# Quantum correlations under decoherence

S. V. Mousavi<EMAIL_ADDRESS>Department of Physics, University of Qom,
Ghadir Blvd., Qom 371614-6611, Iran

###### Abstract

Taking a system of two coupled qubits described by a X-shaped state and
interacting through an anisotropic Heisenberg XY interaction, we examine the
evolution of quantum entanglement and a few quantum correlations beyond
entanglement, local quantum uncertainty and measurement-induced nonlocality,
under the environmental decoherence both for zero and finite temperatures. The
relation between concurrence and log negativity as two well-known quantifiers
of entanglement is argued. It will be proven that measurement-induced
nonlocality equals correlated coherence. The interaction of qubits with the
environment causes quantum entanglement to suddenly die for independent
qubits, but other correlations do not experience this phenomenon. The time of
entanglement sudden death is calculated analytically for zero temperature,
while numerically for finite temperatures. Contrary to its usual destructive
role, the environment plays a constructive role in some situations, inducing
quantum correlations even when the initial quantum correlations are zero. The
steady state quantum correlations, being independent of the initial state, are
found to remain all non-zero for low, finite temperatures. It is found that
quantum correlations beyond entanglement are more robust against temperature
than entanglement. The zero-temperature steady state exhibits less local
quantum uncertainty than the other correlations.

Keywords: Entanglement, Local quantum uncertainty, Uncertainty induced
nonlocality, Correlated Coherence, Environmental Decoherence

##  I Introduction

Quantum states are fundamentally different from classical states. Any
measurements made on one part of a bipartite state will make the results
uncertain. The integral nature of quantum states is the reason for this
randomness, not the measuring device. Only states which can be described by a
classical probability distribution, are invariant under such local
measurements. Perhaps the most striking feature of quantum mechanics is the
quantum entanglement (QE). It is at the heart of quantum information
processing, being a resource for doing tasks like quantum computing, quantum
teleportation and quantum key distribution NiCh-book-2000 . Nonetheless, in
the early age of quantum information science, some researchers questioned
entanglement as the source of power for quantum computation MiLaSaKn-PRA-2002
. QE is not the only correlation which exists between different parts of a
multi-party quantum system. Although pure states can be either uncorrelated or
just entangled AdBrCi-JPA-2016 , it has been proven that certain quantum
separable mixed states exhibit quantum correlations, the so-called quantum
discord OlZu-PRL-2001 ; HeVe-JPA-2001 . In addition to the significant role of
quantum correlations in efficient quantum communication and computational
tasks, they are also important in figuring out cooperative phenomena in multi-
party quantum systems. Among various beyond entanglement quantum correlations
have already been identified, one can mention local quantum uncertainty (LQU)
GiTuAd-PRL-2013 , measurement-induced nonlocality (MIN) and correlated quantum
coherence LuFu-PRL-2011 ; HuFa-NJP-2015 . While the former is introduced as
the minimum skew information WiYa-PNAS-1963  achievable on a single local
measurement, the latter is interpreted as the maximum global effect caused by
locally invariant measurements.

Beside quantum correlations, quantum coherence BaPl-PRL-2014 , being
cornerstone of quantum computation and quantum communication theory, has a
fundamental and practical significance for developing the quantum mechanics
Huetal-PR-2018 . Contrary to quantum correlations, being defined for systems
with at least two parties, quantum coherence measures can also be defined for
single systems. However, both arise from quantum superposition principle, thus
being interrelated. In this regard, the link between quantum coherence and
entanglement have already been identified StSiDhBeAd-PRL-2015 ; QiGaYa-
JPA-2017 . Furthermore, the concept of “correlated coherence”(CC) Taetal-
PRA-2016  has been introduced and its connection to quantum discord and
entanglement has been argued. The idea is splitting the coherence contained in
the global state into two portions, those stored locally in the marginal
(reduced) states and that stored non-locally only in the correlated state, and
then subtracting local portions from the correlated one. For X states, we’ll
find that MIN equals CC.

Since a quantum system cannot be completely isolated from its environment, its
influence on the system’s dynamics must be taken into account. Two approaches
have been developed to study dynamics of open quantum systems: the quantum
operations formalism and the approach of master equations which each one has
its own place NiCh-book-2000 . The latter strategy employs a differential
equation, which in addition to the conventional unitary term for an isolated
system, includes a non-unitary term responsible for interaction with the
surrounding environment. In the decoherence process, one of the most striking
results is the complete loss of entanglement in a finite time, the so-called
“entanglement sudden death” (ESD) implying complete termination of
entanglement after a finite time interval without a long-lasting smooth
decrease YuEb-PRL-2004 ; YuEb-PRL-2006 ; EbYu-Sci-2007 . Mathematically, ESD
is based on the cutoff in the definition of concurrence, but it is difficult
to find a physical interpretation. However, its connection to energy transfer
between the system and its surroundings has been discussed CuLiYi-PLA-2007 .
On contrary, the environment-induced FiTa-PRA-2006  or decoherence-induced
entanglement DaLi-JPB-2007  has been worked out. Here, environment plays a
constructive role, in contrast to its customary destructive role, inducing QE.
Furthermore, entanglement re-birth or revival has also been reported.

In this work, we examine a system of two localized coupled qubits that that
are in interaction with an external magnetic field. The interaction with the
environment is introduced by a non-unitary term in the Lindblad equation. We
are going to solve this equation analytically first for a number of initial
states in the form X and then for a general X-shaped state; and study the
evolution of different quantum correlations for different emission rates as
well as different interaction intensities. It is imperative to determine
whether quantum correlations beyond entanglement experience sudden death or
not. Furthermore, it is important to conduct an investigation into the
phenomenon of correlation induced by the environment. After all this, it will
be time to consider the effect of temperature on quantum correlations. How
does this parameter affect ESD?

The paper is organized as follows. In Section II we concisely address
quantifiers of the mentioned quantum correlations and also the quantum
coherence. In Section III we introduce the model and solve the corresponding
Lindblad equation for some X-shaped states. Section IV includes steady state
solution of the Lindblad equation in a finite temperature for a general X
state. Section V is devoted to computations, both analytical and numerical,
and discussion of the results. Finally, in Section VI some concluding results
are presented.

##  II Some Quantum Correlations; entanglement and beyond

In this section we briefly review QE, LQU and MIN. At the end, we consider CC
and relate it to MIN. Different measures have been introduced for
quantification of QE. One of the most famous measures of entanglement is the
concurrence HiWo-PRL-1997 ; Wo-PRL-1998  which for a two-qubit state $\rho$ is
defined as

$\displaystyle\mathcal{C}(\rho)$ $\displaystyle=$
$\displaystyle\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$
(1)

