# Intrinsic decoherence effects on correlated coherence and quantum discord in
XXZ Heisenberg model

Zakaria Dahbi<EMAIL_ADDRESS>Lab of High Energy Physics-Modeling
and Simulation, Faculty of Sciences,
Mohammed V University in Rabat, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat,
Morocco  Mansoura Oumennana<EMAIL_ADDRESS>Mostafa Mansour
<EMAIL_ADDRESS>Laboratory of High Energy Physics and Condensed
Matter, Department of Physics, Faculty of Sciences Ain Chock, Hassan II
University, P.O. Box 5366 Maarif, Casablanca 20100, Morocco

###### Abstract

Spin qubits are at the heart of technological advances in quantum processors
and offer an excellent framework for quantum information processing. This work
characterizes the time evolution of coherence and nonclassical correlations in
a two-spin XXZ Heisenberg model, from which a two-qubit system is realized. We
study the effects of intrinsic decoherence on coherence (correlated coherence)
and nonclassical correlations (quantum discord), taking into consideration the
combined impact of an external magnetic field, Dzyaloshinsky-Moriya (DM) and
Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions. To fully
understand the effects of intrinsic decoherence, we suppose that the system
can be prepared in one of the two well-known extended Werner-like (EWL)
states. The findings show that intrinsic decoherence leads the coherence and
quantum correlations to decay and that the behavior of the aforementioned
quantum resources relies strongly on the initial EWL state parameters. We,
likewise, found that the two-spin correlated coherence and quantum discord;
become more robust against intrinsic decoherence depending on the type of the
initial state. These outcomes shed light on how a quantum system should be
engineered to achieve quantum advantages.

intrinsic decoherence, spin qubits, quantum discord, correlated coherence

.

##  I Introduction

Studying solid-state physical systems under multiple interactions has recently
attracted particular focus. The unprecedented possibility of taking advantage
of these systems has opened the door to constructing new quantum-based
technologies [1]. Quantum superposition and entanglement [2] are two
surprising aspects of the quantum theory and are found to be imperative
resources for achieving speedup in information processing [3] and for
realizing several non-local tasks and classically unattainable applications
[4, 5, 6, 7] within the realm of classical physics. Further, it has been
demonstrated that even when entanglement is lost, quantum information
processing tasks may still be performed in the case of particular mixed states
due to the presence of nonclassical correlations beyond entanglement [8, 9].
On the other side, decoherence [10] effects present a grave challenge to the
beneficial applications of quantum mechanics because it prevents the retention
and controllable handling of qubits.

Previously many studies focused on measuring and characterizing quantum
resources in the framework of intrinsic and standard decoherence models in
different quantum systems [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24]. Although many methods were introduced to measure quantum systems,
assessing the quantumness of multipartite systems is still a challenging task
[8, 25, 26]. For specific quantum systems, quantum discord (QD) [27] was the
first quantifier introduced to capture nonclassical correlations. QD measures
the difference between total correlations and classical correlations in a
quantum system. However, it is a strenuous measure to calculate, and
analytical expressions of QD were only obtained for two-qubit states [28, 29]
and quantum X-states ( see for instance the works [30, 31, 32, 33] and
references therein). A rigorous mathematical characterization on the
computational difficulties of quantum discord is provided in [34],
demonstrating that computing quantum discord is an NP-complete problem.
Quantum correlations have been studied in many quantum systems, such as
Heisenberg spin chain models (for instance, spin-1 Heisenberg chains [35],
anisotropic spin-1/2 XY chain in transverse magnetic field [36]) and quantum
dot systems such as two coupled double quantum dots systems [37, 38] and
double quantum dot system with single electron under Rashba interaction [39]
). Recently, considerable attention has been dedicated to studying the
influence of the Dzyaloshinsky-Moriya (DM) interaction and the Kaplan-
Shekhtman-Entin-Wohlman-Aharony (KSEA) interaction on the quantum features of
specific quantum systems [40, 41, 42, 43].

Quantum coherence is another quantum feature that one should deal with. This
latter emanates from the superposition principle of quantum states and is a
valuable resource to be preserved due to its pivotal role in quantum
information processing [44] and quantum thermodynamics [45]. It was indeed
proven that long-lasting quantum coherence is vital for overcoming classical
limitations of measurement accuracy in quantum metrology [46]. Moreover,
growing interest is accorded to the role played by quantum coherence in some
biological mechanisms [47], such as photosynthesis [48] and bird navigation
[49]. Similarly to quantum correlations and entanglement, several quantifiers
were introduced to capture quantum coherence in quantum systems appropriately,
e.g. relative entropy of coherence and $l_{1}$-norm of coherence [50] as well
as intrinsic randomness [51].

In this article, we investigate the evolution of correlated coherence and
quantum correlations quantified by quantum discord in a two-qubit XXZ
Heisenberg spin chain under intrinsic decoherence effects with an applied
magnetic field, KSEA and DM interactions using the Milburn’s decoherence model
[52]. We mainly focus on how the Hamiltonian parameters and the initial state
affect the dynamical behavior of correlated coherence and quantum discord. We
look at two different extended Werner-like states to show how the choice of
the initial state is essential to suppress the effects of specific
interactions.

The rest of this work is as follows. In Sec. II, we give a short overview of
the quantum resources indicators used in this study. The Hamiltonian of the
considered physical model is introduced in Sec. III, and the evolved density
matrix corresponding to the system is derived for the parameterized initial
state considered. In Sec. IV we present the results obtained for the dynamics
of correlated coherence and quantum discord under the influence of intrinsic
decoherence with the combined impact of KSEA and DM interactions. Finally, we
conclude the main findings from the proposed model in Sec. V.

##  II Quantum information indicators

This part defines the correlated coherence, used as a quantum coherence
quantifier, and the quantum discord operated to quantify nonclassical
correlations.

