# Quantum teleportation and dynamics of quantum coherence and metrological
non-classical correlations for open two-qubit systems: A study of Markovian
and non-Markovian regimes

Yassine Dakir  LPHE-Modeling and Simulation, Faculty of Sciences, Mohammed V
University in Rabat, Rabat, Morocco.  Abdallah Slaoui  0000-0002-5284-3240
Corresponding author<EMAIL_ADDRESS>LPHE-Modeling and
Simulation, Faculty of Sciences, Mohammed V University in Rabat, Rabat,
Morocco.  Centre of Physics and Mathematics, CPM, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  Abdel-Baset A. Mohamed
Department of Mathematics, College of Science and Humanities in Al-Aflaj,
Prince Sattam bin Abdulaziz University, Saudi Arabia.  Department of
Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.  Rachid Ahl
Laamara  LPHE-Modeling and Simulation, Faculty of Sciences, Mohammed V
University in Rabat, Rabat, Morocco.  Centre of Physics and Mathematics, CPM,
Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco.  Hichem
Eleuch  Department of Applied Physics and Astronomy, University of Sharjah,
Sharjah 27272, United Arab Emirates.  College of Arts and Sciences, Abu Dhabi
University, Abu Dhabi 59911, United Arab Emirates.  Institute for Quantum
Science and Engineering, Texas A&M University, College Station, TX 77843, USA

###### Abstract

We investigate the dynamics of non-classical correlations and quantum
coherence in open quantum systems by employing metrics like local quantum
Fisher information, local quantum uncertainty, and quantum Jensen-Shannon
divergence. Our focus here is on a system of two qubits in two distinct
physical situations: the first one when the two qubits are coupled to a
single-mode cavity, while the second consists of two qubits immersed in
dephasing reservoirs. Our study places significant emphasis on how the
evolution of these quantum criterion is influenced by the initial state’s
purity (whether pure or mixed) and the nature of the environment (whether
Markovian or non-Markovian). We observe that a decrease in the initial state’s
purity corresponds to a reduction in both quantum correlations and quantum
coherence, whereas higher purity enhances these quantumness. Furthermore, we
establish a quantum teleportation strategy based on the two different physical
scenarios. In this approach, the resulting state of the two qubits functions
as a quantum channel integrated into a quantum teleportation protocol. We also
analyze how the purity of the initial state and the Markovian or non-Markovian
regimes impact the quantum teleportation process.

Keywords: Metrological non-classical correlations, Open quantum systems,
Quantum coherence, Markovian and non-Markovien model, Quantum teleportation.

##  I Introduction

One of the most interesting topics in the microscopic world involving quantum
mechanics is quantum entanglement (QE), a particular property of quantum
systems with no classical analogue [1, 2, 3]. The concept of QE shows how two
or more objects can be bound to each other contrary to what our common sense
would predict. If an ensemble of entangled particles is individually measured,
then while the particles are physically separated by a large distance, in some
ways they behave as a single object rather than as two separate objects and
the resulting results may exhibit ”non-local” effects [4]. Indeed, QE has been
seen as a concrete physical resource with many important applications [5, 6],
ranging from cryptography [7] to teleportation [8, 9, 10] to quantum computers
[11, 12, 13]. Thus was born the quantum information sector which developed
surprisingly. Besides, the further growth of the joint entanglement of a pair
of qubits exposed to local noisy environments has recently attracted a lot of
attention. The reason is related to the discovery of Yu and Eberly [14] that,
quite unexpectedly, the Markovian dynamics of joint entanglement of qubits and
decoherence of a single qubit for this system can be quite different. In fact,
quantum systems can be classified into two types; closed and open. A closed
quantum system is completely isolated from its environment and does not
interact with it, while an open quantum system exchanges energy, matter or
information with its external environment. Over time, an open quantum system
can undergo decoherence, which results in a loss of coherence between the
system and its environment, and a loss of information about the initial state
of the system.

The investigations concerning the decoherence started mainly through the works
of Zeh [15] and Zurek [16], with the initial aim of proposing a mechanism that
could explain the emergence of classical properties from a purely quantum
context. Furthermore, decoherence theory allows us to understand the evolution
of correlations existing in a quantum system and could therefore help us to
develop strategies to combat noise effects. Overcoming decoherence is a
prerequisite for the very attractive idea of coding information in quantum
systems. For this purpose, the possibility of QE disappearing completely at
limited times has attracted most of the interest. This phenomenon of
”entanglement sudden death” has been demonstrated in a quantum optics
experiment [17]. Further, we keep discovering new situations in which
dissipation play a crucial role in manipulating open quantum systems. In many
circumstances, it is no longer possible to assume a significant boundary
between the time scales of the system and the environment, resulting in non-
markovian behavior and ultimately feedback from the environment to the system
[22, 26]. It is therefore essential to create a description of the system-
environment interaction that is accurate and efficient. In this present work,
we focus specifically on two scenarios of open quantum systems; The first one
concerns a Jaynes-Cummings model (JCM) [20] and the second scenario deals with
a non-markovian dephasing model [21]. Basically, the JCM extensively employed
in quantum optics, which serves as a fundamental framework for elucidating the
interaction between two-level atoms and quantized electromagnetic fields in
the dipole approximation. This model can be precisely solved when applying the
rotating wave approximation, wherein non-energy-conserving terms are
eliminated from the Hamiltonian. Consequently, the JCM has been a focal point
of considerable theoretical and experimental investigations in recent decades.
On the other hand, non-markovian dephasing model are designed to capture the
effects of memory in the evolution of an quantum system. Non-markovian
dynamics can capture the implications of memory in the evolution of an open
quantum systems and can be used to study a wide range of phenomena, including
quantum memory and entanglement dynamics [22, 23, 24, 25, 26]. The
differentiation between markovian and non-markovian models lies in how they
depict the temporal evolution of the system. A markovian model assumes that
the quantum state of the system at a particular moment is solely determined by
its immediate preceding state. Consequently, the properties of the system at
time $t+1$ can be completely ascertained by knowing its state at time $t$. In
other words, the Markovian model postulates that the future is independent of
the past, given the system’s current state. Conversely, a non-markovian model
incorporates longer-term influences and permits non-local dependencies over
time [27]. This implies that the quantum state of the system at a specific
moment may depend on its past states over several time intervals, rather than
solely relying on its immediately preceding state.

The characterization of non-classical correlations poses a significant
challenge in this research field, and numerous measures have been introduced
in the literature to quantify them in quantum systems [28, 29, 30]. Measures
such as entanglement of formation and concurrence serve as quantitative
indicators of entanglement [31]. Another measure, known as quantum discord
(QD), based on the von Neumann entropy, provides a means to assess quantum
correlations in bipartite quantum systems, even those that are separable. It
was initially introduced by Ollivier and Zurek [32] and Henderson and Vedral
[33]. However, computing QD based on von Neumann entropy is generally a
complex task. In fact, it has been proven to be NP-complete [34], and only
partial results are available for certain special two-qubit states. In
response to these obstacles, scientists have put forth a geometric approach
for quantifying quantum discord that employs Schatten $p$-norms. The initial
formulation of this geometric quantum discord measure was introduced in [35],
where the Hilbert-Schmidt norm was employed. Despite being computationally
tractable [36, 37, 38], the geometric QD is not an ideal indicator of non-
classical correlations. This occurs because the geometric QD, measured using
the Hilbert-Schmidt distance, can actually increase when local quantum
operations are performed on the unmeasured qubit [39]. This undesirable
characteristic stems from the absence of contractivity which is a vital
property for any quantum correlations quantifier. It is now established that
the trace norm is the only Schatten $p$-norm suitable for the geometric
measure of QD [43, 41, 42, 43]. Recently, a new measure called local quantum
uncertainty (LQU) [44] has been introduced to examine pairwise quantum
correlations of the QD type in multipartite systems. It possesses all the
desirable properties expected of a reliable quantum correlations quantifier.
This measure is based on the concept of Skew information, initially introduced
by Wigner to determine the uncertainty in measuring an observable [45, 46].
The advantage of the LQU is its analytical computability for any qubit-qudit
system. Furthermore, it is worth noting that this new quantum correlation
quantifier is closely linked to quantum Fisher information, which is commonly
employed in the field of quantum metrology [47]. In a recent study [48], the
introduction of local quantum Fisher information (LQFI) provided a means to
quantify non-classical correlations using QFI. This powerful quantifier, akin
to quantum discord, involves minimizing QFI over a locally informative
observable related to a specific subsystem. Moreover, LQFI holds great
potential as a tool for comprehending the impact of quantum correlations,
beyond entanglement, in improving the precision and efficiency in quantum
estimation protocols.

The purpose of this study is to investigate the behavior of non-classical
correlations (LQU and LQFI) and quantum coherence in two distinct open quantum
systems. The first system consists of two qubits coupled with a single-mode
cavity field, and it is analyzed under the rotating wave approximation. The
second system involves two qubits interacting with dephasing reservoirs. A
comparative analysis of their time evolution is conducted, and particular
attention is given to examining the impact of both Markovian and non-Markovian
environments on their quantumness. Both models share the same dependence on
the initial state, enabling a meaningful comparison between them.
Additionally, we investigate the quantum teleportation of a two-qubit system
in an arbitrary pure entangled state through the two considered models as a
quantum channel [49, 50]. The output average fidelity is examined to validate
the effectiveness of the teleportation process. This article is organized as
follows: Sec.II provides an overview of LQFI, LQU, and QC, which are used to
quantify the degree of quantumness in our models of interest. Sec.III is
dedicated to analyzing the behavior of LQU, LQFI, and QC in the Jaynes-
Cummings model as well as in a non-Markovian dephasing model. Furthermore, in
Sec.IV, we discuss the success of quantum teleportation in both models by
measuring the average fidelity. Finally, we sum up our results in Sec.V.

