# Probing a hybrid channel for the dynamics of non-local features

Atta ur Rahman1, Ma-Cheng Yang1, S. M. Zangi2, Cong-Feng Qiao
<EMAIL_ADDRESS>School of Physics, University of Chinese Academy of
Science, Yuquan Road 19A, Beijing 100049, China
2School of Physics and Astronomy, Yunnan University, P.O. Box 650500 Kunming,
China

###### Abstract

Effective information transmission is a central element in quantum information
protocols, but the quest for optimal efficiency in channels with symmetrical
characteristics remains a prominent challenge in quantum information science.
In light of this challenge, we introduce a hybrid channel that encompasses
thermal, magnetic, and local components, each simultaneously endowed with
characteristics that enhance and diminish quantum correlations. To investigate
the symmetry of this hybrid channel, we explore the quantum correlations of a
simple two-qubit Heisenberg spin state, quantified using measures such as
negativity, $\ell_{1}$-norm coherence, entropic uncertainty, and entropy
functions. Our findings reveal that the hybrid channel can be adeptly tailored
to preserve quantum correlations, surpassing the capabilities of its
individual components. We also identify optimal parameterizations to attain
maximum entanglement from mixed-entangled/separable states, even in the
presence of local dephasing. Notably, various parameters and quantum features,
including non-Markovianity, exhibit distinct behaviors in the context of this
hybrid channel. Ultimately, we discuss potential experimental applications of
this configuration.

_Keywords_ thermal reservoir, magnetic field, classical dephasing, static
noise, quantum characteristics

##  I Introduction

The transmission of information in quantum information processing is done
using various types of communicating means which include local and non-local
channels [1]. The designing of such transmitting channels is important because
the resourcefulness of the quantum systems may be lost if the inclusive
channels have disorders or no optimal characteristics. In the previous era,
various types of channels with various characteristics have been treated
separately. However, this would be far from reality, as the characteristics of
these various channels can be found at the same place simultaneously, for
example, the thermal, magnetic, and dephasing effects. In this regard, the
thermal interaction picture of the channels has been studied using the concept
of Gibb’s density matrix operation [2]. Besides this, magnetic fields have
also been used to demonstrate the quantum correlations dynamics explicitly
[3]. Most importantly, the local channels are successfully used for decades
for the transmission of information [4]. On the contrary, non-local channels
have been investigated and a lot of efforts for their practical empowerment
have been made recently [5]. Compared to the non-local ones, the classical
ones have the advantage of being easily implemented, as they do not need any
complex design. Pure classical channels, nonetheless, are influenced by
certain types of flaws and disorders, such as the impact of surface charge
carriers [6], electronic currents [7], thermal fluctuations [8, 9] and so
forth. One of these disorders causes static noise, which will be considered to
influence classical channels in the current situation [10]. Previously, static
noise has been widely studied in independent classical channels and has been
found to completely degrade quantum correlations [8, 9, 10].

We are planning to demonstrate the effectiveness of a complex mixed channel,
deployed for information transmission, therefore, we will consider simple two-
qubit correlations to evaluate our goal. In the emerging discipline of quantum
information science, non-classical correlations are now the primary priority
of foundational scientific study [11]. A crucial non-local resource known as
quantum entanglement explains a particular kind of non-classical correlation
among subsystems of a quantum composite state [12]. When a group of particles
interacts spatially such that the quantum states of each particle are mutually
exclusive, this is referred to as quantum entanglement. This non-local
correlation causes measurements on one part of an entangled pairing and
immediately affects the results of measurements on the other. Quantum
computers can perform computations that are not conceivable on classical
computers because of the advantage that non-local systems can exist in
multiple states simultaneously [13]. Researchers are actively developing a
quantum internet and quantum encrypted communications using entanglement [14],
which would allow for new varieties of telescopes [15], sensors [16], and
ultra-secure broadcasting [17]. In addition, the information encoded in qubits
can be processed more quickly and with less processing power because of the
quantum entanglement phenomenon. Recently, it has been demonstrated that
quantum coherence and entanglement, two fundamentally dissimilar aspects of
quantum theory, are functionally equivalent and made a variety of quantum
technologies possible, and therefore, both would remain part of the current
study [18].

The uncertainty principle holds that the results of simultaneous measurements
of non-commuting observables commonly have an intrinsic lower bound on the
uncertainty [19]. This assumption both illustrates how the classical and
quantum realms differ and provides the rationale for the ambiguity of quantum
mechanics. In essence, it states that you cannot begin preparing a quantum
particle whose location and momentum are predictable with certainty,
simultaneously [20]. Kennard and Robertson revisited the principle of
uncertainty after it had taken the form of a standard deviation and was based
on the combined variance of two observables [21]. For example, for two
observables $\mathcal{X}$ and $\mathcal{Y}$, the associated uncertainty
relation was defined as $\Delta\mathcal{X}\Delta\mathcal{Y}\geq
1/2|\langle[\mathcal{X},\mathcal{Y}]\rangle|$. Furthermore, Deutsch
established the well-known version of the entropic uncertainty (EU) relation
in 1983 and proposed the uncertainty principle in terms of Shannon entropy
[22]. Afterward, Kraus first proposed another simplified version of the EU
relation which was verified in 1988 by Maassen and Uffink [23, 24]. Besides,
for different quantum systems and situations, various EU relations have been
established The study of the EU will provide an important aspect of the
current hybrid channel to induce uncertainty in the process of quantum
information processing [25, 26, 27].

In this work, we consider examining a two-qubit spin state influenced by a
thermal and magnetic field [28]. The spin system is further assumed to be
characterized by the Dzyaloshinskii-Moriya interaction (DM), Kaplan,
Shekhtman, Entin-Wohlman, and Aharony (KSEA) and anisotropy parameters
interactions [29, 30]. The DM interaction, an antisymmetric exchange
interaction that controls chiral spin configurations, is brought on by
inversion symmetry breaking in non-centrosymmetric crystal lattice interfaces
[31]. On the other hand, the KSEA interaction is caused by the symmetric
helical interaction [30]. Besides, the generated quantum correlations need
interaction with external channels to be transmitted. As it is known that
classical channels are highly destructive for quantum correlations, therefore,
we become motivated to investigate the impact of the joint implications of the
external thermal-magnetic-classical channel (TMCC) on quantum correlation
dynamics. The reason behind designing such a configuration is to devise a
reliable and physically constructional transmitting channel which we believe
will be helpful in quantum correlations preservation during communication
using non-local systems. The entanglement measurement in the spin state is
done by using negativity (NG), which is one of the well-known bipartite
entanglement monotones [32]. As stated before that entanglement and coherence
are two fundamental correlations of quantum systems. Motivated by this, we
also analyze the dynamics of coherence using $\ell_{1}$-norm of coherence in
the spin system [33]. Quantum systems are always influenced by uncertainty,
therefore, we also consider assessing the degree of uncertainty between
different observables of the system using EU relations [34]. Besides
uncertainty, mixedness disorder is a commonly found phenomenon where most of
the quantum correlations in quantum systems are lost because of its emergence.
The degree of disorder in the spin state when subjected to the hybrid channel
with thermal, magnetic, and local dephasing parts will be computed utilizing
linear entropy (EN) [35]. One of the basic motivations behind the simultaneous
measurement of entanglement, coherence, entropic uncertainty, and mixedness
disorder is to find out the reaction of the assumed hybrid channel towards
different non-classical characteristics.

This work is presented as follows: In Sec. II, we provide the details of the
physical model of the assumed system and channel, along with an introduction
to the measurement of two-qubit correlations. Sec. III gives a detailed
analysis of the results obtained, and in Sec. V, we summarize this work.

