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# Physically Accessible and Inaccessible Quantum Correlations of Dirac Fields
in Schwarzschild Spacetime

Samira Elghaayda  0000-0002-6655-0465<EMAIL_ADDRESS>Laboratory of High Energy Physics and Condensed Matter, Department of Physics,
Faculty of Sciences of Aïn Chock, Hassan II University,
P.O. Box 5366 Maarif, Casablanca 20100, Morocco.  Asad Ali
0000-0001-9243-417X<EMAIL_ADDRESS>Qatar Centre for Quantum Computing,
College of Science and Engineering, Hamad Bin Khalifa University, Qatar
Foundation, Doha, Qatar.  Saif Al-Kuwari  0000-0002-4402-7710
<EMAIL_ADDRESS>Qatar Centre for Quantum Computing, College of Science
and Engineering, Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar.
Mostafa Mansour  0000-0003-0821-0582<EMAIL_ADDRESS>Laboratory
of High Energy Physics and Condensed Matter, Department of Physics,
Faculty of Sciences of Aïn Chock, Hassan II University,
P.O. Box 5366 Maarif, Casablanca 20100, Morocco.

###### Abstract

In this study, we investigate the influence of Hawking decoherence on the
quantum correlations of Dirac fields between Alice and Bob. Initially, they
share a Gisin state near the Schwarzschild black hole (SBH) in an
asymptotically flat region. Then, Alice remains stationary in this region,
while Bob hovers near the event horizon (EH) of the SBH. We expect that Bob,
using his excited detector, will detect a thermal Fermi-Dirac particle
distribution. We assess the quantum correlations in the evolved Gisin state
using quantum consonance and uncertainty-induced non-locality across
physically accessible, physically inaccessible, and spacetime regions. Our
investigation examines how these measures vary with Hawking temperature, Dirac
particle frequency, and the parameters of the initial Gisin state.
Additionally, we analyze the distribution of these quantum correlation
measures across all possible regions, noting a redistribution towards the
physically inaccessible region. Our findings demonstrate that Hawking
decoherence reduces the quantum correlations of Dirac fields in the physically
accessible region, with the extent of reduction depending on the initial state
parameters. Moreover, as Hawking decoherence intensifies in the physically
inaccessible and spacetime regions, the quantum correlations of Dirac fields
reemerge and ultimately converge to specific values at infinite Hawking
temperature. These results contribute to our understanding of quantum
correlation dynamics within the framework of relativistic quantum information
(RQI).

##  I Introduction

From Einstein’s perspective, the gravitational collapse of sufficiently
massive stars gives rise to intriguing black holes (BHs), which stand as
captivating entities in our universe [1, 2]. According to the no-hair theorem,
BHs preserve only information concerning their mass, charge, and angular
momentum, while concealing all other specifics [3]. Recent advancements in
astronomy have provided both direct and indirect evidence of these enigmatic
objects. Notably, the detection of gravitational waves by advanced LIGO and
Virgo detectors marked a significant milestone, revealing the merger of binary
BH systems [4]. Additionally, the Event Horizon Telescope (EHT) achieved a
monumental feat by capturing the first-ever image of a supermassive BH at the
center of the massive elliptical galaxy $M87$ [5, 6, 7, 8, 9, 10].
Furthermore, EHT also photographed $Sgr$ $A^{*}$, another supermassive BH
[11]. Simulations have shown growing interest in modeling Hawking radiation
from BHs and generating cosmological particles in quantum systems [12, 13, 14,
15]. The Hawking effect has generated significant interest, resulting in
extensive research that investigates its implications in various physical
contexts [16, 17, 18, 19, 20, 21, 22, 23].

Modern physics is built upon the foundational theories of quantum mechanics
and general relativity. However, the challenge of unifying these theories
remains. The emerging field of RQI seeks to bridge this gap [16]. Through its
exploration, RQI not only enhances our understanding of quantum mechanics but
also provides new insights into long-standing puzzles, such as the BH
information paradox. This paradox revolves around the unitary evaporation of
BHs [24, 2, 25, 26]. Understanding how relativistic particles become entangled
in both flat and curved spacetimes is not only intriguing for quantum
information but also holds significant implications for BH physics [27, 28,
17, 18, 19, 20]. Extracting information about entanglement from quantum field
vacua has thus become crucial for comprehending both the vacuum itself and the
underlying spacetime structure [29, 30]. This interest has spurred research
into quantum information within BH backgrounds and the dynamics of
entanglement [21, 22, 23, 31, 32, 33, 34, 35, 36, 37, 38, 39]. For instance,
Pan et al. explored entanglement in scalar fields near BHs [40], while Deng et
al. investigated the effect of Hawking decoherence and prepared states on the
distillability of entanglement in Dirac fields [41]. Xu et al. extended this
study to analyze the impact of Hawking radiation on multipartite entanglement
in Schwarzschild spacetime (SST) [42]. Furthermore, Hawking effect-induced
entanglement degradation has been shown to increase the entropic uncertainty
bound [43]. Recently, Pan et al. achieved a significant milestone in the study
of RQI by using the ”Micius” quantum communication satellite to experimentally
test gravitational decoherence effects [44].

Quantum resource theories provide a foundational framework for understanding
the essential characteristics of physical systems, encompassing non-locality
[45, 46, 47], quantum coherence [48, 49, 50], entanglement [51, 52, 53, 54,
55], and quantum correlations [56, 57, 58]. By offering an operational
interpretation of these quantum phenomena, this theory illuminates the
mechanisms underlying complex systems [59, 60, 61]. The Hawking effect in BHs
has been observed to consistently degrade the quantum resources of quantum
fields in curved spacetime [62], akin to environmental decoherence that
constrains certain quantum information processes. Therefore, gaining a
quantitative understanding of this decoherence in non-inertial frames is
crucial for discussing specific quantum information tasks. While the
distribution of quantum correlations in a relativistic framework has been
explored within the realm of RQI, there are still unanswered questions that
require further investigation. We aim to explore whether the Hawking
decoherence uniformly diminishes the quantum correlations of Dirac fields, and
if its impact varies across different measures of quantum correlations. To
investigate this, we analyze the dynamics of quantum correlations of Dirac
fields in distinct regions of spacetime — accessible, inaccessible, and
spacetime boundaries. Our evaluation employs the uncertainty-induced non-
locality measure [63, 47, 64] to assess quantum correlations and utilizes the
concept of quantum consonance to quantify both separable and entangled
correlations [65, 66, 67, 38].