##  II Quantifiers of non-classical correlations

###  II.1 Non-classical correlations measured by local quantum uncertainty

Using local measurements on a part of the given quantum state $\rho_{AB}$,
local quantum uncertainty (LQU) is a quantifier of non-classical correlations
that captures a fully quantum part of the bipartite state $\rho_{AB}$ [44] and
which goes beyond entanglement. It can be computed as

${\cal U}_{Q}\left(\rho_{AB}\right)=\min_{{\cal K}_{A}}{\cal
I}\left(\rho_{AB},{\cal K}_{A}\otimes\openone_{B}\right),$ (1)

with the minimum is optimized over all local observables ${\cal K}_{A}$ on the
subsystem $A$, and ${\cal I}\left(\rho,{\cal K}_{A}\otimes\openone_{B}\right)$
is the Wigner–Yanase skew information [45, 51] that quantifies the information
content and the noncommutativity in the state $\rho_{AB}$ with regard to the
local observable ${\cal K}_{A}$, it writes as

${\cal I}\left(\rho_{AB},{\cal
K}_{A}\otimes\openone_{B}\right)=-\frac{1}{2}{\rm
Tr}\left(\left[\sqrt{\rho_{AB}},{\cal
K}_{A}\otimes\openone_{B}\right]^{2}\right).$ (2)

Therefore, the minimum quantum uncertainty associated with a single
measurement on a single subsystem is referred to as LQU. Importantly, LQU is
one of the most adapted quantifiers of non-classical correlations and has been
shown to satisfy all the necessary physical requirements for such a discord-
like quantifier. For instance, LQU is non-increasing under local operations on
B, it vanishes if and only if the quantum state is a zero-discord state, it is
invariant under local unitary operations for bipartite quantum systems, and
lastly, it reduced to a monotone entanglement for pure states. For bipartite
$2\otimes d$-quantum states, a closed from of LQU can be derived as

${\cal U}_{Q}\left(\rho_{AB}\right)=1-\Lambda_{\max}\left({\cal W}\right),$
(3)

$\Lambda_{\max}$ is the largest eigenvalue of the $3\times 3$ symmetric matrix
${\cal W}\equiv[{\cal W}_{ij}]$, whose components are calculated using the
following formula

${\cal W}_{ij}={\rm
Tr}\left\\{\sqrt{\rho_{AB}}\left(\sigma_{i}\otimes\openone_{B}\right)\sqrt{\rho_{AB}}\left(\sigma_{j}\otimes\openone_{B}\right)\right\\},$
(4)

with $\sigma_{i,j}$ (where $i,j=1,2,3$) are the standard Pauli matrices acting
on the part $A$.

###  II.2 Non-classical correlations quantified by local quantum Fisher
information

Here we employ local quantum Fisher information (LQFI) as a second quantifier
of non-classical correlations based on quantum uncertainty. Briefly, QFI
$\mathcal{F}$ is the most commonly applied quantity to characterize the
ultimate accuracy in the parameter estimation protocols via the quantum
Cramér-Rao bound, where $\Delta\theta\geq 1/\sqrt{\mathcal{F}}$ [52, 53].
Recently, numerous efforts have been done to establish the relevance of non-
classical correlations in quantum metrology, resulting in showing how quantum
discord can significantly improve the precision of estimated parameter
$\theta$ in unitary evolution processes, i.e in the case of unitary dynamics
$\rho_{\theta}={\cal U}_{\theta}^{\dagger}\rho{\cal U}_{\theta}$ with ${\cal
U}_{\theta}=\exp[i{\cal K}\theta]$. According to estimation theory, we can
derive the QFI by optimizing the classical FI over all possible measurements
and can be reformulated as

$\mathcal{F}\left(\rho_{\theta}\right)=\frac{1}{4}{\rm
Tr}\left(\rho_{\theta}{\cal L}_{\theta}^{2}\right),$ (5)

where ${\cal L}_{\theta}$ is the symmetric logarithmic derivative (SLD)
associated to the estimated parameter $\theta$, which was determined by
solving the equation

$\frac{\partial\rho_{\theta}}{\partial\theta}=\frac{1}{2}\left[\rho_{\theta}{\cal
L}_{\theta}+{\cal L}_{\theta}\rho_{\theta}\right].$ (6)

Applying this result to the spectrum decomposition
$\rho_{\theta}=\sum_{i=1}^{N}\eta_{i}|{\psi_{i}}\rangle\langle{\psi_{i}}|$
with $\eta_{i}\geq 0$ and $\sum_{i=1}^{N}\eta_{i}=1$, the SLD can be computed
explicitly as

${\cal
L}_{\theta}=2\sum_{i,j}\frac{\left\langle\psi_{j}\right|\partial_{\theta}\rho_{\theta}\left|\psi_{i}\right\rangle}{\eta_{i}+\eta_{j}}\left|\psi_{j}\right\rangle\left\langle\psi_{i}\right|,$
(7)

and the explicit formula for the QFI is obtained by reporting Eq. (7) into
Eq.(5) as

$\mathcal{F}\left(\rho_{\theta}\right)=\frac{1}{2}\sum_{i\neq
j}\frac{\left(\eta_{i}-\eta_{j}\right)^{2}}{\eta_{i}+\eta_{j}}\arrowvert\langle{\psi_{i}}|{\cal
K}|{\psi_{j}}\rangle\arrowvert^{2}.$ (8)

It should be noted that in the case of unitary dynamics, where ${\cal K}$ is a
fixed Hermitian operator on the system $\rho$, QFI becomes independent of the
estimated parameter $\theta$ as shown in [46]. If $\rho_{AB}$ is a $2\otimes
d$-bipartite quantum state, and if the dynamics of the first subsystem is
controlled by the local phase shift transformation $e^{-i\theta{\cal K}_{A}}$,
with ${\cal K}={\cal K}_{A}\otimes\openone_{B}$, The QFI with local
Hamiltonians becomes

$\mathcal{F}\left(\rho,{\cal K}_{A}\right)={\rm Tr}\left(\rho{\cal
K}_{A}^{2}\right)-\sum_{i\neq
j}\frac{2\eta_{i}\eta_{j}}{\eta_{i}+\eta_{j}}\arrowvert\langle{\psi_{i}}|{\cal
K}_{A}|{\psi_{j}}\rangle\arrowvert^{2}.$ (9)

Similar to LQU (1), the local Fisher quantum information (LQFI) is defined as
the worst-case QFI on all local Hamiltonians ${\cal K}_{A}$ acting on the
subsystem $A$ [48] as

$\mathcal{Q}_{F}\left(\rho\right)=\underset{{\cal
K}_{A}}{\text{min}}\mathcal{F}\left(\rho,{\cal K}_{A}\right)$ (10)

It is another analytically computable measure of discord-like correlations and
has all the desirable features that any acceptable non-classical correlation
quantifier should fulfill. The local Hamiltonian ${\cal K}_{A}$ can be taken
as ${\cal K}_{A}=\vec{\sigma}.\vec{r}$ for minimizing the above equation (10)
for any qubit-qudit system, with $\arrowvert\vec{r}\arrowvert=1$ and
$\vec{\sigma}=\left(\sigma_{1},\sigma_{2},\sigma_{3}\right)$. In this case, it
is easy to prove that ${\rm Tr}\left(\rho{\cal K}_{A}^{2}\right)=1$ and the
second term in equation (9) can be formulated as

$\displaystyle\sum_{i\neq j}\frac{2\eta_{i}\eta_{j}}{\eta_{i}+\eta_{j}}$
$\displaystyle\arrowvert\langle{\psi_{i}}|{\cal
K}_{A}|{\psi_{j}}\rangle\arrowvert^{2}=\vec{r}^{\dagger}.{\cal M}.\vec{r},$
(11)

with the matrix elements of the symmetric matrix ${\cal M}\equiv[{\cal
M}_{kl}]$ are given by

${\cal M}_{kl}=\sum_{i\neq
j}\frac{2\eta_{i}\eta_{j}}{\eta_{i}+\eta_{j}}\langle{\psi_{i}}|\sigma_{k}\otimes\openone_{B}|{\psi_{j}}\rangle\langle{\psi_{j}}|\sigma_{l}\otimes\openone_{B}|{\psi_{i}}\rangle.$
(12)

Applying the minimization process over all possible local observables, as we
already made by LQU (3), we obtain an explicit closed form of LQFI as follows

${\cal Q}_{F}\left(\rho\right)=1-\xi_{\max}[{\cal M}],$ (13)

where $\xi_{\max}[{\cal M}]$ denotes the largest eigenvalue of the symmetric
matrix ${\cal M}$ defined by (12).

###  II.3 Quantum coherence

Quantum coherence (QC) is one of the latest correlations formally introduced
[54, 55, 56], although its concept has been mentioned since the origins of
quantum mechanics. Actually, QC is the basis of entanglement, multi-particle
interference and other types of non-classical correlations, and plays a
central role in quantum information processing [57, 58, 59, 60]. Therefore, in
order to explore the QC in the context of quantum resource theory, quantum
information science proposes rigorous notions and techniques, and several
quantifiers have been proposed in finite dimensional systems. Radhakrishnan
and his co-authors [61] have proposed a new quantifier of quantum coherence,
combining both entropic and metric properties and satisfying all known
criteria for a QC quantifier. It is defined as the square root of the quantum
Jensen-Shannon divergence ${\cal J}\left(\rho,\rho_{d}\right)$ as

${\cal Q}_{C}\left(\rho\right)=\sqrt{{\cal J}\left(\rho,\rho_{d}\right)},$
(14)

where the quantum Jensen-Shannon divergence, as a distance measure between the
state $\rho$ and the closest incoherent state $\rho_{d}$, is given by

${\cal
J}\left(\rho,\rho_{d}\right)=S\left(\frac{\rho+\rho_{d}}{2}\right)-\frac{1}{2}\left[S\left(\rho\right)+S\left(\rho_{d}\right)\right],$

with $\rho_{d}$ is the diagonal part of quantum state $\rho$ in the
computational basis and $S\left(\rho\right)=-{\rm
Tr}\left(\rho\log_{2}\rho\right)$ is the von Neumann entropy.