###  II.1 Correlated coherence

Quantum coherence is a basis-dependent property of a quantum state that arises
from the superposition of system states reflected by the non-diagonal
components of the density matrix on a given basis. Here, we consider a
conveniently computable coherence measure, specifically the $l_{1}$–norm of
coherence [50]. The $l_{1}$-norm of coherence is a trustworthy key indicator
that meets the requirements of a good coherence measure and can be obtained
for a given density matrix $\varrho$. Baumgratz $et$ $al.$ [50] has
demonstrated that a quantum system’s coherence is given as

$\mathcal{C}_{l_{1}}(\varrho)=\min_{\eta\in\mathfrak{I}}|\varrho-\eta|,$ (1)

where $\mathfrak{I}$ is a set containing all incoherent states, that is
$\mathcal{C}_{l_{1}}(\eta)=0$ for all $\eta\in\mathfrak{I}$. The $l_{1}$-norm
coherence can be obtained by means of the non-diagonal entries of $\varrho$ as

$\mathcal{C}_{l_{1}}(\varrho)=\sum_{i\neq
j}\sqrt{\varrho_{ij}\varrho_{ij}^{*}},$ (2)

where ∗ refers to the complex conjugate. It should be noted that the total
$l_{1}$-norm coherence of a state $\varrho$, living in Hilbert space of
dimension $d$, should not exceed $d-1$. Correlated coherence is yet another
metric of coherence that can provide information about the quantumness of a
particular state. For any given bipartite state $\varrho$, correlated
coherence is defined as the total coherence subtracting local coherences. The
definition of the $l_{1}$-norm based correlated coherence [50] reads as

$\mathcal{C}_{cc}(\varrho):=\mathcal{C}_{l_{1}}(\varrho)-\mathcal{C}_{l_{1}}(\varrho_{A})-\mathcal{C}_{l_{1}}(\varrho_{B}),$
(3)

where $\varrho_{A}=tr_{B}\varrho$ and $\varrho_{B}=tr_{A}\varrho$ are the
reduced density matrices of local subsystems.

###  II.2 Quantum discord

QD [27] quantifies quantum correlations inhibited in a composite system. In a
two-qubit system, QD is defined by removing the existing classical
correlations from the quantum mutual information of the system as

${\mathcal{D}}(\varrho)={\mathcal{I}}(\varrho)-{\mathcal{C}}(\varrho),$ (4)

with

${\mathcal{I}}(\varrho)={\mathcal{S}}(\varrho_{B})+{\mathcal{S}}(\varrho_{A})-{\mathcal{S}}(\varrho)$
(5)

and

${\mathcal{C}}(\varrho)={\mathcal{S}}(\varrho_{A})-min_{\\{\pi_{i}^{B}\\}}\sum_{i}p_{i}{\mathcal{S}}(\varrho_{A|i}).$
(6)

${\mathcal{S}}(\varrho)=-tr(\varrho\log_{2}\varrho)$ is the von Neumann
entropy. $\varrho_{A}$ and $\varrho_{B}$ are the reduced density operators
corresponding, respectively, to the subsystems $A$ and $B$.
$\/{\left\\{\pi_{B}^{i}\right\\}}={|i_{B}\rangle\langle i_{B}|}$ is the
complete ensemble of orthonormal projectors acting only on the second
subsystem $B$.
$\varrho_{A|i}=tr_{B}(\pi_{B}^{i}\varrho\pi_{B}^{i})/p_{i}$ is the resulting
state of the first subsystem $A$ after obtaining the result $i$ on B, and
$p_{i}=tr_{AB}(\pi_{B}^{i}\varrho\pi_{B}^{i})$ is the probability of having
$i$ as a result. Following the relations (5) and (6), the quantum discord (4)
can be rewritten as [53, 54]

$\displaystyle{\mathcal{D}}(\varrho)=min_{\left\\{\pi_{B}^{i}\right\\}}[{\mathcal{S}}(\varrho/{\left\\{\pi_{B}^{i}\right\\}}]+{\mathcal{S}}(\varrho_{B})-{\mathcal{S}}(\varrho).$
(7)

In general, for arbitrary quantum states, finding analytical formulas for QD
is complicated due to the minimization process required for conditional
entropy. It was only possible to obtain approximate analytical expressions
[55] for a limited number of states, such as the X-states. For $X$-states

$\varrho={\begin{bmatrix}{{\varrho_{11}}}&0&0&{{\varrho^{*}_{14}}}\\\
0&{{\varrho_{22}}}&{{\varrho^{*}_{23}}}&0\\\
0&{\varrho_{23}}&{{\varrho_{33}}}&0\\\
{\varrho_{14}}&0&0&{{\varrho_{44}}}\end{bmatrix}}$ (8)

the quantum discord is redefined by the succeeding expression [56, 57]

${\mathcal{D}}(\varrho)=\min\\{\mathcal{QD}_{1},\mathcal{QD}_{2}\\},$ (9)

with

$\mathcal{QD}_{i}=f(\varrho_{11}+\varrho_{33})+\sum_{k=1}^{4}\lambda_{k}\log_{2}(\lambda_{k})+{\mathcal{D}}_{i}.$

${\mathcal{D}}_{1}=f(\Lambda)$,
${\mathcal{D}}_{2}=-\sum_{n=1}^{4}\varrho_{nn}\log_{2}(\varrho_{nn})-f(\varrho_{11}+\varrho_{33})$,
$\Lambda=\frac{1}{2}\Big{(}1+\Big{(}(1-2(\varrho_{33}+\varrho_{44}))^{2}+4(|\varrho_{14}|+|\varrho_{23}|)^{2}\Big{)}^{1/2}\Big{)}$
and $f(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$ is the binary Shannon entropy.
$\lambda_{k}$’s denote the eigenvalues of the density matrix $\varrho$.