##  II Physical model

###  II.1 Thermal and magnetic interaction

We assume a two spin-$1/2$ XXZ type Heisenberg system when exposed to an
external homogeneous magnetic field characterized by the DM ($D_{z}$) and KSEA
$(K_{z})$ coupling interaction oriented along the $z$-axis. This configuration
has the Hamiltonian model given by [36]

$\textbf{H}=\Delta_{z}\Big{(}S_{1}^{z}S_{2}^{z}\Big{)}+D_{z}\Big{(}S_{1}^{x}S_{2}^{y}-S_{1}^{y}S_{2}^{x}\Big{)}+J\Big{(}S_{1}^{x}S_{2}^{x}+S_{1}^{y}S_{2}^{y}\Big{)}+K_{z}\Big{(}S_{1}^{x}S_{2}^{y}+S_{1}^{y}S_{2}^{x}\Big{)}+B\Big{(}S_{1}^{z}+S_{2}^{z}\Big{)},$
(1)

Figure 1: Shows the physical model of the hybrid channel with thermal,
magnetic, and classical dephasing parts controlled by static noise employed
for the dynamics of two-qubit Heisenberg spin state characterized by various
parameters, such as spin-spin, DM, and KSEA interaction.

where $S_{f}^{g}$ with $(f=\\{1,~{}2\\},~{}g=\\{x,~{}y,~{}z\\})$ are the
spin-$1/2$ Pauli matrices of the spin $f$, $\Delta_{z}$ is the real anisotropy
coupling constant describing the symmetric exchange spin-spin interaction in
$z$-direction, $D_{z}$ is the DM interaction strength regulating the spin-
orbit antisymmetric coupling, $J$ is the Heisenberg exchange interaction
between the spin particles. Note that for $J>0$($J<0$) correspond to the
antiferromagnetic(ferromagnetic) interaction between the spins sites,
respectively. Besides, $K_{z}$ represents the KSEA interaction strength
oriented along $z-$axis responsible for symmetric spin-orbit coupling while
$B$ is the homogeneous part of the considered magnetic field. The Hamiltonian
expression given in Eq. (2) in the computational basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ takes the form

$\textbf{H}=\left[\begin{array}[]{cccc}2B+\Delta_{z}&0&0&-2iK_{z}\\\\[5.69046pt]
0&-\Delta_{z}&2iD_{z}+2J&0\\\\[5.69046pt]
0&-2iD_{z}+2J&-\Delta_{z}&0\\\\[5.69046pt]
2iK_{z}&0&0&-2B+\Delta_{z}\end{array}\right].$ (2)

The corresponding eigenvectors of the Hamiltonian H are

$\displaystyle|\Theta_{1^{+},4^{-}}\rangle$
$\displaystyle=\sqrt{\frac{\Lambda\pm
2B}{2\Lambda}}\left(|00\rangle\pm\frac{2iK_{z}}{{\Lambda\pm
2B}}|11\rangle\right),$ $\displaystyle|\Theta_{2^{+},3^{-}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|01\rangle\pm\frac{2J\pm
2iD_{z}}{\upsilon}|10\rangle\right),$

with $\Lambda=\sqrt{4B^{2}+4K_{z}^{2}}$ and
$\upsilon=\sqrt{4J^{2}+4D_{z}^{2}}$. Besides, the eigenvalues of Eq. (2) are
$E_{1^{+},4^{-}}=\Delta_{z}\pm\Lambda$, $E_{2}=-\Delta_{z}+\upsilon$,
Using Gibbs’s density operator, the thermal state density matrix for the
Hamiltonian given in Eq. (2) in equilibrium with a thermal reservoir at
temperature $T$ has the form

$\displaystyle\rho(0,T)=\frac{1}{{Z}}\exp\Big{\\{}\frac{-\textbf{H}}{k_{B}T}\Big{\\}}=\left[\begin{array}[]{llll}\rho_{11}&0&0&\rho_{14}\\\\[5.69046pt]
0&\rho_{22}&\rho_{23}&0\\\\[5.69046pt]
0&\rho_{32}^{\ast}&\rho_{33}&0\\\\[5.69046pt]
\rho_{41}^{\ast}&0&0&\rho_{44}\end{array}\right],$ (7)

where $Z={\rm
Tr}[\exp\\{\frac{-\textbf{H}}{T}\\}]=\sum^{4}_{i=1}\exp\\{\frac{-E_{i}}{T}\\}$
is the partition function with $k_{B}=1$ is the Boltzmann’s constant. The
corresponding entries are given by

$\displaystyle\rho_{11}$
$\displaystyle=\frac{1}{Z}e^{-\frac{\Delta_{z}}{T}}\left(\cosh(\varphi)-\frac{B\sinh(\varphi)}{\sqrt{B^{2}+K_{z}^{2}}}\right),$
$\displaystyle\rho_{14}$
$\displaystyle=\rho_{41}^{*}=\frac{1}{Z\sqrt{B^{2}+K_{z}^{2}}}iK_{z}e^{-\frac{\Delta_{z}}{T}}\sinh(\varphi),$
$\displaystyle\rho_{22}$
$\displaystyle=\rho_{33}=\frac{1}{Z}e^{\Delta_{z}}/T\cosh(\varpi),$
$\displaystyle\rho_{23}$
$\displaystyle=\rho_{32}^{*}=\frac{1}{Z\sqrt{D_{z}^{2}+J^{2}}}(-J-iD_{z})e^{\Delta_{z}}/T\sinh(\varpi),$
$\displaystyle\rho_{44}$
$\displaystyle=\frac{1}{Z}e^{-\frac{\Delta_{z}}{T}}\left(\frac{B\sinh(\varphi)}{\sqrt{B^{2}+K_{z}^{2}}}+\cosh(\varphi)\right),$
(8)

where $\varphi=\frac{1}{T}(2\sqrt{B^{2}+K_{z}^{2}})$,
$\varpi=\frac{1}{T}(2\sqrt{\text{Dz}^{2}+J^{2}})$ and partition function
$Z=2e^{-\frac{\Delta_{z}}{T}}\left(e^{\frac{2\Delta_{z}}{T}}\cosh(\varpi)+\cosh(\varphi)\right).$

The eigenvalues of the two-qubit density matrix $\rho(0,T)$ are given by

$\displaystyle E_{1^{+},2^{-}}=$
$\displaystyle\frac{1}{2}\left(\pm\sqrt{4\rho_{14}\rho_{14}^{*}+(\rho_{11}-\rho_{44})^{2}}+\rho_{11}+\rho_{44}\right),$
$\displaystyle E_{3^{+},4^{-}}=$
$\displaystyle\frac{1}{2}\left(\pm\sqrt{4\rho_{23}\rho_{23}^{*}+(\rho_{22}-\rho_{33})^{2}}+\rho_{22}+\rho_{33}\right).$
(9)

###  II.2 The exposure to a classical channel

Here we provide the exposure of the two-qubit spin state to a common classical
environment driven by static noise. In the present case, the Hamiltonian,
which governs the current physical model is written as [37]

${\rm\bf H}_{XY}={\rm\bf H}_{X}\otimes{\rm I}_{Y}+{\rm I}_{X}\otimes{\rm\bf
H}_{Y},~{}~{}\text{with}~{}~{}{\rm\bf
H}_{\mathcal{P}}=\left[\begin{array}[]{cc}\Delta_{\mathcal{P}}\lambda+\epsilon&0\\\
0&\epsilon-\Delta_{\mathcal{P}}\lambda\\\ \end{array}\right],$ (10)

where ${\rm\bf H}_{\mathcal{P}}~{}(\mathcal{P}=X,Y)$ denotes the Hamiltonian
state of the sub-system $\mathcal{P}$, $\epsilon$ is the equal energy
splitting between the sub-systems, $I$ is the $2\times 2$ identity matrix,
$\lambda$ is the coupling constant, $\Delta_{\mathcal{P}}$ regulates the
stochastic behavior of the classical field and is flipping between $\pm 1$,
and $S^{z}$ is the spin-Pauli matrix.