##  III Theoretical Model

###  III.1 The field interacts with two non-interacting qubits

Many physical phenomena can be modeled by two-level systems interacting with a
harmonic oscillator, including atoms with electromagnetic fields [20, 62],
nuclear spin interactions with magnetic fields [63, 64], electrons with
crystal lattice phonon modes [65], superconducting LC circuits [66, 67],
superconducting qubits with nanomechanical resonators [68, 69], among others.
The Rabi Hamiltonian controls the behavior of all these systems. Analytical
solutions for its eigenvalues and eigenvectors are still lacking despite the
fact that it has been the subject of much investigation since it was
originally presented within the framework of nuclear magnetic spin resonance.
When the qubits are nearly resonant with the oscillator and the coupling
forces are much smaller than the frequencies of the oscillator (i.e., in the
rotating wave approximation), it is possible to use the counter-rotation terms
to get the Jaynes-Cummings model having the Hamiltonian [20],

$\hat{H}=w\hat{a}^{\dagger}\hat{a}+\sum\limits_{j=A,B}\frac{w_{0}}{2}\sigma_{j}^{z}+\gamma\sum\limits_{j=A,B}\left(\hat{a}\sigma_{j}^{+}+\hat{a}^{\dagger}\sigma_{j}^{-}\right).$
(15)

In this context, $\omega_{0}$ denotes the transition frequency that
characterizes the two levels of the atoms. The parameter $\gamma$ represents
the constant of coupling between the atoms and the fields. Additionally,
$\omega$ corresponds to the angular frequency of the single-mode field, while
$\hat{a}^{\dagger}$ and $\hat{a}$ symbolize the creation and annihilation
operators for the cavity mode. The atomic transition operators
$\sigma_{j}^{\pm}$ and inversion operator $\sigma_{j}^{z}$ are defined as

$\sigma_{j}^{+}=|{0}\rangle_{j}\langle{1}|,\quad\sigma_{j}^{-}=|{1}\rangle_{j}\langle{0}|,\quad\sigma_{j}^{z}=|{0}\rangle_{j}\langle{0}|-|{1}\rangle_{j}\langle{1}|,$

with $|{0}\rangle_{j}$ and $|{1}\rangle_{j}$ representing respectively the
ground and excited states of the $j$-th qubit. In fact, the interaction
between the system and field influences how the whole system evolves. However,
the energy transfer between the system and the field occurs from the
perspective of the qubits, and this is defined by the backaction term
$\hat{a}\sigma_{j}^{+}$ and the relaxation term
$\hat{a}^{\dagger}\sigma_{j}^{-}$. we assume that the particles A and B are
resonantly coupled to the single-mode cavity field and that they are initially
prepared in Werner-like states [70] as
$\rho_{AB}=\varsigma|{\phi}\rangle\langle{\phi}|+(1-\varsigma)/4\openone$ with
$|{\phi}\rangle=\sin\alpha|{00}\rangle+\cos\alpha|{11}\rangle$, described in
terms of the purity $\varsigma$ and mixing $\alpha$. However, the initial
state of the cavity field is adjusted to the vacuum state, i.e.
$\rho_{F}=|{0}\rangle\langle{0}|$. The density matrix operator of the system
evolves to $\varrho_{AB}\left(t\right)={\rm Tr}_{F}\left[{\cal
U}\left(t\right)\rho_{ABF}{\cal U}^{\dagger}\left(t\right)\right]$, where the
time evolution operator ${\cal U}\left(t\right)=\exp\left(-iHt\right)$. In the
standard basis
$\left\\{|{00}\rangle,|{01}\rangle,|{10}\rangle,|{11}\rangle\right\\}$, the
density matrix $\varrho_{AB}\left(t\right)$ is given by

$\displaystyle\varrho_{AB}\left(t\right)=$ $\displaystyle
u|{00}\rangle\langle{00}|+y\left[|{01}\rangle\langle{01}|+|{10}\rangle\langle{10}|\right]$
$\displaystyle+z\left[|{01}\rangle\langle{10}|+|{10}\rangle\langle{01}|\right]+v|{11}\rangle\langle{11}|$
$\displaystyle+w\left[|{00}\rangle\langle{11}|+|{11}\rangle\langle{00}|\right],$
(16)

with the elements

$\displaystyle
u=\frac{1-\varsigma}{4}+\frac{\varsigma\sin^{2}\alpha}{9}\left(\cos\left(\delta
t\right)+2\right)^{2},$ $\displaystyle
v=\frac{1-\varsigma}{4}+\varsigma\cos^{2}\alpha+\frac{2\varsigma}{9}\left(\cos\left(\delta
t\right)-1\right)^{2}\sin^{2}\alpha,$ $\displaystyle w=\frac{\varsigma\sin
2\alpha}{6}\left(\cos\left(\delta t\right)+2\right),$ $\displaystyle
z=\frac{\varsigma}{6}\sin^{2}\left(\delta t\right)\sin^{2}\alpha,\hskip
28.45274pty=\frac{1-\varsigma}{4}+z,$

where $\delta=\sqrt{6}\gamma$. We need to determine the components provided by
Eq.(4) to determine the analytical formula of LQU. Following some
simplifications, we arrive at

${\cal
W}_{11}=\frac{2\left(y+\sqrt{y^{2}-z^{2}}\right)\left(u+v+2\sqrt{uv-w^{2}}\right)+4zw+uv^{2}}{\sqrt{2\left(y+\sqrt{y^{2}-z^{2}}\right)\left(u+v+2\sqrt{uv-w^{2}}\right)}},$
(17) ${\cal
W}_{22}=\frac{2\left(y+\sqrt{y^{2}-z^{2}}\right)\left(u+v+2\sqrt{uv-w^{2}}\right)-4zw-
uv^{2}}{\sqrt{2\left(y+\sqrt{y^{2}-z^{2}}\right)\left(u+v+2\sqrt{uv-w^{2}}\right)}},$
(18) $\displaystyle{\cal W}_{33}=$
$\displaystyle\frac{u^{2}-4w^{2}}{2\left(u+v+2\sqrt{uv-w^{2}}\right)}+\frac{v^{2}-4z^{2}}{4\left(y+\sqrt{y^{2}-z^{2}}\right)}$
$\displaystyle+\frac{u+v}{4}+\sqrt{uv-w^{2}}+\sqrt{y^{2}-z^{2}}+y^{2},$ (19)

Thus, it is simple to check that ${\cal W}_{11}\geq{\cal W}_{22}$, then LQU is
given in term of

${\cal U}_{Q}\left(\varrho_{AB}\right)=1-\max\left\\{{\cal W}_{11},{\cal
W}_{33}\right\\}.$ (20)

To derive the explicit expression for LQFI, we first compute the eigenvalues
and eigenstates of the density matrix (16). The eigenvalues are explicitly
given by

$\displaystyle\eta_{1}=\frac{1-\varsigma}{4},\hskip
42.67912pt\eta_{2}=\eta_{1}+\frac{\varsigma\sin^{2}\alpha\sin^{2}\left(\delta
t\right)}{3},$ (21)
$\displaystyle\eta_{3,4}=\frac{1}{2}\left[u+v\mp\frac{1}{3}\sqrt{9\left(u-v\right)^{2}+\left(2+\cos\left(\delta
t\right)\right)^{2}\varsigma^{2}\sin^{2}2\alpha}\right],$ (22)

and the corresponding eigenstates are given by

$\displaystyle|{\psi_{1,2}}\rangle=\frac{1}{\sqrt{2}}\left[|{01}\rangle\mp|{10}\rangle\right],$
$\displaystyle|{\psi_{3,4}}\rangle=\frac{1}{\sqrt{\chi_{\mp}^{2}+1}}\left[\chi_{\mp}|{00}\rangle+|{11}\rangle\right],$
(23)

where the quantities $\chi_{\mp}$ are

$\chi_{\mp}=\frac{3\left(u-v\right)\mp\sqrt{9\left(u-v\right)^{2}+16\varsigma^{2}\left(2+\cos\delta
t\right)^{2}\sin^{2}2\alpha}}{4\varsigma\left(2+\cos\delta t\right)\sin
2\alpha}.$ (24)

Using (12), the non-zero components of the symmetric matrix ${\cal M}$ are the
diagonal elements given by

$\displaystyle{\cal M}_{11}$
$\displaystyle=\frac{2\eta_{1}\eta_{3}\left(\chi_{-}-1\right)^{2}}{\left(\eta_{1}+\eta_{3}\right)\left(\chi_{-}^{2}+1\right)}+\frac{2\eta_{1}\eta_{4}\left(\chi_{+}-1\right)^{2}}{\left(\eta_{1}+\eta_{4}\right)\left(\chi_{+}^{2}+1\right)}$
$\displaystyle+\frac{2\eta_{2}\eta_{3}\left(\chi_{-}+1\right)^{2}}{\left(\eta_{2}+\eta_{3}\right)\left(\chi_{-}^{2}+1\right)}+\frac{2\eta_{2}\eta_{4}\left(\chi_{+}+1\right)^{2}}{\left(\eta_{2}+\eta_{4}\right)\left(\chi_{+}^{2}+1\right)},$
(25) $\displaystyle{\cal M}_{22}$
$\displaystyle=\frac{2\eta_{1}\eta_{3}\left(\chi_{-}+1\right)^{2}}{\left(\eta_{1}+\eta_{3}\right)\left(\chi_{-}^{2}+1\right)}+\frac{2\eta_{1}\eta_{4}\left(\chi_{+}+1\right)^{2}}{\left(\eta_{1}+\eta_{4}\right)\left(\chi_{+}^{2}+1\right)}$
$\displaystyle+\frac{2\eta_{2}\eta_{3}\left(\chi_{-}-1\right)^{2}}{\left(\eta_{2}+\eta_{3}\right)\left(\chi_{-}^{2}+1\right)}+\frac{2\eta_{2}\eta_{4}\left(\chi_{+}-1\right)^{2}}{\left(\eta_{2}+\eta_{4}\right)\left(\chi_{+}^{2}+1\right)},$
(26) ${\cal
M}_{33}=\frac{2\eta_{3}\eta_{4}\left(\chi_{-}\chi_{+}-1\right)^{2}}{\left(\eta_{3}+\eta_{4}\right)\left(\chi_{-}^{2}+1\right)\left(\chi_{+}^{2}+1\right)}.$
(27)