$\lambda_{i}$s being the eigenvalues of the Hermitian matrix
$\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ in non-increasing order. Here,
$\tilde{\rho}$ is the the spin-flipped Hermitian density matrix given by
$(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$;
$\sigma_{y}$ being the second Pauli matrix and $\rho^{*}$ the complex
conjugate of $\rho$ which due to the Hermiticity of $\rho$ equals its
transpose, $\rho^{*}=\rho^{T}$. Two matrices
$S=\sqrt{\rho}\tilde{\rho}\sqrt{\rho}$ and $R=\rho\tilde{\rho}$ are connected
through the invertible matrix $\sqrt{\rho}$ i.e.,
$R=(\sqrt{\rho})S(\sqrt{\rho})^{-1}$. Thus, they are similar, having the same
spectrum. In this way, one sees $\lambda_{i}$s are the square roots of the
eigenvalues of $S$ and thus the non-Hermitian matrix $R$. Note that since
$\sigma_{y}\otimes\sigma_{y}$ is its own inverse,
$(\sigma_{y}\otimes\sigma_{y})^{-1}=\sigma_{y}\otimes\sigma_{y}$ then
$\tilde{\rho}$ is similar to $\rho^{*}$ which its eigenvalues are just those
of $\rho$. Therefore, both $\rho^{*}$ and its similar matrix $\tilde{\rho}$
are non-negative. Now, since both $\rho$ and $\tilde{\rho}$ are non-negative,
eigenvalues of the their product $R=\rho\tilde{\rho}$ are non-negative Bh-
book-1997 111See page 256.. Concurrence, defined in this way, becomes zero for
separated states but takes the value one for maximally entangled states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ systems Dhetal-book-2017 . It has been
argued that the concurrence is related to the entanglement of formation which
itself is an entropic entanglement monotone HiWo-PRL-1997 ; Wo-PRL-1998 . In
the basis $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, a X-state has
the form

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho_{32}&\rho_{33}&0\\\
\rho_{41}&0&0&\rho_{44}\end{pmatrix}$ (2)

with constraints

$\displaystyle\qquad\sum_{i=1}^{4}\rho_{ii}=1,\quad\rho_{22}\rho_{33}\geq|\rho_{23}|^{2},\quad\rho_{11}\rho_{44}\geq|\rho_{14}|^{2},$
(3)

coming from properties of the density operator i.e., unit trace, Hermiticity
and positivity. The concurrence (1) for the X-state (2) is then Waetal-
JPB-2006

$\displaystyle\mathcal{C}$ $\displaystyle=$ $\displaystyle
2\max\bigg{\\{}0,|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}},|\rho_{32}|-\sqrt{\rho_{11}\rho_{44}}\bigg{\\}}.$
(4)

As this equation show, if and only if either
$\rho_{22}\rho_{33}<|\rho_{14}|^{2}$ or $\rho_{11}\rho_{44}<|\rho_{23}|^{2}$
then the state (2) is entangled. Both conditions cannot be fulfilled
simultaneously AlRaAl-PRA-2010 .

As another natural and widely used entanglement measure, negativity for a
bipartite state $\rho$ is defined through the trace norm of the partially
transposed density matrix ViWe-PRA-2002 ,

$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle\frac{||\rho^{T_{B}}||_{1}-1}{2},$ (5)

where $||X||_{1}=\text{tr}(\sqrt{X^{\dagger}X})$ is the trace norm and
$\rho^{T_{B}}$ denotes the partial transpose of the density matrix with
respect to the party $B$,

$\displaystyle\rho^{T_{B}}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\sum_{k,l}\rho_{ij,kl}|i\rangle\langle
j|\otimes|l\rangle\langle k|,$ (6)

where $\\{|i\rangle\\}$ and $\\{|k\rangle\\}$ are given local orthonormal
basis for subsystems $A$ and $B$ respectively. It is noteable that the
spectrum of the partial transposition of the density matrix is independent
both of the choice of the local basis and also whether the partial
transposition is performed over the party $A$ or $B$ PlVi-bookchap-2014 . One
can easily check that $\rho^{T_{B}}$ is Hermitian

$\displaystyle\langle i,k|(\rho^{T_{B}})^{\dagger}|j,l\rangle=\langle
j,l|\rho^{T_{B}}|i,k\rangle^{*}=\langle j,k|\rho|i,l\rangle^{*}=\langle
i,l|\rho|j,k\rangle=\langle i,k|\rho^{T_{B}}|j,l\rangle,$

where in the third equality we have used Hermiticity of the density operator
$\rho$. Now from Hermiticity of $\rho^{T_{B}}$ and the fact that trace is
independent of basis, one has that

$\displaystyle||\rho^{T_{B}}||_{1}$ $\displaystyle=$
$\displaystyle\text{tr}\bigg{(}\sqrt{(\rho^{T_{B}})^{2}}\bigg{)}=\sum_{i}\sqrt{\lambda_{i}^{2}}=\sum_{i}|\lambda_{i}|,$
(7)

$\lambda_{i}$ being eigenvalues of $\rho^{T_{B}}$. From Eqs. (5) and (7), and
the fact that $\sum_{i}\lambda_{i}=\text{tr}(\rho^{T_{B}})=\text{tr}(\rho)=1$
one obtains

$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{|\lambda_{i}|-\lambda_{i}}{2}=-\sum_{\lambda_{i}<0}\lambda_{i}=\begin{cases}0&\rho^{T_{B}}\text{~{}is
positive definite}\\\ -\lambda_{\min}&\text{only one negative
eigenvalue}\end{cases}$ (8) $\displaystyle=$
$\displaystyle\max\\{0,-\lambda_{\min}\\},$

where in the third equality we have used the fact that for entangled two qubit
states the partiality transposed state $\rho^{T_{B}}$ has only one negative
eigenvalue MiGr-JPB-2004  which has been denoted by $\lambda_{\min}$ in Eq.
(8) as it is the minimum eigenvalue. Some authors have defined negativity of
two-qubit states as

$\displaystyle\mathcal{N}^{\prime}$ $\displaystyle=$ $\displaystyle
2\mathcal{N}=\max\\{0,-2\lambda_{\min}\\}$ (9)

which for maximally entangled states where the spectrum of $\lambda$ is
$\\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\\}$, takes the largest
value 1 ZyHoSaLe-PRA-1998 ; VeAuDeMo-JPA-2001 . For a mixed state of two
qubits, the subject of our study, it has been proved that the negativity
$\mathcal{N}^{\prime}$ of a state can never exceed its concurrence and is
always greater than
$\sqrt{(1-\mathcal{C})^{2}+\mathcal{C}^{2}}-(1-\mathcal{C})$; $\mathcal{C}$
being the concurrence of the state VeAuDeMo-JPA-2001 . Now, from Eqs. (8) and
(9) we have that $\mathcal{N}=\mathcal{N}^{\prime}/2$ and thus from the
results of VeAuDeMo-JPA-2001  we have that

$\displaystyle\frac{\sqrt{[1-\mathcal{C}]^{2}+\mathcal{C}^{2}(t)}-(1-\mathcal{C})}{2}\leq\mathcal{N}\leq\frac{\mathcal{C}}{2}.$
(10)

The logarithmic negativity (log-negativity) as an appropriate quantifier of QE
in a bipartite quantum system is defined as ViWe-PRA-2002 ; Pl-PRL-2005

$\displaystyle\mathcal{L_{N}}$ $\displaystyle=$
$\displaystyle\log_{2}(||\rho^{T_{B}}||_{1})=\log_{2}(2\mathcal{N}+1),$ (11)

where in the second equality we have used (5). From Eqs. (10) and (11), and
since function $\log(x)$ is an increasing function of its argument, we get

$\displaystyle\log_{2}\bigg{(}\sqrt{[1-\mathcal{C}]^{2}+\mathcal{C}^{2}}+\mathcal{C}\bigg{)}\leq\mathcal{L_{N}}\leq\log_{2}[\mathcal{C}+1].$
(12)

Now we consider quantum correlations beyond QE. Based of the notion of skew
information, authors of GiTuAd-PRL-2013  have proposed a discord-like scheme
for characterizing non-classical correlation via LQU. In this way, LQU has
been introduced as the minimum skew information achievable on a single local
measurement on the subsystem $A$ and it has been obtained as