For the time evolution of the system, we use the following time-unitary
operator where

$U_{XY}(t)=\exp\Big{\\{}-i\int^{t}_{t_{0}}H(z)dz\Big{\\}}=\left[\begin{array}[]{cccc}U_{11}&U_{12}&U_{13}&U_{14}\\\\[5.69046pt]
U_{12}&U_{11}&U_{14}&U_{13}\\\\[5.69046pt]
U_{13}&U_{14}&U_{11}&U_{12}\\\\[5.69046pt] U_{14}&U_{13}&U_{12}&U_{11}\\\
\end{array}\right],$ (11)

which is the time-unitary matrix with $\hbar=1$ and the matrix entries are

$\displaystyle U_{11}=$
$\displaystyle\exp\\{-2it\epsilon\\}\cos(\Delta_{X}\lambda
t)\cos(\Delta_{Y}\lambda t),$ $\displaystyle U_{12}=$
$\displaystyle-i\exp\\{-2it\epsilon\\}\cos(\Delta_{X}\lambda
t)\sin(\Delta_{Y}\lambda t),$ $\displaystyle U_{13}=$
$\displaystyle-i\exp\\{-2it\epsilon\\}\sin(\Delta_{X}\lambda
t)\cos(\Delta_{Y}\lambda t),$ $\displaystyle U_{14}=$
$\displaystyle-\exp\\{-2it\epsilon\\}\sin(\Delta_{X}\lambda
t)\sin(\Delta_{Y}\lambda t).$

To obtain the time-evolved state of the system for the initial thermal state
density matrix $\rho(0,T)$ given in Eq. (8) when exposed to identical channel
i.e. $\Delta_{X}=\Delta_{Y}$, we use [37]

$\rho_{X,Y}(t,T)=U_{XY}(t)\rho(0,T)U_{XY}(t)^{\dagger}\rightarrow\rho(t,T)=U_{XX}(t)\rho(0,T)U_{XX}(t)^{\dagger},$
(12)

and explicitly, the above equation takes the following shape:

$\displaystyle\rho(t,T)=\left[\begin{array}[]{cccc}\rho_{11}&0&0&e^{-4i\Delta_{X}\lambda
t}\rho_{14}\\\\[5.69046pt] 0&\rho_{22}&\rho_{23}&0\\\\[5.69046pt]
0&\rho_{32}^{*}&\rho_{33}&0\\\\[5.69046pt] \rho_{41}^{*}e^{4i\Delta_{X}\lambda
t}&0&0&\rho_{44}\\\ \end{array}\right].$ (17)

####  II.2.1 The influence of classical static noise disorder

Next, we provide the application and influence of the static noise on the
time-evolved state of the spin system. In this regard, static noise is
primarily characterized by $\Delta_{Q}$, namely, the disorder parameter. This
noise has the probability distribution function
$\mathcal{O}(\delta)=1/\Delta_{Q}$ and exhibits the range
$|\delta-\delta_{o}|\leq\Delta_{Q}/2$ where $\delta_{o}$ denotes the mean
value of the probability distribution function [38]. The static noise auto-
correlation expression is written as
$\langle\delta\Delta(t)\Delta(0)\rangle=\Delta^{2}_{Q}/12$. As a result, this
kind of noise tends to have a characteristic duration that is significantly
longer than the coupling between the system and its environment. To evaluate
the impact of the static noise on the dynamics of the spin state, the time-
evolved state density matrix is averaged over all possible noise
configurations. Therefore, we integrate the matrix in Eq. (17) between
$r^{+}=\delta_{o}-\Delta_{Q}/2$ and $r^{-}=\delta_{o}+\Delta_{Q}/2$ as [10]

$\rho_{st}(t,T)=\int^{r^{+}}_{r^{-}}\frac{1}{\Delta_{Q}}\rho(t,T)d\Delta_{X}=\left[\begin{array}[]{cccc}\rho_{11}&0&0&\hat{\rho}_{14}\\\\[5.69046pt]
0&\rho_{22}&\rho_{23}&0\\\\[5.69046pt]
0&\rho_{23}^{*}&\rho_{33}&0\\\\[5.69046pt]
\hat{\rho}_{41}^{*}&0&0&\rho_{44}\\\ \end{array}\right],$ (18)

where $\rho_{ii}(ii=11,22,33,44,23)$ remains the same as given in Eq. (8).
However,
$\hat{\rho}_{14}=\hat{\rho}_{41}^{*}=\frac{\rho_{14}e^{-4i\delta_{o}\lambda
t}\sin(2\Delta_{Q}\lambda t)}{2\Delta_{Q}\lambda t}.$

Note that the final matrix given in Eq. (18) represents the two-spin system
simultaneously influenced by a thermal, magnetic, and classical channel. Note
that the off-diagonal matrix elements are included now with dephasing terms
hence, coherence loss of the systems is included. Therefore, the loss of
entanglement, rise of the uncertainty, and mixedness in the state with time
would give us a realistic model in comparison, if one would only consider the
case of magnetic and thermal interaction.

###  II.3 Quantum criteria quantifiers

####  II.3.1 Bipartite Negativity

Multiple quantitative factors have been developed that demonstrate the level
of entanglement in a quantum state. For example, negativity (${\rm NG}$) has
been investigated as a valuable and calculable witness to entanglement for any
pure and mixed states. The Peres-Horodecki separability criterion serves as
the foundation for this criterion. On a rescaled scale, NG for a statistical
ensemble $\rho_{XY}$ is defined as [39]

$\displaystyle{\rm NG}=2\sum_{i}|\mathcal{Q}_{i}|,$ (19)

where $\mathcal{Q}_{i}$ are the negative eigenvalue(s) of the partially
transposed density matrix $\rho_{XY}^{T_{X}}$ with respect to sub-system $X$.
The state is maximally correlated if $NG=1$ while for $NG=0$, the state will
become separable [37].

####  II.3.2 The entropic uncertainty measure

Consider Bob and Alice to be the two users. Alice receives a qubit in the
desired quantum state created by Bob. Alice must now select one of the two
measurements and notify Bob of her selection. We can now lower the result’s
uncertainty by using Bob’s measurement data [21]. The uncertainty standard
deviation for two observables $\mathcal{X}$ and $\mathcal{Y}$, can be
represented as:
$\Delta\mathcal{X}\Delta\mathcal{Y}\geq\frac{1}{2}|\langle[\mathcal{X},~{}\mathcal{Y}]\rangle|$
[22, 25, 26, 27].

Instead of utilizing the standard deviation to describe uncertainty, Deutsch
suggested the entropic uncertainty relation for every pair of observables, as
shown in the above equation. Based on Deutsch’s method, Maassen and Uffink
created a tighter entropic uncertainty formulation, which can be represented
as $S(\mathcal{X})+S(\mathcal{Y})\geq\log_{2}c(\mathcal{X};\mathcal{Y}),$ [40]
where $S(K)(K=\mathcal{X},\mathcal{Y})$ is known as the Shannon entropy
denoting the probability distribution of measuring the observable $K$ while
$c(\mathcal{X};\mathcal{Y})=\max_{a,~{}b}|\langle\psi|\phi\rangle|^{2}$ is the
maximal overlap between the eigenvectors $\psi$ and $\phi$ of the two non-
degenerate observables.

A new definition presented by Renes et al. and Berta et al. considered the
entropic uncertainity relations in composite systems and proposed a quantum-
memory-assisted entropic uncertainty relation [41, 42]. Therefore, the
quantum-memory assisted entropic uncertainty regarding system $A$ is
reproduced corresponding to q quantum memory $B$ with a tighter bound, and has
the form

$\displaystyle S(\mathcal{X}|B)+S(\mathcal{Y}|B)\geq
S(A|B)-\log_{2}c(\mathcal{X};\mathcal{Y})),$ (20)

where $S(\mathcal{X}|B)=S(\rho_{\mathcal{X}B})-S(\rho_{B})$ is the conditional
von Neumann entropy of the system after the unilateral measurement is applied
on $\mathcal{X}$ on subsystem $A$. Note that the same procedure can be carried
our for $\mathcal{Y}$ too. The associated post-measurement state now can be
written as

$\rho_{\mathcal{X}B}=\sum_{i}(|\psi\rangle\langle\psi|\otimes
I)\rho_{AB}(|\psi\rangle\langle\psi|\otimes I).$ (21)

The conditional entropy $S(A|B)$ computes the correlation between subsystem
$B$ owned by its memory system $A$.

Eq. (20) comprises the entropic uncertainty (EU) on the left-hand side and
entropic uncertainty lower bound on the right-hand side. In the current case,
we only emphasize measuring the left-hand side of the uncertainty relations.

####  II.3.3 $\ell_{1}$-norm of coherence

Different metrics, including the distance metric [43], the relative entropy of
coherence [44] and purity [8] can be used to evaluate the coherence. The
$\ell_{1}$-norm is a reliable coherence monotone and a coherence criterion
[44]. Besides, the $\ell_{1}$-norm coherence for a two-qubit state
$\rho=\sum_{X,Y}\rho_{XY}|X\rangle\langle Y|$ is the sum of all the off-
diagonal entries as [45]

$LC(t)=\sum_{X\neq Y}|\rho_{XY}|,$ (22)

where $|\rho_{XY}|$ is the absolute value of the elements of the density
matrix $\rho_{XY}$.