Similar to LQU, it is simple to see that ${\cal M}_{11}\geq{\cal M}_{22}$ and
then the LQFI writes

${\cal Q}_{F}\left(\varrho_{AB}\right)=1-\max\left\\{{\cal M}_{11},{\cal
M}_{33}\right\\}.$ (28)

Subsequently, we determined the amount of quantum coherence existing in the
system using the quantum Jensen–Shannon divergence. For the density matrix
(16), the analytical expression of the quantum Jensen-Shannon divergence takes
the form

$\displaystyle{\cal
J}\left(\varrho_{AB},\varrho_{d}\right)=\frac{1}{2\log\left(2\right)}$
$\displaystyle\left[\sum_{i=1}^{4}\eta_{i}\log\eta_{i}+u\log u+v\log v\right.$
$\displaystyle\left.-2\sum_{j=1}^{4}x_{j}\log x_{j}+2y\log y\right],$

and therefore the quantum coherence is ${\cal
Q}_{C}\left(\varrho_{AB}\right)=\sqrt{{\cal
J}\left(\varrho_{AB},\varrho_{d}\right)}$, where $x_{i}$ are eigenvalues of
the matrix $\left(\varrho_{AB}+\varrho_{d}\right)/2$.

$\left(a\right)$

$\left(b\right)$

$\left(c\right)$

$\left(d\right)$

$\left(e\right)$

$\left(f\right)$

Figure 1: Time evolutions of local quantum uncertainty, quantum coherence, and
local quantum Fisher information versus the coupling $\gamma t$ for different
value of $\alpha$. The top row corresponds to $\varsigma=1$ and the bottom row
corresponds to $\varsigma=0.5$.

We can observe that the parameters of the initial state, purity $\varsigma$
and mixing $\alpha$ control the evolution of local quantum Fisher information,
local quantum uncertainty and quantum coherence in the Jaynes-Cumming model.
At $\varsigma=1$, the behavior of the LQFI is observed to be periodic, where
it initially takes its maximum value and then decreases over time to its
minimum value before rising again. This behavior repeats itself while
maintaining the same amplitude and expansion. The control of the $\alpha$
value contributes to the minimization of quantum correlation collapses,
whereby a smaller parameter value leads to fewer collapses, and vice versa, as
depicted in Fig.1(a). Similarly, LQU exhibits the same periodic behavior in
terms of both behavior and quantity, as illustrated in Fig.1(b).

On the other hand, QC behaves differently than LQFI and LQU, starting from the
lowest value and gradually increasing to its maximum value. Raising the value
of $\alpha$ also contributes to an increase in the amount of quantum
coherence, opposite to LQU, and LQFI (see Fig.1). If we reduce the purity
value to $0.5$, the same behavior is maintained, but the quantity is different
$\mathcal{Q_{C}}|_{\varsigma=1}>\mathcal{Q_{C}}|_{\varsigma=0.5}$ the same for
LQFI and LQU (see fig.1(d),(e) and (f)). The degradation of quantum
correlations can be attributed to the decoherence of qubits resulting from
their interaction with the environment. As time progresses, the two qubits
exchange energy with the environment, which leads to a loss of coherence and
an increase in the number of classical correlations. Moreover, the eigenstate
of the system also contributes to the degradation of quantum correlations.

###  III.2 Non-Markovian dephasing model for a double two-qubit system

In this part, we explore the dynamics of the aforementioned quantum criteria
in the case of the system-environment interactions with memory, but without
taking into account the Born-Markov approximation which assumes that the
system-environment correlation time is infinitely short in order to neglect
the memory effects [21, 71]. Firstly, we focus on a colored noise dephasing
model where its dissipative dynamics is described by the following master
equation

$\dot{\rho}\left(t\right)=KL\rho,$ (29)

$K$ stands for a time-dependent integral operator whose action on the system
is given by
$K\varphi=\int_{0}^{t}k\left(t-t^{,}\right)\varphi\left(t^{,}\right)dt^{,}$,
$L$ is a Lindblad superoperator, $\rho$ is the density operator of the
principal system and $k\left(t-t^{,}\right)$ represents the kernel function
allowing to determine the type of memory in a specific environment. Indeed,
even in the absence of the integral operator $K$ in Eq.(29), we typically
obtain the master equation with Markovian approximation. If a two-level
quantum system interacts with a reservoir that exhibits random telegraph
signal noise characteristics, then this type of master equation could arise.
The time-dependent Hamiltonian

$H\left(t\right)=\hbar\sum\limits_{i=1}^{3}\Gamma_{i}\left(t\right)\sigma_{i},$
(30)

$\Gamma_{i}\left(t\right)$ are independent random variables complying with the
characteristics of a random telegraph signal and $\sigma_{i}$ are the standard
Pauli matrices. The random variables can be specifically stated as
$\Gamma_{i}\left(t\right)=a_{i}{n_{i}\left(t\right)}$, with
$n_{i}\left(t\right)$ has a Poisson distribution with a mean of $t/2\tau_{i}$
and $a_{i}$ is an independent random variable having values $\pm a_{i}$. The
random telegraph signal is described by a broadly defined stationary
stochastic process with zero mean. This concept can be used for any two-level
quantum system that interacts with a random telegraph signal noise. This can
be applied to a two-level atom exposed to a fluctuating laser field with
random jump-like phase noise [72, 73]. The parameters $a_{i}$ determine the
degree of coupling of the system with the external impact. The inverse of
$\tau_{i}$ is the flipping or fluctuation rate. Applying the von Neumann
equation of motion
$\dot{\rho}\left(t\right)=-i\sum_{i}\Gamma\left(t\right)\left[\sigma_{i},\rho\right]$,
the matrix density of a two-level system can be written as

$\rho\left(t\right)=\rho\left(0\right)-i\int_{0}^{t}dk\sum_{i}\Gamma\left(k\right)\left[\sigma_{i},\rho\left(k\right)\right].$
(31)

Substituting Eq.(31) into the von Neumann equation and carrying out the
stochastic averages, we find ourselves with

$\dot{\rho}\left(t\right)=-\int_{0}^{t}\sum\limits_{k}\exp\left(-\frac{\mid
t-t^{,}\mid}{\tau_{k}}\right)a_{k}^{2}\left[\sigma_{k},\left[\sigma_{k},\rho\left(t^{,}\right)\right]\right]dt^{,}.$
(32)

The memory kernel derived from the correlation functions of the random
telegraph signal takes the form

$<\Gamma_{j}\left(t\right)\Gamma_{k}\left(t^{,}\right)>=a_{k}^{2}\exp\left(-\frac{\arrowvert
t-t^{,}\arrowvert}{\tau_{k}}\right)\delta_{jk}.$ (33)

The model proposed above generates a non-Markovian time evolution when one
takes into account some recently developed measures of non-markovianity [74,
75]. The dynamic evolution produced by Eq.(32) appears to be entirely positive
if two of the $a_{k}$ are zero. This would be analogous to a physical scenario
in which noise only comes from one direction. Notably, the system dynamics is
those of a dephasing with colored noise if $a_{1}=a_{2}=0$ and $a_{3}=a$, and
in such case, the Kraus operators which is related to the dynamics of two-
level system are [21]

$F_{1}\left(t\right)=\sqrt{\frac{1+\Lambda\left(\xi,\kappa\right)}{2}}\openone_{2\times
2},\hskip
14.22636ptF_{2}\left(t\right)=\sqrt{\frac{1-\Lambda\left(\xi,\kappa\right)}{2}}\sigma_{3}.$

where

$\displaystyle\Lambda\left(\xi,\kappa\right)=e^{-\xi}\left(\cos(\kappa\xi)+\sin(\kappa\xi)/\kappa\right),$
$\displaystyle\kappa=\sqrt{\left(4a\tau\right)^{2}-1},\hskip
28.45274pt\xi=\frac{t}{2\tau},$ (34)

and the Kraus operators $F_{i}\left(t\right)$ satisfying the normalization
condition $\sum_{i}F_{i}^{\dagger}\left(t\right)F_{i}\left(t\right)=\openone$.
For bipartite systems, the time evolution of an initial density operator is
described as follows

$\rho_{AB}\left(t\right)=\sum_{i,j}\left(F_{i}^{A}\otimes
F_{i}^{B}\right)\rho_{AB}\left(0\right)\left(F_{i}^{A}\otimes
F_{i}^{B}\right)^{\dagger}$ (35)

where the operators $F_{i}^{A}$ and $F_{i}^{B}$ respectively act on the first
and second qubits, while $\rho_{AB}\left(0\right)$ denotes the initial state
of the two-qubit system. In this part, we select the same initial state as in
the first model, and after damping, the density operator takes the form

$\displaystyle\rho_{AB}\left(t\right)$
$\displaystyle=A|{00}\rangle\langle{00}|+D\left(|{01}\rangle\langle{01}|+|{10}\rangle\langle{10}|\right)$
$\displaystyle+C\left(|{00}\rangle\langle{11}|+|{11}\rangle\langle{00}|\right)+B|{11}\rangle\langle{11}|,$
(36)

with the entries notations are adopted as

$\displaystyle A=\frac{1-\varsigma}{4}+\varsigma\sin^{2}\alpha,\quad
B=\frac{1-\varsigma}{4}+\varsigma\cos^{2}\alpha,$ $\displaystyle
C=\frac{\varsigma}{2}\Lambda\left(\xi,\kappa\right)\sin 2\alpha,\hskip
28.45274ptD=\frac{1-\varsigma}{4}.$

