# Nonclassical characteristics in spin-1/2 Heisenberg XYZ model with added DM
and KSEA interactions under sinusoidal magnetic field: Hierarchy of quantum
resources

A. Ali  0000-0001-9243-417X<EMAIL_ADDRESS>Qatar Centre for Quantum
Computing (QC2), College of Science and Engineering, Hamad Bin Khalifa
University, Qatar Foundation, Doha, Qatar  S. Al-Kuwari  0000-0002-4402-7710
Qatar Centre for Quantum Computing (QC2), College of Science and Engineering,
Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar  M. T. Rahim
0000-0003-1529-928X Qatar Centre for Quantum Computing (QC2), College of
Science and Engineering, Hamad Bin Khalifa University, Qatar Foundation, Doha,
Qatar  M. Ghominejad  0000-0002-0136-7838 Faculty of Physics, Semnan
University, P.O. Box 35195-363, Semnan, Iran  H. Ali  0000-0003-1957-3629
Department of Physics, Abbottabad University of Science and Technology, P.O.
Box 22500 Havellian KP, Pakistan  S. Haddadi  0000-0002-1596-0763
<EMAIL_ADDRESS>Faculty of Physics, Semnan University, P.O. Box
35195-363, Semnan, Iran

###### Abstract

We investigate the behavior of various measures of quantum coherence and
quantum correlation in the spin-1/2 Heisenberg XYZ model with added
Dzyaloshinsky-Moriya (DM) and Kaplan–Shekhtman–Entin-Wohlman–Aharony (KSEA)
interactions at a thermal regime described by a Gibbs density operator. We aim
to understand the restricted hierarchical classification of different quantum
resources, where quantum coherence $\supseteq$ quantum discord $\supseteq$
quantum entanglement $\supseteq$ quantum steering $\supseteq$ Bell
nonlocality. In order to enhance quantum coherence, quantum correlation, and
fidelity of teleportation, our analysis encompasses the effects of
independently provided sinusoidal magnetic field control as well as DM and
KSEA interactions on the considered system. The results reveal that enhancing
the entanglement or quantum correlation of the channel does not always
guarantee successful teleportation or even an improvement in teleportation
fidelity. Thus, the relationship between teleportation fidelity and the
channel’s underlying quantum properties is intricate. Our study provides
valuable insights into the complex interplay of quantum coherence and
correlation hierarchy, offering potential applications for quantum
communication and information processing technologies.

Heisenberg XYZ model, quantum coherence, quantum correlation, teleportation

##  I INTRODUCTION

Quantum coherence is a broad concept that encompasses all kinds of quantum
correlations within a multipartite quantum system [1, 2]. When the off-
diagonal elements of a density matrix are non-zero, it indicates the presence
of quantum coherence. Baumgratz _et al._ [2] introduced a quantum resource
theory of quantum coherence, where the $l_{1}$-norm of coherence effectively
quantifies this notion. This quantum coherence can stem from all forms of
separable and non-separable correlations, such as nonlocality, quantum
steering (QS), entanglement, and quantum discord (QD)[3, 4, 5, 6, 7]. Figure 1
illustrates the hierarchy of quantum correlations, where each layer represents
a subset of the one encompassing it, with quantum coherence being the largest
and most inclusive, and nonlocality being the most specific and restrictive.
This hierarchical structure visually represents how each type of quantum
correlations may build upon the previous ones, showcasing their interrelations
and dependencies [8, 9, 10].

Entanglement refers to the remarkable nonseparable quantum correlations
between particles, where the state of one particle instantaneously influences
the state of another, regardless of distance. QD extends beyond entanglement,
encompassing all quantum correlations residing in both separable and
nonseparable states shared between subsystems of a quantum system [5, 6, 7].
For pure nonseparable states, QD and entanglement are equivalent [7, 6], but
this is not the case for mixed states.

QS [11, 12, 13, 14] represents a unique form of quantum correlations. In a
scenario where two distant observers share an entangled quantum state, QS
describes the ability of one observer to remotely influence the state of the
other observer’s system. While entanglement entails quantum correlations
between quantum particles regardless of distance, QS introduces an asymmetry
in the observer’s roles. In a QS scenario, one observer can manipulate the
state of the other observer’s system, but the reverse may not always hold.
This fundamental asymmetry sets QS apart and raises intriguing questions about
the relationship between QS, entanglement, and nonlocality. However, the
quantum correlation measures discussed above are highly complex and
intricately linked, making it challenging to definitively distinguish between
them, even in the context of two- and three-qubit systems. In essence, these
measures exhibit intricate interconnections, reflecting the nuanced nature of
quantum correlations.

Nonlocality is a concept that arises from the violation of certain
inequalities known as Bell inequalities [15, 16]. These inequalities describe
limits on quantum correlations that can occur between distant systems within a
classical framework. When these inequalities are violated, it implies that the
quantum correlations between the systems cannot be explained by local hidden
variables (underlying properties or information that determine the outcome of
measurements)and suggests a form of nonlocality in quantum mechanics. From
Bell inequalities, we now understand that entanglement and nonlocality are
synonymous in a pure state, meaning that entanglement implies nonlocality and
vice versa. However, the situation is not clear and simple for mixed states,
even in two-qubit systems[17, 18, 19, 20].

