space-time.

Motivated to investigate the relationship between these two important
concepts, we considered a practical model in which a particle residing in the
flat region interacts with its surroundings while the other particle
experiences free fall in the vicinity of the event horizon of a Schwarzschild
black hole. Based on this scenario, we studied the quantum correlations,
quantum coherence, and non-locality under the collective effects of Hawking
radiation and two decohering channels (PD and De).

As expected, the quantumness (quantum correlations, quantum coherence, and
non-locality) of the considered system consisting of two maximally non-
classically correlated qubits in the background of the Schwarzschild black
hole decreases with increasing decoherence parameter $p$. It is worthwhile to
note that in the case of the PD channel, quantum correlations and quantum
coherence tend to zero when $p\to 1$. However, in the case of the De channel,
although they decrease, they do not go to zero. Interestingly, quantum
correlations experience abrupt transitions in certain values of $p$ at fixed
temperatures when qubit $A$ is exposed to the PD channel, but they decrease
with increasing $p$ in a monotonic way when Alice’s qubit is subjected to the
De channel.

Regarding the effect of Hawking radiation, it is not an exciting result that
increasing the Hawking temperature reduces quantum correlations, quantum
coherence, and non-locality. But the intriguing result is that at high
temperatures ($T\to\infty$), corresponding to the case of the black hole
approximating to evaporate completely, we observe that the two-qubit system
still preserves its quantumness, which is of great importance in quantum
information theory for practical goals. Notably, when the Bell non-locality is
present, its strength diminishes by enlarging the intensity of the Hawking
effect.

These results not only explain how quantum correlations and quantum coherence
decay but also provide better insight into quantum non-locality near black
holes. Hence, our findings are helpful to guide us in choosing suitable
quantum channels and quantum states to tackle relativistic quantum information
processing tasks.

## Acknowledgments

M.A. thanks Saeed’s Quantum Information Group (SSQIG) for providing
hospitality during the course of this work. M.A.Y. was supported in part by a
state task, the state registration number of the Russian Federation is
AAAAA19-119071190017-7.

## Disclosures

The authors declare that they have no known competing financial interests.

## Data availability

No datasets were generated or analyzed during the current study.

##  Appendix A LQU and LQFI at high and low temperatures

In this appendix, we provide the analytical expressions of
$\mathcal{W}_{11(33)}$ and $\mathcal{M}_{11(33)}$ in two cases (high- and low-
temperature limits) for PD and De channels.

From expression (IV.0.1), we obtain

$\small\lim_{T\to\infty}\mathcal{W}_{11}^{\textmd{PD}}(T)=\frac{2+\sqrt{2p}}{2\sqrt{3+2\sqrt{2p}}},$
(36)
$\displaystyle\lim_{T\to\infty}\mathcal{W}_{33}^{\textmd{PD}}(T)=\frac{(1+\sqrt{2p})^{2}}{3+2\sqrt{2p}},$
(37)

and

$\small\lim_{T\to 0}\mathcal{W}_{11}^{\textmd{PD}}(T)=0,$ (38)
$\displaystyle\lim_{T\to 0}\mathcal{W}_{33}^{\textmd{PD}}(T)=\sqrt{p},$ (39)

Likewise, based on Eq. (21) we have

$\displaystyle\lim_{T\to\infty}\mathcal{M}_{11}^{\textmd{PD}}(T)=\frac{1+p}{2+p},$
(40)
$\small\lim_{T\to\infty}\mathcal{M}_{33}^{\textmd{PD}}(T)=\frac{1+2p}{3},$
(41)

and

$\small\lim_{T\to 0}\mathcal{M}_{11}^{\textmd{PD}}(T)=0,$ (42)

$\small\lim_{T\to 0}\mathcal{M}_{33}^{\textmd{PD}}(T)=p.$ (43)