The eigenvalues of the density operator (36) are

$\displaystyle\lambda_{1,2}=\frac{1-\varsigma}{4},$
$\displaystyle\lambda_{3,4}=\frac{1}{4}\left(1+\varsigma\mp\sqrt{2\varsigma^{2}\left(1+\cos
4\alpha\right)+4\Lambda\left(\xi,\kappa\right)^{2}\sin^{2}2\alpha}\right),$
(37)

and the corresponding eigenstates are given by

$\displaystyle|{\psi_{1}}\rangle=|{10}\rangle,\quad|{\psi_{2}}\rangle=|{01}\rangle,$
$\displaystyle|{\psi_{3,4}}\rangle=\frac{1}{\sqrt{\varepsilon_{\mp}^{2}+1}}\left(\varepsilon_{\mp}|{00}\rangle+|{11}\rangle\right),$
(38)

with

$\varepsilon_{\mp}=\frac{1}{\sin\left(2\alpha\right)\Lambda\left(\xi,\kappa\right)}\left(-\cos
2\alpha\mp\varsigma\sqrt{\Lambda\left(\xi,\kappa\right)^{2}\sin^{2}2\alpha+\cos^{2}2\alpha}\right).$
(39)

Based on the above formalism, the elements of the $3\texttimes 3$ symmetric
matrix ${\cal M}$ are calculated analytically from Eq.(12) and found to be as
follows

$\displaystyle{\cal M}_{11}={\cal
M}_{22}=4\left(\frac{\lambda_{1}\lambda_{3}}{\lambda_{1}+\lambda_{3}}+\frac{\lambda_{1}\lambda_{4}}{\lambda_{1}+\lambda_{4}}\right),$
$\displaystyle{\cal
M}_{33}=\frac{4\lambda_{3}\lambda_{4}}{\lambda_{3}+\lambda_{4}}\frac{\left(\varepsilon_{+}\varepsilon_{-}-1\right)^{2}}{\left(\varepsilon_{+}^{2}+1\right)\left(\varepsilon_{-}^{2}+1\right)},$
(40)

and to obtain the analytical form of the local quantum Fisher information
(13), it is necessary to treat separately two cases, i.e. ${\cal
M}_{11}\geq{\cal M}_{33}$ or ${\cal M}_{33}\geq{\cal M}_{22}$ with which

${\cal Q}_{F}\left(\rho_{AB}\right)=1-\max\left\\{{\cal M}_{11},{\cal
M}_{33}\right\\}.$ (41)

Likewise, the explicit expressions of the matrix elements ${\cal W}_{ij}$ (4)
can be found as

$\displaystyle{\cal W}_{11}={\cal
W}_{22}=\frac{\Theta^{2}+\varsigma^{2}\Lambda\left(\xi,\kappa\right)^{2}\sin^{2}(2\alpha)}{4\sqrt{\Theta}},$
$\displaystyle{\cal
W}_{33}=\frac{\left(3-\varsigma+2\sqrt{\Sigma}\right)^{2}+8\varsigma^{2}\Lambda\left(\xi,\kappa\right)^{2}\sin^{2}(2\alpha)}{8\left(3-\varsigma+2\sqrt{\Sigma}\right)},$
(42)

where

$\displaystyle\Theta=$ $\displaystyle
2\left(\varsigma-1\right)\sqrt{1+\left(2-3\varsigma\right)\varsigma-4\varsigma^{2}\left(\Lambda\left(\xi,\kappa\right)^{2}-1\right)\sin^{2}(2\alpha)}$
$\displaystyle+2\left(\varsigma^{2}-1\right),$ (43)
$\displaystyle\Sigma=1+\left(2-3\varsigma\right)\varsigma-4\varsigma^{2}\left(\Lambda\left(\xi,\kappa\right)^{2}-1\right)\sin^{2}(2\alpha).$
(44)

Then the local quantum uncertainty is derived by

${\cal U}_{Q}\left(\rho_{AB}\right)=1-\max\left\\{{\cal W}_{11},{\cal
W}_{33}\right\\}.$ (45)

On the other side, we determined the analytic evolution of the quantum
coherence after the damping by means of Eq.(14). Following a straightforward
calculation, the analytical expression of the quantum coherence for the non-
Markovian dephasing model described by the density matrix (36) is

$\displaystyle{\cal Q}_{C}\left(\rho_{AB}\right)$
$\displaystyle=\left[\frac{1}{16\log
16}\left(\varsigma-1-2\varsigma\cos^{2}\alpha+\left(1+\varsigma\right)\log
240\right)\right.$ $\displaystyle\left.+\left(\frac{1+\varsigma\mp
2\varsigma\cos 2\alpha}{2\log 16}\right)\log\left(\frac{1+\varsigma\mp
2\varsigma\cos 2\alpha}{4}\right)\right.$
$\displaystyle\left.-\frac{\varsigma\sin^{2}\alpha}{2}-\left(\frac{2+2\varsigma\mp\sqrt{\varpi}}{2\log
16}\right)\log\left(\frac{2+2\varsigma\mp\sqrt{\varpi}}{4}\right)\right.$
$\displaystyle\left.+\left(\frac{1+\varsigma\mp\sqrt{\Delta}}{\log
16}\right)\log\left(\frac{1+\varsigma\mp\sqrt{\Delta}}{4}\right)\right]^{\frac{1}{2}},$

where

$\displaystyle\varpi=2\varsigma^{2}\left(4+\Lambda\left(\xi,\kappa\right)^{2}+\left(4-\Lambda\left(\xi,\kappa\right)^{2}\right)\cos(4\alpha)\right),$
$\displaystyle\Delta=2\varsigma^{2}\left(1+\Lambda\left(\xi,\kappa\right)^{2}+\left(1-\Lambda\left(\xi,\kappa\right)^{2}\right)\cos(4\alpha)\right).$
(46)

$\left(a\right)$

$\left(b\right)$

$\left(c\right)$

$\left(d\right)$

$\left(e\right)$

$\left(f\right)$

Figure 2: Dynamics of LQFI, LQU and QC as a function of $t/2\tau$ for
different values of the degree of non-Markovianity $\tau$. Panels
($a$)-($b$)-($c$) corresponds to the pure initial state with $\varsigma=1$,
while panels ($d$)-($e$)-($f$) correspond to the mixed initial state with
$\varsigma=0.5$. Parameters $a$ and $\alpha$ are fixed as $a=1$ and
$\alpha=\pi/4$.

To investigate the dynamic behavior of local quantum uncertainty and quantum
coherence, as well as local quantum Fisher information, and explore how both
markovian and non-markovian environments influence their evolution, we
depicted their dynamics in Fig.(2) for a two-qubit system interacting with
independent colored dephasing reservoirs. The parameter $t/2\tau$ was employed
as the variable. It is worth noting that in Fig.(2), where $\varsigma=1$ (i.e.
pure initial state) is fixed for panels ($a$-$b$-$c$) and $\varsigma=0.5$
(i.e. mixed initial state) is fixed for panels ($d$-$e$-$f$), the values
$\tau=0.5$ and $\tau=1$ represent the markovian case, while $\tau=3$ and
$\tau=5$ correspond to the non-markovian regime.

For the non-markovian regime (i.e., $\tau\geq 2$), the dynamics of non-
classical correlations (LQU and LQFI) and quantum coherence exhibit a
phenomenon of death and revival, where their amplitudes undergo continuous
damping. This behavior arises from the effects of environmental memory, which
facilitate the back-flow of information. In contrast, for the markovian regime
(i.e., $\tau<2$), the QC, LQU and LQFI asymptotically approach zero as time
increases, with a little revival. This trend can be attributed to the rapid
outflow of quantum information from the system to the environment, facilitated
by weak system-environment coupling and the transfer of information without
memory. Furthermore, a decrease in the degree of non-markovianity $\tau$ leads
to a reduction in the frequency and delay of oscillations in the correlation
function, as well as a decrease in its amplitude. Additionally, decreasing the
purity results in a decrease in the amount of quantum criteria. Moreover, as
depicted in Fig.(2), all aspects of quantumness (LQU, LQFI, and QC) evolve
simultaneously and at the same rate. Also, in the non-Markovian dephasing
model, the quantumness is higher when the initial state is pure compared to
when it is mixed. This observation aligns with previous findings in the
Jaynes-Cummings model.

As time progresses in the non-Markovian dephasing model, the correlation
amplitudes of LQFI and LQU decrease due to the strong influence of the
environment. However, it is noteworthy that the qualitative agreement between
the LQFI and LQU functions remains intact. Notably, the LQFI exhibits higher
correlation than the LQU, implying that the dynamics of LQFI and LQU align
with the principles of quantum-Fisher and Wigner-Yanase skew information
theories [47]. Specifically, the amplitudes of LQFI are consistently much
larger than those of the LQU. Besides, QC exhibits greater robustness compared
to LQFI and LQU. In fact, while the non-classical correlations vanished, the
QC remained non-zero. This clear distinction indicates that under a colored
noise dephasing model, quantum coherence is more resilient and robust in
comparison to LQFI and LQU.

##  IV Quantum teleportation scheme

Quantum teleportation is a captivating phenomenon in the realm of quantum
physics, challenging our classical understanding [8, 76, 77]. In contrast to
the instantaneous matter transfer of science fiction, quantum teleportation
revolves around exchanging quantum information between distant particles. This
extraordinary feat is achievable through quantum entanglement. Its
significance lies in its vital role in quantum computing, secure quantum
communication, and the profound exploration of quantum reality’s essence.
Despite its current limitation in transporting physical objects or individuals
across long distances, quantum teleportation paves the way for thrilling
prospects in the realms of quantum physics and advanced quantum technologies.
To achieve quantum teleportation, cientists exploit the fascinating phenomenon
of entanglement between two particles, called entangled particles. The idea is
to have a third particle, which we want to teleport, and a fourth particle
entangled with the third and located where we want to ”teleport” the
information [78, 79, 80]. By performing complex quantum measurements on the
two entangled particles and the particle to be teleported, we can transfer the
quantum state of the original particle to the destination particle. This
happens without any physical transport of matter, as only the quantum state is
transferred.