$\displaystyle\mathcal{U}$ $\displaystyle=$ $\displaystyle
1-\lambda_{\max}\\{W\\},$ (13)

where $\lambda_{\max}\\{W\\}$ shows the maximum eigenvalue of the matrix

$\displaystyle W_{ij}$ $\displaystyle=$
$\displaystyle\text{tr}(\sqrt{\rho}~{}\sigma^{(A)}_{i}\otimes\mathds{1}^{(B)}~{}\sqrt{\rho(t)}~{}\sigma^{(A)}_{j}\otimes\mathds{1}^{(B)}).$
(14)

Due to the cyclic property of the trace operation one can see that $W$ is a
symmetric matrix i.e., $W_{ji}=W_{ij}$. Structure of this matrix for the X
state (2) is

$\displaystyle W$ $\displaystyle=$
$\displaystyle\begin{pmatrix}W_{11}&W_{12}&0\\\ W_{12}&W_{22}&0\\\
0&0&W_{33}\end{pmatrix},$ (15)

and thus

$\displaystyle\mathcal{U}$ $\displaystyle=$ $\displaystyle
1-\max\bigg{\\{}\frac{W_{11}+W_{22}\pm\sqrt{[W_{11}-W_{22}]^{2}+4W_{12}^{2}}}{2},W_{33}\bigg{\\}}.$
(16)

MIN as a measure of nonlocality, a unique property of quantum mechanics, is a
description of the maximum global effect caused by locally invariant
measurements LuFu-PRL-2011 . Originally, MIN was defined in terms of the
Hilbert-Schmidt norm $||X||_{2}=\sqrt{\text{tr}(X^{\dagger}X)}$ but, then, it
was argued that this definition suffers the ancilla and noncontractivity
problems. Thus, it was defined based on the trace norm HuFa-NJP-2015 , Wigner-
Yanase skew information Lietal-EPL-2016  and fidelity MuSa-PLA-2017 . Trace-
norm based MIN avoids problems was mentioned above and is defined via

$\displaystyle N_{1}(\rho)$ $\displaystyle=$
$\displaystyle\max_{\Pi^{(A)}}||\rho-\Pi^{(A)}(\rho)||_{1},\qquad||X||_{1}=\text{tr}\sqrt{X^{\dagger}X}.$
(17)

Here, the maximum is taken over the complete set of local projective
measurements $\Pi^{(A)}=\\{\Pi_{i}^{(A)}\\}$, which do not disturb the reduced
state $\rho^{(A)}=\text{tr}_{B}(\rho)$ i.e.,
$\sum_{i}\Pi_{i}^{(A)}\rho^{(A)}\Pi_{i}^{(A)}=\rho^{(A)}$. Trace norm is
invariant under local unitary transformation CiTuGi-NJP-2014 . By using proper
local unitary transformations, one transforms the X state (2) into

$\displaystyle\rho(t)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\rho_{11}&0&0&|\rho_{14}|\\\
0&\rho_{22}&|\rho_{23}|&0\\\ 0&|\rho_{32}|&\rho_{33}&0\\\
|\rho_{41}|&0&0&\rho_{44}\end{pmatrix}$ (18)

which is still X-shaped but now with real elements EsKhMaDa-OQE-2022 . For
such type of X-shaped states, it has been shown that the trace-norm MIN (17)
recasts QiLi-QIP-2016

$\displaystyle N_{1}$ $\displaystyle=$
$\displaystyle\begin{cases}2(|\rho_{14}|+|\rho_{23}|)&x\neq 0,\\\ \\\
\max\\{|u_{1}|,|u_{2}|,|u_{3}|\\}&x=0,\end{cases}$ (19)

where

$\displaystyle x=\rho_{11}+\rho_{22}-(\rho_{33}+\rho_{44}),$ (20)
$\displaystyle u_{1}=2(|\rho_{14}|+|\rho_{23}|),$ (21) $\displaystyle
u_{2}=2(-|\rho_{14}|+|\rho_{23}|),$ (22) $\displaystyle
u_{3}=\rho_{11}-\rho_{22}-\rho_{33}+\rho_{44}.$ (23)

Another measure of quantum correlations is correlated coherence Taetal-
PRA-2016 ; Huetal-PR-2018  which is defied as

$\displaystyle C_{cc}$ $\displaystyle=$ $\displaystyle
C(\rho)-C(\rho^{(A)})-C(\rho^{(B)}),$ (24)

where $\rho^{(A)}=\text{tr}_{B}(\rho)$ and $\rho^{(B)}=\text{tr}_{A}(\rho)$
are reduced states describing respectively subsystems $A$ and $B$; and
$C(\rho)$ is the “quantum coherence” BaPl-PRL-2014  of the state $\rho$
quantified by e.g., $\ell_{1}-$norm which is the sum of the magnitudes of all
off-diagonal elements. The key idea in this definition is to split the total
coherence contained in the state $\rho$ into the local ones, stored locally in
the subsystems, and the nonlocal one stored only in the correlated states.
Connection of this notion to quantum discord and entanglement has been argued
Taetal-PRA-2016 . For the X state (2), the reduced states are diagonal,

$\displaystyle\rho^{(A)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\rho_{11}+\rho_{22}&0\\\
0&\rho_{33}+\rho_{44}\end{pmatrix}$ (25) $\displaystyle\rho^{(B)}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}\rho_{11}+\rho_{33}&0\\\
0&\rho_{22}+\rho_{44}\end{pmatrix}$ (26)

having no coherence. Therefore, for a X-shaped state CC is just the quantum
coherence of the global state $\rho$;

$\displaystyle C_{cc}$ $\displaystyle=$ $\displaystyle\sum_{i}\sum_{j\neq
i}|\rho_{ij}|=2\bigg{(}|\rho_{14}|+|\rho_{23}|\bigg{)}.$ (27)

Comparison of (19) with (27) reveals that the trace-norm MIN for $x\neq 0$ is
just the CC of the state.

##  III The model, the master equation and its solution

Our model system is a Heisenberg chain with only two localised qubits having
the spin-spin interaction
$J_{x}S_{x}^{(1)}S_{x}^{(2)}+J_{y}S_{y}^{(1)}S_{y}^{(2)}+J_{z}S_{z}^{(1)}S_{z}^{(2)}$
and in interaction with an applied magnetic field along $z$-direction. The
Hamiltonian for the $XY(J_{z}=0)$ model then recasts

$\displaystyle H$ $\displaystyle=$ $\displaystyle J(S_{+}^{(1)}\otimes
S_{-}^{(2)}+S_{-}^{(1)}\otimes S_{+}^{(2)})+\Delta(S_{+}^{(1)}\otimes
S_{+}^{(2)}+S_{-}^{(1)}\otimes S_{-}^{(2)})$ (28) $\displaystyle+$
$\displaystyle\omega(S_{z}^{(1)}\otimes\mathds{1}^{(2)}+\mathds{1}^{(1)}\otimes
S_{z}^{(2)}),$

where $S_{\pm}^{(i)}=S_{x}^{(i)}\pm iS_{y}^{(i)}$ are the spin raising
(lowering) operators for the $i^{th}$ qubit and $\omega$ is the intensity of
the applied magnetic field. Coupling between qubits have been denoted as
$J=(J_{x}+J_{y})/4$ and $\Delta=(J_{x}-J_{y})/4$ where the latter quantifies
the anisotropicity of the interaction. It is noteworthy that systems with two
coupled qubits have been utilized extensively in various contexts such as
cavity QED GyMeAw-PRB-2006  and nuclear spin systems with the hope of possible
applications in quantum information schemes and quantum computing NiCh-
book-2000 ; MiLaSaKn-PRA-2002 .