Through meticulous analysis, we investigate how Hawking decoherence affects
the dynamics of quantum correlations among Dirac fields across different
regions: accessible, inaccessible, and spacetime. Our study reveals that
Hawking decoherence consistently diminishes these correlations in the
accessible region. Surprisingly, in inaccessible regions, these correlations
experience a resurgence as Hawking decoherence intensifies. This resurgence
persists until the correlations asymptotically approach certain constants at
extremely high Hawking temperatures. This phenomenon stems from the formation
of quantum correlations across the event horizon for Dirac fields. Finally, we
explore the distribution of these correlations among subsystems, aiming to
comprehensively grasp the evolving dynamics of quantum correlations among
Dirac fields.

The rest of this paper is organized as follows: Section II provides the
definitions of the metrics used to assess quantum correlations of Dirac fields
within accessible, inaccessible, and spacetime regions. Next, in Section III,
we revisit the vacuum structure and Hawking decoherence for Dirac fields in
SST. By using the Kruskal basis formulation, we derive a pure state that is
shared between particles inside and outside the EH. The findings and
subsequent discussions are presented in Section IV, and concluding remarks are
provided in Section V.

##  II Quantum correlations measures

In this section, we provide a brief overview of the fundamental concepts of
quantum correlations in the current system. We will specifically focus on
quantum consonance and uncertainty-induced nonlocality, as these are commonly
used as quantifiers for measuring quantum correlations.

###  II.1 Quantum consonance

Quantum consonance is a quantum measurement that integrates entanglement and
other quantum correlations by eliminating local coherence from global
coherence using local unitary operations [65, 66, 67]. It serves as a metric
for quantifying quantum correlations, mathematically represented as a
composite of entanglement and specific other quantum correlations [65]. This
formulation arises from applying local unitary transformations that isolate
the local coherence contribution from the overall coherence.

$\displaystyle\mathcal{C_{AB}}(\varrho)=\sum_{ijmn}\left|\varrho_{ijmn}^{q}(1-\delta_{im})(1-\delta_{jn})\right|,$
(1)

The state
$\varrho^{q}=(\mathcal{W}_{1}\otimes\mathcal{W}_{2})\varrho(\mathcal{W}_{1}\otimes\mathcal{W}_{2})^{\dagger}$
is derived by applying local unitary operations $\mathcal{W}_{1}$ and
$\mathcal{W}_{2}$ to the original state $\varrho$. For a two-qubit X-state,
it’s noteworthy that quantum consonance equals the sum of the off-diagonal
elements of the density operator.

###  II.2 Uncertainty-induced nonlocality

The concept of uncertainty-induced nonlocality, as introduced in [63],
quantifies nonlocal quantum correlations differently from measurement-induced
nonlocality (MIN). MIN is originally defined as the maximal change in a
bipartite quantum state due to a projective measurement, but suffers from non-
contractivity issues and is absent in states lacking MIN. Initially, MIN is
computed as the maximum Hilbert-Schmidt distance between the quantum state and
its post-measurement counterpart on a subsystem, affecting the state globally
without altering local reduced density operators [68, 69]. In contrast,
uncertainty-induced nonlocality is defined as the maximum skew information,
and serves as a revised measure that resolves these issues and assesses the
maximum skew information correlations of any bipartite state $\varrho$ [63].

$\mathcal{U_{AB}}(\varrho)=\max_{\zeta^{C}}\mathcal{I}(\varrho,\zeta^{C}),$
(2)

where,

$\mathcal{I}(\varrho,\zeta)=-\frac{1}{2}\mathrm{Tr}\left[\sqrt{\varrho},\zeta\right]^{2},$
(3)

Skew information, as introduced by Wigner and Yanase [70, 71], quantifies the
uncertainty resulting from the observable $\zeta$ acting on the subsystem
$\varrho$. In Eq. (2), the maximization is performed over all locally
maximally informative commuting observables
$\zeta^{C}=\zeta_{A}^{C}\otimes\mathbb{I}_{B}$, where $\zeta^{C}_{A}$ is a
Hermitian operator acting on qubit $A$ with distinct eigenvalues, and
$\mathbb{I}_{B}$ denotes the identity operator on qubit $B$. The precise form
of the uncertainty-induced nonlocality is explicitly detailed for any
($2\otimes d$)-dimensional quantum state $\varrho$ in [63].

$\mathcal{U_{AB}}(\varrho)=\left\\{\begin{array}[]{cc}1-n_{min}(\mathcal{N}),&\quad\boldsymbol{v}=\boldsymbol{0}\\\
1-\frac{1}{|\boldsymbol{v}|^{2}}\,\boldsymbol{v}\,\mathcal{N}\,\boldsymbol{v}^{T},&\quad\boldsymbol{v}\neq\boldsymbol{0}\end{array}\right.$
(4)

The notation $\boldsymbol{v}^{T}$ denotes the transpose of the Bloch vector
$\boldsymbol{v}$, and $n_{\min}(\mathcal{N})$ represents the smallest
eigenvalue of the symmetric real matrix $\mathcal{N}_{3\times 3}$ defined by
its elements.

$(\mathcal{N})_{ij}=\mathrm{Tr}\left\\{\varrho^{1/2}(\hat{\sigma}_{i}^{A}\otimes\mathbb{I}_{B})\varrho^{1/2}(\hat{\sigma}_{j}^{A}\otimes\mathbb{I}_{B})\right\\},$
(5)

where $\\{\hat{\sigma}_{i}^{A}\\}_{i=x,y,z}$ represents the conventional Pauli
matrices that operate on qubit $A$.

##  III Gisin state in SST

In this part, we aim to delve into the concepts of fermionic fields and vacuum
structure in SST, particularly from the perspective of observers of an initial
Gisin state. This state notably manifests complete entanglement for an
observer in free fall towards a SBH.