Figure 3: Quantum teleportation protocol for the transmission of quantum
information from Alice to Bob through the entanglement of qubits: Alice
possesses a qubit $A$ in an initial state $\rho_{in}$ that she wishes to
teleport to Bob. To facilitate the process, Alice and Bob share an ancillary
entangled pair of qubits ($B$ and $C$), generated by an Entangled partial
source. Alice performs a joint Bell state measurement on qubit $A$ and qubit
$B$, projecting them onto one of the four orthogonal entangled states known as
the Bell states ($|{\psi^{\mp}}\rangle$, $|{\phi^{\mp}}\rangle$). The
measurement result is then sent to Bob through classical communication.
Depending on the outcome, Bob either keeps qubit $C$ (case
$|{\phi^{-}}\rangle$) unchanged (case $|{\psi^{-}}\rangle$) or applies a
unitary transformation ($U$) to qubit $C$ (cases $|{\psi^{+}}\rangle$ and
$|{\phi^{+}}\rangle$). This transformation ensures that the output state
$\varrho_{out}$ (or $\rho_{out}$) of qubit $C$ is identical to the input state
$\rho_{in}$ of the original qubit $A$, achieving successful quantum
teleportation.

In this scenario, we employ a teleportation protocol proposed by Cola and
Paris [81] to transfer a two-qubit system using only a single quantum channel.
The protocol utilizes a single entangled two-qubit pair, the transmission of
three classical bits, and an additional qubit introduced by the receiver (see
Fig.(3)). Standard teleportation can be regarded as a general depolarizing
channel [82]. In the two models discussed earlier (J-C and non-Markovian
dephasing model), we are examining how the initial state impacts the
efficiency of teleportation. In this context, Alice and Bob share a partially
entangled state described by Eq.(16) (for the JC model) and Eq.(36) (for the
non-Markovian dephasing model). Alice’s unknown state is given in the form
$\rho_{in}=\left|\psi_{in}\right\rangle\left\langle\psi_{in}\right|$, where

$|{\psi_{in}}\rangle=\cos\left(\frac{\theta}{2}\right)|{10}\rangle+e^{-i\varphi}\sin\left(\frac{\theta}{2}\right)|{01}\rangle,$
(47)

with $0\leq\theta\leq\pi$ and $0\leq\varphi\leq 2\pi$. Here, $\theta$
describes an arbitrary state and $\varphi$ is the corresponding phase of this
state. At the concluding step of the teleportation protocol, Bob will receive
the state

$\rho_{out}=\sum\limits_{kl=\left\\{0,1,2,3\right\\}}p_{kl}\left(\sigma_{k}\otimes\sigma_{l}\right)\rho_{in}\left(\sigma_{k}\otimes\sigma_{l}\right),$
(48)

where $p_{kl}={\rm Tr}\left[K^{k}\rho_{ch}\right]{\rm
Tr}\left[K^{l}\rho_{ch}\right]$, $\sigma_{n}$ ($n=k,l$) are the Pauli
operators, $\sum_{kl}p_{kl}=1$ and $\rho_{ch}$ and A stands for the resource
state linking Alice and Bob. Here, the different projection operators $K^{l}$
are defined as

$\displaystyle K^{0,3}=|{\psi^{\mp}}\rangle\langle{\psi^{\mp}}|,\hskip
14.22636ptK^{1,2}=|{\phi^{\mp}}\rangle\langle{\phi^{\mp}}|,$
$\displaystyle|{\phi^{\pm}}\rangle=\frac{1}{\sqrt{2}}\left(|{00}\rangle\pm|{11}\rangle\right),\hskip
14.22636pt|{\psi^{\pm}}\rangle=\frac{1}{\sqrt{2}}\left(|{01}\rangle\pm|{10}\rangle\right).$
(49)

The Bell states mentioned above (49) exhibit maximum entanglement, implying an
inseparable connection between their component qubits. Such states have been
utilized as quantum channels to experimentally transmit an arbitrary state
from one distant location to another. Nonetheless, in the two models under
consideration, it is evident that the states may not be maximally entangled
due to the presence of decoherence arising from the interaction between the
system and its environment. However, our findings demonstrate that by
manipulating the parameters within the reduced density operator, it becomes
possible to attain a significant level of quantum correlations. To evaluate
the effectiveness of the teleportation process, it is valuable to examine the
fidelity between $\rho_{in}$ and $\rho_{out}$, which measures how close the
final state is to the initial state. When the input state is a pure state,
fidelity serves as a beneficial metric for assessing the teleportation
performance of a quantum channel quantifier [83]. It is defined as

$F\left(\rho_{in},\rho_{out}\right)=\left\\{{\rm
Tr}\left[\sqrt{\sqrt{\rho_{in}}\rho_{out}\sqrt{\rho_{in}}}\right]\right\\}^{2}.$
(50)

As the specific state to be teleported is generally unknown, it proves more
beneficial to compute the average fidelity, which represents the mean value of
fidelity for the four outcomes resulting from Bell state measurements [84].
This average fidelity is expressed as

$F_{av}\left(\rho_{in},\rho_{out}\right)=\frac{1}{4\pi}\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}F\left(\rho_{in},\rho_{out}\right)\sin\theta
d\theta,$ (51)

where $4\pi$ is the solid angle. If Alice and Bob share the quantum state
$\varrho_{AB}\left(t\right)$ that describes the Jaynes-Cummings model, i.e.
$\rho_{ch}=\varrho_{AB}\left(t\right)$, the resulting output state can be
expressed as follows:

$\displaystyle\varrho_{out}\left(t\right)=$
$\displaystyle\kappa\left(|{00}\rangle\langle{00}|+|{11}\rangle\langle{11}|\right)+\chi|{01}\rangle\langle{01}|$
$\displaystyle+\vartheta\left(|{00}\rangle\langle{11}|+|{11}\rangle\langle{00}|\right)+\zeta|{10}\rangle\langle{10}|$
$\displaystyle+\Delta|{01}\rangle\langle{10}|+\Theta|{10}\rangle\langle{01}|,$
(52)

with their entries are given by

$\displaystyle\kappa=2y\left(u+v\right),\hskip
28.45274pt\vartheta=4zw\cos\varphi\sin\theta,$
$\displaystyle\chi=\left(4y^{2}+\left(u+v\right)^{2}\right)\sin^{2}\left(\theta/2\right),$
$\displaystyle\zeta=\left(4y^{2}+\left(u+v\right)^{2}\right)\cos^{2}\left(\theta/2\right),$
$\displaystyle\Delta=2\varsigma
e^{-i\varphi}\left(z^{2}+w^{2}e^{2i\varphi}\right)\sin\theta,$
$\displaystyle\Theta=2\varsigma
e^{-i\varphi}\left(w^{2}+z^{2}e^{2i\varphi}\right)\sin\theta,$ (53)

which are written according to the entries of the entangled resource state
$\varrho_{AB}\left(t\right)$ (Eq.16). After some calculations, the fidelity of
Bob’s state takes the form

$\displaystyle F\left(\varrho_{out}\right)$
$\displaystyle=\frac{1}{188}\Bigg{[}\Big{(}6+5\varsigma+\varsigma\cos
2\alpha+2\varsigma\cos\Big{(}2\delta t\Big{)}\sin^{2}\alpha\Big{)}^{2}\times$
$\displaystyle\Big{(}1+\cos^{2}\alpha\Big{)}+4\sin^{2}\theta\Big{(}4\varsigma^{2}\Big{(}2+\cos\left(\delta
t\right)\Big{)}^{2}\sin^{2}2\alpha$
$\displaystyle+\Big{(}3-3\varsigma+2\varsigma\sin^{2}\left(\delta
t\right)\sin^{2}\alpha\Big{)}^{2}\Big{)}\Bigg{]}.$ (54)

By inserting equation (54) into equation (51), the result yields the average
fidelity

$\displaystyle F_{av}\left(\varrho_{out}\right)$
$\displaystyle=\varsigma\Bigg{[}\frac{4}{9}\cos\left(\delta
t\right)\sin^{2}2\alpha-\frac{5\varsigma}{64}\cos 4\alpha+$
$\displaystyle\frac{1}{108}\Big{(}6+19\varsigma+7\varsigma\cos
2\alpha\Big{)}+\frac{\varsigma}{72}\cos\left(4\delta
t\right)\sin^{4}\alpha\Bigg{]}$
$\displaystyle+\frac{1}{4}+\frac{\varsigma}{144}\left(4+9\varsigma\right)\cos
2\alpha+\frac{\varsigma}{576}\Big{(}80+135\varsigma\Big{)}$
$\displaystyle+\frac{\varsigma^{2}}{18}\sin^{2}\left(\delta
t\right)\sin^{6}\alpha\cos 2\varphi.$ (55)

If Alice and Bob share the state $\rho_{AB}\left(t\right)$ (eq.36) with
$\rho_{ch}=\rho_{AB}\left(t\right)$, which characterizes the non-Markovian
dephasing model, then the resulting output state is given by

$\displaystyle\rho_{out}\left(t\right)=$
$\displaystyle\varpi\left[|{00}\rangle\langle{00}|+|{11}\rangle\langle{11}|\right]+\Sigma\left[|{10}\rangle\langle{01}|+|{01}\rangle\langle{10}|\right]$
$\displaystyle+\Upsilon\left[|{01}\rangle\langle{01}|+|{10}\rangle\langle{10}|\right],$
(56)

where the elements $\varpi$, $\Sigma$ and $\Upsilon$ are given by

$\displaystyle\varpi=2d\left(a+b\right),\hskip
28.45274pt\Sigma=2c^{2}e^{-i\varphi}\sin\theta,$
$\displaystyle\Upsilon=4d^{2}\cos^{2}\left(\theta/2\right)+\left(a+b\right)\sin^{2}\left(\theta/2\right).$

According to equation (50), the teleportation fidelity turns out to be

$\displaystyle F\left(\rho_{out}\right)=$
$\displaystyle\left(\left(\frac{1-\varsigma}{4}\right)^{2}+\frac{\varsigma^{2}\Lambda\left(\xi,\kappa\right)^{2}}{2}\sin^{2}2\alpha\right)\sin^{2}\theta$
$\displaystyle+\left(\frac{1+\varsigma}{4}\right)^{2}\left(1+\cos^{2}\theta\right),$
(57)

and subsequently, the average fidelity is provided as

$F_{av}\left(\rho_{out}\right)=\frac{2}{3}\left(\frac{1+\varsigma}{4}\right)^{2}+\frac{1}{3}\left(\left(\frac{1-\varsigma}{4}\right)^{2}+\frac{\varsigma^{2}\Lambda\left(\xi,\kappa\right)^{2}}{2}\sin^{2}2\alpha\right).$
(58)

$\left(a\right)$ Jaynes-Cummings model

$\left(b\right)$ Markovian regime

$\left(c\right)$ non-Markovian regime

Figure 4: The average fidelity $F_{av}$ as a function of the temps $t/2\tau$
for different values of the purity $\varsigma$ with $\alpha=\pi/4$; panel
($a$) corresponds to the Jaynes-Cumming model, panel ($d$) reflects the
Markovian regime for a non-Markovian dephasing model with $a=1$ and
$\tau=0.5$, while panel ($c$) for non-Markovian regime with $\tau=5$.