##  III Two-qubit XXZ model

We consider the model as a two-qubit system operating spin polarization of two
nearest spin-$1/2$ XXZ particles. The particles are exposed to the interplay
of an external homogeneous magnetic field and the combination of DM and KSEA
interactions along the $z$-axis. The associated Hamiltonian is solvable and is
represented as [40]

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle
J_{z}\sigma_{1}^{z}\sigma_{2}^{z}+J(\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{1}^{y}\sigma_{2}^{y})+D_{z}\left(\sigma_{1}^{x}\sigma_{2}^{y}-\sigma_{1}^{y}\sigma_{2}^{x}\right)$
(10) $\displaystyle+$
$\displaystyle\Gamma_{z}\left(\sigma_{1}^{x}\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{x}\right)+B(\sigma_{1}^{z}+\sigma_{2}^{z}),$

where $\sigma_{\mu}^{i=x,y,z}$ $(\mu=1,2)$ are the typical Pauli matrices
corresponding to the spin $\mu$, while $J_{z}$ and $J$ are real coupling
coefficients denoting, respectively, the anisotropy coupling constant defining
the symmetric spin-spin exchange interaction in the $z$-direction, and the
interaction coupling constant. The $B$ is a parameter, restricted to $B\geq
0$, that indicates the strength of the magnetic field. Besides, the
$\Gamma_{z}$ and $D_{z}$ parameters reflect the $z$-KSEA and $z$-DM
interactions, which result in symmetric and anti-symmetric spin-orbit coupling
contributions, respectively. We assume that the two-spin XXZ model behaves as
in the antiferromagnetic case, $J>0$ and $J_{z}>0$. It is worth stating that
all parameters are considered to be dimensionless. The Hamiltonian (10) can be
written on the two-qubit basis,
$\\{|\Downarrow\Downarrow\rangle,|\Downarrow\Uparrow\rangle,|\Uparrow\Downarrow\rangle,|\Uparrow\Uparrow\rangle\\}$,
as

$\displaystyle\hat{H}=\left(\begin{array}[]{cccc}J_{z}+2B&0&0&-2i{\rm\Gamma}_{z}\\\
0&-J_{z}&2J+2iD_{z}&0\\\ 0&-2iD_{z}+2J&-J_{z}&0\\\
2i{\rm\Gamma}_{z}&0&0&-2B+J_{z}\end{array}\right).$ (11)

The eigenvalues and the associated eigenvectors of the aforementioned
Hamiltonian $\hat{H}$ are

$\displaystyle\mathcal{V}_{1}$
$\displaystyle=J_{z}+\chi,\quad\mathcal{V}_{2}=-J_{z}+\omega,$
$\displaystyle\mathcal{V}_{3}=-J_{z}-\omega,\quad\mathcal{V}_{4}=J_{z}-\chi,$

$\displaystyle|u_{1}\rangle$
$\displaystyle=\sqrt{\frac{\chi+2B}{2\chi}}\left(|\Downarrow\Downarrow\rangle+\frac{2i{\rm\Gamma}_{z}}{{\chi+2B}}|\Uparrow\Uparrow\rangle\right),$
$\displaystyle|u_{2}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\Downarrow\Uparrow\rangle+\frac{2J-2iD_{z}}{\omega}|\Uparrow\Downarrow\rangle\right),$
$\displaystyle|u_{3}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\Downarrow\Uparrow\rangle-\frac{2J-2iD_{z}}{\omega}|\Uparrow\Downarrow\rangle\right),$
$\displaystyle|u_{4}\rangle$
$\displaystyle=\sqrt{\frac{\chi-2B}{2\chi}}\left(|\Downarrow\Downarrow\rangle-\frac{2i{\rm\Gamma}_{z}}{\chi-2B}|\Uparrow\Uparrow\rangle\right),$

with $\chi=2(B^{2}+\Gamma_{z}^{2})^{1/2}$ and
$\omega=2(J^{2}+D_{z}^{2})^{1/2}$. To introduce intrinsic decoherence, we use
Milburn’s decoherence model [52], which assumes that quantum systems evolve
continually in an arbitrary sequence of identical unitary transformations
instead of unitary evolution. The following equation describes such evolution
[52]

$\frac{d\varrho_{t}}{dt}=\frac{1}{\gamma}\Bigl{(}\exp(-i\gamma\hat{H})\varrho_{t}\exp(i\gamma\hat{H})-\varrho_{t}\Bigr{)},$
(12)

here, $\varrho_{t}$ is the density matrix associated to the Hamiltonian
$\hat{H}$, and $\gamma$ is the intrinsic decoherence constant. Thus in the
limit of $\gamma^{-1}\rightarrow\infty$, there is no intrinsic decoherence,
and Eq. (12) is reduced to the typical von Neumann equation characterizing an
isolated quantum system. Milburn altered the Schrödinger equation in order for
quantum coherence to be spontaneously destroyed throughout the evolution of
the quantum system. The ensuing equation is obtained

$\frac{d\varrho_{t}}{dt}=-\frac{\gamma}{2}[\hat{H},[\hat{H},\varrho_{t}]]-i[\hat{H},\varrho_{t}].$
(13)

where $\frac{\gamma}{2}[\hat{H},[\hat{H},\varrho_{t}]]$ designates the non-
unitary evolution under the intrinsic decoherence in our considered two-qubit
system. A proper solution for the equation (13) is obtained using the Kraus
operators $\mathcal{\hat{M}}_{l}$ [52]

$\varrho_{t}=\sum^{\infty}_{l=0}\mathcal{\hat{M}}_{l}(t)\varrho^{t=0}\mathcal{\hat{M}}^{\dagger}_{l}{}(t)$
(14)

with $\varrho^{t=0}$ being the density matrix at $t=0$ of the considered
system and $\mathcal{\hat{M}}_{l}$ are given by

$\displaystyle\displaystyle\mathcal{\hat{M}}_{l}(t)=\Big{(}\frac{\gamma^{l}t^{l}}{l!}\Big{)}^{1/2}{\hat{H}}^{l}\exp\left(-i{\hat{H}}t\right)\exp\left(-\frac{\gamma
t}{2}{\hat{H}}^{2}\right),$

with
$\sum^{\infty}_{l=0}\mathcal{\hat{M}}_{l}(t)\mathcal{\hat{M}}^{\dagger}_{l}{}(t)=\mathbb{I}$.
Finally, the evolved state of the two-spin XXZ quantum system described by
$\hat{H}$ under intrinsic decoherence effects can be obtained by [52]