###  III.1 Gisin state

Gisin states, introduced by Gisin in [72], represent a class of states notable
for their unique properties. These states encompass combinations of pure,
entangled, and separable mixed states, and have been instrumental in
demonstrating hidden nonlocality. Specifically, certain Gisin states initially
exhibit local behavior but can lose this characteristic after undergoing
purely local operations [72, 73, 74, 59]. Mathematically, the Gisin states can
be expressed as:

$\varrho_{\lambda,\psi}(0)=\lambda|\phi_{\psi}\rangle\langle\phi_{\psi}|+\frac{1-\lambda}{2}\left(|00\rangle\langle
00|+|11\rangle\langle 11|\right),$ (6)

The mixing parameter $\lambda\in[0,1]$ defines the degree of mixture, while
the state $|\phi_{\psi}\rangle=\sin(\psi)|01\rangle+\cos(\psi)|10\rangle$
represents a Bell-like state with $\psi\in[0,\pi/2]$. At $\psi=\frac{\pi}{4}$,
$|\phi_{\psi}\rangle$ becomes the familiar Bell state
$|\phi^{+}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$. Gisin states are
distinct from Weyl states but intersect non-trivially when locally maximally
mixed at $\psi=\frac{\pi}{4}$. For $\lambda<1$, the Gisin state is fully
mixed. However, at $\lambda=1$, it simplifies to the pure entangled state
$\left|\phi_{\psi}\right\rangle\left\langle\phi_{\psi}\right|$. To understand
how $\lambda$ and $\psi$ influence the correlation measures $\mathcal{U_{AB}}$
and $\mathcal{C_{AB}}$, we present the results in Figure (1) below.

Figure 1: Quantum correlations $\mathcal{U_{AB}}$ 1 and $\mathcal{C_{AB}}$ 1
as a function of the mixing parameter $\lambda$ for different angle values
$\psi$ for the initial Gisin state $\varrho_{\lambda,\psi}(0)$.

From Figure (1), it is evident that both $\mathcal{U_{AB}}$ and
$\mathcal{C_{AB}}$ consistently increase with the mixing parameter. Initially,
$\mathcal{C_{AB}}$ and $\mathcal{U_{AB}}$ share a common minimum value of
$\mathcal{U_{AB}}=\mathcal{C_{AB}}=0$, due to the separable and incoherent
nature of the Gisin state when $\lambda\approx 0$. Both measures then steadily
grow, reaching their maximum degree of correlation at $\lambda=1$, signifying
a transition from a partially correlated state to a maximally correlated
state. Additionally, it is notable that
$\mathcal{U_{AB}}(\psi=\frac{\pi}{4})>\mathcal{U_{AB}}(\psi=\frac{\pi}{5})$
and
$\mathcal{C_{AB}}(\psi=\frac{\pi}{4})>\mathcal{C_{AB}}(\psi=\frac{\pi}{5})$.
This implies that the Gisin state exhibits the highest level of quantum
correlations when $\psi=\frac{\pi}{4}$. Interestingly, for
$\psi<\frac{\pi}{4}$ and $\lambda=1$, $\mathcal{C_{AB}}$ bounds the
$\mathcal{U_{AB}}$ quantifier. This indicates that the parameters $\psi$ and
$\lambda$ significantly influence the behavior of the $\mathcal{U_{AB}}$ and
$\mathcal{C_{AB}}$ quantifiers.

###  III.2 Dirac quantum fields in SST

In asymptotically flat spacetime, the Dirac equation represents a spinor field
$\chi$ with a wave function, as shown in [75, 76]:

$[i\gamma^{a}e^{\mu}_{a}(\partial_{\mu}+\Gamma_{\mu})]\chi=0,$ (7)

The symbol $\gamma^{a}$ denotes the Dirac gamma matrix, $e^{\mu}_{a}$
represents the inverse of the tetrad $e_{\mu}^{a}$, and $\Gamma_{\mu}$ is
equal to $\frac{1}{8}\left[\gamma^{a},\,\gamma^{b}\right]$. The expression
$e^{\nu}_{a}e_{b\nu,\mu}$ denotes the spin connection coefficient. In our
further discussion, we will adopt natural units by setting the values of $G$,
$c$, $\hbar$, and $k_{B}$ to be equal to 1. In addition, we will be examining
the asymptotically flat SST, which may be characterized by the following
metric element:

$ds^{2}=-(1-\frac{2M_{H}}{r})dt^{2}+(1-\frac{2M_{H}}{r})^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\alpha^{2}),$ (8)

with $M_{H}$ denotes the BH mass, the Dirac equation in SST may be rewritten
by combining Eq. (7) with Eq. (8).

$\displaystyle-\Xi\frac{\partial\chi}{\partial t}$ $\displaystyle+$
$\displaystyle\gamma_{1}\sqrt{1-\frac{2M_{H}}{r}}\left[\frac{\partial}{\partial
r}+\frac{1}{r}+\frac{M_{H}}{2r(r-2M_{H})}\right]\chi$ (9) $\displaystyle+$
$\displaystyle\frac{\gamma_{2}}{r}(\frac{\partial}{\partial\theta}+\frac{\cot\theta}{2})\chi+\frac{\gamma_{3}}{r\sin\theta}\frac{\partial\chi}{\partial\alpha}=0.$

with $\Xi=\frac{\gamma_{0}}{\sqrt{1-\frac{2M_{H}}{r}}}$. To solve the Dirac
equation (9) near the EH, it is essential to extend the spinor field $\chi$ by
utilizing positive and negative frequency solutions for the outside and inside
of the EH, respectively. You can find detailed descriptions of these solutions
in [42, 77, 78, 79].

$\chi^{I+}_{\varsigma_{i}}(r>r_{+})=\mathfrak{P}e^{-i\omega_{i}\mathfrak{u}},\qquad\chi^{II+}_{\varsigma_{i}}(r<r_{+})=\mathfrak{P}e^{i\omega_{i}\mathfrak{u}},\quad\forall
i,$ (10)

where $\mathfrak{P}$ represents a 4-component Dirac spinor [80, 41],
$\omega_{i}$ indicates the monochromatic frequency of the Dirac field, and
$\mathfrak{u}=t-r-2M_{H}\ln\left[\frac{r-2M_{H}}{2M_{H}}\right]$ acts as the
tortoise coordinate [81]. Since $\chi^{I+}_{\varsigma_{i}}$ and
$\chi^{II+}_{\varsigma_{i}}$ are analytic outside and inside the EH
respectively, they form a complete orthogonal basis. Here, $I$ is the exterior
region and $II$ is the interior region. Therefore, the Dirac field
$\chi_{out}$ can be quantized in SST.