Prior to commencing our analysis of the results displayed in Fig.(4), it is
crucial to bear in mind some previous investigations. These studies evaluated
the effectiveness of quantum teleportation by measuring the average fidelity.
A value of $F_{av}<2/3$ indicates a failure in quantum teleportation,
$F_{av}>2/3$ signifies an improvement in quantum teleportation, approaching
the upper limit achievable through classical protocols. Furthermore, a value
of $F_{av}=1$ represents the highest level of teleportation achievement.
Fig.(4) depicts the behavior of the average fidelity when the resource state
that connects Alice and Bob corresponds to the Jaynes-Cumming model
represented by state (16) (i.e., panel ($a$)), as well as when it is a non-
Markovian dephasing model described by state (36) (i.e., panels ($b$)-($c$).
In both cases, the teleported state is initially prepared in state (47), and
various values of purity $\varsigma$ are taken into consideration.

When Alice and Bob share the Jaynes-Cummings model state, as depicted in
Fig.4.($a$), we observe a distinct periodic pattern in the average fidelity of
the teleported state. This behavior is influenced by the purity of the shared
state, denoted by $\varsigma$. As the purity increases, the efficiency of
teleportation improves, leading to higher fidelity in the teleported state.
Conversely, as the purity decreases, the efficiency of teleportation
diminishes, resulting in lower fidelity. To explain further, the average
fidelity serves as a measure of how well the quantum information is preserved
during teleportation. A value of less than $2/3$ indicates that teleportation
fails, and the state is not effectively teleported. On the other hand, when
the average fidelity reaches $F_{av}=1$, it signifies a perfect teleportation
process, where the teleported state matches the original state with utmost
accuracy. By analyzing our results, we can confidently assert that
teleportation significantly improves with higher values of purity, as
evidenced by the increase in average fidelity. In essence, the higher the
purity of the shared state, the better the teleportation outcome.
Additionally, referring back to our previous findings, as shown in Fig.(1), we
notice a similar trend. The average fidelity of the teleported state achieves
its maximum when the degree of quantumness (LQFI, LQU, and QC) of the quantum
resource state is maximized. In essence, the more quantum properties the
resource state possesses, the better the fidelity of the teleported state
during the teleportation process. This further supports the significance of
quantumness in achieving efficient and accurate quantum teleportation.

The results displayed in panels ($b$)-($c$) of Fig.(4) depict the evolution of
the average fidelity of the teleported state when Alice and Bob share the non-
Markovian dephasing model state. We investigate two specific scenarios: the
Markovian regime (Fig.4$b$) and the non-Markovian regime (Fig.4$c$). The study
highlights the critical role of state purity in determining the success of
quantum teleportation, with pure states demonstrating higher average fidelity
compared to mixed states. In the Markovian regime, as seen in Fig.(4$b$), the
average fidelity shows a gradual decrease in its maximum value over time,
eventually stabilizing at a particular level. This behavior suggests that the
system exhibits long-term memory. However, the Markovian regime allows for a
relatively straightforward transfer of quantum information through
teleportation since it does not retain any memory of its previous states. On
the other hand, Fig.(4$c$) displays the average fidelity of the non-Markovian
regime with a specific parameter $\tau=5$, revealing more pronounced short-
term fluctuations. The average fidelity exhibits a gradual decrease in its
maximum value, followed by an increase after a brief period, reaching a peak
lower than the previous one. These fluctuations arise due to transient
transitions between different states of the system, leading to temporary
variations in average fidelity.

These findings underscore the challenges of transferring quantum information
in the non-Markovian regime due to memory effects. The system retains
information about its previous states, which can impact the quantumness
between particles involved in the teleportation process. Consequently, the
memory effects in the non-Markovian regime introduce additional complexity and
fluctuations, resulting in a less stable and predictable average fidelity
during quantum teleportation. In contrast, the lack of memory in the Markovian
regime ensures a smoother and more efficient transfer of quantum information.
Accordingly, our analysis shows that state purity, as well as the presence or
absence of memory effects, play pivotal roles in influencing the success and
efficiency of quantum teleportation. Pure states and Markovian regimes offer
more favorable conditions for achieving high average fidelity in the
teleported state.

##  V Conclusion

To conclude, we have explored the significance of the quantifiers of non-
classical correlations, specifically examining LQU and LQFI, along with
quantum coherence. These factors play a crucial role in enhancing our
comprehension of quantum information processing and the intricate dynamics
within open quantum systems. Our focus was on a comparative analysis of their
dynamic characteristics using two distinct physical situations. The first
involves a system of two qubits coupled with a single-mode cavity field, while
the second centers around the interaction of two qubits with dephasing
reservoirs. Our research uncovers that the dynamics of these quantum
characteristics are strongly influenced by the initial conditions
($\varsigma$,$\alpha$) and the extent of non-Markovian behavior denoted by
$\tau$ within the system. Specifically, our observations highlight that the
Jaynes-Cummings model showcases periodic oscillations and distinct sudden-
death and revival phenomena, which can be regulated by manipulating the
initial state parameters. In contrast, the non-Markovian dephasing model
manifests a fundamentally divergent behavior compared to the Markovian system.
Within the Markovian context, quantum correlations and coherence prove to be
more fragile. Subsequently, we introduced a quantum teleportation scheme
founded upon the two models under consideration, in which the resulting two-
qubit state from these models is employed as a quantum channel within the
framework of a quantum teleportation protocol. The results substantiate that
the proposed scheme can be regarded as a reliable protocol for quantum
teleportation, utilizing both models as quantum channels.