A one-dimensional Heisenberg spin chain is a theoretical model generally used
in condensed matter physics to describe a linear array of interacting quantum
spins [21, 22, 23, 24, 25, 26, 27, 28]. Each spin interacts with its nearest
neighbors, exhibiting phenomena such as quantum phase transitions and quantum
correlations. In quantum technologies, Heisenberg spin chains offer potential
for quantum simulation [29, 30], computing [31, 32], metrology and sensing
[33, 34, 35], and communication [36, 37]. Entanglement properties of spin
chains can be harnessed for quantum communication protocols, where one can use
spin chains as quantum channels to transfer quantum states between distant
parties securely and with higher fidelity compared to classical communication
channels [38].

In this work, we take the most general two-qubit Heisenberg XYZ model under a
sinusoidally controlled magnetic field applied to an individual spin site with
asymmetric spin–orbit coupling interaction called Dzyaloshinsky-Moriya (DM)
interaction and symmetric exchange coupling known as Kaplan–Shekhtman–Entin-
Wohlman–Aharony (KSEA) interaction, and study the behavior of all the
aforementioned quantifiers with changes in the model parameters.

To elucidate the occurrence of weak ferromagnetism observed in certain
rhombohedral antiferromagnets, Dzyaloshinsky [39, 40, 41] developed a
phenomenological approach grounded in the Landau theory of second-order phase
transitions. This approach highlighted that the appearance of a nonzero net
magnetization in the system is attributed to the antisymmetric mixed term in
the expansion of the thermodynamic potential. Additionally, Dzyaloshinsky
noticed that in antiferromagnetic crystals with tetragonal lattices, weak
ferromagnetism can be induced by the symmetric mixed term in the corresponding
thermodynamic potential. In 1960, Moriya [42] improved the understanding by
formulating a microscopic theory of anisotropic superexchange interaction,
extending the Anderson theory of superexchange to include spin–orbit coupling.
Through perturbation theory, Moriya identified the primary anisotropy
contribution to the interaction between neighboring spins as the DM
interaction. Moriya also revealed a second-order correction term involving a
symmetric traceless tensor. While historically considered negligible, Kaplan
and subsequently Shekhtman _et al._ [43, 44, 42] argued for the significance
of the symmetric term. This term, called KSEA interaction as mentioned before,
can restore the O(3) invariance of the isotropic Heisenberg system, a property
disrupted by the DM term. This reevaluation of the interaction emphasizes its
importance in understanding weak ferromagnetism in antiferromagnetic systems.

Figure 1: Conventional hierarchy breakdown of quantum coherence and
correlations, i.e. quantum coherence $\supseteq$ QD $\supseteq$ entanglement
$\supseteq$ QS $\supseteq$ nonlocality.

Finally, we also consider a teleportation protocol by employing the two-qubit
Heisenberg spin chain as a channel to study how the channel parameters can
impact the fidelity of the teleported state. Heisenberg XYZ model has already
been experimentally studied, such as in Rydberg or dipolar atoms, through a
combination of dipole interactions with engineered optical pumping [45].
Floquet-engineered XXZ spin coupling in bulk systems and optical tweezer
arrays have been demonstrated in [46, 47]. Similarly, this model has been
recently realized in multiple experiments, notably in dipolar-octupolar (DO)
Kramers compounds Ce2(Sn, Zr)2O7 and Nd2Zr2O7 [48, 49, 50]. Moreover, the XYZ
model has been realized in Weyl-Heisenberg ferromagnets [51].

###  I.1 Contribution

The contribution of this work is to explore the hierarchical classification of
different quantum resources and their impact on teleportation fidelity. We
analyze how magnetic field control as well as DM and KSEA interactions affect
quantum coherence, correlations, and teleportation, revealing the nuanced
relationship between teleportation fidelity and the channel’s quantum
properties. The work sheds light on the robustness of quantum resources under
temperature variations and control parameters, offering crucial insights for
developing quantum communication and information processing technologies.

###  I.2 Organization

The scheme of this paper is as follows: in Sec. II, we define and explain the
different notions of quantum coherence and quantum correlation measures. In
Sec. III, we briefly introduce the spin-1/2 Heisenberg XYZ model, diagonalize
the system, and come up with the thermal state of the model. In Sec. IV, we
present a simple teleportation scheme based on the considered model to be
implemented. Section V presents the results and discussion, and finally, Sec.
VI concludes this paper.

##  II Quantum resources

###  II.1 $l_{1}$-norm quantum coherence

Quantum coherence is a fundamental concept that arises from the superposition
principle in quantum mechanics. A rigorous framework to quantify coherence as
a resource, known as the resource theory of quantum coherence, has been
previously developed [1, 2]. This theory identifies the set of incoherent
states $\mathcal{I}$ which are diagonal in a reference basis
$\\{|i\rangle\\}$:

$\delta\in\mathcal{I}\iff\delta=\sum_{i}\delta_{i}|i\rangle\langle i|.$ (1)

The free operations are the incoherent operations that map incoherent states
to incoherent states. Revealing and quantifying quantum coherence is essential
to enable quantum correlations and information processing.

Accordingly, Baumgratz _et al._ [2] proposed the $l_{1}$-norm of quantum
coherence as a quantifier of coherence, given by

$Q(\rho)=\sum_{i\neq j}|\langle
i|\rho|j\rangle|=\sum_{i,j}|\rho_{ij}|-\sum_{i}|\rho_{ii}|,$ (2)

where $\rho$ is the density operator of the considered system.