At zero temperature, $T=0$, the state of the system evolves under the
Lindbladian master equation Waetal-JPB-2006

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle-i[H,\rho(t)]+\gamma~{}\mathcal{D}[S_{-}^{(1)}\otimes\mathds{1}^{(2)}]\rho(t)+\gamma~{}\mathcal{D}[\mathds{1}^{(1)}\otimes
S_{-}^{(2)}]\rho(t),$ (29)

where $\mathcal{D}[A]B=ABA^{\dagger}-\\{A^{\dagger}A,B\\}/2$ is the Lindblad
super-operator and $\\{X,Y\\}$ denotes the anti-commutator of operators $X$
and $Y$. In our open quantum system this Lindblad equation describes the
population relaxation of the upper state of each qubit with the relaxation
rate $\gamma$ which for simplicity it has been taken equal for both qubits. In
other words, interaction of qubits with the environment is assumed to be the
same. As discussed in Waetal-JPB-2006 , this assumption is reasonable when the
rotating wave approximation remains valid. The Lindblad equation (29)
maintains the shape X of the state i.e., if the initial state is a X-shaped
i.e., of the form (2) then the evolved state $\rho(t)$ is too.

In the following, we first examine the evolution of two special X-shaped
states, a mixture and the Werner state. Then, we present the steady state
solution of the Lindblad equation (29) for the most general X-shaped state
(2). It is seen the evolved Werner state and the general steady state are both
independent of the interaction intensity $J$. Furthermore, the steady solution
is independent of the initial state.

Consider the initial state to be an equal mixture of states $|01\rangle$ and
$\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ i.e.,

$\displaystyle\rho_{\text{mixture}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}|01\rangle\langle
01|+\frac{1}{4}(|00\rangle+|11\rangle)(\langle 00|+\langle 11|)=$ (30)
$\displaystyle=$ $\displaystyle\begin{pmatrix}\frac{1}{4}&0&0&\frac{1}{4}\\\
0&\frac{1}{2}&0&0\\\ 0&0&0&0\\\ \frac{1}{4}&0&0&\frac{1}{4}\end{pmatrix}$

Then, the solution of the Lindblad equation (29) is given by

$\displaystyle\rho_{11}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\Omega(\gamma^{2}+4\Omega^{2})}\bigg{\\{}4\Omega\Delta^{2}+e^{-2\gamma
t}\Omega\bigg{[}\gamma^{2}+4\omega(\Delta+\omega)\bigg{]}$ (31)
$\displaystyle+2\Delta e^{-\gamma t}\bigg{[}\gamma(\omega-2\Delta)\sin(2\Omega
t)-2\omega\Omega\cos(2\Omega t)\bigg{]}\bigg{\\}}$ $\displaystyle\rho_{14}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{4\Omega^{3}(\gamma^{2}+4\Omega^{2})}\bigg{\\{}-8\Delta\omega\Omega^{3}+e^{-\gamma
t}\omega\Omega\bigg{(}\gamma^{2}(\omega-2\Delta)+4\omega\Omega^{2}\bigg{)}\cos(2\Omega
t)$ (32) $\displaystyle+\Delta\Omega e^{-\gamma
t}\bigg{(}(\gamma^{2}+4\Omega^{2})(\Delta+2\omega)+4\gamma\omega\Omega\sin(2\Omega
t)\bigg{)}\bigg{\\}}$ $\displaystyle-i\frac{4\gamma\Delta\Omega+e^{-\gamma
t}\bigg{[}\bigg{(}\gamma^{2}(\omega-2\Delta)+4\omega\Omega^{2}\bigg{)}\sin(2\Omega
t)-4\gamma\Delta\Omega\cos(2\Omega
t)\bigg{]}}{4\Omega(\gamma^{2}+4\Omega^{2})}$ $\displaystyle\rho_{22}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{4\Omega^{3}(\gamma^{2}+4\Omega^{2})}\bigg{\\{}4\Delta^{2}\Omega^{3}+e^{-\gamma
t}\bigg{[}\Omega^{3}(\gamma^{2}+4\Omega^{2})\cos(2Jt)$ (33)
$\displaystyle+\gamma\Delta\bigg{(}\gamma\Omega(2\Delta-\omega)\cos(2\Omega
t)-2\omega\Omega^{2}\sin(2\Omega t)\bigg{)}\bigg{]}$ $\displaystyle+e^{-\gamma
t}\Omega\bigg{[}-\Omega^{2}(\gamma^{2}+4\omega(\Delta+\omega))+\omega(\gamma^{2}+4\Omega^{2})(\Delta+2\omega)e^{\gamma
t}\bigg{]}\bigg{\\}}$ $\displaystyle\rho_{23}(t)$ $\displaystyle=$
$\displaystyle\rho_{32}^{*}(t)=\frac{i}{4}e^{-\gamma t}~{}\sin(2Jt)$ (34)
$\displaystyle\rho_{33}(t)$ $\displaystyle=$
$\displaystyle\rho_{22}(t)-\frac{1}{2}e^{-\gamma t}\cos(2Jt)$ (35)
$\displaystyle\rho_{41}(t)$ $\displaystyle=$ $\displaystyle\rho_{14}^{*}(t)$
(36) $\displaystyle\rho_{44}(t)$ $\displaystyle=$ $\displaystyle
1-[\rho_{11}(t)+\rho_{22}(t)+\rho_{33}(t)]$ (37)

where we have defined

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\sqrt{\Delta^{2}+\omega^{2}}.$ (38)

One notes that only elements $\rho_{22}(t)$, $\rho_{23}(t)$,$\rho_{32}(t)$ and
$\rho_{33}(t)$ depends on the parameter $J$ which due to the damping factor
$e^{-\gamma t}$, this dependence dies at the limit $t\to\infty$.

Take the initial state to be now the Werner state,

$\displaystyle\rho_{W}$ $\displaystyle=$ $\displaystyle
p|\Psi_{-}\rangle\langle\Psi_{-}|+\frac{1-p}{4}\mathds{1}\otimes\mathds{1}=\begin{pmatrix}\frac{1-p}{4}&0&0&0\\\
0&\frac{1+p}{4}&-\frac{p}{2}&0\\\ 0&-\frac{p}{2}&\frac{1+p}{4}&0\\\
0&0&0&\frac{1-p}{4}\end{pmatrix}$ (39)

where $|\Psi_{-}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$ is one of
the maximally entangled Bell states, and $p$ a real number with constraint
$p\in[-\frac{1}{3},1]$ which comes from the non-negativity requirement of the
density matrix. For $p\in[-\frac{1}{3},\frac{1}{3}]$ the Werner state is
separable and for $p\in[-\frac{1}{3},\frac{1}{\sqrt{2}}]$ it is local Luc-
thesis-2019 . From this one sees that the Werner state is entangled but local
for $p\in(\frac{1}{3},\frac{1}{\sqrt{2}}]$ i.e., entanglement does not imply
nonlocality. The purity of the Werner state (39) is
$\text{tr}(\rho_{W}^{2})=(1+3p^{2})/4$ being one for $p=1$ and $1/4$ for
$p=0$. The former implies the pure ensemble $|\Psi_{-}\rangle\langle\Psi_{-}|$
while the latter corresponds to the completely random ensemble. The solution
of the Lindblad equation (29) this time is given by