## References

  * [1] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev, 47 (1935) 777.
  * [2] J.S. Bell, On the Einstein Podolsky Rosen paradox, Physics, 1 (1964) 195.
  * [3] S. Hill, W.K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett, 78 (1997) 5022.
  * [4] M. Genovese and M. Gramegna, Quantum correlations and quantum non-locality: A review and a few new ideas, Applied Sciences, 9 (2019) 5406.
  * [5] A. Peres, Quantum theory: concepts and methods, Dordrecht: Kluwer Academic, (1997).
  * [6] Fuchs, C. A. (2011). Coming of age with quantum information. Coming of Age With Quantum Information.
  * [7] A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett, 67 (1991) 661.
  * [8] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W.K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett, 70 (1993) 1895.
  * [9] M. Asjad, M. Qasymeh and H. Eleuch, Continuous-variable quantum teleportation using a microwave-enabled plasmonic graphene waveguide, Phys. Rev. Applied, 16 (2021) 034046.
  * [10] M. El Kirdi, A. Slaoui, N. Ikken, M. Daoud and R. A. Laamara, Controlled quantum teleportation between discrete and continuous physical systems, Physica Scripta, 98 (2023) 025101.
  * [11] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe and J. L. O’Brien, Quantum computers, nature, 464 (2010) 45-53.
  * [12] C. Boudreault, H. Eleuch, M. Hilke and R. MacKenzie, Universal quantum computation with symmetric qubit clusters coupled to an environment, Phys. Rev. A, 106 (2022) 062610.
  * [13] M. A. Nielsen and I. L. Chuang: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000).
  * [14] T. Yu and J. H. Eberly, Finite-time disentanglement via spontaneous emission, Phys. Rev. Lett, 93 (2004) 140404.
  * [15] H. D. Zeh, The meaning of decoherence, Lect. Notes Phys, 538 (2000) 19-42.
  * [16] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys, 75 (2003) 715.
  * [17] M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. S. Ribeiro and L. Davidovich, Environment-induced sudden death of entanglement, science, 316 (2007) 579-582.
  * [18] H. P. Breuer, E. M. Laine, J. Piilo and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys, 88 (2016) 021002.
  * [19] I. De Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys, 89 (2017) 015001.
  * [20] E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE, 51 (1963) 89.
  * [21] S. Daffer, K. Wódkiewicz, J. D. Cresser and J. K. McIver, Depolarizing channel as a completely positive map with memory, Phys. Rev. A, 70 (2004) 010304.
  * [22] H. P. Breuer, E. M. Laine, J. Piilo and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys, 88 (2016) 021002.
  * [23] M. Zidan, A. H. Abdel-Aty, A. Khalil, M. Abdel-Aty and H. Eleuch, A novel efficient quantum random access memory, IEEE Access, 9 (2021) 151775-151780.
  * [24] H. El Euch, M. Zidan, M. Abdelaty, A.H. Abdel-Aty, A. Khalil, Quantum Random Access Memory, US Patent 11093850 B2, August 17, 2021.
  * [25] H. El Euch, M. Zidan, M. Abdelaty, A.H. Abdel-Aty, A. Khalil, Quantum Random Access Memory System, US Patent 11651 266 B2, May 16, 2023.
  * [26] I. De Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys, 89 (2017) 015001.
  * [27] M. M. Wolf, J. Eisert, T. S. Cubitt and J. I. Cirac, Assessing non-Markovian quantum dynamics, Phys. Rev. Lett, 101 (2008) 150402.
  * [28] V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys, 74 (2002) 197.
  * [29] K. Modi, A. Brodutch, H. Cable, T. Paterek and V. Vedral, The classical-quantum boundary for correlations: Discord and related measures, Rev. Mod. Phys, 84 (2012) 1655.
  * [30] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys, 81 (2009) 865.
  * [31] W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett, 80 (1998) 2245.
  * [32] H. Ollivier and W. H. Zurek, Quantum discord: a measure of the quantumness of correlations, Phys. Rev. Lett, 88 (2001) 017901.
  * [33] L. Henderson and V. Vedral, Classical, quantum and total correlations, J. Phys. A, 34 (2001) 6899.
  * [34] Y. Huang, Computing quantum discord is NP-complete. New journal of physics, New J. Phys, 16 (2014) 033027.
  * [35] B. Dakić, V. Vedral, Č. Brukner, Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett, 105 (2010) 190502.
  * [36] B. Bellomo, G.L. Giorgi, F. Galve, R. Lo Franco, G. Compagno and R. Zambrini, Unified view of correlations using the square-norm distance, Phys. Rev. A, 85 (2012) 032104.
  * [37] E. G. Brown, K. Cormier, E. Martin-Martinez and R. B. Mann, Vanishing geometric discord in noninertial frames, Phys. Rev. A, 86 (2012) 032108.
  * [38] B. Bellomo, R. Lo Franco and G. Compagno, Dynamics of geometric and entropic quantifiers of correlations in open quantum systems, Phys. Rev. A, 86 (2012) 012312.
  * [39] M. Piani, Problem with geometric discord, Phys. Rev. A, 86 (2012) 034101.
  * [40] F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, Geometric quantum discord through the Schatten 1-norm. Physical Review A, Phys. Rev. A, 87 (2013) 064101.
  * [41] A. B. Mohamed, H. Eleuch and C. R. Ooi, Non-locality correlation in two driven qubits inside an open coherent cavity: Trace norm distance and maximum bell function, Sci Rep, 9 (2019) 19632.
  * [42] A. Slaoui, M. I. Shaukat, M. Daoud and R. A. Laamara, Universal evolution of non-classical correlations due to collective spontaneous emission, Eur. Phys. J. Plus, 133 (2018) 413.
  * [43] F. M. Paula, J. D. Montealegre, A. Saguia, T. R. de Oliveira and M. S. Sarandy, Geometric classical and total correlations via trace distance, EPL, 103 (2013) 50008.
  * [44] D. Girolami, T. Tufarelli and G. Adesso, Characterizing nonclassical correlations via local quantum uncertainty, Phys. Rev. Lett, 110 (2013) 240402.
  * [45] E. P. Wigner and M. M. Yanase, Information contents of distributions, Proc. Nat. Acad. Sci. USA, 49 (1963) 910-918.
  * [46] S. Luo, Wigner-Yanase skew information vs. quantum Fisher information, Proceedings of the American Mathematical Society, 132 (2004) 885-890.
  * [47] A. Slaoui, L. Bakmou, M. Daoud and R. A. Laamara, A comparative study of local quantum Fisher information and local quantum uncertainty in Heisenberg XY model, Phys. Lett. A, 383 (2019) 2241-2247.
  * [48] S. Kim, L. Li, A. Kumar and J. Wu, Characterizing nonclassical correlations via local quantum Fisher information, Phys. Rev. A, 97 (2018) 032326.
  * [49] P. F. Yu, J. G. Cai, J. M. Liu and G. T. Shen, Teleportation via a two-qubit Heisenberg XYZ model in the presence of phase decoherence, Phys. A, 387 (2008) 4723.
  * [50] J. M. Liu, B. Weng and Y. Xia, Scheme for teleportation of atomic states within cavities in thermal states, J. Opt. Soc. Am. B, 23 (2006) 1499.
  * [51] S. Luo, Wigner-Yanase skew information and uncertainty relations, Phys. Rev. Lett, 91 (2003) 180403.
  * [52] M. G. Paris, Quantum estimation for quantum technology, Int. J. Quan. Infor, 7 (2009) 125-137.
  * [53] M. Szczykulska, T. Baumgratz and A. Datta, Multi-parameter quantum metrology, Advances in Physics X, 1 (2016) 621-639.
  * [54] T. Baumgratz, M. Cramer and M. B. Plenio, Quantifying coherence. Phys. Rev. Lett, 113 (2014) 140401.
  * [55] A. Streltsov, G. Adesso and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys, 89 (2017) 041003.
  * [56] F. Levi and F. Mintert, A quantitative theory of coherent delocalization, New J. Phys, 16 (2014) 033007.
  * [57] H.-L. Shi, S.-Y. Liu, X.-H. Wang, W.-L. Yang, Z.-Y. Yang, and H. Fan, Coherence depletion in the Grover quantum search algorithm, Phys. Rev. A, 95 (2017) 032307.
  * [58] A. Olaya-Castro, C. F. Lee, F. F. Olsen, and N. F. Johnson, Efficiency of energy transfer in a light-harvesting system under quantum coherence, Phys. Rev. B, 78 (2008) 085115.
  * [59] A. Slaoui, A. Salah and M. Daoud, Influence of Stark-shift on quantum coherence and non-classical correlations for two two-level atoms interacting with a single-mode cavity field, Physica A: Statistical Mechanics and its Applications, 558 (2020) 124946.
  * [60] A.-B.A. Mohamed and H. Eleuch, Coherence and information dynamics of a $\Lambda$-type three-level atom interacting with a damped cavity field, Eur. Phys. J. Plus, 132 (2017) 75.
  * [61] C. Radhakrishnan, M. Parthasarathy, S. Jambulingam and T. Byrnes, Distribution of quantum coherence in multipartite systems, Phys. Rev. Lett, 116 (2016) 150504.
  * [62] L. Allen and J.H. Eberly, Optical resonance and two-level atoms, Courier Corporation, 28 (1987)
  * [63] I. I. Rabi, On the process of space quantization, Phys. Rev, 49 (1936) 324.
  * [64] I. I. Rabi, Space quantization in a gyrating magnetic field, Phys. Rev, 51 (1937) 652.
  * [65] T. Holstein, Studies of polaron motion: Part I. The molecular-crystal model, Ann. Phys, 8 (1959) 325-342.
  * [66] J. Johansson, S. Saito, T. Meno, H. Nakano, M. Ueda, K. Semba, and H. Takayanagi, Vacuum Rabi Oscillations in a Macroscopic Superconducting Qubit LC Oscillator System, Phys. Rev. Lett, 96 (2006) 127006.
  * [67] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C.J.P.M. Harmans, and J.E. Mooij, Coherent dynamics of a flux qubit coupled to a harmonic oscillator, Nature, 431 (2004) 159-162.
  * [68] K. C. Schwab and M. L. Roukes, Putting mechanics into quantum mechanics, Phys. Today, 58 (2005) 36-42.
  * [69] E. K. Irish and K. Schwab, Quantum measurement of a coupled nanomechanical resonator–Cooper-pair box system, Phys. Rev. B, 68 (2003) 155311.
  * [70] R.F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A, 40 (1989) 4277.
  * [71] J. P. Pinto, G. Karpat and F. F. Fanchini, Sudden change of quantum discord for a system of two qubits, Phys. Rev. A, 88 (2013) 034304.
  * [72] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1992).
  * [73] J. H. Eberly, K. Wódkiewicz, and B. W. Shore, Noise in strong laser-atom interactions: phase telegraph noise, Phys. Rev. A, 30 (1984) 2381.
  * [74] H. P. Breuer, E. M. Laine and J. Piilo, Measure for the degree of non-Markovian behavior of quantum processes in open systems, Phys. Rev. Lett, 103 (2009) 210401.
  * [75] S. Luo, S. Fu and H. Song, Quantifying non-Markovianity via correlations, Phys. Rev. A, 86 (2012) 044101.
  * [76] G. Bowen and S. Bose, Teleportation as a depolarizing quantum channel, relative entropy, and classical capacity, Phys. Rev. Lett, 87 (2001) 267901.
  * [77] R. Ji-Gang, X. Ping, Y. Hai-Lin, et al, Ground-to-satellite quantum teleportation, Nature, 549 (2017) 70-73.
  * [78] M.E. Kirdi, A. Slaoui, H.E. Hadfi and M. Daoud, Improving the probabilistic quantum teleportation efficiency of arbitrary superposed coherent state using multipartite even and odd j-spin coherent states as resource, Appl. Phys. B, 129 (2023) 94.
  * [79] N. Ikken, A. Slaoui, R. A. Laamara and L. B. Drissi, Bidirectional quantum teleportation of even and odd coherent states through the multipartite Glauber coherent state: Theory and implementation, (2023) preprint arXiv:2306.00505.
  * [80] M.E. Kirdi, A. Slaoui, H.E. Hadfi and M. Daoud, Efficient Quantum Controlled Teleportation of an Arbitrary Three-Qubit State Using Two GHZ Entangled States and One Bell Entangled State, J Russ Laser Res, 44 (2023) 121-134.
  * [81] M.M. Cola and M.G.A. Paris, Teleportation of bipartite states using a single entangled pair, Phys. Lett. A, 337 (2005) 10-16.
  * [82] G. Bowen and S. Bose, Teleportation as a depolarizing quantum channel, relative entropy, and classical capacity, Phys. Rev. Lett, 87 (2001) 267901.
  * [83] R. Jozsa, Fidelity for mixed quantum states, Journal of Modern Optics, 41 (1994) 2315-2323.
  * [84] M. D. Bowdrey, D. K. Oi, A. J. Short, K. Banaszek and J. A. Jones, Fidelity of single qubit maps, Phys. Lett. A, 294 (2002) 258-260.