####  II.3.4 Linear entropy

Linear entropy, a simple to evaluate scalar field, is used to measure the
mixedness of quantum states. For example, by approximating the
$\log_{2}\rho_{XY}$ of $S(\rho_{XY})$ with the first-order term i.e.,
$\rho_{XY}-1$ in the Mercator series given in Ref. [8], one may get

$\displaystyle EN(t)=$
$\displaystyle-\operatorname{Tr}[\rho_{XY}\log_{2}\rho_{XY}].$ (23)

For the next-to-last equality, the density matrix’s unit trace feature
($\operatorname{Tr}[\rho_{XY}]=1$) is used.

##  III Results

This section is devoted to exploring the numerical results obtained for the
dynamics of the two-qubit spin state influenced by an external TMCC
configuration. Additionally, the impact of various parameters such as
anisotropy, spin coupling strength, DM and KSEA interactions as well as the
external noise parameters of the static noise is studied. Utilizing $NG$,
$EU$, $LC$, and $EN$ functions given in Eqs. (19), (20), (22), and (23), we
explore the dynamics of entanglement, quantum memory, coherence, and entropy
in the spin system. All the functions are evaluated for the final density
matrix obtained in Eq. (18).

($a$)($b$)($c$)($d$)

Figure 2: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of static
noise disorder parameter $\Delta_{Q}$ against time in a two-spin system
influenced by an external TMCC. For all the plots, we have set $\lambda=0.1$,
$K_{z}=5$, and $J/T/D_{z}/J_{z}/\Delta_{z}/B/J/=1$.

In Fig. 2, we analyze the dynamics of entanglement, entropic uncertainty,
coherence, and entropy in a two-spin state when coupled with an external TMCC.
Initially, the state remains maximally entangled and coherent as seen that
$NG=LC=1$. The depicted entropic uncertainty and entropy in the state remain
zero initially, as $EU=EN=0$. After the onset, the action of the joint TMCC
appears and as a result, entanglement as well as coherence becomes easily
lost. In agreement, the $EU$ and $EN$ remained increasing functions of the
entropic uncertainty and disorder in the system. The overall dynamical maps of
the spin state correlations remained non-Markovian, however, the degree of
non-Markovianity shown by each measure differs. For example, the $NG$ and $LC$
functions show a large number of entanglement and coherence revivals. This
suggests that the two-qubit system and coupled fields strongly support
information exchange between them. On the other hand, the non-Markovian
behavior shown by the $EU$ and $EN$ function remains weaker. Therefore, $NG$
and $LC$ functions are more sensitive than the $EU$ and $EN$ functions and
record the least exchange of attributes between the state and the field.
Besides, the $NG$ and $LC$ functions show anti-correlation with $EU$ and $EN$
functions, as both pairs evolve in opposite directions to each other.
Furthermore, for the increasing values of the disorder parameter $\Delta_{Q}$,
the entanglement, and coherence remain fragile and become easily lost in the
state. On the other hand, the uncertainty between observables of the particles
and associated mixedness disorder in the state is enhanced by the increasing
strength of $\Delta_{Q}$. However, it is noticeable that the rate of revivals
is directly dependent upon $\Delta_{Q}$. The phenomenon of sudden death and
birth of entanglement occurs repeatedly. Finally, entanglement and coherence
functions seem completely decay with time while the uncertainty and entropy
functions achieve a final higher saturation level.

($a$)($b$)($c$)($d$)

Figure 3: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of
classical field’s coupling strength $\lambda$ against time in a two-spin state
influenced by an external TMCC. For all the plots, we have set $\Delta_{Q}=2$,
$K_{z}=5$, and $J/T/D_{z}/J_{z}/\Delta_{z}/B/J=1$.

In Fig. 3, the impact of weak as well as strong coupling strength $\lambda$ of
the classical channel on the dynamics of entanglement, coherence, entropic
uncertainty, and entropy disorder in a two-spin system when exposed to an
external magnetic field is studied. Initially, the two-qubit spin system
remains maximally entangled and coherent as the functions are $NG=LC=1$. On
the contrary, the $EU=EN=0$ suggests that the system is free of entropic
uncertainty and mixedness. As the interaction between the external TMCC and
the two-qubit spin system starts, the initial maximal correlations in the
state decrease. The speed of decay of entanglement and coherence is regulated
by the coupling strength of the classical field. The strong coupling of the
spin state with the classical environment results in a quicker decay of
correlations and vice versa. Besides, the non-Markovian behavior of the
classical fields is highly enhanced in the strong coupling regimes while
becoming negligible at the weaker coupling strength end. Likewise, the
uncertainty and entropy disorder increases with higher speeds as $\lambda$
increases. Finally, the speed of entropic uncertainty and entropy is directly
related to the decay rate of the entanglement and coherence. Therefore,
showing an anti-correlation between the two pairs of phenomena. The phenomenon
of sudden death and birth of entanglement varies with the varying strengths of
coupling strength regimes of the classical field. Compared to Fig. 2, the
decay observed in the current case has the least values. Hence, quantum
correlation decay can be highly controlled by tuning the coupling intensity
between the classical field and the two-qubit state. However, the fact remains
consistent that both the parameters, namely the disorder $\Delta_{Q}$ and
coupling parameter $\lambda$ regulated the rate of sudden death and birth of
the entanglement and coherence. Finally, for the higher $\lambda$ values, the
entanglement and coherence become completely lost, hence, suggesting maximum
entropy disorder and absolute separability in the state.

($a$)($b$)($c$)($d$)

Figure 4: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of
temperature $T$ against time in a two-spin system influenced by an external
TMCC. For all the plots, we have set $\Delta_{Q}=2$, $\lambda=0.1$, $K_{z}=5$
and $J/D_{z}/J_{z}/\Delta_{z}/B/J=1$.

The temperature influence on the dynamics of entanglement, coherence, entropic
uncertainty, and entropy is analyzed in a system of two-qubit spin state when
subjected jointly to TMCC in Fig. 4. The influence is taken into account when
the temperature of coupled fields is assumed at different fixed higher and
lower values. The difference between the dynamics of the entanglement,
coherence, uncertainty, and disorder functions is significant. As seen for the
higher temperature values $(T=7)$, the initial entanglement and coherence
values decrease while the original entropic uncertainty and mixedness in the
state increase. Hence, contradicting the characteristic properties of disorder
parameter $\Delta_{Q}$ (Fig. 2) and $\lambda$ (Fig. 3) which do not influence
the initial values of the inclusive functions. Besides this, the dynamical
maps of entanglement comprise repeated sudden death and birth revivals for the
lower temperature values ($T=0.1$). On the contrary, the entanglement decays
exponentially with time for the higher temperature values. The coherence
function $LC$ deviates from the entanglement $NG$ function and shows revivals
of coherence, even at the higher temperature values. Hence, suggesting the
strengthened nature of coherence in the spin state compared to the associated
entanglement. Unlike the cases in Fig. 2 and 3, entanglement and coherence
remain preserved for longer interval of time in the current case, especially,
for the higher temperature values. It is interesting to note that for the
higher temperature values, the initial values of the entanglement and
coherence functions decrease but remain largely preserved for the latter
interval of time. Likewise, the entropic uncertainty and entropy functions are
initially directly affected by the temperature, however, for the higher
temperature values, the uncertainty and mixedness in the state become reduced
at the latter notes of time. However, for the lower entropic uncertainty and
mixedness, one must keep the temperature at minimum values. In comparison, the
$NG$ function’s revival character seems very susceptible to the increasing
values of temperature and becomes easily dissipated.

($a$)($b$)($c$)($d$)

Figure 5: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of KSEA
interaction along $z$-axis $K_{z}$ against time in a two-spin state influenced
by an external TMCC. For all the plots, we have set $\Delta_{Q}=2$,
$\lambda=0.1$, and $J/D_{z}/J_{z}/\Delta_{z}/B/J=1$.