Further, using Eqs. (IV.0.2) and (27), we get

$\displaystyle\lim_{T\to\infty}\mathcal{W}_{11}^{\textmd{De}}(T)=$
$\displaystyle\frac{1}{144}\bigg{[}\frac{24(4p(3-5p)+9)}{\left(w_{1-}+w_{1+}\right)\left(w_{2-}+w_{2+}\right)}$
$\displaystyle+6\left(w_{1-}+w_{1+}\right)\left(w_{2-}+w_{2+}\right)\bigg{]},$
(44) $\displaystyle\lim_{T\to\infty}\mathcal{W}_{33}^{\textmd{De}}(T)=$
$\displaystyle\frac{1}{48}\bigg{[}-\frac{4(4p(p+3)-9)}{\left(w_{1-}+w_{1+}\right){}^{2}}+\left(w_{1-}+w_{1+}\right){}^{2}$
$\displaystyle-\frac{4(4p(17p-33)+63)}{\left(w_{2-}+w_{2+}\right){}^{2}}+\left(w_{2-}+w_{2+}\right){}^{2}\bigg{]},$
(45)

where

$w_{1\pm}=\sqrt{3+2p\pm\sqrt{3}\sqrt{4p(p-1)+3}},$
$w_{2\pm}=\sqrt{9-2p\pm\sqrt{4p(19p-39)+81}},$

and

$\displaystyle\lim_{T\to 0}\mathcal{W}_{11}^{\textmd{De}}(T)=\lim_{T\to
0}\mathcal{W}_{33}^{\textmd{De}}(T)=\frac{2}{3}\sqrt{p(3-2p)}.$ (46)

Besides

$\displaystyle\lim_{T\to\infty}\mathcal{M}_{11}^{\textmd{De}}(T)=\frac{(8p(2p-3)-9)(p(4p+3)-18)}{6[p(4p(2p-9)+9)+54]},$
(47)
$\displaystyle\lim_{T\to\infty}\mathcal{M}_{33}^{\textmd{De}}(T)=1-\frac{2}{3}\frac{[16p^{3}+9(3-4p)]}{(3+2p)(9-2p)},$
(48)

and

$\displaystyle\lim_{T\to 0}\mathcal{M}_{11}^{\textmd{De}}(T)=\lim_{T\to
0}\mathcal{M}_{33}^{\textmd{De}}(T)=\frac{4p(3-2p)}{3(3-p)}.$ (49)

##  Appendix B Quantum coherence and Bell non-locality at high and low
temperatures

Here, we give the analytical expressions of $C(\varrho)$ and
$B_{\max}(\varrho)$ in two cases (high- and low-temperature limits) for PD and
De channels.

From expressions (29) and (30), we get

$\small\lim_{T\to\infty}C[\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}}](T)=\sqrt{\frac{1-p}{2}},$
(50) $\small\lim_{T\to
0}C[\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}}](T)=\sqrt{1-p},$ (51)

and

$\small\lim_{T\to\infty}C[\varrho_{AB_{\mathrm{I}}}^{\textmd{De}}](T)=\frac{3-2p}{3\sqrt{2}},$
(52)

$\small\lim_{T\to
0}C[\varrho_{AB_{\mathrm{I}}}^{\textmd{De}}](T)=\frac{3-2p}{3}.$ (53)

Moreover, according to Eqs. (32) and (34) we obtain

$\small\lim_{T\to\infty}B_{\max}[\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}}](T)=2\sqrt{1-p},$
(54) $\small\lim_{T\to
0}B_{\max}[\varrho_{AB_{\mathrm{I}}}^{\textmd{PD}}](T)=2\sqrt{2-p},$ (55)

and

$\small\lim_{T\to\infty}B_{\max}[\varrho_{AB_{\mathrm{I}}}^{\textmd{De}}](T)=\frac{2}{3}\sqrt{2p(5p-9)+9},$
(56) $\small\lim_{T\to
0}B_{\max}[\varrho_{AB_{\mathrm{I}}}^{\textmd{De}}](T)=\frac{2}{3}\sqrt{2}(3-2p).$
(57)

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[8pc]  Mockup of an Android lock screen using a reactive security message that
reports the last unlock time. In addition, the reason why the user has to
enter the PIN instead of using the fingerprint is displayed. Copyright is held
by the author/owner. Permission to make digital or hard copies of all or part
of this work for personal or classroom use is granted without fee. Poster
presented at the 15th Symposium on Usable Privacy and Security (SOUPS 2019).