$\chi_{out}=\sum_{i,\vartheta}\int
dk\left(a^{\vartheta}_{\varsigma_{i}}\chi^{\vartheta+}_{\varsigma_{i}}+b^{\vartheta+}_{\varsigma_{i}}\chi^{\vartheta-}_{\varsigma_{i}}\right),$
(11)

where $a^{\vartheta}_{\varsigma_{i}}$ and $b^{\vartheta+}_{\varsigma_{i}}$
represent the fermionic annihilation operator and anti-fermionic creation
operator, respectively, with $\vartheta=(I,II)$. On the other hand, the
viewpoint of an accelerating observer in an asymptotically flat space-time may
be correctly characterized using the Kruskal coordinates [82]. Similarly, for
SST, the generalized light-like Kruskal coordinates, denoted by $\mathfrak{U}$
and $\mathfrak{V}$, are introduced as
$\mathfrak{u}=-4M_{H}\ln\left[\frac{-\mathfrak{U}}{4M_{H}}\right],\qquad\mathfrak{v}=4M_{H}\ln\left[\frac{\mathfrak{V}}{4M_{H}}\right],\quad\text{if}\quad
r>r_{+},$ (12)
$\mathfrak{u}=-4M_{H}\ln\left[\frac{\mathfrak{U}}{4M_{H}}\right],\qquad\mathfrak{v}=4M_{H}\ln\left[\frac{\mathfrak{V}}{4M_{H}}\right],\quad\text{if}\quad
r<r_{+},$ (13) Based on [83], it is possible to perform an analytic
continuation for $\chi^{I+}_{\varsigma_{i}}$ and $\chi^{II+}_{\varsigma_{i}}$.
Furthermore, it is possible to find the complete bases for positive energy
modes that are analytic for all real values of $\mathfrak{U}$ and
$\mathfrak{V}$. Specifically,

$\Upsilon^{I+}_{\varsigma_{i}}=e^{2\pi
M_{H}\omega_{i}}\chi^{I+}_{\varsigma_{i}}+e^{-2\pi
M_{H}\omega_{i}}\chi^{II-}_{\varsigma_{i}},$ (14)
$\Upsilon^{II+}_{\varsigma_{i}}=e^{-2\pi
M_{H}\omega_{i}}\chi^{I-}_{-\varsigma_{i}}+e^{2\pi
M_{H}\omega_{i}}\chi^{II+}_{\varsigma_{i}},$ (15)

The Dirac field in Kruskal spacetime can then be decomposed using the
aforementioned complete basis.

$\chi_{out}=\sum_{i,\vartheta}\int dk\left[2\cosh(4\pi
M_{H}\omega_{i})\right]^{-1/2}\left(c^{\vartheta}_{\varsigma_{i}}\Upsilon^{\vartheta+}_{\varsigma_{i}}+d^{\vartheta+}_{\varsigma_{i}}\Upsilon^{\vartheta-}_{\varsigma_{i}}\right),$
(16)

where $c^{\vartheta}_{\varsigma_{i}}$ and $d^{\vartheta+}_{\varsigma_{i}}$
represent the annihilation and creation operators that act on the Kruskal
vacuum. It is evident that the two quantization procedures shown in Eqs. (11)
and (16) are not the same. The relationship between the creation and
annihilation operators in the Schwarzschild and Kruskal coordinates is
reflected in the Bogoliubov transformations [84]. Since the Bogoliubov
transformations are diagonal, we can express each annihilation operator
$c^{I}_{\varsigma_{i}}$ as a combination of Schwarzschild particle operators
with a single Schwarzschild frequency $\omega_{i}$ [85, 86]. More
specifically,

$c^{I}_{\varsigma_{i}}=\left(e^{-8\pi
M_{H}\omega_{i}}+1\right)^{-1/2}a^{I}_{\varsigma_{i}}-\left(e^{8\pi
M_{H}\omega_{i}}+1\right)^{-1/2}b^{II+}_{\varsigma_{i}}.$ (17)

The vacuum and excited states of the Kruskal particle for mode $\Lambda$ in
the Schwarzschild Fock space bases can be determined through a series of
calculations. The representations of these states are as follows:
$\left|1\right\rangle_{\Lambda}=\otimes_{i}\left|1_{\omega_{i}}\right\rangle_{\Lambda}$
and
$\left|0\right\rangle_{\Lambda}=\otimes_{i}\left|0_{\omega_{i}}\right\rangle_{\Lambda}$.
$\begin{array}[]{cc}\begin{aligned}
\left|0\right\rangle_{\Lambda}&=\left(e^{-\omega_{i}/T_{H}}+1\right)^{-1/2}\exp\left[e^{-\omega_{i}/2T_{H}}a^{I+}_{\varsigma_{i}}b^{II+}_{-\varsigma_{i}}\right]\left|0_{\varsigma_{i}}\right\rangle_{I}^{+}\left|0_{-\varsigma_{i}}\right\rangle_{II}^{-}\\\
&=\left(e^{-\omega_{i}/T_{H}}+1\right)^{-1/2}\left|0_{\varsigma_{i}}\right\rangle_{I}^{+}\left|0_{-\varsigma_{i}}\right\rangle_{II}^{-}+\left(e^{\omega_{i}/T_{H}}+1\right)^{-1/2}\left|1_{\varsigma_{i}}\right\rangle_{I}^{+}\left|1_{-\varsigma_{i}}\right\rangle_{II}^{-}\end{aligned}\end{array},$
(18) and

$\left|1\right\rangle_{\Lambda}=c^{I+}_{\varsigma_{i}}\left|0\right\rangle_{\Lambda}=\left|1_{\varsigma_{i}}\right\rangle_{I}^{+}\left|0_{-\varsigma_{i}}\right\rangle_{II}^{-},$
(19)

where $T_{H}=\frac{1}{8\pi M_{H}}$ represents the Hawking temperature, we
simplify $\left\\{\left|p_{\varsigma_{i}}\right\rangle^{+}\right\\}$ by
$\left\\{\left|p\right\rangle_{I}\right\\}$ and
$\left\\{\left|p_{-\varsigma_{i}}\right\rangle^{-}\right\\}$ by
$\left\\{\left|p\right\rangle_{II}\right\\}$. These represent the orthonormal
bases for the outside and inside of the EH, respectively. We note that in a
fully quantum framework, the processes of BH formation and evaporation involve
system entanglement. The definitions of the Kruskal particle’s vacuum state
(18) and excited state (19) clarify this idea. Furthermore, in the context of
quantum information theory, Hawking decoherence can also be understood as a
quantum noisy channel [87, 88]. For example, in Eqs. (18) and (19), the
Hawking decoherence functions as an amplification quantum channel and is
associated with the observable
$\left(e^{\omega_{i}/T_{H}}+1\right)^{-1/2}$[89, 90].