###  II.2 Quantum discord

QD measures nonclassical correlations between subsystems in a quantum system,
capturing quantumness not explained by classical means. QD quantifies
differences between quantum and classical mutual information.

QD indicates the difference between total and classical correlation for a two-
qubit system [7, 5, 6, 52]. That is, QD for a bipartite quantum X state,
$\rho_{X}$ is expressed as

$D(\rho_{X})=\min\\{q_{1},~{}q_{2}\\},$ (3)

where

$q_{j}=H(\rho_{11}+\rho_{33})+\sum^{4}_{i=1}\lambda_{i}\log_{2}\lambda_{i}+w_{j},$
(4)

here, $\lambda_{i}$’s represent the eigenvalues of density matrix $\rho_{X}$,
and

$w_{1}=H(\xi),$ (5)
$w_{2}=-\sum_{i}\rho_{ii}\log_{2}\rho_{ii}-H(\rho_{11}+\rho_{33}),$ (6)

with

$H(\xi)=-\xi\log_{2}\xi-(1-\xi)\log_{2}(1-\xi),$ (7)

and

$\xi=\frac{1}{2}\left\\{1+\sqrt{[1-2(\rho_{33}+\rho_{44})]^{2}+4(|\rho_{14}|+|\rho_{23}|)^{2}}\right\\}.$
(8)

###  II.3 Concurrence

Concurrence is an entanglement monotone used to quantify the degree of
entanglement in arbitrary two-qubit states [4, 53]. For a given density matrix
$\rho$, concurrence $C(\rho)$ of two qubits is defined as follows

$C(\rho)=\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}.$ (9)

where $\lambda_{i}$’s are the square roots of the eigenvalues of the matrix
(non-Hermitian)
$\rho(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$ in
decreasing order. As before, $\sigma_{y}$ represents the $y$-component of
Pauli matrices, and $\rho^{*}$ denotes the complex conjugate of $\rho$. Note
that the measure (9) takes values between 0 and 1, indicating the absence of
entanglement when $C(\rho)=0$ and maximal entanglement when $C(\rho)=1$.

###  II.4 Quantum steering

Erwin Schrödinger proposed steering as an extension of the Einstein-Podolsky-
Rosen paradox [11]. Though initially overlooked as nonlocal correlations, it
later sparked significant advancements. Steerable states exhibit quantum
advantages in device-independent quantum cryptography [54], secure
teleportation [55], randomness generation [56], and subchannel discrimination
[57]. The three-setting linear steering inequality [58, 59] is based on the
assumption that either Alice or Bob are allowed three measurement observables
on their respective subsystems. It is a crucial tool for detecting and
quantifying steering in a state, particularly in a two-qubit system which
reads

$\frac{1}{\sqrt{3}}\sum_{i=1}^{3}\mathrm{Tr}(A_{i}\otimes B_{i}\rho)\leq 1,$
(10)

where $A_{i}=a_{i}\cdot\sigma$ and $B_{i}=b_{i}\cdot\sigma$ are Hermitian
operators acting on qubits $A$ and $B$, respectively. Here,
$a_{i},b_{i}\in\mathbb{R}^{3}$ are two unit vectors with
$\\{b_{1},b_{2},b_{3}\\}$ being orthogonal to each other, and Pauli matrices
$\sigma=(\sigma_{1}=\sigma_{x},\sigma_{2}=\sigma_{y},\sigma_{3}=\sigma_{z})$.
Any violation of the inequality (10) implies that $\rho$ is steerable, and the
maximal violation gives a measure of steering [59]. Explicitly,

$S(\rho)=\max_{\\{A_{i},B_{i}\\}}\left\\{\frac{1}{\sqrt{3}}\sum_{i=1}^{3}\mathrm{Tr}(A_{i}\otimes
B_{i}\rho)\right\\}$ (11)

represents the maximum violation of the three-setting linear steering
inequality.

###  II.5 Bell nonlocality

In order to quantify the degree of quantum nonlocality and enable a more
thorough comparison of quantum correlations, one can use the Bell inequality
violation. The normalized form of Bell nonlocality function can be written as
[60, 61, 62, 63, 64]

$\mathcal{B}(\rho)=\max\left\\{0,\frac{\mathcal{B}_{\textmd{CHSH}}-2}{\mathcal{B}_{\text{max}}-2}\right\\},$
(12)

where $0\leq\mathcal{B}(\rho)\leq 1$ since
$\mathcal{B}_{\textmd{CHSH}}\leq\mathcal{B}_{\text{max}}=2\sqrt{2}$ for a two-
qubit system $\rho$. Notice, $\mathcal{B}_{\textmd{CHSH}}$ is the maximum
violation of Bell Clauser–Horne–Shimony–Holt (CHSH) inequality.