Fig. 5 discloses the dynamics of entanglement, coherence, entropic
uncertainty, and mixedness in a two-qubit spin system coupled with TMCC. In
particular, the impact of KSEA interaction $(K_{z})$ fixed to different
strengths, is evaluated on the dynamics of the system. The dynamical maps of
$NG$, $EU$, $LC$, and $EN$ functions seem increasingly different from those
observed in Figs. 2-4. As can be seen that a higher degree of entanglement and
coherence preservation limit is achieved for lower KSEA interaction strength
$K_{z}=1.0$. In close connection, a lower limit of entropic uncertainty and
mixedness in the two-qubit state has been observed at $K_{z}=1.0$. However,
for the increasing strength of the KSEA interaction ($K_{z}=5$), the
entanglement, as well as the coherence function, suffers a greater decay. This
contradicts most of the previous findings obtained in Refs. [44, 46, 47, 48]
where for the increasing strength of $K_{z}$, quantum correlations in the
state becomes more preserved. However, it is noticeable that the increasing
KSEA interaction strength improves the revival character of the $NG$, $EU$,
$LC$, and $EN$ functions. Particularly, for the $K_{z}=5.0$, the entanglement
and coherence functions start facing sudden deaths and births with decreasing
amplitudes. Besides, entanglement for $K_{z}=3.0$ dissipates completely and
only agrees with the entropy disorder in the state. The coherence function on
the other hand remains non-zero for $K_{z}=3.0$ and agrees with the entropic
uncertainty function. As can be seen that for $K_{z}=3.0$, the $EU$ function
shows minimal uncertainty between the observables of the state that that seen
at $K_{z}=5.0$. Hence, the KSEA interaction of the external magnetic field
most likely differently affects different quantum criteria. The revivals in
entanglement and coherence functions suggest strong information exchange
between the two-qubit spin state and coupled fields. Therefore, with the
increasing KSEA interaction, one can reverse the information lost from the
system and the conversion of free-classical states into resourceful non-local
states is feasible. Finally, entanglement and coherence seem indefinitely
preserved in the two-qubit state for the lower $K_{z}$ values which may
greatly benefit the quantum information processing protocols. Besides, the
decay rates of entanglement and coherence are found proportional to the
increasing speed of entropic uncertainty and mixedness in the state,
therefore, showing an inverse relation between them. In comparison, the $LC$
followed by $NG$ functions catch a greater number of revivals compared to the
$EU$ and $EN$ functions. This means that $NG$ and $LC$ are highly sensitive to
the stimuli caused by the external TMCC.

($a$)($b$)($c$)($d$)

Figure 6: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of magnetic
field strength $B$ against time in a two-spin state influenced by an external
TMCC. For all the plots, we have set $\Delta_{Q}=2$, $\lambda=0.1$, $K_{z}=5$,
and $J/D_{z}/J_{z}/\Delta_{z}/J=1$.

In Fig. 6, we probe the influence of different fixed values of magnetic field
strength parameter $B$ on the time evolution of the two-spin system when
connected with TMCC. At the onset, the state preserves different values of
initial entanglement and coherence. For the higher magnetic field strengths,
$B=9.0$, the initially encoded entanglement and coherence decrease while the
entropic uncertainty and mixedness in the state increase. Therefore, the
external magnetic field negatively affects the preservation of quantum
correlations in the spin state. Besides the initial level, the time evolution
of the entanglement and coherence functions is also negatively affected by the
magnetic field. As seen for the higher magnetic field strength $B=9.0$,
entanglement and coherence revive at a lower level while the entropic
uncertainty and mixedness functions reach a higher saturation level. On the
contrary, for the lower magnetic field strength $B=1.0$, the entanglement and
coherence functions seem more preserved and the reverse can be seen for the
entropic uncertainty mixedness in the state. The revival rate, on the other
hand, is not disturbed by the different strengths of the external magnetic
field and remains the same. However, for the lower $B$ values, the revivals in
entanglement and coherence get a higher amplitude, therefore, predicting the
larger information flow between the two qubits and coupled fields. In
comparison, the impact of magnetic field parameter $B$ matches with that of
the disorder parameter $\Delta_{Q}$ and coupling constant $\lambda$
(especially, the higher $\lambda$ values) of the classical field illustrated
in Figs. 2 and 3, respectively. As can be seen that in the mentioned cases,
the entanglement functions quickly decay while the emergence of uncertainty
disorder in the state occurs faster. Therefore, for the optimal longer and
greater degree of quantum correlations in the two-qubit state, one must tune
the external magnetic field to the least values. Moreover, in agreement with
Figs. 2-5, we find an anti-correlation between the growing $EU$ and $EN$
functions with the decaying $NG$ and $LC$ functions. Finally, the decreasing
amplitudes of the revivals in entanglement and coherence show that the state
at the final notes of time will become fully separable with a higher degree of
uncertainty and disorder, depending upon the strength of the external magnetic
field.

($a$)($b$)($c$)($d$)

Figure 7: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of DM
interaction strength along $z$-axis $D_{z}$ against time in a two-spin state
influenced by an external TMCC. For all the plots, we have set $\Delta_{Q}=2$,
$\lambda=0.1$, $K_{z}=5$, and $J/D_{z}/\Delta_{z}/B/J=1$.

When coupled with TMCC configuration, the impact of different fixed values of
the DM interaction strength $D_{z}$ on the dynamics of entanglement,
coherence, entropic uncertainty, and entropy functions in a two-qubit spin
state is investigated in Fig. 7. The role of $D_{z}$ parameter in the dynamics
of entanglement, coherence, uncertainty, and mixedness is found to be
contradicting all the cases studied in Figs. 2-6. In all the previous figures,
the parameters associated with the classical as well as the magnetic field
seem negatively affecting the initially encoded quantum correlations and
positively accelerated the uncertainty and mixedness function. However, the
opposite occurs in the current case where for the increasing strength of
$D_{z}$, entanglement and coherence functions get enhanced while the
uncertainty and entropy functions get suppressed. Precisely, the two-qubit
state for the initial as well as for the later interval of time remains
maximally entangled and coherent for the $D_{z}=6.0$. In agreement, for
$D_{z}=6.0$, the state remains indefinitely free of the uncertainty and
mixedness disorder which is interesting. However, for $D_{z}<5$, the agreement
between the entanglement and coherence functions vanishes where the prior one
becomes non-maximal while the latter one remains maximal. It is also
interesting that for $D_{z}=3.5$, entanglement shows a quick drop and seems
completely dissipating while the coherence yet remains preserved with a non-
zero value. The initial, as well as the later dynamical map of entanglement
functions, seems in close connection with the entropy function. On the other
hand, the $LC$ function and the associated dynamical outlooks seem similar to
that obtained for the uncertainty in the state. As seen for different values
of $D_{z}$, the initial level of entanglement and entropy differs. However,
for a similar situation, the initial levels of coherence and uncertainty
correspond to each other and both functions start from maximal coherence and
zero uncertainty points, respectively. The revival characteristic of the
entanglement and coherence appears enhanced for the lower $D_{z}$ values and
so the opposite. Finally, by regulating the $D_{z}$ to higher values, the
state may be kept maximally entangled and coherent while free from uncertainty
and entropy disorder. While detuning of the $D_{z}$ may cause the initial as
well as the latter preserved levels of quantum correlations to be decreased
while enhancement in the uncertainty and entropy in the state may occur.
Besides this, the fact remains consistent that there is an anti-correlation
between the entanglement and coherence pair with the uncertainty and entropy
pair.

($a$)($b$)($c$)($d$)

Figure 8: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of the
symmetric exchange spin-spin interaction strength in $z$-direction along
$\Delta_{z}$ against time in a two-spin state influenced by an external TMCC.
For all the plots, we have set $\Delta_{Q}=2$, $\lambda=0.1$, $K_{z}=5$, and
$J/D_{z}/B/J=1$.

In Fig. 8, we address the dynamics of entanglement, coherence, entropic
uncertainty, and entropy mixedness in a two-spin system system when exposed to
an external TMCC. The influence of different fixed values of the anisotropy
factor of the two-spin system systems on the dynamics of the two-qubit
correlations is obtained in detail. The dynamical outlook obtained for
different values of $\Delta_{z}$ seems in agreement with that obtained in Fig.
7 for different fixed values of the DM-interaction. However, the results
obtained contradict the results obtained in Figs. 2-6 where strong
entanglement and coherence decay has been detected. For the increasing
anisotropy strengths, entanglement and coherence remained highly preserved.
For instance, see the slopes of entanglement and coherence at $\Delta_{z}=6.0$
where the state remains maximally entangled and coherent indefinitely. In
close connection, for $\Delta_{z}=6.0$, the $EU$ and $EN$ functions remain
zero, hence, predicting zero-entropic uncertainty and disorder in the spin
state. It is interesting to note that for $\Delta_{z}=3.5$, coherence is
improved while entanglement has been witnessed to decay completely in a short
time and the same has been also witnessed in Fig. 7 against $D_{z}$.
Entanglement and coherence show repeated sudden death and birth revivals at
$\Delta_{z}=1.0$, therefore, predicting strong information exchange between
the two-qubit state and coupled fields. The entropy function $EN$ agrees with
entanglement function $NG$ (where entanglement completely decays) and shows
that for $\Delta_{z}=3.5$, entropy in the system becomes maximum. On the other
hand, the $LC$ and $EU$ functions agree and show that minimum coherence decay
and uncertainty rise occurs for $\Delta_{z}=3.5$ when compared to
$\Delta_{z}=1.0$. The entanglement and coherence functions stay in inverse
relation with the entropic uncertainty/entropy functions, as can be seen, that
both grow in opposite directions.