Figure 2: A Penrose diagram is useful for illustrating the trajectories of Bob
and Alice in a SBH. It shows how they share an initial Gisin state in the
asymptotically flat region of SST. In this diagram, Bob is shown as free-
falling towards the SBH, while Alice remains static. The interior of the BH is
represented by region $III$, and the exterior is divided into regions $I$ and
$II$. Region $IV$ represents the location of the white hole.

We will investigate the quantum correlations of Dirac fields near a SBH, both
in physically accessible and inaccessible regions, by examining the behavior
of Dirac particles. From an inertial standpoint, the particle exists in a
distinct superposition of Kruskal monochromatic modes. In SST, the observer
detects the field in states defined by a single mode. For our analysis, let’s
consider two observers, Bob and Alice. Initially, they are both in an
asymptotically flat region of SST, sharing an entangled Gisin state. Bob’s
detector is sensitive to mode $\left|p\right\rangle_{B}$, while Alice’s
detector is intended to detect mode $\left|p\right\rangle_{A}$. After their
first encounter, Alice remains in the flat region that approaches infinity,
whereas Bob freely falls towards a SBH with constant acceleration, as shown in
Figure (2), eventually arriving near its EH. To describe Bob’s point of view,
we must expand the initial Gisin state corresponding to mode $\Lambda$, taking
into account both the interior and exterior regions of the EH. The Kruskal
states $\left|0\right\rangle_{\Lambda}$ and $\left|1\right\rangle_{\Lambda}$
correspond to two-mode states within the SST, as described by Eqs. (18) and
(19). By rewriting the initial Gisin state, we obtain the evolved state.
$\displaystyle\varrho_{AB_{I}B_{II}}$ $\displaystyle=$
$\displaystyle\frac{1-\lambda}{2}\left(\varpi^{2}|000\rangle\langle
000|+\varpi\varepsilon(|000\rangle\langle 011|+|011\rangle\langle
000|)+\varepsilon^{2}|011\rangle\langle 011|+|110\rangle\langle 110|\right)$
(20) $\displaystyle+$
$\displaystyle\lambda\cos^{2}\psi\left(\varpi^{2}|100\rangle\langle
100|+\varpi\varepsilon(|100\rangle\langle 111|+|111\rangle\langle
100|+\varepsilon^{2}|111\rangle\langle
111|)\right)+\lambda\sin^{2}\psi|010\rangle\langle 010|$ $\displaystyle+$
$\displaystyle\lambda\sin\psi\cos\psi\left(\varpi(|010\rangle\langle
100|+|100\rangle\langle 010|)+\varepsilon(|010\rangle\langle
111|+|111\rangle\langle 010|)\right).$ where we define
$\varpi=\frac{1}{\left(e^{-\omega/T_{H}}+1\right)^{1/2}}$,
$\varepsilon=\frac{1}{\left(e^{\omega/T_{H}}+1\right)^{1/2}}$, and
$|opq\rangle=|o\rangle_{A}|p\rangle_{B_{I}}|q\rangle_{B_{II}}$. By tracing
over one degree of freedom in the density operator $\varrho_{AB_{I}B_{II}}$,
three subsystems can be obtained: a physically accessible subsystem
$\varrho_{AB_{I}}$, a physically inaccessible subsystem $\varrho_{AB_{II}}$,
and a spacetime subsystem $\varrho_{B_{I}B_{II}}$.

##  IV Results and Discussion

Following the above setups, we now study the quantum correlations of Dirac
fields that are present in every subsystem. Although the established global
quantum state consists of fermionic modes both inside and outside the EH, it
is a pure state. However, due to the causal disconnection between the interior
and exterior regions, an observer or detector outside the SBH cannot access
all the information. By using the partial trace operation, one can distinguish
between physically accessible and inaccessible scenarios. Despite the
inaccessibility of certain scenarios to experimental observation or detection,
the global quantum state remains unitary and pure. With the prevailing
consensus among physicists, including Hawking, now in favor of the
conservation of quantum information, we can theoretically investigate both
accessible and inaccessible scenarios. The following subsections will explore
these scenarios in detail.

###  IV.1 Physically accessible scenario

As previously mentioned, the exterior and interior regions of SBH are
completely disconnected. Therefore, the information can only be accessed
within the subsystem $\varrho_{AB_{I}}$. Within this subsystem, the
corresponding reduced density operator is as follows:

$\displaystyle\varrho_{AB_{I}}$ $\displaystyle=$
$\displaystyle\frac{1-\lambda}{2}\varpi^{2}|00\rangle\langle
00|+(\frac{1-\lambda}{2}\varepsilon^{2}+\lambda\sin^{2}\psi)|01\rangle\langle
01|$ (21) $\displaystyle+$
$\displaystyle\lambda\cos^{2}\psi\varpi^{2}|10\rangle\langle
10|+\lambda\sin\psi\cos\psi\varpi(|01\rangle\langle 10|+|10\rangle\langle
01|)$ $\displaystyle+$
$\displaystyle(\frac{1-\lambda}{2}+\lambda\cos^{2}\psi\varepsilon^{2})|11\rangle\langle
11|.$

Figure 3: Quantum correlations of Dirac fields $\mathcal{U_{AB_{I}}}$ 3–3 and
$\mathcal{C_{AB_{I}}}$ 3–3 as a function of the Hawking temperature $T_{H}$
for different state parameters $\lambda$ and $\psi$ in the physically
accessible system $\varrho_{AB_{I}}$.