# “Please enter your PIN” - On the Risk of Bypass Attacks on Biometric
Authentication on Mobile Devices

Christian Tiefenau

Maximilian Häring

Mohamed Khamis

Emanuel von Zezschwitz

University of Bonn  Bonn, Germany<EMAIL_ADDRESS>Fraunhofer FKIE
Bonn, Germany<EMAIL_ADDRESS>University of Glasgow  Glasgow, UK
mohamed.khamis@
glasgow.ac.uk University of Bonn, Fraunhofer FKIE  Bonn, Germany
<EMAIL_ADDRESS>
###### Abstract

Nowadays, most mobile devices support biometric authentication schemes like
fingerprint or face unlock. However, these probabilistic mechanisms can only
be activated in combination with a second alternative factor, usually
knowledge-based authentication. In this paper, we show that this aspect can be
exploited in a bypass attack. In this bypass attack, the attacker forces the
user to “bypass” the biometric authentication by, for example, resetting the
phone. This forces the user to enter an easy-to-observe passcode instead. We
present the threat model and provide preliminary results of an online survey.
Based on our results, we discuss potential countermeasures. We conclude that
better feedback design and security-optimized fallback mechanisms can help
further improve the overall security of mobile unlock mechanisms while
preserving usability.

##  1 Introduction and Background

Mobile devices store a lot of potentially sensitive information, e.g., in the
form of e-mails, pictures, or contact data. A lock screen, which requires user
authentication, is often the last barrier of protection from unauthorized
access. While knowledge-based approaches (e.g., PIN) have been the quasi-
standard for many years, biometric approaches (e.g., fingerprint) are
increasingly employed as a more usable alternative. This can be explained by
recent research which shows that users unlock their mobile devices several
times an hour [8], which underlines the importance of usability in terms of
speed and error rate. Indeed, in comparison to their to knowledge-based
counterparts, biometric authentication schemes are faster and easier to use
[4]. Biometric authentication is also superior in terms of security as it
resists many attacks that knowledge-based schemes are vulnerable to, such as
observation [10], smudge [3] and thermal attacks [2].

Notification of Android 8 and iOS 12.1.2 on an iPhone 5s and X if the
fingerprint was not correctly identified 5 times in a row or the device was
unable to identify the users’ face. We assume that a more detailed message
(e.g., “fingerprint authentication failed 5 times in a row in the last 5
minutes”) could improve both the usability of the system and the security
consciousness of the user.

Nevertheless, biometric authentication does not come without limitations.
Previous work showed that biometric authentication can be vulnerable to
sophisticated camera-based smudge attacks, and sometimes suffers from
recognition errors when the user’s features are distorted or occluded (e.g.,
using TouchID with dirty or wet fingers, or using FaceID when only part of the
face is visible).

We identify an additional vulnerability in biometric authentication that has
not been investigated in prior research. Namely, because of their
probabilistic nature, mobile biometric authentication always requires a second
deterministic mode, which in turn reduces security. For example, Android and
iOS require users to set a knowledge-based password (e.g., PIN or Lock
Pattern) whenever they choose biometric authentication as the main unlocking
mechanism111iOS:
https://www.apple.com/business/site/docs/iOS_Security_Guide.pdf
Android: https://source.android.com/security/encryption/full-disk.html. We
argue that this is a security issue, as lock screens that support multiple
authentication modes can only be as secure as the weakest mechanism provided.

In this paper, we show that the current lock screen implementations are still
prone to observation attacks, as users can easily be tricked into using the
knowledge-based factor. For example, an attacker could repeatedly push their
finger against a user’s fingerprint sensor to force the system to require a
knowledge-based password, which in turn can be easily uncovered via
observation, thermal or smudge attacks. Therefore, we argue that there is
little need for educated attacks on biometrics, as long as the provided
alternative mode (e.g., PIN) is still vulnerable.