($a$)($b$)
($c$)($d$)

Figure 9: Dynamics of negativity (a), entropic uncertainty (b),
$\ell_{1}$-norm coherence (c), and linear entropy (d) as functions of the
Heisenberg exchange interaction strength $J$ against time in a two-spin state
influenced by an external TMCC. For all the plots, we have set $\Delta_{Q}=2$,
$\lambda=0.1$, $K_{z}=5$, $T=0.5$ and $J/D_{z}/B/J=1$.

In Fig. 9, we investigate the impact of Heisenberg exchange interaction
parameter $J$ on the preservation of entanglement/coherence and generation of
entropic uncertainty and entropy disorder in the spin state exposed to the
TMCC. We have demonstrated the results for the ferromagnetic ($J<0$) as well
as for the antiferromagnetic ($J>0$) regimes. The impact of both the
ferromagnetic and antiferromagnetic regimes have been found similarly
affecting the generation of entanglement and suppression of the entropic
uncertainty and disorder in the state. As can be seen that for both the
positive and negative regimes of the Heisenberg exchange interaction, the
entanglement and coherence functions gradually generate while the entropic
uncertainty and entropy functions decay with time. Notice that for certain
critical ranges, such as $\pm 3.5<J<\pm 4.5$, the state seems initially
partially entangled while having non-zero uncertainty and mixedness.
Interestingly, coherence in the state is unaffected by this range and the
state remains maximally entangled and coherent. Therefore, suggesting the
strengthened nature of coherence compared to the entanglement. In the range
$+4.5<J<-4.5$, the dynamics of the state exhibit either zero- or partially
preserved entanglement and coherence regimes at the latter intervals of time.
However, for the range, $\pm 4.5<J=\pm 6.0$, the state remains maximally
entangled and coherent while encountering no decay. In agreement, for the
forenamed region, the state remains completely free of uncertainty and
entropy. In comparison, the coherence function exhibits a larger number of
revivals of coherence compared to the other inclusive functions. As seen that
all other functions exhibit a lower number of revivals or completely dissipate
after a short interval of time.

###  III.1 Fidelity of the state

Finally, we give a brief time to cover how much the state becomes
distinguished from the one originally assumed. In this regard, the notion of
fidelity can be used to determine the discrimination between any two given
states. Let one of the states is $\rho_{st}(t,T)$ given in Eq. (18) and the
other one be any arbitrary state $\sigma$. Further, the $\sigma$ state is
assumed into two catteries: the first given in the Eq. (7) namely,
$\rho(0,T)$. In the second case, let us consider a maximally entangled state
$\psi=\frac{1}{\sqrt{2}}(\ket{00}+\ket{00})$, therefore, the initial density
matrix of the state becomes as $\rho_{0}=\ket{\psi}\bra{\psi}$. Therefore,
$\sigma\in\\{\rho(0,T),\rho_{0})\\}$. Using such a comparison would lead us to
clarify the distance between the resultant, initial thermal, or maximally
entangled two-qubit state.
In this case, fidelity for the two matrices $\rho_{st}(t,T)$ and $\sigma$ can
be written as

$FID_{X}=Tr[\rho_{st}(t,T)\sigma]+2\sqrt{\det[\rho_{st}(t,T)]\det[\sigma]},$
(24)

where fidelity between state $\rho_{st}(t,T)$ and $\rho(0,T)$ becomes as

$FID_{1}=X1+X2+X3$ (25)

where

$\displaystyle X1=$
$\displaystyle\rho_{11}^{2}+\rho_{22}^{2}+\rho_{33}^{2}+\rho_{44}^{2}+2\rho_{23}\rho_{23}^{*},$
$\displaystyle X2=$
$\displaystyle\frac{\rho_{14}\rho_{14}^{*}\sin(2\Delta_{Q}\lambda
t)\cos(4\Delta_{o}\lambda t)}{\Delta_{Q}\lambda t},$ $\displaystyle X3=$
$\displaystyle\sqrt{\frac{\left(|\rho_{14}|^{2}-\rho_{11}\rho_{44}\right)\left(\rho_{22}\rho_{33}-\rho_{23}\rho_{23}^{*}\right)^{2}\left(\rho_{14}\rho_{14}^{*}\sin^{2}(2\Delta_{Q}\lambda
t)-4\Delta_{Q}^{2}\lambda^{2}\rho_{11}\rho_{44}t^{2}\right)}{\Delta_{Q}^{2}\lambda^{2}t^{2}}}.$

Besides, fidelity between the $\rho_{st}(t,T)$ and maximally entangled state
has the form

$FID_{2}=\frac{1}{4}\left(2(\rho_{11}+\rho_{44})+\frac{e^{-4i\Delta_{o}\lambda
t}\sin(2\Delta_{Q}\lambda
t)\left(\rho_{14}+\rho_{14}^{*}e^{8i\Delta_{o}\lambda
t}\right)}{\Delta_{Q}\lambda t}\right)$ (26)

($a$)

Figure 10: (a)Dynamics of fidelity between states $\rho_{st}(t,T)$ and
$\rho(0,T)$ for the two-qubit case when exposed to the hybrid channel against
various strengths of classical dephasing while setting $\lambda=0.1$,
$K_{z}=5$, $T=0.5$ and $J/D_{z}/B/J=1$. (b) Same as (a) but for
$\rho_{st}(t,T)$ and $\rho_{0}$.

In Fig. 10, the fidelity dynamics for between the states
$\rho_{st}(t,T)-\rho(0,T)$ ($FID_{1}$) and $\rho_{st}(t,T)-\rho_{0}$
($FID_{2}$) is presented against various static noisy dephasing parameter.
This would enable us to predict how much the resultant state deviates from the
originally considered state under strong and weak dephasing limits. In Fig.
10(a), the two states seem completely comparable, however, with time they
become distinguished more and more. However, the fidelity loss rate is mostly
concentrated on the dephasing strength introduced in the system. For example,
for $\Delta_{Q}=5.0$, the fidelity loss rate is enough quicker, and so the
opposite can be seen at the weak dephasing limits. However, for the weak
dephasing limits, the minimum achieved by the green slopes is deeper than that
seen for the strong dephasing limits. This shows that the states become more
distinguishable at a specific duration. Besides, for the stronger dephasing
strengths, the slopes show quicker revivals as compared to that at the weak
dephasing end. After a long time, all the slopes finally seem to achieve a
similar saturation level, hence, predicting, a similar amount of fidelity even
for the different $\Delta_{Q}$ values. In the second case Fig. 10(b),
initially, the state remains more distinguishable, however. the fidelity
increases between the state $\rho_{st}(t,T)$ and $\rho_{0}$. The rate of
fidelity increase remains higher for the weak dephasing strengths and
decreased for the higher dephasing strengths. However, with time, the slopes
for the different values of $\Delta_{Q}$ seem to accumulate the same
saturation level. This shows that there would remain a constant amount of
distinguishability between the maximally entangled and our considered state
$\rho_{st}(t,T)$ in the current given conditions.

Furthermore, in the current work, we have found that entropic uncertainty and
entropic disorder negatively affect the degree of quantum correlations in the
state. As seen that when the entropic uncertainty and disorder rise,
entanglement and coherence decay. It is noticeable that the rate of entropic
uncertainty and mixedness remains higher than the decay rate of entanglement
and coherence. This suggests that the entropic uncertainty and disorder in the
two-qubit spin states lead while the opposite involved functions lag. Hence,
illustrating the rise in uncertainty and entropy as the major causes of the
loss of quantum correlations in the spin systems. Besides, we believe that the
current configuration can be successfully employed for the transmission of
quantum information in associated protocols. As seen that this configuration
can be readily exploitable to induce longer quantum correlations preservation.
Moreover, the cases of implementation of individual classical channels have
been found damaging for the quantum correlations preservation compared to when
the classical channel is employed jointly with the external magnetic field [8,
9, 10, 37]. In addition, the current results also appraise that the same
configuration with individual external magnetic fields and thermal fields
exhibit lesser and shorter preserved quantum correlations compared to when is
jointly employed with classical field [44].