##  III Heisenberg XYZ model

The Hamiltonian describing a general Heisenberg XYZ model under the influence
of an inhomogeneous magnetic field, including DM and KSEA interactions, is
given by

$\hat{\mathcal{H}}_{N}=\hat{\mathcal{H}}_{H}+\hat{\mathcal{H}}_{B}+\hat{\mathcal{H}}_{\textmd{DM}}+\hat{\mathcal{H}}_{\textmd{KSEA}},$
(13)

The initial term $\hat{\mathcal{H}}_{H}$ accounts for Heisenberg spin exchange
interactions, with $\hat{\mathcal{H}}_{B}$ as the Zeeman Hamiltonian
representing the influence of the inhomogeneous external magnetic field on the
system. Additionally, $\hat{\mathcal{H}}_{\textmd{DM}}$ and
$\hat{\mathcal{H}}_{\textmd{KSEA}}$ represent the DM interaction and KSEA
interaction, respectively. This total Hamiltonian (13) can be explicitly
written as [65, 66]

$\begin{split}\hat{\mathcal{H}}_{N}=&\sum_{i=1}^{N-1}\big{\\{}J_{x}\hat{\sigma}_{i}^{x}\otimes\hat{\sigma}_{i+1}^{x}+J_{y}\hat{\sigma}_{i}^{y}\otimes\hat{\sigma}_{i+1}^{y}+J_{z}\hat{\sigma}_{i}^{z}\otimes\hat{\sigma}_{i+1}^{z}\\\
&+\overrightarrow{\mathbf{\mathcal{B}}}_{i}\cdot(\overrightarrow{\mathbf{\mathcal{\sigma}}}_{i}\otimes\mathbb{I}_{2})+\overrightarrow{\mathbf{\mathcal{B}}}_{i+1}\cdot(\mathbb{I}_{2}\otimes\overrightarrow{\mathcal{\sigma}}_{i+1})\\\
&+\overrightarrow{\mathbf{\mathcal{D}}}\cdot(\overrightarrow{\mathbf{\mathcal{\sigma}}}_{i}\times\overrightarrow{\mathbf{\mathcal{\sigma}}}_{i+1})+\overrightarrow{\mathbf{\mathcal{\sigma}}}_{i}\cdot\Gamma\cdot\overrightarrow{\mathbf{\mathcal{\sigma}}}_{i+1}\big{\\}},\end{split}$
(14)

here,

$\Gamma=\begin{pmatrix}0&\Gamma_{z}&\Gamma_{y}\\\ \Gamma_{z}&0&\Gamma_{x}\\\
\Gamma_{y}&\Gamma_{x}&0\\\ \end{pmatrix}.$ (15)

The interaction between neighboring spin sites is determined by the real
spin–spin coupling coefficients along the $x$, $y$, and $z$ directions,
denoted as $J_{k}$ for $k=x,y,z$. Antiferromagnetic coupling is indicated by
positive values of $J_{k}$, while negative values represent ferromagnetic
coupling. The standard Pauli spin operators, denoted as
$\hat{\sigma}_{i}^{k}$, define the spin vector on the $i$th site as
$\overrightarrow{\mathbf{\mathcal{\sigma}}}_{i}=(\hat{\sigma}_{i}^{x},\hat{\sigma}_{i}^{y},\hat{\sigma}_{i}^{z})$.
Moreover,
$\overrightarrow{\mathbf{\mathcal{B}}}_{i}=(B_{i}^{x},B_{i}^{y},B_{i}^{z})$
represents the magnetic flux density contributing to Zeeman splitting at the
$i$th spin site. The DM interaction vector is defined by
$\overrightarrow{\mathbf{\mathcal{D}}}=(D_{x},D_{y},D_{z})$ and serves as the
anti-symmetric exchange coupling responsible for spin–orbit coupling in
specific materials. Note that the KSEA interaction ($\Gamma$) introduces an
anisotropic symmetric interaction [65, 66, 67, 68].

Given the complexity of the model involving numerous parameters, we find it
cumbersome to maintain full generality. Instead, we simplify the scenario by
focusing on a situation involving two adjacent spin sites. In this context,
let $\overrightarrow{\mathbf{\mathcal{B}}}_{1}=(0,0,B\cos{\theta})$,
$\overrightarrow{\mathbf{\mathcal{B}}}_{2}=(0,0,B\sin{\theta})$,
$\overrightarrow{\mathbf{\mathcal{D}}}=(0,0,D_{z})$, and specify that
$\Gamma_{x}=\Gamma_{y}=0$ while $\Gamma_{z}\neq 0$. Consequently, our
Hamiltonian (14) can be expressed in a modified form applicable to any pair of
consecutive spin sites as

$\begin{split}\hat{\mathcal{H}}=&J_{x}~{}\hat{\sigma}_{1}^{x}\otimes\hat{\sigma}_{2}^{x}+J_{y}~{}\hat{\sigma}_{1}^{y}\otimes\hat{\sigma}_{2}^{y}+J_{z}~{}\hat{\sigma}_{1}^{z}\otimes\hat{\sigma}_{2}^{z}\\\
&+B~{}[\cos{\theta}(\hat{\sigma}_{1}^{z}\otimes\mathbb{I}_{2})+\sin{\theta}(\mathbb{I}_{2}\otimes\hat{\sigma}_{2}^{z})]\\\
&+D_{z}~{}(\hat{\sigma}_{1}^{x}\otimes\hat{\sigma}_{2}^{y}-\hat{\sigma}_{1}^{y}\otimes\hat{\sigma}_{2}^{x})\\\
&+\Gamma_{z}~{}(\hat{\sigma}_{1}^{x}\otimes\hat{\sigma}_{2}^{y}+\hat{\sigma}_{1}^{y}\otimes\hat{\sigma}_{2}^{x}).\end{split}$
(16)

As we know, this Hamiltonian (16) has a standard X-structured matrix form.
Using the standard computational basis
$\\{\ket{00},\ket{01},\ket{10},\ket{11}\\}$, it can be expressed in a
simplified form