$\displaystyle\displaystyle\varrho_{t}$ $\displaystyle=$
$\displaystyle\sum_{j,k}\exp\left(-\frac{\gamma
t}{2}({\mathcal{V}}_{j}-{\mathcal{V}}_{k})^{2}-i({\mathcal{V}}_{j}-{\mathcal{V}}_{k})t\right)$
(15) $\displaystyle\times$ $\displaystyle\langle
u_{j}|\varrho^{t=0}|u_{k}\rangle|u_{j}\rangle\langle u_{k}|,$

where ${\mathcal{V}}_{j,k}$ and $\ket{u_{j,k}}$ are, respectively, the
eigenvalues of the Hamiltonian $\hat{H}$ (11) and their corresponding
eigenstates. Eq. (15) allows us to describe how the system resources change as
the state of the system is evolving under the intrinsic decoherence effects.
Following, we use correlated coherence and quantum discord to study how
intrinsic decoherence affects coherence and nonclassical correlations in the
two-spin XXZ Heisenberg model under the different supposed interactions. For a
complete characterization of the influence of intrinsic decoherence on the
system, we take into account two possible scenarios of the initial state

$\displaystyle\ket{\Psi_{1}}$ $\displaystyle=$
$\displaystyle\cos\left(\theta/2\right)\ket{\Downarrow\Downarrow}+\sin\left(\theta/2\right)\ket{\Uparrow\Uparrow},$
(16) $\displaystyle\ket{\Psi_{2}}$ $\displaystyle=$
$\displaystyle\cos\left(\theta/2\right)\ket{\Downarrow\Uparrow}+\sin\left(\theta/2\right)\ket{\Uparrow\Downarrow}.$
(17)

where $\ket{0}\equiv\ket{\Downarrow}$ and $\ket{1}\equiv\ket{\Uparrow}$,
denoting spin-down and spin-up, respectively. We shall study intrinsic
decoherence according to two samples of Bell-like initial states. To do that,
we combine the states in Eq. (16)-(17) using a parametrization and then solve
the dynamical equation. The results for each sample initial state can be
obtained by dealing with the parametrization parameters. Thus, the initial
state looks as follows

$\varrho^{t=0}=\alpha_{1}\varrho^{t=0}_{1}+\alpha_{2}\varrho^{t=0}_{2},$ (18)

where
$\varrho^{t=0}_{i}=p\ket{\Psi_{i}}\bra{\Psi_{i}}+(1-p)/4\leavevmode\nobreak\
I_{4}$, $0\leq p\leq 1$ denoting the level of purity in the initial state,
$0\leq\theta<\pi$ and $\alpha_{1}$ and $\alpha_{2}$ are the parametrization
parameters taking either zero or one in each of the two scenarios. We note
that when Bell-like states reduce to Bell states, those of EWL states reduce
to Werner states [58]. The parameterized initial state is as follows

$\displaystyle\displaystyle\varrho^{t=0}=\left(\begin{array}[]{cccc}\frac{1}{4}\left(4\alpha_{1}^{2}p\cos^{2}\left(\frac{\theta}{2}\right)-p+1\right)&0&0&\alpha_{1}^{2}p\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)\\\
0&\frac{1}{4}\left(4\alpha_{2}^{2}p\cos^{2}\left(\frac{\theta}{2}\right)-p+1\right)&\alpha_{2}^{2}p\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)&0\\\
0&\alpha_{2}^{2}p\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)&\frac{1}{4}\left(4\alpha_{2}^{2}p\sin^{2}\left(\frac{\theta}{2}\right)-p+1\right)&0\\\
\alpha_{1}^{2}p\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)&0&0&\frac{1}{4}\left(4\alpha_{1}^{2}p\sin^{2}\left(\frac{\theta}{2}\right)-p+1\right)\end{array}\right).$

To simplify calculations, all terms that correspond to the overlapping
$\alpha_{1}\alpha_{2}$ in $\varrho^{t=0}$ cancel in both situations and are
thus omitted. By solving Milburn’s equation, one finds that the time-dependent
density matrix encoding both two cases takes the form

$\varrho^{t}=\begin{pmatrix}\varrho_{11}&0&0&\varrho_{14}\\\
0&\varrho_{22}&\varrho_{23}&0\\\ 0&\varrho_{23}^{\ast}&\varrho_{33}&0\\\
\varrho_{14}^{*}&0&0&\varrho_{44}\end{pmatrix}.$ (19)

The time-evolution preserves the X-structure of the initial extended Werner-
like (EWL) state for any choice between the two initial states. The following
table gives the properties of the density matrix for each case.

$\ket{\Psi_{i}}$ $\Downarrow\Downarrow$ | $(\alpha_{1},\alpha_{2})$ | $\varrho_{ij}\neq 0$
---|---|---
$\ket{\Psi_{1}}$ $\Downarrow\Downarrow$ | $(1,0)$ | ($\varrho_{11},\varrho_{14},\varrho_{22},\varrho_{44})$, with $\varrho_{33}=\varrho_{22},\varrho_{41}=\varrho_{14}^{*}$
$\ket{\Psi_{2}}$ $\Downarrow\Downarrow$ | $(0,1)$ | ($\varrho_{11},\varrho_{22},\varrho_{23},\varrho_{33})$, with $\varrho_{11}=\varrho_{44},\varrho_{32}=\varrho_{23}^{*}$

For the combined initial state in (18), we get the following density matrix
entries