$\displaystyle\rho_{11}(t)$ $\displaystyle=$
$\displaystyle\frac{4\Delta^{2}\Omega-4\Delta^{2}\gamma e^{-\gamma
t}\sin(2\Omega t)+\Omega e^{-2\gamma
t}(4\omega^{2}-4\Omega^{2}p+\gamma^{2}(1-p))}{4\Omega(\gamma^{2}+4\Omega^{2})}$
(40) $\displaystyle\rho_{14}(t)$ $\displaystyle=$
$\displaystyle\frac{i\Delta}{2\Omega^{3}(\gamma^{2}+4\Omega^{2})}\bigg{\\{}-2(\gamma-2i\omega)\Omega^{3}+e^{-\gamma
t}\bigg{[}-4i\omega\Omega^{3}-2i\gamma^{2}\omega\Omega\sin^{2}(\Omega t)$ (41)
$\displaystyle+\gamma^{2}\Omega^{2}\sin(2\Omega
t)+2\gamma\Omega^{2}\bigg{(}\Omega\cos(2\Omega t)-i\omega\sin(2\Omega
t)\bigg{)}\bigg{]}\bigg{\\}}$ $\displaystyle\rho_{22}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\Omega^{2}(\gamma^{2}+4\Omega^{2})}\bigg{\\{}4\Omega^{2}\bigg{[}\Delta^{2}+e^{-2\gamma
t}(p\Delta^{2}+(p-1)\omega^{2})+2\omega^{2}e^{-\gamma t}\bigg{]}$ (42)
$\displaystyle+\gamma^{2}\bigg{[}(p-1)\Omega^{2}e^{-2\gamma t}+2e^{-\gamma
t}(\Delta^{2}\cos(2\Omega t)+\omega^{2})\bigg{]}\bigg{\\}}$
$\displaystyle\rho_{23}(t)$ $\displaystyle=$
$\displaystyle\rho_{32}(t)=-\frac{p}{2}e^{-\gamma t}$ (43)
$\displaystyle\rho_{33}(t)$ $\displaystyle=$ $\displaystyle\rho_{22}(t)$ (44)
$\displaystyle\rho_{41}(t)$ $\displaystyle=$ $\displaystyle\rho_{14}^{*}(t)$
(45) $\displaystyle\rho_{44}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\Omega^{3}(\gamma^{2}+4\Omega^{2})}\bigg{\\{}4\Omega^{3}(\gamma^{2}+3\omega^{2}+\Omega^{2})$
(46) $\displaystyle-\Omega^{3}e^{-2\gamma
t}\bigg{[}(\gamma^{2}+4\Omega^{2})p-(\gamma^{2}+4\omega^{2})\bigg{]}-4\omega^{2}\Omega(\gamma^{2}+4\Omega^{2})e^{-\gamma
t}$ $\displaystyle+4\gamma\Delta^{2}e^{-\gamma
t}\bigg{[}\Omega^{2}\sin(2\Omega t)-\gamma\Omega\cos(2\Omega
t)\bigg{]}\bigg{\\}}$

where $\Omega$ is given by (38). As one sees the state $\rho_{W}(t)$ is
independent of the parameter $J$ appearing in the Hamiltonian being
responsible for the isotropic part of the spin-spin interaction. An
interesting point about the Werner state is that for $\gamma=0$, density
operator is independent of time which is the manifestation of the fact that
the Werner state (39) commutes with the Hamiltonian (28) of the system,
$[H,\rho_{W}(t)]=0$.

Although the master equation (29) can be analytically solved for the general
X-shaped state (2) but since the relations are lengthy we only present the
steady state solution i.e., the state at $t\to\infty$ which is

$\displaystyle\rho_{\infty}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{\Delta^{2}}{\gamma^{2}+4\Omega^{2}}&0&0&-\frac{\Delta(2\omega+i\gamma)}{\gamma^{2}+4\Omega^{2}}\\\
0&\frac{\Delta^{2}}{\gamma^{2}+4\Omega^{2}}&0&0\\\
0&0&\frac{\Delta^{2}}{\gamma^{2}+4\Omega^{2}}&0\\\
-\frac{\Delta(2\omega-i\gamma)}{\gamma^{2}+4\Omega^{2}}&0&0&\frac{\gamma^{2}+3\omega^{2}+\Omega^{2}}{\gamma^{2}+4\Omega^{2}}\end{pmatrix}.$
(47)

This result show that the steady state is the same for any X-shaped state and
thus all such states have the same steady state quantum correlations. Note
that the condition $\rho_{22}\rho_{33}<|\rho_{14}|^{2}$ then yields to

$\displaystyle|\Delta|<\sqrt{\gamma^{2}+4\omega^{2}}$ (48)

for the steady state to be entangled. As we saw, reduced density operators for
a X-shaped state are both always diagonal with zero quantum coherence.
However, Eq. (47) shows that the global steady state $\ell_{1}$-norm coherence
is

$\displaystyle C_{\ell_{1}}(\rho_{\infty})$ $\displaystyle=$
$\displaystyle\sum_{i}\sum_{j\neq
i}|\rho_{\infty,ij}|=\frac{|\Delta|\sqrt{4\omega^{2}+\gamma^{2}}}{\gamma^{2}+4\Omega^{2}},$
(49)

i.e., decoherence does not eliminate the off-diagonal elements of the global
state.

##  IV Temperature dependence of the steady state solution

Now we aim to consider the behaviour of steady state quantum correlations with
temperature. To this end, we assume both qubits interact with the same thermal
bath and consider the master equation

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle-i[H,\rho(t)]+\gamma(\bar{n}+1)~{}\mathcal{D}[S_{-}^{(1)}\otimes\mathds{1}^{(2)}]\rho(t)+\gamma(\bar{n}+1)~{}\mathcal{D}[\mathds{1}^{(1)}\otimes
S_{-}^{(2)}]\rho(t)$ (50)
$\displaystyle+\gamma\bar{n}~{}\mathcal{D}[S_{+}^{(1)}\otimes\mathds{1}^{(2)}]\rho(t)+\gamma\bar{n}~{}\mathcal{D}[\mathds{1}^{(1)}\otimes
S_{+}^{(2)}]\rho(t),$

$\bar{n}$ being the average excitation of the thermal bath parametrizing the
temperature: being zero for $T=0$ Waetal-JPB-2006 . This equation is known as
quantum optical master equation where $\gamma$ is the rate for spontaneous
emission and $\gamma\bar{n}$ describes the rate of thermally induced emission
and absorption processes BrPe-book-2002 222page: 148, equation (3.219). Eq.
(50) can be analytically solved for the X-state (2). Again, one sees that this
master equation preserves the shape of the state. Here, we are only interested
in the steady state, $t\to\infty$, solution which is given by

$\displaystyle\rho_{11}(\infty)=\frac{(2\bar{n}+1)^{2}(\Delta^{2}+\gamma^{2}\bar{n}^{2})+4\bar{n}^{2}\omega^{2}}{(2\bar{n}+1)^{2}[4\Omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}]},$
(51)
$\displaystyle\rho_{14}(\infty)=-\frac{2\omega\Delta+i\gamma\Delta(2\bar{n}+1)}{(2\bar{n}+1)[4\Omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}]},$
(52)
$\displaystyle\rho_{22}(\infty)=\frac{\Delta^{2}+\bar{n}(\bar{n}+1)(4\Omega^{2}+\gamma^{2}(2\bar{n}+1)^{2})}{(2\bar{n}+1)^{2}[4\Omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}]},$
(53) $\displaystyle\rho_{23}(\infty)=\rho_{32}(\infty)=0,$ (54)
$\displaystyle\rho_{33}(\infty)=\rho_{22}(\infty),$ (55)
$\displaystyle\rho_{41}(\infty)=\rho_{14}^{*}(\infty),$ (56)
$\displaystyle\rho_{44}(\infty)=\frac{(2\bar{n}+1)^{2}(\Delta^{2}+\gamma^{2}(\bar{n}+1)^{2})+4(\bar{n}+1)^{2}\omega^{2}}{(2\bar{n}+1)^{2}[4\Omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}]}.$
(57)