##  IV Experimental feasibility

Any configuration, without its experimental nature, would be assumed not to be
beneficial. For this reason, we give some experimental prospects for the
current studied configuration. We find that classical channels have been
experimentally well adopted with quantum channels previously for information
processing, for example, see Refs. [49, 50, 51, 52]. In Ref. [49], by keeping
a spacing of $1.6$nm between the local and non-local channels while tuning the
frequency to $200$GHz at $-24$dBm, the authors disclosed a resourceful joint
channel. They found that the total coexistence power in single mode fiber
(SMF), where the secret key rate (SKR) was recorded to drop by 73 $\%$.
Besides this, at $0$dBm total coexistence power in hollow core nested
antiresonant nodeless (HC-NAN) fiber ($250$ times greater power saving than
that achieved for SMF), the SKR remains more preserved. The co-deployment of a
discrete variable-quantum key distribution channel with $8\times 200$Gbps
classical channels was resourcefully demonstrated using a $2$km long HC-NAN
fiber with a high speedy transmission of $1.6$Tbps [50]. Here, the authors
claimed the coexistence of classical-quantum channels are recorder to have a
reduced decay and even with more power saving (nearly 40 times) compared to
the individual utilization of quantum channels. In relevance to our study, the
authors in Ref. [51] experimentally probed counter-propagation of local-
quantum channels over a $1$km-long $7-$core, multicore fiber and found that
they show high tolerance to the noise compared to the individual use of the
counterparts, hence, showing agreement with our results. The authors in Ref.
[52] realized a quantum–classical channel over a $7$-core multicore fiber,
based on space division multiplexing with the highest launch power of $25$dBm.
By looking into the above studies, it can be readily deduced that classical
channels can be coined together with quantum channels such as the ones studied
here. Moreover, we propose that if the current configuration is utilized by
applying certain procedures, then it might turn out to be a resourceful way
for the practical transmission of quantum information, quantum devices, and
associated quantum operations.

##  V Conclusion

This study disclosed the characterization of a hybrid channel comprising
thermal, magnetic, and classical parts characterized by various quantum
correlations strengthening and weakening characteristics. The case of a two-
qubit for the sake of simplicity is owned to investigate the time evolution of
entanglement, coherence, entropic uncertainty, and entropy disorder under the
considered channel. various quantum tools are used to characterize the
parameters of the configuration, and the resultant dynamical maps of the
assumed quantum criteria have been explicitly studied. Finally, we provided
the optimal parameter setting which best suits quantum correlations
preservation and entropic uncertainty as well as disorder suppression under
the action of the hybrid channel.

We demonstrate that the collective symmetry of the current configuration
shared by the thermal, magnetic, and classical fields continues to be a
crucial factor in maintaining the preservation of quantum correlations.. There
exist certain aspects in this joint external field setup can be usefully used
to avoid the emergence of entropic uncertainty and entropy disorder in the
state. For example, in the weaker coupling regimes of the classical field,
certain values of DM, spin-spin, and anisotropy interaction strengths, the
state remains maximally entangled and coherent initially as well as for the
latter interval of time. In close connection, for the given conditions, the
state can be kept completely free of uncertainty and entropy disorder. The
entanglement and coherence decay has been found completely depending upon the
emergence of the entropic uncertainty and entropy disorder in the two-qubit
spin state system. Certain aspects of the coupled fields, such as the coupling
strengths, disorder parameter of the static noise, magnetic field strength,
and temperature of the hybrid channel have been found responsible for the
emergence of uncertainty and entropy as well as for the decay of the
entanglement and coherence. Interestingly, previously the KSEA interaction has
been found highly influential for the preservation of quantum correlations,
however, the opposite has been witnessed when imposed with the current hybrid
channel. Finally, we believe that the intersection of the thermal, classical,
and magnetic fields along with certain state parameters remains a vital choice
that shows the capacity to be controlled easily for the preservation of
quantum data compared to that when they are considered individually.

## Author contributions

Conceptualization, Atta ur Rahman; Data curation, Ma-Cheng Yang and Cong-Feng
Qiao ; Formal analysis, Ma-Cheng Yang, S. M. Zangi and Cong-Feng Qiao ;
Funding acquisition, Cong-Feng Qiao ; Investigation, Atta ur Rahman;
Methodology, S. M. Zangi and Cong-Feng Qiao ; Project administration, Cong-
Feng Qiao ; Resources, Atta ur Rahman and Cong-Feng Qiao ; Software, Atta ur
Rahman; Supervision, Cong-Feng Qiao ; Validation, Ma-Cheng Yang, S. M. Zangi
and Cong-Feng Qiao ; Visualization, Atta ur Rahman; Writing-original draft,
Atta ur Rahman and Cong-Feng Qiao ; Writing-review and editing, Atta ur
Rahman.

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China (NSFC) under the Grants 11975236, 12235008 and by the University of
Chinese Academy of Sciences.

## Data availability

No datasets were generated or analyzed during the current study.

## Competing interests

The authors declare no competing interests.

## ORCID iDs

Atta ur Rahman https://orcid.org/0000-0001-7058-5671
Ma-Cheng Yang https://orcid.org/———
S. M. Zangi https://orcid.org/0000-0002-4601-4681
Cong-Feng Qiao https://orcid.org/0000-0002-9174-7307

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$\\{X_{\alpha}\\}_{\alpha\in\mathcal{A}}$ forms a strongly compatible scale if
there exists a Banach space $\tilde{X}$ such that
$X_{\alpha}\overset{1}{\hookrightarrow}\tilde{X}$, $\alpha\in\mathcal{A}$.

Let $\\{X_{\alpha}\\}_{\alpha\in\mathcal{A}}$ be a strongly compatible scale.
We set

$\Delta(X_{\alpha})_{\alpha\in\mathcal{A}}:=\\{x\in\cap_{\alpha\in\mathcal{A}}X_{\alpha}:\
\|x\|_{\Delta(X_{\alpha})}:=\sup_{\alpha\in\mathcal{A}}\|x\|_{X_{\alpha}}<\infty\\}.$

Then,
$(\Delta(X_{\alpha})_{\alpha\in\mathcal{A}},\|\cdot\|_{\Delta(X_{\alpha})})$
is a Banach space with the following properties:

(i)
$\Delta(X_{\alpha})_{\alpha\in\mathcal{A}}\overset{1}{\hookrightarrow}X_{\alpha}$,
$\forall\alpha\in\mathcal{A}$;

(ii) If $F$ is a Banach space such that
$F\overset{1}{\hookrightarrow}X_{\alpha}$, $\forall\alpha\in\mathcal{A}$, then
$F\overset{1}{\hookrightarrow}\Delta(X_{\alpha})_{\alpha\in\mathcal{A}}$.

Let $M$ be an Orlicz function, $M_{v}(u):=M(uv)/M(v)$, $u\geq 0$, $v>0$. We
consider the family of the Musielak-Orlicz sequence spaces
$\\{l_{M_{\bar{\beta}}}\\}_{\bar{\beta}\in\mathcal{B}}$, where $\mathcal{B}$
is the set of all sequences $\bar{\beta}=(\beta_{k})_{k=1}^{\infty}$ such that
$\sum_{k=1}^{\infty}{1}/{M(\beta_{k})}\leq 1$. Recall that the norm of the
Musielak-Orlicz sequence space $l_{M_{\bar{\beta}}}$ is defined by

$\|(a_{k})\|_{l_{M_{\bar{\beta}}}}:=\Big{\\{}\lambda>0:\,\sum_{k=1}^{\infty}M_{\beta_{k}}\Big{(}\frac{a_{k}}{\lambda}\Big{)}\leq
1\Big{\\}}$

(see e.g. [23, 31]). One can easily check that
$l_{M_{\bar{\beta}}}\overset{1}{\hookrightarrow}l_{\infty}$, for each
$\bar{\beta}\in\mathcal{B},$ and hence this family is a strongly compatible
scale.