$\displaystyle\displaystyle\varrho_{11}$
$\displaystyle=\frac{\alpha_{1}}{\chi^{2}}\left(B^{2}(2p\cos(\theta)+p+1)+p\Gamma_{z}e^{-2\gamma
t\chi^{2}}\left(2\cos(\theta)\Gamma_{z}\cos(2t\chi)-\chi\sin(\theta)\sin(2t\chi)\right)+(p+1)\Gamma_{z}^{2}\right)-\frac{1}{4}\alpha_{2}(p-1),$
$\displaystyle\varrho_{14}$ $\displaystyle=\frac{\alpha_{1}pe^{-2\gamma
t\chi^{2}}}{2\chi^{2}}\left(\chi\sin(\theta)(\chi\cos(2t\chi)-2iB\sin(2t\chi))+2\cos(\theta)\Gamma_{z}\left(-2iBe^{2\gamma
t\chi^{2}}+2iB\cos(2t\chi)+\chi\sin(2t\chi)\right)\right),$
$\displaystyle\varrho_{22}$
$\displaystyle=\frac{\alpha_{2}}{4\omega}\Big{(}\left(2pe^{-2\gamma
t\omega^{2}}\Big{(}\left(2D_{z}\sin(\theta)\sin(2t\omega)+\omega\cos(\theta)\cos(2t\omega)\right)+(p+1)\omega\right)-\alpha_{1}(p-1)\omega\Big{)}\Big{)},$
$\displaystyle\varrho_{23}$ $\displaystyle=\frac{\alpha_{2}pe^{-2\gamma
t\omega^{2}}}{\omega^{2}}\left(J-iD_{z}\right)\left(-2iD_{z}\sin(\theta)\cos(2t\omega)+2J\sin(\theta)e^{2\gamma
t\omega^{2}}+i\omega\cos(\theta)\sin(2t\omega)\right),$
$\displaystyle\varrho_{33}$
$\displaystyle=\frac{\alpha_{2}}{4\omega}\Big{(}\left((p+1)\omega-2pe^{-2\gamma
t\omega^{2}}\left(2D_{z}\sin(\theta)\sin(2t\omega)+\omega\cos(\theta)\cos(2t\omega)\right)\right)-\alpha_{1}(p-1)\omega\Big{)},$
$\displaystyle\varrho_{44}$
$\displaystyle=\frac{\alpha_{1}}{\chi^{2}}\left(B^{2}(-2p\cos(\theta)+p+1)+p\Gamma_{z}e^{-2\gamma
t\chi^{2}}\left(\chi\sin(\theta)\sin(2t\chi)-2\cos(\theta)\Gamma_{z}\cos(2t\chi)\right)+(p+1)\Gamma_{z}^{2}\right)-\frac{1}{4}\alpha_{2}(p-1).$

The populations of the system combining both scenarios are given implicitly as

$\displaystyle\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\sqrt{4\left|\varrho_{23}\right|{}^{2}+\left(\varrho_{22}-\varrho_{33}\right){}^{2}}+\varrho_{22}+\varrho_{33}\right),$
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sqrt{4\left|\varrho_{23}\right|{}^{2}+\left(\varrho_{22}-\varrho_{33}\right){}^{2}}+\varrho_{22}+\varrho_{33}\right),$
$\displaystyle\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\sqrt{4\left|\varrho_{14}\right|{}^{2}+\left(\varrho_{11}-\varrho_{44}\right){}^{2}}+\varrho_{11}+\varrho_{44}\right),$
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sqrt{4\left|\varrho_{14}\right|{}^{2}+\left(\varrho_{11}-\varrho_{44}\right){}^{2}}+\varrho_{11}+\varrho_{44}\right).$

We use the following notation to locate the matrix density and populations for
each case:

$\varrho^{t}=\begin{cases}\varrho^{t}_{1},&\text{if}\
\alpha_{1}=1,\alpha_{2}=0\\\ \varrho^{t}_{2},&\text{if}\
\alpha_{1}=0,\alpha_{2}=1\end{cases}.$ (21)

By combining the two cases in (21), correlated coherence writes as

$\displaystyle\displaystyle\mathcal{C}_{cc}(\varrho^{t})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}pe^{-2\gamma
t\chi^{2}}}{\chi^{2}}\Bigg{[}\Big{(}2B\chi\sin(\theta)\sin(2t\chi)+4B\cos(\theta)$
(22) $\displaystyle\times$ $\displaystyle\Gamma_{z}\left(e^{2\gamma
t\chi^{2}}-\cos(2t\chi)\right)\Big{)}{}^{2}$ $\displaystyle+$
$\displaystyle\left(\chi^{2}\sin(\theta)\cos(2t\chi)+2\chi\cos(\theta)\Gamma_{z}\sin(2t\chi)\right){}^{2}\Bigg{]}^{1/2}$
$\displaystyle+$ $\displaystyle\frac{\alpha_{2}pe^{-2\gamma
t\omega^{2}}}{\omega}\Bigg{[}\Big{(}\omega\cos(\theta)\sin(2t\omega)-2D_{z}\sin(\theta)$
$\displaystyle\times$
$\displaystyle\cos(2t\omega)\Big{)}{}^{2}+4J^{2}\sin^{2}(\theta)e^{4\gamma
t\omega^{2}}\Bigg{]}^{1/2}.$

One can check that $\varrho_{14}$ and $\varrho_{23}$ are the sole non-zero
off-diagonal elements, storing all information about the system’s coherence
for each studied case. As well, one can easily verify that local coherence,
($\mathcal{C}^{L}(\varrho)=\mathcal{C}_{l_{1}}(\varrho_{A})+\mathcal{C}_{l_{1}}(\varrho_{B})=0$),
is always zero and thus correlated coherence is the system’s total coherence.
Similarly, quantum discord is obtained implicitly as