To investigate the impact of various factors on $\mathcal{U_{AB_{I}}}$ and
$\mathcal{C_{AB_{I}}}$, namely the Hawking temperature $T_{H}$, the Gisin
state parameters $\lambda$ and $\psi$, we analyze their dynamic behaviors in
relation to $T_{H}$. Figure (3) demonstrates how $T_{H}$ affects
$\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{I}}}$ while keeping other
parameters, such as the mode frequency $\omega$, constant. Figures 3–3
consistently show a decrease and eventual disappearance of quantum
correlations as $T_{H}$ approaches infinity. At $T_{H}=0$, quantum
correlations reach their maximum and remain unaffected by the Hawking effect.
However, as $T_{H}$ increases, decoherence occurs due to the interaction
between the Gisin state system and Hawking decoherence. The strength of the
quantum correlations depends on the state parameters $\lambda$ and $\psi$,
even at finite $T_{H}$. Reducing the mixing parameter $\lambda$ decreases both
$\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{I}}}$, regardless of the presence
of the Hawking effect. Similarly, Figures 3–3, illustrate that when
$\psi=\frac{\pi}{4}$, both $\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{I}}}$
reach their maximum values, with or without the Hawking effect. The values of
$\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{I}}}$ for Dirac particles close to
a SBH are determined by the state parameters $\lambda$ and $\psi$. Increasing
$\lambda$ enhances the quantum correlations, while increasing the Hawking
temperature $T_{H}$ gradually suppresses them. When $T_{H}$ approaches
infinity, corresponding to the complete evaporation of the BH, residual
quantum correlations only exist for higher values of $\lambda$. As $\psi$
increases, the values of $\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{I}}}$
also increase. However, further increasing $\psi$ from $\frac{\pi}{8}$ to
$\frac{\pi}{4}$ results in an increase in these measures, ultimately leading
to a separable classical state at $\psi=\frac{\pi}{8}$. In addition, as
$T_{H}$ increases, quantum correlations in Figures 3–3 deteriorate rapidly
compared to Figures 3–3, shifting towards the right. The figures illustrate a
linear increase in quantum correlations with increasing values of $\lambda$
and $\psi$. Moreover, further increases in $\lambda$ and $\psi$ lead to a
gradual enhancement of quantum correlations, peaking at $\lambda=0.9$. This
finding aligns with [39], suggesting that quantum correlations in this region
serve as a resource for quantum information tasks in SST.

Figure 4: Quantum correlations of Dirac fields $\mathcal{U_{AB_{I}}}$ 4 and
$\mathcal{C_{AB_{I}}}$ 4 as a function of the Hawking temperature $T_{H}$ for
different frequency modes $\omega$ in the physically accessible system
$\varrho_{AB_{I}}$.

The effect of different frequencies $\omega$ of observed or detected Dirac
particles on $\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{I}}}$ is shown in
Figure 4, while keeping all other parameters constant. As shown in Figure 4,
the quantum correlations that can be accessed always decrease as $T_{H}$
increases. The presence of Hawking decoherence seems to speed up this
reduction, potentially even leading to the sudden death of
$\mathcal{U_{AB_{I}}}$ under certain conditions. In the limit
$T_{H}\rightarrow\infty$, which corresponds to a BH with negligible mass,
$\mathcal{U_{AB_{I}}}$ has a degree of zero, indicating a local state. For any
finite Hawking temperature, the state becomes nonlocal, with lower Hawking
temperatures resulting in greater nonlocal quantum correlations. As
$T_{H}\rightarrow 0$, which corresponds to no BH evaporation, the degree of
quantum correlations approaches 0.9, indicating a maximally correlated state.
This demonstrates that only Dirac particles with high frequencies
($\omega\rightarrow\infty$) exhibit high correlations. Specifically,
$\mathcal{C_{AB_{I}}}$ in a physically accessible region does not approach
zero, regardless of the evaporation state of the BH, with a minimum value of
0.65 for $\omega=10$. Additionally, the Pauli exclusion principle in Fermi-
Dirac statistics prevents the quantum states of Fermi particles from being
infinitely excited by the Hawking effect. This explains why the accessible
correlations $\mathcal{C_{AB_{I}}}$ do not completely vanish, even when the
Hawking temperature increases indefinitely. As $\omega$ decreases, the
negative effects of Hawking decoherence become more noticeable. The maximum
quantum correlations are observed for all frequencies at $T_{H}=0$. However, a
higher value of $\omega$ improves the resilience of the quantum correlations
of the shared Gisin state against Hawking decoherence. We observe that the
value of $\mathcal{C_{AB_{I}}}$ is always nonzero. This indicates that
$\mathcal{C_{AB_{I}}}$ is more robust than $\mathcal{U_{AB_{I}}}$ and can
provide more information about the given system. In other words,
$\mathcal{C_{AB_{I}}}$ can be considered a more effective resource for
implementing RQI in practical applications.

###  IV.2 Physically inaccessible scenario

We will now investigate how Hawking decoherence affects the properties of the
shared Gisin state in the physically inaccessible subsystem
$\varrho_{AB_{II}}$. To comprehend the whereabouts of the lost quantum
correlations, we assume that these vanished quantum correlations are
redistributed to physically inaccessible regions. This assumption is based on
the fact that Hawking decoherence splits Bob’s mode into mode
$\left|p\right\rangle_{B_{I}}$ and mode $\left|p\right\rangle_{B_{II}}$. Our
focus will be on the remaining two subsystems, where we will explore the
quantum correlations in the physically inaccessible region. The subsystem
$\varrho_{AB_{II}}$ is given by

$\displaystyle\varrho_{AB_{II}}$ $\displaystyle=$
$\displaystyle(\frac{1-\lambda}{2}\varpi^{2}+\lambda\sin^{2}\psi)|00\rangle\langle
00|+\frac{1-\lambda}{2}\varepsilon^{2}|01\rangle\langle 01|$ (22)
$\displaystyle+$
$\displaystyle(\frac{1-\lambda}{2}+\lambda\cos^{2}\psi\varpi^{2})|10\rangle\langle
10|+\lambda\cos^{2}\psi\varepsilon^{2}|11\rangle\langle 11|$ $\displaystyle+$
$\displaystyle\lambda\sin\psi\cos\psi\varepsilon(|00\rangle\langle
11|+|11\rangle\langle 00|)$

Figure 5: Quantum correlations of Dirac fields $\mathcal{U_{AB_{II}}}$ 5–5 and
$\mathcal{C_{AB_{II}}}$ 5–5 as a function of the Hawking temperature $T_{H}$
for different state parameters $\lambda$ and $\psi$ in the physically
inaccessible system $\varrho_{AB_{II}}$.