We discuss our threat model and present several attack vectors which exploit
the human factor in combination with flawed interface design (i.e., the lack
of feedback). We demonstrate that current mobile phones can be quickly
manipulated in a way that bypasses biometric authentication and thereby
forcing users to enter their passcode. The results of an online survey (N=200)
indicate, that most users do not question switches from biometric to
knowledge-based authentication, which suggests that users are vulnerable to
said threat. Based on our results, we discuss potential countermeasures and
illustrate directions for future work. We propose small changes to the user
interface (i.e., provide feedback on mode switches) which could prevent the
bypass attack. Furthermore, we highlight the need to revisit previously
discussed security-optimized (e.g., observation-resistant) authentication
mechanisms in the light of the new context (i.e., fallback authentication).

##  2 Threat Model

In our threat model, we assume that the attacker is in the vicinity of the
victim and has short-term access to the victim’s mobile device. The victim’s
device is configured with a biometric lock screen such as fingerprint, as a
fallback mechanism, the victim has configured PIN.

The attacker does not know the victim’s PIN. Once the victim’s device is
unattended, the attacker prepares the device in a way that a PIN-based
authentication will be required for the next unlock. This can be done by
triggering 5 unsuccessful biometric authentication
attempts222https://support.apple.com/en-us/HT204587. As soon as the victim
tries to unlock their phone, the system asks for the PIN. Without questioning
the reason, the victim provides their PIN-code. The attacker then observes the
entered PIN, gets hold of the device (e.g., when the user leaves it
unattended), and then unlocks the device after switching to the fallback
screen. In this scenario, there is no need to break the biometric factor.
Since the device does not provide any information about switches between
factors, the victim remains unaware that someone tampered with their phone.

##  3 Practical Execution of the Bypass Attack

In contrast to previous elaborated attacks on biometric sensors, we
investigated non-detectable low-cost approaches which force the victim to
bypass the biometric unlock and enter the fallback authentication code. Based
on the review of the currently deployed hardware and current lock screen
implementations, we outline software-based and hardware-based attacks.

[-32pc]  Home button covered with lip balm to prevent the fingerprint sensor
from recognizing the users’ fingerprint.

[-12pc]  iPhone Android prim. sec. prim. sec. PIN 27 51 30 56 Pattern 19 31 0
7 Password 6 32 8 22 Fingerprint 42 21 55 23 Face unlock 2 5 6 9 Other 4 4 1 2
Total 100 144 100 119 The distribution of both device-groups divided by the
primary and secondary method as reported by the participants.

Likert scale answers of the survey.

###  3.1 Software-based Attacks

Software-based attacks exploit the implementation of current lock screens. For
example, current iPhones require the passcode 1) after reboot, and 2) after
five failed biometric attempts. This aspect can be exploited by an attacker to
actively require users to enter their passcode; the attacker can switch off /
restart the device or perform five failed attempts using their own finger or
face (depending on the used mechanism). As illustrated in Figure 1, current
mobile devices provide only limited feedback on mode switches. Therefore,
users are likely to remain unaware of the attack.

###  3.2 Hardware-based Attacks

Attackers can actively manipulate the biometric sensor. We performed a number
of experiments with the iPhone 5s and explored different ways. We found that
applying a) greasy lip balm or b) some drops of water prevents the sensors
from working. While this method may be more eye-catching, it is an easy and
direct way to force users to enter their passcode.

##  4 User Behavior and Risk Perception

To gather preliminary results on the risk of the presented threat model, we
used Amazons MTurk to survey 200 US-citizens (100 Android, 100 iPhone) about
their unlock mechanisms. The survey took about 5 minutes and was compensated
with 0.75$. We were interested in the participants’ understanding of mode
switches. This means, if they understand why their device is requiring a
passcode. In addition, we asked how often they leave their smartphone
unattended and if the use of biometrics affects the choice of passcodes.