$\displaystyle\normalsize\mathcal{QD}_{1}$ $\displaystyle=$
$\displaystyle\log_{2}\left(\frac{(1-\Lambda)^{\Lambda}}{\Lambda^{\Lambda}-\Lambda^{\Lambda+1}}\prod_{i=1}^{4}\lambda_{i}^{\lambda_{i}}\left(\frac{\beta}{1-\beta}\right){}^{-\beta}\right)$
(23) $\displaystyle-$ $\displaystyle\log_{2}\left(-\beta+1\right),$
$\displaystyle\mathcal{QD}_{2}$ $\displaystyle=$
$\displaystyle\log_{2}\left(\prod_{i=1}^{4}\lambda_{i}^{\lambda_{i}}\times\prod_{j=1}^{4}\varrho_{jj}^{-\varrho_{jj}}\right),$
(24)

where
$\Lambda=\frac{1}{2}\left(\sqrt{4\left(\varrho_{14}+\varrho_{23}\right){}^{2}+\left(1-2\left(\varrho_{33}+\varrho_{44}\right)\right){}^{2}}+1\right)$,
$\beta=\varrho_{11}+\varrho_{33}$ and $\lambda_{i}$’s are the populations of
the system. Quantum discord is then computed by plugging Eq. (23) and Eq. (24)
in Eq. (9).

##  IV Results and analysis

This section presents the obtained findings regarding correlated coherence and
quantum discord. To sweeten our learning of the dynamical aspect of quantum
correlations and coherence in the two-qubit XXZ Heisenberg model, we
considered two kinds of EWL states. We then depict the behavior of coherence
and quantum correlations as a function of time and all the parameters in the
Hamiltonian as well as the level of purity of the initial state, the Bloch
angle, and the decoherence rate.

Figure 1: Time-evolution of coherence (correlated coherence) 1 and quantum
correlations (quantum discord) 1. Case 1 ($\varrho^{t}\equiv\varrho^{t}_{1}$)
and case 2 ($\varrho^{t}\equiv\varrho^{t}_{2}$), versus decoherence rates
$\gamma$.

In Fig. 1 we plot the correlated coherence and quantum discord as functions of
time $t$ for both initial EWL states $\varrho_{1}^{t=0}$ and
$\varrho_{2}^{t=0}$ for some decoherence rates $\gamma$. We notice that both
measures show oscillatory behavior due to intrinsic decoherence. At the
initial time $t=0$, the two quantifiers start from the same equal nonzero
value for both initial states $\varrho_{1}^{t=0}$ and $\varrho_{2}^{t=0}$,
that is
$\mathcal{C}_{cc}(\varrho_{1}^{t=0})=\mathcal{C}_{cc}(\varrho_{2}^{t=0})\approx
0.495$ and
$\mathcal{D}(\varrho_{1}^{t=0})=\mathcal{D}(\varrho_{2}^{t=0})\approx 0.262$.
Without intrinsic decoherence $\gamma=0$, both measures display non-damping
oscillations and fluctuates between their minimum and maximum values for an
infinite time scale. For $\gamma>0$, as time increases, correlated coherence
and quantum discord experience damped oscillations and get closer to a stable
value after an adequate period suggesting that the system has evolved to a
steady state.

The attained steady state is not dependent on $\gamma$ but on the initial
Werner-like state and the Hamiltonian parameters, namely $p$ and $\theta$.
Also, it is marked that a more considerable $\gamma$ drives the quick decline
of correlated coherence and discord; it causes the number and size of their
oscillations to drop at some point, causing the system’s state to quickly
transition into a steady state. The coherence and correlation deterioration
are due to the damping terms $e^{-2\gamma t\chi^{2}}$ (in the case of
$\varrho_{1}^{t}$) and $e^{-2\gamma t\omega^{2}}$ (in the case of
$\varrho_{2}^{t}$); these functions reduce the coherence and quantum
correlations amplitude after each incrementing $t$ or $\gamma$. Similarly, we
can mark identical routines for all the different cases.

In the asymptotic limit $t\to\infty$, quantum coherence and quantum
correlations reach their stable value. The higher the steady-state
correlation, the more the system will be appropriate for achieving quantum-
based tasks. In the same conditions of
$\Gamma_{z}=J=0.5,B=D_{z}=0.1,p=0.7,\theta=\pi/4$ and $\gamma>0$, we
numerically find the value of correlated coherence as
$\mathcal{C}_{cc}(\varrho_{1}^{t\to\infty})\approx 0.095$ and
$\mathcal{C}_{cc}(\varrho_{2}^{t\to\infty})\approx 0.485$. Similarly, for
quantum discord, we find $\mathcal{D}(\varrho_{1}^{t\to\infty})\approx 0.007$
and $\mathcal{D}(\varrho_{2}^{t\to\infty})\approx 0.212$.

Remarkably, we encountered that
$\mathcal{C}_{cc}(\varrho_{1}^{t\to\infty})<\mathcal{C}_{cc}(\varrho_{2}^{t\to\infty})$
and
$\mathcal{D}(\varrho_{1}^{t\to\infty})<\mathcal{D}(\varrho_{2}^{t\to\infty})$,
this means that the system sustains more superposition and correlations when
it is prepared in the initial state $\varrho_{2}^{t=0}$. On the other hand,
when the system is initially prepared in the Werner-like state
$\varrho_{1}^{t=0}$, its steady state is still a weakly correlated mixed state
that sustains low superposition.

Figure 2: Time-evolution of coherence (correlated coherence) 2 and quantum
correlations (quantum discord) 2. Case 1 ($\varrho^{t}\equiv\varrho^{t}_{1}$):
versus $B$. Case 2 ($\varrho^{t}\equiv\varrho^{t}_{2}$): versus $D_{z}$.

Fig.2 shows the dynamics of correlated coherence and quantum discord versus
time $t$ for various values of the DM interaction strength $D_{z}$ and the
homogeneous magnetic field $B$. It is worth noting, as it was shown
analytically above that the DM interaction does not affect the system when the
considered initial state is $\varrho_{1}^{t=0}$ as $D_{z}$ does not appear in
the expressions of the entries of the evolved density matrix
$\varrho_{1}^{t}$. Likewise, the effect of the magnetic field $B$ can not be
studied for $\varrho_{2}^{t}$ since the structure of the initial EWL state
$\varrho_{2}^{t=0}$ totally suppresses its contribution.