$\displaystyle\hat{\mathcal{H}}=\begin{pmatrix}J_{z}+B(\cos{\theta}+\sin{\theta})&0&0&-2i\Gamma_{z}+J_{x}-J_{y}\\\
0&-J_{z}+B(\cos{\theta}-\sin{\theta})&2iD_{z}+J_{x}+J_{y}&0\\\
0&-2iD_{z}+J_{x}+J_{y}&-J_{z}-B(\cos{\theta}-\sin{\theta})&0\\\
2i\Gamma_{z}+J_{x}-J_{y}&0&0&J_{z}-B(\cos{\theta}+\sin{\theta})\end{pmatrix}.$
(17)

The diagonalization of $\hat{\mathcal{H}}$ (17) results in four eigenvalues,
given by

$\displaystyle E_{1,2}$ $\displaystyle=\pm{\mathcal{K}_{1}}+J_{z},$ (18)
$\displaystyle E_{3,4}$ $\displaystyle=\pm{\mathcal{K}_{2}}-J_{z},$

with expressions

$\mathcal{K}_{1}=\sqrt{k_{1}^{2}+B^{2}\left(\cos{\theta}+\sin{\theta}\right)^{2}}$

and

$\mathcal{K}_{2}=\sqrt{k_{2}^{2}+B^{2}\left(\cos{\theta}-\sin{\theta}\right)^{2}},$

where

$k_{1}=\sqrt{\left(J_{x}-J_{y}\right)^{2}+4\Gamma_{z}^{2}}$

and

$k_{2}=\sqrt{\left(J_{x}+J_{y}\right)^{2}+4D_{z}^{2}}.$

The state of a system in the thermal equilibrium with the absolute temperature
$T$ is described by the Gibbs density operator

$\rho_{T}=\frac{1}{\mathcal{Z}}e^{-\hat{\mathcal{H}}/\kappa_{B}T},$ (19)

where $\mathcal{Z}=\mathrm{Tr}[\exp(-\hat{\mathcal{H}}/\kappa_{B}T]$ is
partition function with the Boltzmann constant $\kappa_{B}$ (considered as
$\kappa_{B}=1$ for simplification purpose).

Using now Hamiltonian (17) and density operator (19), we obtain the following
density matrix at thermal equilibrium

$\rho_{T}=\left(\begin{array}[]{cccc}a^{-}&0&0&c\\\ 0&b^{-}&d&0\\\
0&d^{*}&b^{+}&0\\\ c^{*}&0&0&a^{+}\end{array}\right),$ (20)

where the nonzero elements read

$\displaystyle
a^{\pm}=\frac{e^{-J_{z}/T}}{\mathcal{Z}}\left\\{\cosh\left(\mathcal{K}_{1}/T\right)\pm\left[B(\cos{\theta}+\sin{\theta})/\mathcal{K}_{1}\right]\Theta_{1}\right\\},$
(21) $\displaystyle
b^{\pm}=\frac{e^{J_{z}/T}}{\mathcal{Z}}\left\\{\cosh\left(\mathcal{K}_{2}/T\right)\pm\left[B(\cos{\theta}-\sin{\theta})/\mathcal{K}_{2}\right]\Theta_{2}\right\\},$
$\displaystyle
c=-\frac{e^{-J_{z}/T}}{\mathcal{Z}}\left[\left(-2i\Gamma_{z}+J_{x}-J_{y}\right)/\mathcal{K}_{1}\right]\Theta_{1},$
$\displaystyle
d=-\frac{e^{J_{z}/T}}{\mathcal{Z}}\left[\left(2iD_{z}+J_{x}+J_{y}\right)/\mathcal{K}_{2}\right]\Theta_{2},$

where $\Theta_{(1)2}=\sinh\left(\mathcal{K}_{(1)2}/T\right)$, and

$Z=2\left[e^{-J_{z}/T}\cosh\left(\mathcal{K}_{1}/T\right)+e^{J_{z}/T}\cosh\left(\mathcal{K}_{2}/T\right)\right],$
(22)

The state presented in Eq. (20) has an X structure. Thus, the explicit
expressions of the aforementioned quantifiers based on thermal state elements
(21) can be derived straightforwardly.

For example, the $l_{1}$-norm of quantum coherence (2) can be obtained as

$Q(\rho_{T})=|c|+|c^{*}|+|d|+|d^{*}|.$ (23)

Moreover, the analytical expression of QD for our thermal state (20) would be
given by Eq. (3) with considering $\rho_{11}=a^{-}$, $\rho_{22}=b^{-}$,
$\rho_{33}=b^{+}$, $\rho_{44}=a^{+}$, $\rho_{14}=c$, and $\rho_{23}=d$.

Regarding the analytical form of QS (11), one needs to simply obtain
$\mathrm{Tr}(A_{i}\otimes B_{i}\rho_{T})$. Explicitly, we have

$\begin{split}S(\rho_{T})=&\big{\\{}8(cc^{*}+dd^{*})-4a^{-}(b^{-}+b^{+})\\\
&-4a^{+}(b^{-}-b^{+})+1\big{\\}}^{1/2}.\end{split}$ (24)

Besides, the concurrence (9) can be derived as

$C(\rho_{T})=2\max\left\\{0,t_{1},t_{2}\right\\},$ (25)

where $t_{1}=|d|-\sqrt{a^{-}a^{+}}$ and $t_{2}=|c|-\sqrt{b^{-}b^{+}}$.