From these equations one obtains

$\displaystyle\mathcal{C}(\infty)$ $\displaystyle=$ $\displaystyle
2\max\bigg{\\{}0,\frac{(2\bar{n}+1)|\Delta|\sqrt{4\omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}}-\Delta^{2}}{(2\bar{n}+1)^{2}[4\Omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}]}-\frac{\bar{n}(\bar{n}+1)}{(2\bar{n}+1)^{2}}\bigg{\\}},$
(58) $\displaystyle C_{cc}(\infty)$ $\displaystyle=$ $\displaystyle
2\frac{|\Delta|\sqrt{4\omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}}}{(2\bar{n}+1)[4\Omega^{2}+\gamma^{2}(2\bar{n}+1)^{2}]},$
(59) $\displaystyle N_{1}(\infty)$ $\displaystyle=$ $\displaystyle
C_{cc}(\infty),$ (60)

respectively for the steady state concurrence, CC and MIN. In this case the
$W$ matrix (14) is diagonal with $W_{11}(\infty)=W_{22}(\infty)$. Thus, for
the steady state LQU one has that

$\displaystyle\mathcal{U}(\infty)$ $\displaystyle=$
$\displaystyle\max\\{W_{11}(\infty),W_{33}(\infty)\\}$ (61)

For $\bar{n}=0$ one obtains

$\displaystyle W_{11}(\infty)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}|\Delta|}{\gamma^{2}+4\Omega^{2}}\bigg{\\{}\sqrt{\gamma^{2}+2\omega^{2}+2\Omega^{2}-\sqrt{(\gamma^{2}+4\omega^{2})(\gamma^{2}+4\Omega^{2})}}$
(62)
$\displaystyle+\sqrt{\gamma^{2}+2\omega^{2}+2\Omega^{2}+\sqrt{(\gamma^{2}+4\omega^{2})(\gamma^{2}+4\Omega^{2})}}\bigg{\\}},$
$\displaystyle W_{33}(\infty)$ $\displaystyle=$
$\displaystyle\frac{\gamma^{4}+4\gamma^{2}(\omega^{2}+\Omega^{2})+16(\Delta^{4}+\omega^{2}\Omega^{2})}{(\gamma^{2}+4\Omega^{2})^{2}}.$
(63)

##  V Results and discussion

In this section, we aim to consider the impact of the relaxation rate
$\gamma$, responsible for the interaction of our qubit-qubit Heisenberg XY
system with the environment, on the dynamics of quantum correlations by fixing
other parameters of the problem including coupling strengths $J$ and $\Delta$;
and magnetic field intensity $\omega$. However, as wee see in the previous
section, Werner states and thus corresponding quantum correlations are
independent of $J$. The same is valid for the general X-shaped state in the
steady situation. Furthermore, in some calculations we fix all parameters
other than $\Delta$ to study anisotropicity dependence of quantum
correlations. Our investigation will primarily focus on examining how the
zero-temperature steady state quantum correlations appear for different
initial X states, including a mixture and the Werner state. After, the
interesting phenomenon of entanglement sudden death, for independent qubits,
will be considered for two initially entangled and afterwards not interacting
qubits both for zero and finite temperatures. The time of ESD will be obtained
analytically (numerically) for zero (finite) temperatures. Environment-induced
quantum correlations is another phenomenon which will be considered. At the
end finite temperature quantum correlations will be examined in the steady
state case by studying their behaviour with temperature.

Figure 1 depicts evolution of different quantum correlations for $\omega=1$,
$J=0.1$ and $\Delta=0.5$ for three different values of the relaxation rate
$\gamma=0.1$, $\gamma=0.15$ and $\gamma=0.2$. Here, system is initially
described by the entangled mixture (30). Initial values of the quantum
correlations, being independent of $\gamma$, are $\mathcal{C}(0)=0.5$,
$\mathcal{L_{N}}(0)=0.2716$, $\mathcal{U}(0)=0.5$ and $C_{cc}(0)=0.5$. As this
figure reveals, both entanglement quantifiers, concurrence and log negativity,
exhibit several dark and revival periods FiTa-PRA-2006  before reaching the
steady state. Because of the spontaneous emission, the initial entanglement
diminishes with time, disappears in a given finite time which itself depends
of the value of $\gamma$ and after a dark while it revives. The number and
length of these dark intervals both depend on the value of $\gamma$. The first
dark period is the longest one which itself decreases with $\gamma$. The
difference between entanglement quantifiers, $\mathcal{C}$ and
$\mathcal{L_{N}}$, and quantum correlations beyond entanglement, $\mathcal{U}$
and $C_{cc}$, is that the former ones revive after zero entanglement, while
the others can revive immediately several times without completely vanishing
for a period of time; there are no dark periods for quantum correlations
beyond QE. Moreover, LQU captures less quantum correlations than the other
quantifiers, with the former converging to a stationary value of $0.1597$ for
$\gamma=0.1$. As one expects, the transition to the steady state occurs sooner
for higher values of $\gamma$ and as one can check analytically from Eq. (47)
the steady state value of quantum correlation although is independent of the
initial state but depends on the value of $\gamma$.

Figure 1:  Evolution of the quantum correlations, concurrence (left top
panel), log negativity (right top panel), LQU (left bottom panel) and CC
(right bottom panel), for $\gamma=0.1$ (red curves), $\gamma=0.15$ (green
curves), $\gamma=0.2$ (blue curves) for $\omega=1$, $J=0.1$ and $\Delta=0.5$.
The quantum system is initially described by the mixture (30).

We consider now a system of two independent non-interacting qubits,
$J=\Delta=0$, which initially is described by the mixture

$\displaystyle\rho_{\text{mix}}$ $\displaystyle=$ $\displaystyle
w|01\rangle\langle 01|+\frac{1-w}{2}(|00\rangle+|11\rangle)(\langle
00|+\langle 11|),\qquad 0\leq w\leq 1.$ (64)

Then one has that $1-\text{tr}(\rho_{\text{mix}}^{2})=2w(1-w)$ for the purity
of the state. Time dependent density matrix obtained from the solution of the
Lindblad equation (29) with $J=\Delta=0$ and the initial condition (64) is
then,

$\displaystyle\rho_{11}(t)=\frac{1-w}{2}e^{-2\gamma t}$ (65)
$\displaystyle\rho_{14}(t)=\frac{1-w}{2}e^{-(\gamma+2i\omega)t}$ (66)
$\displaystyle\rho_{22}(t)=e^{-3\gamma t/2}\bigg{(}\sinh(\frac{\gamma
t}{2})+w\cosh(\frac{\gamma t}{2})\bigg{)}$ (67)
$\displaystyle\rho_{23}(t)=\rho_{32}(t)=0$ (68)
$\displaystyle\rho_{33}(t)=(1-w)e^{-3\gamma t/2}\sinh(\frac{\gamma t}{2})$
(69) $\displaystyle\rho_{41}(t)=\frac{1-w}{2}e^{-(\gamma-2i\omega)t}$ (70)
$\displaystyle\rho_{44}(t)=\frac{1-w}{2}e^{-2\gamma t}+2e^{-\gamma
t/2}\sinh(\frac{\gamma t}{2})$ (71)

From the definition (4) one obtains

$\mathcal{C}(t)=\max\bigg{\\{}0,e^{-\gamma t}\bigg{[}1-w-2e^{-\gamma
t/2}\sqrt{(1-w)\sinh(\frac{\gamma t}{2})\bigg{(}\sinh(\frac{\gamma
t}{2})+w\cosh(\frac{\gamma t}{2})\bigg{)}}\bigg{]}\bigg{\\}}$ (72)

for the concurrence of this mixture. This equation shows that this state is
always separable for $w=1$ as it is obvious from (64) for the initial state.
For $0\leq w<1$ which the initial state is entangled, Eq. (72) shows that
there is a finite time where the entanglement disappears, the signature of
ESD. This time is obtained as

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma}\ln\bigg{(}\frac{1+\sqrt{1-2w(1-w)}}{2w}\bigg{)}$
(73)

which is infinite for $w=0$ meaning that the entanglement does not die
suddenly for the maximally entangled state
$\frac{1}{2}(|00\rangle+|11\rangle)(\langle 00|+\langle 11|)$. The age of
entanglement is shorter for higher values of both $\gamma$ and $w$. See figure
2.