In Fig. 2, we notice that for growing intensities of the magnetic field $B$,
the oscillatory behavior of correlated coherence within $\varrho_{1}^{t}$
swiftly declines as the amplitudes are shrinking, and
$\mathcal{C}_{cc}(\varrho_{1}^{t})$ rapidly stabilizes on a nonzero frozen
state. Furthermore, the oscillations are slightly phase-shifted, and when
$B\neq 0$, the peaks presenting the maximal values reached over time are
broader for more diminutive intensities of $B$. We again observe that the
lower bound of the oscillations is unsteady as it can increase or decrease
over time. When the external magnetic field is non-existing ($B=0$),
correlations are damped over time due to intrinsic decoherence, and we regard
the collapse and regeneration phenomena. On the other hand, quantum discord
(Fig. 2) exhibits erratic oscillations; they are indeed damped over time but
present successive small and irregular spikes. Physically, since the Milburn
model is characterized by the associated magnetic field $B$, the irregular
oscillatory behavior of correlated coherence and discord in the case
$\varrho^{t}\equiv\varrho_{1}^{t}$, could be due to an ongoing information
flow between the magnetic field and the two-qubit XXZ spin system. For the
second density matrix $\varrho^{t}\equiv\varrho_{2}^{t}$, we see that the same
behavior is manifested by both quantum measures with the mere difference that
oscillations are regular and that correlated coherence records higher values
than quantum discord. At $t=0$, the quantum correlations and correlated
coherence amounts are nonzero. Their initial recorded quantities do not depend
on the DM interaction’s strength $D_{z}$ since they are fixed when the initial
state is given. In the absence of the DM interaction $D_{z}=0$, both measures
record the highest values. In contrast, for increasing $D_{z}$ values, the
oscillations’ frequency increases despite being quickly damped. The
oscillations get damped and saturate to a stable value for a finite time
value. This might be explained through the many-body localization-
delocalization viewpoint [59]. Meanwhile, for
$\varrho^{t}\equiv\varrho_{1}^{t}$ and $\gamma>0$, the steady states of
correlated coherence and discord are closely related to both $B$ and
$\Gamma_{z}$, it turns out that there is a competition between $B$ and
$\Gamma_{z}$. In more specific terms, the steady states increase/decrease by
increasing/decreasing either $B$ or $\Gamma_{z}$ until $B=\Gamma_{z}$, the
point at which the steady state reaches its maximum value. This result
indicates that $B$ and $\Gamma_{z}$ cancel each other so that the steady
states of correlated coherence and discord remain dependent only on $p$ and
$\theta$. We note that for all $\gamma>0$, the maximum steady state recorded
for $\theta=\pi/4$ and $p=0.7$ remains constant for all nonzero intensities
$B=\Gamma_{z}$, numerically,
$\mathcal{C}_{cc}(\varrho_{1}^{t\to\infty})\approx 0.2474$ and
$\mathcal{D}(\varrho_{1}^{t\to\infty})\approx 0.0543$.

For the second configuration $\varrho^{t}\equiv\varrho_{2}^{t}$, we see that
as the DM interaction strength grows, the value of the steady state decreases.
In contrast, as the spin-spin interaction coupling becomes strong, the steady
state of correlated coherence and quantum discord improves. For $\gamma>0$ and
$J=D_{z}\neq 0$, the steady state can be modified by changing $p$ and
$\theta$. These results highlight the impact of the initial state on the whole
evolution of the system and suggest that its quantum resources could be
tweaked and enhanced through the modification of certain parameters.

Figure 3: Time-evolution of coherence 3 and quantum correlations 3. Case 1
($\varrho^{t}\equiv\varrho^{t}_{1}$): versus $\Gamma_{z}$. Case 2
($\varrho^{t}\equiv\varrho^{t}_{2}$): versus $J$. In Fig. 3, we depict the
influence of the strength of the KSEA interaction ($\Gamma_{z}$) and the spin-
spin coupling $J$ on correlated coherence and quantum discord for both
considered initial states. We specify again that the effect of the first
parameter is examined only in the first case where
$\varrho^{t}\equiv\varrho_{1}^{t}$, while the influence of the anisotropy
parameter $J$ can solely be examined when $\varrho^{t}\equiv\varrho_{2}^{t}$.

The first salient observation regarding the evolution of correlated coherence
and quantum discord in the first considered case
$\varrho^{t}\equiv\varrho_{1}^{t}$, is that both appear to be stable for
$\Gamma_{z}=0$ for short periods. However, they eventually decrease for
sufficiently long periods. In the absence of the KSEA interaction, the
expression of correlated coherence reduces to
$\mathcal{C}_{cc}(\varrho_{1}^{t})=pe^{-2\gamma t\times 4B^{2}}\sin{\theta}$,
and we can numerically perceive its time evolution. Correlated coherence
exhibits the same oscillatory pattern in the presence of the KSEA interaction
($\Gamma_{z}\neq 0$). However, as $\Gamma_{z}$ decreases, the lower bound of
oscillations is higher, and the minimum values shift to the right. As KSEA
interaction strengthens, the frequency of oscillations decreases, and
coherence becomes less erratic. Quantum discord displays identical behavior as
correlated coherence with lower quantities. In distinction, we spot in the
dynamical behavior of QD the resurgence of the collapse and revival phenomena
and the previously mentioned pattern of minor successive increases and
decreases when oscillations reach their minimum values.

In the second case $\varrho^{t}\equiv\varrho_{2}^{t}$, we observe that for
increasing interaction coupling constant $J$, the frequency of oscillations of
both quantifiers increases, but the amplitudes shrink. For $J=0$, the lower
bounds of both measures nearly drop to zero at the same instants $t$, at
$t\approx 2$ and $t\approx 10$, and we see that correlated coherence displays
sharp peaks at these instants. Once again, it is apparent that the initial
amounts of quantum correlations and correlated coherence are not related to
the parameters that do not figure in the initial state.