In Figure 5, we plot $\mathcal{U_{AB_{II}}}$ and $\mathcal{C_{AB_{II}}}$ as
functions of $T_{H}$ for different values of the state parameters $\lambda$
and $\psi$. The plot shows that in the physically inaccessible subsystem
$\varrho_{AB_{II}}$, $\mathcal{U_{AB_{II}}}$ and $\mathcal{C_{AB_{II}}}$ are
zero between modes $A$ and $B_{II}$ when Hawking decoherence is absent. When
the Hawking temperature increases, the inaccessible quantum correlations
monotonically increase to a nonzero value, in contrast to the $N$-partite
entanglement which has a monotonic and non-monotonic relationship with the
Hawking temperature [92]. It should be noted that the amount of quantum
correlations depends on the initial Gisin state parameters $\lambda$ and
$\psi$. In other words, the initial state parameters control how quantum
information travels between the interior and exterior regions of the SBH and
how the quantum information is influenced by Hawking decoherence. Previous
studies [93, 92, 39] have shown that increasing the state parameters can
mitigate the decoherence effect caused by Hawking radiation. However, within
the inaccessible region, enhancing the state parameters $\lambda$ and $\psi$
can amplify the decay of quantum resources in the system under study. As shown
in Fig. 3, it is evident that the Hawking decoherence environment in the
accessible region leads to a higher loss of quantum correlations compared to
the Hawking decoherence environment in the inaccessible region.

Figure 6: Quantum correlations of Dirac fields $\mathcal{U_{B_{I}B_{II}}}$ 6
and $\mathcal{C_{B_{I}B_{II}}}$ 6 as a function of the Hawking temperature
$T_{H}$ for different frequency modes $\omega$ in the spacetime region
$\varrho_{B_{I}B_{II}}$.

We now analyze how the frequency mode of Dirac particles influences
$\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{II}}}$. To facilitate a direct
comparison between these correlation measures, we present
$\mathcal{U_{AB_{I}}}$ and $\mathcal{C_{AB_{II}}}$ as functions of the Hawking
temperature in Figure (6). In the absence of Hawking decoherence, there are no
quantum correlations between modes $A$ and $B_{II}$, indicating that quantum
correlations of Dirac fields remain finite at finite Hawking temperatures. At
an infinite Hawking temperature ($T_{H}=100$), the system’s state
$\varrho_{AB_{II}}$ shows partial correlation in the physically inaccessible
region that can serve as a resource for specific RQI tasks. In contrast, for
the physically accessible system $\varrho_{AB_{I}}$, quantum correlations
approach zero under this limit. This behavior arises due to statistical
correlations inherent in fermionic fields beyond entanglement, which persist
and cannot be eliminated. This statistical correlation corresponds to the
second quantized manifestation of statistical entanglement discussed in [94].
Notably, these conclusions extend straightforwardly to the scenario of a
Schwarzschild black hole. Therefore, employing the low-frequency mode
($\omega\approx 10$) preserves quantum correlations in this region, while the
high-frequency mode is suitable for handling tasks related to RQI. Our
observations of their distinct frequency dependencies in SST represent a novel
contribution to understanding the Hawking decoherence of BHs.

###  IV.3 Spacetime scenario

Considering the spacetime region, the reduced density operator in this region,
$\varrho_{B_{I}B_{II}}$ is

$\displaystyle\varrho_{B_{I}B_{II}}$ $\displaystyle=$
$\displaystyle(\frac{1-\lambda}{2}+\lambda\cos^{2}\psi)\varpi^{2}|00\rangle\langle
00|+(\frac{1-\lambda}{2}+\lambda\cos^{2}\psi)$ (23) $\displaystyle\times$
$\displaystyle\varpi\varepsilon(|00\rangle\langle 11|+|11\rangle\langle
00|)+(\frac{1-\lambda}{2}+\lambda\sin^{2}\psi)|10\rangle\langle 10|$
$\displaystyle+$
$\displaystyle(\frac{1-\lambda}{2}+\lambda\cos^{2}\psi)\varepsilon^{2}|11\rangle\langle
11|$

Figure 7: Quantum correlations of Dirac fields $\mathcal{U_{B_{I}B_{II}}}$ 7–7
and $\mathcal{C_{B_{I}B_{II}}}$ 7–7 as a function of the Hawking temperature
$T_{H}$ for different state parameters $\lambda$ and $\psi$ in the space–time
region $\varrho_{B_{I}B_{II}}$.

Here, we present a comprehensive examination of quantum correlations within
the region of spacetime affected by Hawking decoherence. In Figure (7), we
illustrate the behaviors of $\mathcal{C_{B_{I}B_{II}}}$ and
$\mathcal{U_{B_{I}B_{II}}}$ as a function of $T_{H}$, for various values of
the state parameters $\lambda$ and $\psi$. It is clear that the quantum
correlations reach their maximum level at higher Hawking temperatures. Within
this region of spacetime, these quantum correlations are able to recover at
zero Hawking temperature and gradually approach a constant value as the
Hawking temperature approaches infinity. It is worth noting that the amount of
quantum correlations decreases dramatically to zero at a temperature value
$T_{H}\approx 10$, which depends on the Gisin state parameters, and then
increases monotonically to reach a stable value. This result suggests the
possibility of Hawking-induced quantum correlations in this region of
spacetime. Additionally, we observe that the quantum correlations increase as
the parameter $\lambda$ increases, particularly for high Hawking temperatures
($T_{H}>40$). Therefore, the Gisin state parameter $\lambda$ can be used to
mitigate the detrimental effect of high Hawking temperature. Furthermore, from
Figure (7), we also observe that at the highest values of $T_{H}$ (i.e.,
$T_{H}>80$), $\mathcal{C_{B_{I}B_{II}}}$ is significantly larger than
$\mathcal{U_{B_{I}B_{II}}}$, indicating that the current system is more
coherent than correlated. In particular, based on Hawking’s initial argument
[25, 2], larger black holes have a lower temperature, which results in weaker
radiation emission compared to smaller black holes. Simply put, the mass of
the black hole directly influences its radiation temperature. Our findings
support Hawking’s assertion by demonstrating that quantum correlations
increase as the temperature of the black hole increases in the spacetime
region. The maximum difference between $\mathcal{C_{B_{I}B_{II}}}$ and
$\mathcal{U_{B_{I}B_{II}}}$ occurs as $T_{H}$ increases. In this case,
$\mathcal{C_{B_{I}B_{II}}}$ sharply increases and then remains constant at an
infinite value of $T_{H}$. On the other side, $\mathcal{U_{B_{I}B_{II}}}$
exhibits similar behavior. Furthermore, when the values of Hawking decoherence
increase, $\mathcal{U_{B_{I}B_{II}}}$ experiences a significant increase
followed by freezing.