Table 3 provides self-reported data on primary and secondary authentication.
In the survey, we asked the participants which methods they “usually used to
unlock” (primary) and “all other activated unlock mechanisms” they have
(secondary). 145 participants (61 iPhone, 84 Android) reported to use
fingerprint or face unlock as _primary_ or _secondary_ method. Please note
that in this particular case “primary” and “secondary” was related to the
participant’s perceived usage of the respective method.

We used a 5-point scale (_1 - Never_ to _5 - Always_) to ask biometric users
“how often they know the reason why their mobile device asked them for their
PIN or passcode instead of accepting their biometric login”. 97 of 145 ( 67%),
as seen in Figure 3 reported, that they do not always understand the reasons
of mode switches, seven iPhone users acknowledged that they can never figure
out the reason. This result indicates that users are likely to remain unaware
of malicious mode switches. In addition, we asked to what extent the
participants agree to the statement _“I chose an easy to remember
PIN/Pattern/Password because I’m using biometric authentication (fingerprint
or face unlock) most of the time.”_. As seen in Figure 3, we found a trend
indicating that users select weaker passwords when they use biometric
authentication. 80 of 145 ( 55%) participants agreed that they deliberately
chose an easy password. We asked how often the participants leave their phone
unattended when e.g., going to the rest rooms. 135 of 200 (67.5%) reported
that they occasionally or often do this, which makes short-term access for the
attacker possible.

##  5 Potential Countermeasures

Based on our preliminary results, we propose to investigate the effectiveness
of the following countermeasures.

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
WARNING:POSSIBLE DNS SPOOFING DETECTED
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
The ECDSA host key for server has
changed, and the key for the
corresponding IP address 192.168.0.1
is unchanged. This could either mean
that DNS SPOOFING is happening or the
IP address for the host and its host
key have changed at the same time.

Warning message of SSH when the IP-address of a known server has changed.
Similarly, we recommend showing users a warning message when biometric
authentication fails to improve their security consciousness.

###  5.1 Improved Interface Design

Most users do not question the fact that a passcode is required. Our
preliminary results indicate that the system state of current lock screens is
not always easy to assess. We argue that future work needs to investigate if
the unlock screen can be improved in a way that better communicates the
current system state. Following the usability heuristics by Jakob Nielsen [9],
we recommend to provide more information about mode switches between primary
and secondary authentication. Figure “Please enter your PIN” - On the Risk of
Bypass Attacks on Biometric Authentication on Mobile Devices shows an example
of how such feedback could look like. We would recommend to inform the user
about the last unlock event, maybe only when it seems necessary, e.g.,
determined by uncharacteristic behavior. We assume that this information could
help the user to discover ongoing security breaches (cf., threat model).
Similar reactive security measures are already in use in OpenSSH (see Figure
5) or in online banking platforms.

###  5.2 Improved Knowledge-based Authentication

Our preliminary findings indicate that biometric authentication schemes are
often used as the primary authentication method. As a consequence, the
knowledge-based approach is used less frequently. We argue that this aspect
changes the requirements for feasible knowledge-based authentication
mechanisms. We assume that the new context of use renders efficiency (i.e.,
input time) less important and makes memorability and effectiveness more
important. Previous research has proposed a plethora of knowledge-based
authentication mechanisms which were optimized to resist different types of
side-channel attacks like observation attacks [6], smudge attacks [3] or
thermal attacks [2]. While these mechanisms (e.g., [5, 7]) were shown to be
more secure, they were often too slow to be used as a primary (i.e., most
used) mobile authentication method. We recommend to review previous work in
the light of the new use case (i.e., as a fallback authentication). We assume
that using such security-optimized authentication schemes becomes feasible in
the new context and would improve the overall security of the device.

##  6 Conclusion and Future Work

In this paper, we discussed a new threat model, a bypass attack. By performing
a bypass attack, the attacker is forcing the user to “bypass” biometric
authentication and to use an easy-to-observe knowledge-based mode instead.
Based on the preliminary results of an online survey, we discussed potential
countermeasures which need to be investigated in future work. We argued that
the overall security of the lock screens could be improved by 1) providing
more feedback to the user and by 2) deploying security-optimized (e.g.,
observation-resistant) knowledge-based mechanisms.

## References

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