Figure 4: Time-evolution of coherence 4 and quantum correlations 4. Case 1
($\varrho^{t}\equiv\varrho^{t}_{1}$) and case 2
($\varrho^{t}\equiv\varrho^{t}_{2}$), versus the Bloch angle $\theta$.

In Fig. 4, we visualize the effect of the angle $\theta$ on correlated
coherence and quantum discord dynamics. Unlike the previous figures where the
initial values, at $t=0$, of $\mathcal{C}_{cc}(\varrho^{t})$ and
${\mathcal{D}}(\varrho^{t})$ are not dependent on the strengths of the
parameters $\gamma$ (Fig. 1), $D_{z}$ and $B$ (Fig. 2), $J$ and $\Gamma_{z}$
(Fig. 3), we notice in Fig. 4 that the initial amounts of quantum correlations
and correlated coherence are related to the value of the angle $\theta$.

Clearly, as the value of $\theta$ rises, not only the initial recorded values
are higher, but also those recorded during the time evolution. The angle
$\theta$ does not affect the frequency of oscillations, but it affects the
quantities of correlated coherence and nonclassical correlations within the
system. Moreover, oscillations are in phase when
$\varrho^{t}\equiv\varrho_{2}^{t}$, unlike the first case
$\varrho^{t}\equiv\varrho_{1}^{t}$ where oscillations are slightly phase-
shifted to the right for decreasing $\theta$ values.

For $\theta=\frac{\pi}{2}$, which is the angle value for which the Bell-like
states (16-17) reduce to the maximally entangled Bell states and the EWL
becomes a Werner state, $\mathcal{C}_{cc}(\varrho^{t})$ and
${\mathcal{D}}(\varrho^{t})$, present the higher quantities but they are
roughly invariant as they present timid fluctuations, when
$\varrho^{t}\equiv\varrho_{2}^{t}$. Another worth noting remark is the
collapse and revival phenomena exhibited in the dynamical behavior of quantum
discord when $\varrho^{t}\equiv\varrho_{1}^{t}$, in addition to the pattern of
the small spikes in the lower bound of QD.

Figure 5: Time-evolution of coherence 5 and quantum correlations 5. Case 1
($\varrho^{t}\equiv\varrho^{t}_{1}$) and case 2
($\varrho^{t}\equiv\varrho^{t}_{2}$), versus the level of purity $p$. In Fig.
5, we show the impact of the level of purity $p$ in the initial state on
correlated coherence and quantum discord for both considered states
$\varrho_{1}^{t}$ and $\varrho_{2}^{t}$. When $p=0$, the system presents zero
coherence and zero discord since the initial state (18) reduces to
$\varrho^{t=0}=\varrho_{1}^{t=0}=\varrho_{2}^{t=0}=\frac{1}{4}I_{4}$ which is
an incoherent separable state. As it is shown in Figs. 5-5, the evolved
density matrix corresponding to $p=0$ does not contain discord-type
correlations
(${\mathcal{D}}(\varrho_{1}^{t})={\mathcal{D}}(\varrho_{2}^{t})=0$) nor
correlated coherence
($\mathcal{C}_{cc}(\varrho_{1}^{t})=\mathcal{C}_{cc}(\varrho_{2}^{t})=0$).
Raising the level of purity, $p$, improves quantum correlations and coherence
within the system, whereas it does not affect the frequency of their
oscillations. Both quantifiers, given both cases
$\varrho^{t=0}=\varrho_{1}^{t=0}$ and $\varrho^{t=0}=\varrho_{2}^{t=0}$,
display damped oscillations. Nevertheless, we recognize in the first case the
collapse and revival phenomenon, which is not manifested when
$\varrho^{t=0}=\varrho_{2}^{t=0}$. As mentioned earlier, the initial amounts
of quantum discord and correlated coherence in the system are those of the
initial state given, so it is only natural that they are exclusively dependent
on the purity level $p$ and the angle $\theta$. In all figures, it can be seen
that the values of quantum coherence and correlations, for both considered
cases, are the same before the system starts to evolve (at $t=0$). This is
because both Bell-like states have the same superposition and the same amount
of entanglement. Furthermore, correlated coherence is always more significant
than the nonclassical correlations measured by QD for both initial states.

##  V Conclusion

By utilizing correlated coherence and quantum discord, the dynamics of
correlated coherence and quantum discord versus intrinsic decoherence rate are
investigated in the two-spin XXZ model under the interplay of two different
types of interactions. We found that correlated coherence and quantum discord
dynamics depend on all system parameters except the anisotropy coupling
$J_{z}$, which has no effects. The two measures go up and down in a pattern
that gets smaller and quieter as time passes. In particular, we find that the
initial state affects the system’s dynamics significantly because it allows it
to avoid specific interactions. We also found that the maximum value of the
steady state gets smaller by increasing $B,\Gamma_{z}$, and $D_{z}$, but
increasing $J$ makes the steady state bigger. We show that it is possible to
have more robust quantum resources by engineering an appropriate initial state
for the system. Moreover, we found that the exchange of information between
the system and the magnetic field causes quantum discord to have irregular
oscillations for the case $\rho^{t}\equiv\rho_{1}^{t}$. We note that the only
part contributing to the correlated coherence dynamic is total coherence
because local coherence is always zero, meaning that the reduced density
matrices of the subsystems are incoherent states. The results of this work
give a more in-depth understanding of the influence of specific interactions
and initial state and their consequences on the coherence and discord dynamics
of the considered quantum system.

## Acknowledgement

Z. D expresses special thanks to the Abdus Salam International Centre for
Theoretical Physics (ICTP) for the hospitality and for providing access to
their research facilities during his visit, which helped accomplish some of
this work. M. O acknowledges the financial support received from the National
Center for Scientific and Technical Research (CNRST) under the Program of
Excellence Grants for Research.

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