Finally, the normalized form of Bell nonlocality function (12) is found to be

$\mathcal{B}(\rho_{T})=\max\left\\{0,(2\max\\{0,m_{1},m_{2}\\}-2)/(2\sqrt{2}-2)\right\\},$
(26)

where $m_{1}=\sqrt{4(|c|+|d|)^{2}+(a^{-}-b^{-}-b^{+}+a^{+})^{2}}$ and
$m_{2}=\sqrt{8(|c|^{2}+|d|^{2})}$.

Figure 2: The variations of $l_{1}$-norm quantum coherence (a), QD (b),
concurrence (c), QS (d), Bell nonlocality (e), and average fidelity (f) as a
function of temperature $T$ at $J_{x}=0.8$, $J_{y}=0.5$, $J_{z}=0.3$,
$D_{z}=0$, $\Gamma_{z}=0$ when the magnetic field is applied to single spin
($\theta=0$), and when a same magnetic field is applied to both spins
($\theta=\pi/4$). Black horizontal lines show $S(\rho_{T})=1$ in (d) and
$\mathcal{F}(\rho_{T})=2/3$ in (f).

##  IV Quantum teleportation

In this section, we examine the teleportation protocol, considering $\rho_{T}$
as the quantum channel state, assuming the input state is an arbitrary unknown
pure state consisting of two qubits such as

$|\psi_{\text{in}}\rangle=\cos(\alpha/2)|10\rangle+e^{i\phi}\sin(\alpha/2)|01\rangle,$
(27)

where $\alpha$ and $\phi$ represent the amplitude and phase of the target
state to be teleported, respectively. The quantum channel is known as a
completely positive and trace-preserving operator. Through this process, an
input state is mapped to an output state.

When a quantum state is teleported via the mixed channel $\rho_{\text{ch}}$,
the resulting output replica state $\rho_{\text{out}}$ is achieved by
performing joint measurements and local unitary transformations on the input
state $\rho_{\text{in}}=|\psi_{\text{in}}\rangle\langle\psi_{\text{in}}|$

$\rho_{\text{out
}}=\sum_{i,j\in\\{0,x,y,z\\}}p_{i}p_{j}\left(\sigma^{i}\otimes\sigma^{j}\right)\rho_{\text{in
}}\left(\sigma^{i}\otimes\sigma^{j}\right),$ (28)

where $\sigma^{0}=\mathbb{I}_{2}$, and
$p_{i}=\operatorname{Tr}\left(q^{i}\rho_{\text{ch}}\right)$ satisfies the
condition $\sum_{i}p_{i}=1$. Moreover, we have
$q^{0}=|\Psi^{-}\rangle\langle\Psi^{-}|$,
$q^{1}=|\Phi^{-}\rangle\langle\Phi^{-}|$,
$q^{2}=|\Psi^{+}\rangle\langle\Psi^{+}|$, and
$q^{3}=|\Phi^{+}\rangle\langle\Phi^{+}|$ in which $|\Psi^{\pm}\rangle$ and
$|\Phi^{\pm}\rangle$ are four Bell states. Here, let us suppose that the
quantum channel is a thermal state (20), namely $\rho_{\mathrm{ch}}=\rho_{T}$.

The quality of the teleported state is determined by the criterion of fidelity
$F(\rho_{\text{in}},\rho_{\text{out}})$, defined as

$F(\rho_{\text{in}},\rho_{\text{out}})=\left(\mathrm{Tr}\sqrt{\sqrt{\rho_{\text{in}}}\rho_{\text{out}}\sqrt{\rho_{\text{in}}}}\right)^{2}.$
(29)

Based on the above definition, the average fidelity of teleportation
$\mathcal{F}$ is given by

$\mathcal{F}=\frac{1}{4\pi}\int_{0}^{2\pi}\text{d}\phi\int_{0}^{\pi}\text{d}\alpha
F(\rho_{\text{in}},\rho_{\text{out}})\sin\alpha.$ (30)

The maximum classical average fidelity threshold is observed at
$\mathcal{F}=2/3$. Beyond this point, we transition into the quantum average
fidelity regime. The proximity of the quantum average fidelity to unity
signifies reduced information leakage, indicating optimal conditions for
quantum teleportation.

In our case (density matrix (20)), the analytical form of average fidelity is
derived by

$\mathcal{F}(\rho_{T})=\frac{2}{3}(b^{-}+b^{+})^{2}+\frac{1}{3}(a^{-}+a^{+})^{2}+\frac{1}{3}(d+d^{*})^{2}.$
(31)

##  V Results and discussion

In this investigation, our objective is to analyze various quantum coherence
and correlation measures within the Heisenberg XYZ model, using it as a
quantum resource channel for teleporting a target quantum state
$|\psi_{\text{in}}\rangle=\cos(\alpha/2)|10\rangle+e^{i\phi}\sin(\alpha/2)|01\rangle$.
The goal is to enhance average quantum teleportation fidelity beyond 2/3 by
adjusting parameters such as magnetic field, bath temperature, and DM and KSEA
interactions, thereby enhancing the channel’s quantum resources. We consider
an antiferromagnetic spin chain characterized by anisotropic spin–spin
coupling constants (a random case): $J_{x}=0.8$, $J_{y}=0.5$, and $J_{z}=0.3$.