######  Theorem 6.14.

For every Orlicz function $M$ such that $M\in\Delta_{2}^{\infty}$ we have
$(L_{M})_{U}=\Delta(l_{M_{\bar{\beta}}})_{{\bar{\beta}}\in\mathcal{B}}$ (with
equivalence of norms). Moreover, the following embeddings hold:

(6.16) $l_{\Phi_{M}}\hookrightarrow(L_{M})_{U}\hookrightarrow l_{p_{M}},$

where $\Phi_{M}$ is the dilation function of $M$ for large values of arguments
and

$p_{M}:=\sup\\{p\geq 1:\,M(uv)\leq Cu^{p}M(v)\;\;\mbox{for
some}\;\;C>0\;\;\mbox{and all}\;\;0<u\leq 1,uv\geq 1\\}.$

###### Proof.

Without loss of generality, we will assume that $a_{k}\geq 0$, $k=1,2,\dots$.

Let $F_{k}\subset[0,1]$, $k=1,2,\dots$, be arbitrary pairwise disjoint sets of
positive measure. Then, setting $\beta_{k}:=M^{-1}(1/m(F_{k}))$,
$k=1,2,\dots$, we see that

$\sum_{k=1}^{\infty}\frac{1}{M(\beta_{k})}=\sum_{k=1}^{\infty}m(F_{k})\leq 1,$

and hence ${\bar{\beta}}=(\beta_{k})_{k=1}^{\infty}\in\mathcal{B}$.
Conversely, for each ${\bar{\beta}}=(\beta_{k})_{k=1}^{\infty}\in\mathcal{B}$
we can find pairwise disjoint sets $F_{k}\subset[0,1]$, $m(F_{k})>0$, such
that $\beta_{k}=M^{-1}(1/m(F_{k}))$, $k=1,2,\dots$ These observations together
with Corollary 6.12 imply, for every $a=\\{a_{k}\\}_{k=1}^{\infty}$, the
following:

$\displaystyle\|a\|_{(L_{M})_{U}}$ $\displaystyle\asymp$
$\displaystyle\sup\left\\{\Big{\|}\sum_{k=1}^{n}a_{k}\frac{\chi_{F_{k}}}{\varphi_{L_{M}}(m(F_{k}))}\Big{\|}_{L_{M}}:\,n\in\mathbb{N},F_{k}\subset[0,1]\;\mbox{pairwise
disjoint}\right\\}$ $\displaystyle=$
$\displaystyle\sup_{n\in\mathbb{N},F_{k}\mbox{\small{pairwise
disjoint}}}\inf\left\\{\lambda>0:\,\sum_{k=1}^{n}M\left(\frac{a_{k}M^{-1}({1}/{m(F_{k})})}{\lambda}\right)m(F_{k})\leq
1\right\\}$ $\displaystyle=$
$\displaystyle\sup_{\bar{\beta}\in\mathcal{B},n\in\mathbb{N}}\inf\left\\{\lambda>0:\,\sum_{k=1}^{n}\frac{M\left(\frac{a_{k}}{\lambda}\beta_{k}\right)}{M(\beta_{k})}\leq
1\right\\}$ $\displaystyle=$
$\displaystyle\sup_{\bar{\beta}\in\mathcal{B},n\in\mathbb{N}}\inf\left\\{\lambda>0:\,\sum_{k=1}^{n}M_{\beta_{k}}(a_{k}/\lambda)\leq
1\right\\}$ $\displaystyle=$
$\displaystyle\|a\|_{\Delta(l_{M_{\bar{\beta}}})_{\bar{\beta}\in\mathcal{B}}}.$

Thus, $(L_{M})_{U}=\Delta(l_{M_{\bar{\beta}}})_{\bar{\beta}\in\mathcal{B}}$,
with equivalence of norms.

To prove the left-hand side embedding in (6.16) we assume that
$\|(a_{k})\|_{l_{\Phi_{M}}}\leq 1$. This implies that
$\sum_{k=1}^{n}\Phi_{M}(a_{k})\leq 1$ for every $n\in\mathbb{N}$. Then, for
each sequence $\bar{\beta}=(\beta_{k})_{k=1}^{\infty}\in\mathcal{B}$ we have

$\sum_{k=1}^{n}\frac{M\left({a_{k}}\beta_{k}\right)}{M(\beta_{k})}=\sum_{k:\;a_{k}\beta_{k}\leq
1}\frac{M\left({a_{k}}\beta_{k}\right)}{M(\beta_{k})}+\sum_{k:\;a_{k}\beta_{k}>1}\frac{M\left({a_{k}}\beta_{k}\right)}{M(\beta_{k})}\leq\sum_{k=1}^{n}\frac{1}{M(\beta_{k})}+\sum_{k=1}^{n}\Phi_{M}(a_{k})\leq
2.$

Thus, $\|(a_{k})\|_{l_{M_{\bar{\beta}}}}\leq 2$ for every
$\bar{\beta}=(\beta_{k})_{k=1}^{\infty}\in\mathcal{B}$ and therefore, by the
first assertion of the theorem, it follows

$\|(a_{k})\|_{(L_{M})_{U}}\leq
C\|(a_{k})\|_{\Delta(l_{M_{\bar{\beta}}})_{\bar{\beta}\in\mathcal{B}}}\leq
2C.$

Furthermore, by [15], $p_{M}$ is the supremum of the set of all $p\geq 1$ such
that $M$ is equivalent to a $p$-convex function on the interval $[1,\infty)$,
or equivalently $p_{M}$ is the supremum of the set of all $p\geq 1$ such that
the Orlicz space $L_{M}[0,1]$ admits an upper $p$-estimate. Thus, by using the
notation of this paper, we have $p_{M}=\delta(L_{M}[0,1])$ and hence the
right-hand side embedding in (6.16) is a consequence of Proposition 3.9(i).
This completes the proof. ∎

######  Remark 6.15.

Informally, the Orlicz space $l_{\Phi_{M}}$ is located rather ”close” to the
space $(L_{M})_{U}$, because the fundamental functions of these spaces are
equivalent. Indeed, let $n\in\mathbb{N}$ and $\varepsilon>0$ be arbitrary.
Then, by definition, $\phi_{l_{\Phi_{M}}}(n)=1/u_{n}$, where $u_{n}$ satisfies
the conditions:

$\frac{M(u_{n}v_{n})}{M(v_{n})}\geq(1-\varepsilon)\frac{1}{n}\;\;\mbox{for
some}\;\;v_{n}\geq
1/u_{n}\;\;\mbox{and}\;\;\frac{M(u_{n}v)}{M(v)}\leq\frac{1}{n}\;\;\mbox{for
all}\;\;v\geq 1/u_{n}.$

In particular, from the last estimate it follows that $M(1/u_{n})\geq n$.
Therefore, since $M^{-1}$ is concave, we get

$u_{n}\geq\frac{M^{-1}((1-\varepsilon)s_{n}/n)}{M^{-1}(s_{n})}\geq(1-\varepsilon)\frac{M^{-1}(s_{n}/n)}{M^{-1}(s_{n})},\;\;\mbox{where}\;\;s_{n}=M(v_{n})\geq
M(1/u_{n}),$

and

$u_{n}\leq\frac{M^{-1}(s/n)}{M^{-1}(s)},\;\;\mbox{for all}\;\;s\geq
M(1/u_{n}).$

Thus,

$\frac{1}{\phi_{l_{\Phi_{M}}}(n)}=\inf_{s\geq
M(1/u_{n})}\frac{M^{-1}(s/n)}{M^{-1}(s)},$

whence

$\phi_{l_{\Phi_{M}}}(n)=\sup_{s\geq
M(1/u_{n})}\frac{M^{-1}(s)}{M^{-1}(s/n)}=\sup_{t\geq
M(1/u_{n})/n}\frac{M^{-1}(tn)}{M^{-1}(t)}.$

Consequently, since $M(1/u_{n})/n\geq 1$, by Corollary 6.13, we have

$\phi_{l_{\Phi_{M}}}(n)\leq\mathcal{M}_{M^{-1}}(n)\preceq\phi_{(L_{M})_{U}}(n),\;\;n\in\mathbb{N}.$

It remains to note that the opposite inequality follows from the left-hand
side embedding (6.16).

Data availability statement. All data generating or analysed during this study
are included in this published article.

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