Figure 2:  Left panel: time of entanglement death $\gamma\tau$ given by Eq.
(73) in terms of the weight $w$ appearing in the mixture (64). Right panel:
evolution of concurrence (72) for $w=0.1$ (black curve), $w=0.3$ (red curve),
$w=0.5$ (green curve) and $w=0.8$ (blue curve). Parameters have been chosen
$\omega=1$ and $\Delta=J=0$ i.e, qubits solely interact with the common
environment not with each other.

Now the natural question arises: Does the phenomenon of death occurs for
correlations beyond entanglement? From, Eq. (27) we have

$\displaystyle C_{cc}(t)$ $\displaystyle=$ $\displaystyle(1-w)e^{-\gamma t}$
(74)

for CC or equivalently for MIN with respect to (19). This equation shows that
$C_{cc}(0)=1-w$, being nonzero for $w\neq 1$, and CC decreases exponentially;
the usual behaviour of decaying systems. Thus, there is no phenomenon of CC
sudden death i.e., this quantity does not disappear in a finite time. For LQU
which is computed by (16) analytical calculations show that the square root
expression appearing in this equation is zero. Thus, LQU for $J=\Delta=0$ is
obtained from

$\displaystyle\mathcal{U}(t)$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}\frac{W_{11}(t)+W_{22}(t)}{2},W_{33}(t)\bigg{\\}}$
(75)

where $W_{ij}$ is given by (14). The initial value of LQU is
$\mathcal{U}(0)=\max\\{1-w,1-\sqrt{w(1-w)}\\}$ being zero for $w=1$. In the
left panel of Fig. 3 we have plotted initial LQU versus the probability
weight. As the numerical computations in the right panel of Fig. 3 show, LQU
decreases first reaching a local minimum then increases to a given value and
breaks downwards and continuously, not abruptly, vanishes. Ref. GuPeZeWa-
QIP-2020  has already reported this sudden change in LQU when one of the
qubits is subjected to local decoherence noise. In any way, there is no LQU
sudden death. Note that breaking points in the curves are due to the “max”
operation in the definition (75).

Figure 3:  Initial value of LQU in terms of the probability weight $w$ (left
panel) and evolution of LQU (right panel) for $w=0.4$ (cyan), $w=0.5$ (red),
$w=0.6$ (green) and $w=0.7$ (blue) for $\omega=1$ and $\Delta=J=0$ i.e, qubits
solely interact with the common environment not with each other. System is
initially described by the mixture (64).

In the next step we analyse ESD for finite temperatures. This time the
Lindblad equation (50) must be considered. This equation can still be solved
analytically, from which the concurrence for $\Delta=J=0$ is obtained as
follows:

$\displaystyle\mathcal{C}(t)$ $\displaystyle=$
$\displaystyle\max\left\\{0,(1-w)e^{-(2\bar{n}+1)\gamma
t}-\frac{\sqrt{f(t)}}{(2\bar{n}+1)^{2}}\right\\}$ (76)

where

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle
a_{0}(t)+a_{1}(t)w+a_{2}(t)w^{2}$ (77) $\displaystyle a_{0}(t)$
$\displaystyle=$ $\displaystyle 4e^{-3(2\bar{n}+1)\gamma
t}\sinh^{2}\left(\frac{(2\bar{n}+1)\gamma t}{2}\right)$ (78)
$\displaystyle\times\left[1+4\bar{n}(\bar{n}+1)e^{(2\bar{n}+1)\gamma
t/2}\cosh\left(\frac{(2\bar{n}+1)\gamma t}{2}\right)\right]^{2}$
$\displaystyle a_{1}(t)$ $\displaystyle=$ $\displaystyle
4(2\bar{n}+1)^{2}e^{-7(2\bar{n}+1)\gamma
t/2}\sinh\left(\frac{(2\bar{n}+1)\gamma t}{2}\right)$ (79)
$\displaystyle\times\left[1+4\bar{n}(\bar{n}+1)e^{(2\bar{n}+1)\gamma
t/2}\cosh\left(\frac{(2\bar{n}+1)\gamma t}{2}\right)\right]$ $\displaystyle
a_{2}(t)$ $\displaystyle=$
$\displaystyle-2(2\bar{n}+1)^{4}e^{-3(2\bar{n}+1)\gamma
t}\sinh[(2\bar{n}+1)\gamma t]$ (80)

Eq. (76) show that entanglement death occurs here. The time of ESD is obtained
from the solution of

$\displaystyle e^{2(2\bar{n}+1)\gamma t}f(t)$ $\displaystyle=$
$\displaystyle(2\bar{n}+1)^{4}(1-w)^{2}$ (81)

which cannot be solved analytically. Numerical solution of this equation for
$w=\frac{1}{2}$, as a function of $\bar{n}$, has been given in figure 4. Right
panel of this figure depicts evolution of concurrence for a few values of
$\bar{n}$. ESD is clearly seen being stronger for higher values of $\bar{n}$.
This means that at finite temperatures, the temperature speeds up ESD. As the
left panel of this figure displays, the age of entanglement decreases with
$\bar{n}$.

Figure 4:  Time of entanglement death, $\gamma\tau$, versus $\bar{n}$ (left
panel) and concurrence (right panel) versus $\gamma t$ for $\bar{n}=0$
(black), $\bar{n}=0.2$ (red), $\bar{n}=0.4$ (green) and $\bar{n}=0.6$ (blue)
for $\omega=1$ and $\Delta=J=0$ i.e., qubits solely interact with the common
environment not with each other. System is initially described by the mixture
(64) with $w=\frac{1}{2}$.