###  V.1 Impact of Magnetic Field

First, our focus is on the impact of applying a magnetic field to either one
spin ($\theta=0$ or $\theta=\pi/2$) or to both spins ($\theta=\pi/4$) with
$B=1$ on quantum resources and teleportation fidelity in the absence of DM and
KSEA interactions. Figure 2 illustrates the variations of quantum resources
and teleportation fidelity with temperature under zero DM and KSEA
interactions, comparing two distinct fixed magnetic field values.

Figure 3:  The behaviors of $l_{1}$-norm quantum coherence (a), QD (b),
concurrence (c), QS (d), Bell nonlocality (e), and average fidelity (f) versus
temperature $T$ at $J_{x}=0.8$, $J_{y}=0.5$, $J_{z}=0.3$, $\theta=\pi/4$, and
$\Gamma_{z}=0$ for different fixed values of $D_{z}$.

In Figs. 2(a) and 2(b), the $l_{1}$ norm of quantum coherence and quantum
discord are shown as functions of temperature. At low temperatures, peak
values are higher when the same magnetic field is applied to both spins
(dashed blue curve) compared to when it is applied to just one spin (solid
green curve). Both measures decrease with increasing temperature. This result
does not surprise us because the temperature can diminish the quantum
coherence and correlation due to thermal fluctuations in the system.

Figures 2(c), 2(d), and 2(e) depict the behaviors of concurrence, QS, and Bell
nonlocality versus $T$, respectively. The entanglement captured by concurrence
disappears after $T\approx 1.9$. Note that at low temperatures, peak values of
concurrence are greater when the same magnetic field is applied to both spins
(dashed blue curve). Our findings show that for $0.7\leq T\leq 0.9$, the
channel state is steerable without nonlocality, as $S(\rho_{T})>1$ in Fig.
2(d). Between $T\approx 0.7$ and $T\approx 1.9$, the channel state remains
entangled [see Fig. 2(c)] but without nonlocality [see Fig. 2(e)]. More
precisely, Bell nonlocality decreases to zero before concurrence and discord
as the channel becomes thermally mixed. Notice, $Q(\rho_{T})$ and
$D(\rho_{T})$ show more robustness compared to $C(\rho_{T})$,
$\mathcal{S}(\rho_{T})$, and $\mathcal{B}(\rho_{T})$. Thus, the mixed
entangled states can exist without violating Bell inequalities or exhibiting
nonlocality. These highlight the hierarchical relationship between the
mentioned quantum resources.

Our results in Fig. 2 indicate that applying the same magnetic field to both
qubits significantly enhances average teleportation fidelity [see Fig. 2(f)],
quantum coherence, and correlations at low temperatures. These enhancements
diminish at higher temperatures due to thermal effects. However, fidelity
remains above the classical limit $\mathcal{F}(\rho_{T})=2/3$ at low
temperatures for both $\theta=0$ and $\theta=\pi/4$, with maximum fidelity
achieved for $\theta=\pi/4$.

From Fig. 2, we conclude that applying the same magnetic field to both qubits
enhances teleportation fidelity and quantum correlations more significantly
across different temperatures compared to applying it to just one qubit. The
hierarchy of quantum resources—quantum coherence $\supseteq$ QD $\supseteq$
entanglement $\supseteq$ QS $\supseteq$ nonlocality—depicted in Figs. 1 and 2,
demonstrates that the Bell nonlocality function is the weakest measure, while
the $l_{1}$-norm of coherence is the strongest indicator of nonclassical
characteristics.

Given that using the same magnetic field control for both spins enhances
quantum coherence and correlations as well as average teleportation fidelity
at low temperatures, we will adopt $\theta=\pi/4$ as the standard setting and
explore other model parameters to improve these measures against the
increasing of temperature.

Figure 4: The behaviors of $l_{1}$-norm quantum coherence (a), QD (b),
concurrence (c), QS (d), Bell nonlocality (e), and average fidelity (f) versus
temperature $T$ at $J_{x}=0.8$, $J_{y}=0.5$, $J_{z}=0.3$, $\theta=\pi/4$, and
$D_{z}=0$ for different fixed values of $\Gamma_{z}$.

###  V.2 Impact of DM Interaction

Next, we examine the impact of different fixed values of DM interaction,
without introducing KSEA interaction, on the quantum resources and the average
fidelity of teleportation. This analysis is presented in Fig. 3 where the
considered functions are evaluated against temperature $T$ for various fixed
values of DM interaction: $D_{z}=0.5$ (dashed black), $D_{z}=1.0$ (solid red),
$D_{z}=1.5$ (dashed green), and $D_{z}=2.0$ (solid blue).

We observe that although DM interaction does not increase the peak values of
these functions, it aids in maintaining high peak values of quantum resources
even at significantly higher temperatures (compare with Fig 2 when $D_{z}=0$).
Thus, we can consider its effect as a positive influence on all quantum
coherence and correlations of the quantum channel. For instance, when
$D_{z}=0.5$, Bell nonlocality diminishes to zero around $T\approx 0.9$,
whereas for $D_{z}=2.0$, it decreases to zero at $T\approx 2.5$, marking more
than a five-fold improvement in the sustainability of Bell nonlocality. A
similar trend is observed for QD and concurrence, with even higher $T$ values.
Notably, among all quantum resources, Bell nonlocality is the most delicate,
followed by QS.