Figure 8: Quantum correlations of Dirac fields $\mathcal{U_{B_{I}B_{II}}}$ 8
and $\mathcal{C_{B_{I}B_{II}}}$ 8 as a function of the Hawking temperature
$T_{H}$ for different frequency modes $\omega$ in the space–time region
$\varrho_{B_{I}B_{II}}$.

We now analyze the influence of the frequency mode $\omega$ of the fermionic
particles on the change of quantum correlations between modes $B_{I}$ and
$B_{II}$ in the presence of Hawking decoherence. In Figure (8),
$\mathcal{C_{B_{I}B_{II}}}$ and $\mathcal{U_{B_{I}B_{II}}}$ are displayed
against $T_{H}$ for different values of the frequency mode $\omega$. From
Figure (8), one can observe that $\mathcal{C_{B_{I}B_{II}}}$ exhibits similar
behavior to $\mathcal{U_{B_{I}B_{II}}}$, but they have different values.
Initially, both measures show zero correlations in the spacetime region, and
then increase to attain a saturation value for sufficiently large
$T_{H}\approx 100$. However, as the frequency mode $\omega$ increases, the
influence of the Hawking temperature $T_{H}$ on the quantum correlations
attenuates. This suggests that by adjusting the frequency mode $\omega$,
quantum correlations can be improved and preserved. Therefore, when preparing
Gisin states in the spacetime region with significant correlations, the
advantageous impact of Hawking temperature $T_{H}$ must be taken into account.
The decline of quantum correlations is more noticeable for finite Hawking
temperatures. This indicates that low levels of external Hawking decoherence
adversely affect the quantum correlations between modes $B_{I}$ and $B_{II}$
in the evolved Gisin state. Consequently, as the Hawking temperature increases
and the mode frequency decreases, the system’s state in the spacetime region
becomes more correlated.

###  IV.4 Quantum correlations redistribution

Based on the analysis presented above, it is clear that the process of Hawking
decoherence in a SBH leads to the redistribution of quantum correlations. This
process shows that Hawking decoherence decreases the presence of accessible
quantum correlations while at the same time increasing the amount of
inaccessible quantum correlations in either a monotonic or non-monotonic way.
Therefore, in order to understand how quantum information is distributed
within a SBH, both in its interior and exterior regions, we will examine the
distribution of quantum correlations in Dirac fields across various regions.
Our main goal will be to analyze how these correlations are distributed among
all subsystems.

Figure 9: Quantum correlations of Dirac fields $\mathcal{U_{AB}}$ 9–9 and
$\mathcal{C_{AB}}$ 9–9 among all the subsystems $\varrho_{AB_{I}}$,
$\varrho_{AB_{II}}$ and $\varrho_{B_{I}B_{II}}$ as functions of Hawking
temperature $T_{H}$.  To thoroughly investigate the quantum correlations of
Dirac fields in the SST, we plotted their distributions in Figure (9). These
distributions demonstrate the effect of Hawking decoherence on quantum
correlations across all possible subsystems. We observe that the initially
accessible quantum correlations decrease and then stabilize as the Hawking
temperature $T_{H}$ increases. On the other hand, the initially inaccessible
quantum correlations remain at zero and then gradually increase and stabilize
as the Hawking decoherence intensifies. Our findings indicate that at low
Hawking temperatures, mode $AB_{I}$ retains the majority of the quantum
correlation, while the correlation is smaller in modes $AB_{II}$ and
$B_{I}B_{II}$. As the Hawking decoherence increases, more accessible
information leaks into the inaccessible regions, causing a greater degree of
correlation between them. In essence, the original quantum information
exchanged between Alice and Bob gets distributed across the modes $A$,
$B_{I}$, and $B_{II}$. Furthermore, when the SBH completely evaporates, the
quantum correlations of subsystems $\varrho_{AB_{I}}$ and
$\varrho_{B_{I}B_{II}}$ will be approximately equal for a maximally entangled
Gisin state ($\lambda=1$ and $\psi=\pi/4$). Additionally, it is worth noting
that physically accessible quantum correlations can be redistributed to a
region that is physically inaccessible. Moreover, we have observed that the
redistribution of quantum correlations does not occur in the same way when
considering a mixed Gisin state. This suggests that the maximally initially
entangled Gisin state is more effective in resisting Hawking decoherence and
is better suited for processing RQI. These findings offer valuable guidance in
choosing suitable quantum states and resources for RQI tasks. For example, the
maximally initially entangled Gisin state can be regarded as the quantum
state, while quantum consonance can be chosen as the quantum resource.

##  V Concluding remarks

In conclusion, we have examined the distribution of physically accessible and
inaccessible quantum correlations of Dirac fields within the SST framework. We
have introduced two measures of quantum correlation: uncertainty-induced non-
locality and quantum consonance. Through our analysis, we have demonstrated
how quantum correlations of Dirac fields are distributed among different
subsystems. Our findings indicate that an increase in Hawking radiation
inevitably leads to a reduction in quantum correlations. This reduction can be
explained as the leakage of information from the initial Gisin state into SST,
stimulated by Hawking decoherence. Specifically, the magnitudes of quantum
correlation that are physically accessible are determined by the initial Gisin
state parameters and consistently diminish with intensifying Hawking
radiation. In the physically inaccessible and spacetime regions, quantum
correlations of Dirac fields are absent at zero Hawking temperature, but
reemerge as Hawking decoherence increases, eventually approaching specific
constants at infinite Hawking temperature. As a result, the growing quantum
correlations of Dirac fields can be understood as the transfer of quantum
information from the physically accessible to the inaccessible region. To
provide a more intuitive understanding of the flow of information, we also
examine their distributions among all subsystems. Our results suggest that in
the absence of Hawking decoherence, mode $AB_{I}$ contains the highest amount
of information, while modes $AB_{II}$ and $B_{I}B_{II}$ contain less
information. As the intensity of Hawking decoherence increases, some of the
accessible information leaks into regions that are inaccessible. Moreover, the
quantum information in $\varrho_{AB_{I}}$ and $\varrho_{AB_{II}}$ is almost
identical when the Hawking temperature approaches infinity. Finally, we
anticipate that the results of our investigation will advance our
understanding of the quantum information paradox in curved spacetime,
particularly with regard to quantum correlations of Dirac fields.

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