Next, we illustrate in figure 5 dynamics of quantum correlations for the
Werner state (39) for different values of $p$. Initial values of quantum
correlations are $\mathcal{C}(0)=\max\\{0,|p|-\frac{1-p}{2}\\}$,
$\mathcal{L_{N}}(0)=\log_{2}\left(\frac{1+p}{2}+\frac{1}{2}\left|\frac{1-p}{2}-|p|\right|+\frac{1}{2}\left|\frac{1-p}{2}+|p|\right|\right)$,
$C_{cc}(0)=|p|$ and $\mathcal{U}(0)=1-\frac{1}{2}[1-p+\sqrt{(1-p)(1+3p)}]$
from which one finds the concurrence is non-zero only for $\frac{1}{3}<p\leq
1$ implying, as is well-known, that in this region the Werner state (39) is
entangled. While QE, quantified by both concurrence and log negativity, is
initially zero for values $p=-\frac{1}{3}$ and $p=0$ after a while it is born
and finally reaches the steady value $0.2999$ and $0.3784$, being independent
of $p$, respectively for concurrence and log negativity for $\gamma=0.1$,
$\omega=1$ and $\Delta=0.5$. Using Eq. (47) one obtains
$\mathcal{C}(\infty)=2\max\left\\{0,\frac{|\Delta|\sqrt{4\omega^{2}+\gamma^{2}}-\Delta^{2}}{4\Omega^{2}+\gamma^{2}}\right\\}$
for the steady state value of quantum concurrence which is non-zero only for
$|\Delta|<\sqrt{4\omega^{2}+\gamma^{2}}$ . As pointed out in the previous
section, for $\gamma=0$, the Werner state (39) does not evolve in time meaning
that all quantum correlations remain constant. From all these we see that if
both conditions $\frac{1}{3}<p\leq 1$ and
$|\Delta|<\sqrt{4\omega^{2}+\gamma^{2}}$ fulfil simultaneously then for
$|p|-\frac{1-p}{2}<\frac{|\Delta|\sqrt{4\omega^{2}+\gamma^{2}}-\Delta^{2}}{4\Omega^{2}+\gamma^{2}}$,
the steady state is more entangled than the initial state. For parameters
$\gamma=0.1$, $\omega=1$ and $\Delta=0.5$ where $\mathcal{C}(\infty)=0.2999$,
this condition fulfils for $\frac{1}{3}<p<0.5333$.

Figure 5:  Evolution of the quantum correlations, concurrence (left top
panel), log negativity (right top panel), LQU (left bottom panel) and CC
(right bottom panel), for $p=-\frac{1}{3}$ (black curves), $p=0$ (red curves),
$p=\frac{1}{\sqrt{3}}$ (green curves), and $p=1$ (blue curves) for
$\gamma=0.1$, $\omega=1$ and $\Delta=0.5$. Note that the Werner state is
independent of the isotropic part of qubits’ interaction i.e., the coefficient
$J$.

In contrast to the entanglement which is generated after a while, as can be
found from the red curve corresponding to $p=0$, both LQU and CC are generated
much more quickly than entanglement. As before, steady state LQU captures less
quantum correlations than the other quantifiers. Note that the Werner state
for $p=0$ is the completely unpolarized state.

Figure 6:  Evolution of the quantum correlations, concurrence (left top
panel), log negativity (right top panel), LQU (left bottom panel) and CC
(right bottom panel) for different values of the anisotropy interaction
coefficient: $\Delta=0$ (brown), $\Delta=0.2$ (magenta) $\Delta=0.3$ (cyan)
and $\Delta=0.4$ (orange); and for $\omega=1$ and $\gamma=0.1$. System is
initially described by the Werner state (39) with $p=0$ (completely random
ensemble; $\rho_{0}=\frac{1}{4}\mathds{1}_{4}$).

After, we highlight the constructive role played by the environment in favour
of quantum correlations; the so-called environment-induced or decoherence-
induced quantum correlations. To this end, we consider our system is initially
described by the Werner state (39) with $p=0$ i.e.,
$\rho_{0}=\frac{1}{4}\mathds{1}_{4}$; the completely random ensemble which
contains no correlations. Then, evolution of quantum correlations is
considered for different values of the interaction strength $\Delta$. Note
that the evolved state and thus quantum correlations are all independent of
the value of $J$. Figure 6 depicts variation of quantum correlations and CC
with time for a given value of $\gamma$ but different values of interaction
strength $\Delta$. For $\Delta=0$ all quantum correlations are always zero but
for $\Delta\neq 0$ quantum correlations are generated as time elapses. From
this point we learn that (i) environment plays a constructive role, in
contrast to its customary destructive role, inducing quantum correlations (ii)
in addition to the interaction of qubits with the environment, the anisotropic
part of their own interaction with each other is necessary for creation of
quantum correlations. As top plots of this figure show, generation of QE from
an initially separable state, the so-called environment-induced entanglement,
requires both interaction between qubits and also their own interaction with
the environment; no one is sufficient on its own. This constructive role
played by the environment in favour of entanglement, in contrast to its
customary destructive role, has already been noticed. See for instance AnWaLu-
PA-2007 ; Hoetala-PRA-2009 . Despite the apparent implication from this figure
that quantum correlations increase with the strength of interaction $\Delta$,
it is important to note that this is not always the case. Consider the steady
state as a example. For our parameters, $\gamma=0.1$ and $\omega=1$, the
steady state concurrence
$\mathcal{C}(\infty)=2\max\left\\{0,\frac{|\Delta|\sqrt{4\omega^{2}+\gamma^{2}}-\Delta^{2}}{4\Omega^{2}+\gamma^{2}}\right\\}$
is nonzero only for $\Delta\in(-2.0025,0)\cup(0,2.0025)$ and is an increasing
function of $\Delta$ only when $-2.0025<\Delta\leq-0.6188$ and $0<\Delta\leq
0.6188$.

Figure 7:  Plots of the finite temperature, steady state quantum correlations,
concurrence (left top panel), log negativity (right top panel), LQU (left
bottom panel) and CC (right bottom panel), in terms of the average excitation
of the thermal bath, $\bar{n}$, for $\gamma=0.01$ (blue curves) and $\gamma=3$
(orange curves) for $\Delta=0.5$ and $\omega=1$. Note that X-shaped steady
state is independent of both the initial state and the isotropic part of
qubits’ interaction i.e., the coefficient $J$.

Finally, in figure 7 we have plotted the finite temperature, steady state
quantum correlations versus the average excitation of the thermal bath,
$\bar{n}$, for two values of the emission rate, $\gamma=0.01$ (blue curves)
and $\gamma=3$ (orange curves), for $\Delta=0.5$ and $\omega=1$. As one sees,
a steady state level of all quantum correlations is always reached for zero or
finite, low temperatures. For nonzero temperatures all quantum correlations
are found to vanish at a finite temperature. This means none of the quantum
correlations are robust against the temperature. However, CC which equals MIN
is more robust than others. Emission rate $\gamma$ speeds up the decay of
quantum correlations.

##  VI Summary and conclusions

In this work, our model was a system of two localized qubits interacting
through their spins, and also, an external magnetic field was applied. The
interaction with the environment was introduced by a non-unitary term in the
Lindblad equation. We solved this equation analytically at zero temperature
for several X-shaped states and studied the evolution of different quantum
correlations for different values of the emission rate $\gamma$. After that,
it was time to bring the temperature into play and consider the influence of
thermal decoherence on quantum correlations. Although the corresponding master
equation could be solved analytically, but due to the length of the solution,
we considered only the stable solution and examined the dependence of
different quantum correlations on temperature in this case. Our main results
can be summarized as follows: (i) MIN is equal to CC. (ii) The sudden death of
quantum correlations beyond entanglement is not possible, maybe a not an
unexpected result with respect to their definition. (iii) Even if the initial
state lacks correlations, the interaction with the environment induces them.
In other words, like entanglement, it is possible to induce other quantum
correlations by the decoherence. (iv) In addition to the interaction of qubits
with the common environment, their own interaction with each other is
necessary to induce quantum correlations. (v) Although the revival and rebirth
of entanglement happens after a finite period of time, the revival of other
correlations is much faster. (vi) In the zero-temperature steady state, LQU
captures less quantum correlations than the other quantifiers. (vii)
Temperature speeds up ESD. (viii) Quantum correlations beyond entanglement are
more resistant to the negative effect of temperature to destroy them.

Acknowledgements: Support from the University of Qom is acknowledged.

Data availability: This manuscript has no associated data.

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