Despite achieving maximal entanglement at near-zero temperatures, the average
teleportation fidelity remains greater than the classical limit of $2/3$,
indicating successful but not maximal teleportation. At low temperatures, the
value of teleportation fidelity slightly decreases with larger DM
interactions. However, this underscores that the maximal entanglement in a
mixed-state situation does not always guarantee ideally successful
teleportation, even at the lowest $T$ values. Thus, increasing the DM
interaction decreases the peak value of the average fidelity of teleportation
while making it stable at higher temperatures. Nevertheless, this shows that
maintaining the nonclassical characteristics at higher temperatures with
increased DM interaction does not guarantee enhanced values of average
teleportation fidelity.

In summary, increasing DM interaction generally enhances all the quantum
correlations and coherence in our work but does not significantly increase the
peak value of average teleportation fidelity, although it helps maintain these
values at higher $T$. Again, the hierarchy of quantum resources, as depicted
in Fig. 1, is held.

###  V.3 Impact of KSEA Interaction

Finally, we explore the impact of various fixed values of KSEA interaction,
without introducing DM interaction, on quantum resources and the average
fidelity of teleportation. This analysis is illustrated in Fig. 4 when we have
plotted all functions against $T$ for different fixed values of KSEA
interaction: $\Gamma_{z}=0.5$ (dashed black), $\Gamma_{z}=1.0$ (solid red),
$\Gamma_{z}=1.5$ (dashed green), and $\Gamma_{z}=2.0$ (solid blue) with zero
value of $D_{z}$. One can observe that, unlike DM interaction, KSEA
interaction increases the peak values of these measures [compare with Fig 3
when $\Gamma_{z}=0$]. Besides, it helps to maintain high peak values of
quantum coherence and quantum correlations even at significantly higher
temperatures. In general, its effect is positive on all quantum resources of
the quantum channel. For instance, when $\Gamma_{z}=0.5$, Bell nonlocality
diminishes to zero around $T\approx 0.2$, whereas for $\Gamma_{z}=2.0$, it
decreases to zero at $T\approx 1.8$, marking a more than nine-fold improvement
in the sustainability of Bell nonlocality. A similar trend is observed for QD
and concurrence. Again, among all quantum resources, Bell nonlocality remains
the most delicate, followed by QS.

Despite achieving maximal entanglement at very low temperatures, the average
teleportation fidelity only remains greater than the classical limit ($2/3$)
when KSEA interaction is very small [look at Fig. 4(f)]. Namely, at
$\Gamma_{z}=0.5$, the average fidelity of teleportation is greater than $2/3$
at near-zero temperatures, whereas increasing KSEA interaction does not help
in achieving fidelity greater than $2/3$ even at low temperatures. This shows
that increasing the KSEA interaction does not enhance teleportation fidelity
and can not guarantee successful teleportation.

Although a large value of KSEA interaction does not enhance the average
fidelity of teleportation, it is beneficial for enhancing quantum coherence
and correlations (quantum resources). Therefore, increasing KSEA interaction
enhances all the correlations and coherence but does not increase the peak
value of average teleportation fidelity. While it helps maintain these values
at higher $T$. Overall, the impact of KSEA interaction on teleportation
fidelity is negative.

Achieving a quantum advantage for teleportation is unsuccessful across all
temperatures with different fixed KSEA interaction values. Although quantum
coherence and correlations generally persist at nonzero levels for even higher
temperatures, indicating the presence of nonlocality, entanglement, discord,
and steering, successful quantum teleportation (greater than
$\mathcal{F}(\rho_{T})=2/3$) is not achieved in this scenario. Additionally,
we find that the channel state exhibits nonlocality and steerability, as well
as entanglement, yet teleportation remains unachievable.

##  VI Conclusion and outlook

In this work, we investigate the behavior of various quantum resources in the
spin-1/2 Heisenberg XYZ model with added Dzyaloshinsky-Moriya (DM) and
Kaplan–Shekhtman–Entin-Wohlman–Aharony (KSEA) interactions under sinusoidal
magnetic field. Our analysis has revealed the intricate interplay between the
measures of nonclassical characteristics, shedding light on the restricted
hierarchical classification of quantum resources: quantum coherence
$\supseteq$ quantum discord $\supseteq$ entanglement $\supseteq$ quantum
steering $\supseteq$ Bell nonlocality. Our results demonstrate that applying
the same magnetic field to both qubits significantly enhances quantum
coherence, correlations, and teleportation fidelity at low temperatures.
However, increasing the DM interaction or KSEA interaction does not
necessarily improve the peak teleportation fidelity, despite enhancing and
sustaining quantum resources at higher temperatures. We found that maximal
entanglement does not guarantee optimal teleportation, even at the lowest
temperatures, highlighting the complex relationship between teleportation
fidelity and the underlying quantum properties of the channel. This study
provides valuable insights into the nuanced interplay between quantum
coherence, correlations, and their practical implications for quantum
communication and information processing technologies. These findings
underscore the importance of carefully considering and optimizing various
quantum resources for specific applications, as enhancements in certain
measures do not necessarily translate to improvements in others.

## Acknowledgements

M.G. and S.H. were supported by Semnan University under Contract No. 21270.

## Disclosures

The authors declare that they have no known competing financial interests.

## Data availability

No datasets were generated or analyzed during the current study.

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