Skew information correlations and bipartite entanglement in hybrid qubit–qutrit system under intrinsic decoherence effect

Abstract

We investigate how bipartite entanglement and skew information correlations are affected by intrinsic decoherence in a hybrid qubit–qutrit Heisenberg XXZ model that incorporates the Dzyaloshinskii–Moriya interaction and is subject to the action of an external inhomogeneous magnetic field. We employ local quantum uncertainty and uncertainty-induced non-locality to estimate skew information correlations in the considered system, while logarithmic negativity is employed to quantify bipartite entanglement. We examine the impact of intrinsic decoherence rate, hybrid qubit–qutrit system parameters, Dzyaloshinskii–Moriya interaction strength, and external magnetic fields intensities on the dynamics of the three indicators of quantum correlations. The results indicate that quantum correlations are compromised by intrinsic decoherence; additionally, the strength of the Dzyaloshinskii–Moriya interaction reduces oscillatory tendencies and attenuates skew information correlations and bipartite entanglement within the hybrid system. However, the adverse impacts of intrinsic decoherence on quantum correlations may be alleviated by modifying hybrid-spin Heisenberg XXZ parameters. Our results show the uncertainty-induced non-locality measure is capable of detecting quantum correlations that other measures fail to detect. Furthermore, we have observed that the skew correlations can be enhanced by adjusting the mixture parameter. On a related note, our findings demonstrate that quantum correlations are reduced by the intervention of the external homogeneous magnetic field.

1 Introduction

Quantum entanglement [1,2,3], a phenomenon that occurs when two or more particles are connected in such a way that the quantum state of each particle cannot be described independently of the others. Entanglement has been widely studied and recognized as an important resource for quantum information processing [4,5,6,7,8,9,10,11]. However, it has been shown that entanglement does not necessarily capture all non-classical correlations, such as those found in separable mixed states [12, 13], which have non-zero quantum correlations but zero entanglement. These states have been explored as potential resources for the implementation of predictable quantum processing using a single qubit [14, 15]. Quantum correlations go beyond entanglement and encompass additional quantifiers, such as measurement-induced non-locality [16] and quantum discord [17], which have emerged as valuable tools for characterizing non-classical correlations. other authentic measures have been developed to assess and compare various types of quantum correlations. These measures include the relative entropy [18], Hilbert–Schmidt norm [19], trace distance [20], Bures metric [21], and Hellinger distance [22]. A notable advantage of these geometric measures is their capability to directly compare quantum and classical correlations [18, 23], as well as their applicability to multipartite states [18, 24].

Nevertheless, the utilization of entanglement and quantum correlations in practical scenarios is greatly hindered by a phenomenon known as decoherence. Decoherence has the potential to modify their characteristics and, in some cases, completely eliminate them [25]. Interestingly, there exists a type of decoherence called intrinsic decoherence, as suggested by Milburn [26], which can manifest even without any external interactions from the environment [27, 28]. Intrinsic decoherence arises from a series of unitary transformations that are random and stochastic, rather than continuous, on short timescales. The effects of intrinsic decoherence on non-classical correlations have been widely studied, particularly in multipartite systems such as qubits and qutrits [29,30,31,32,33,34,35,36,37,38,39,40]. Hybrid quantum states, such as qubit–qutrit systems, have been shown to be more resilient to intrinsic decoherence and better at maintaining quantum correlations than simpler states like bipartite and tripartite systems [41,42,43,44,45]. Thus, there is a growing interest in studying the effects of inhomogeneous magnetic fields and Dzyaloshinskii–Moriya interactions (DMI) [46, 47] on quantum correlations in Heisenberg qubit–qutrit XXZ chains. These systems are particularly useful for modeling quantum computers based on nuclear magnetic resonance [48, 49] and electron behavior in helium [50]. The presence of inhomogeneities, such as magnetic impurities and defects that are frequently present in solid-state structures, can cause disturbances in the magnetic field. Thus, it is crucial to consider these factors while examining the effects of intrinsic decoherence on quantum correlations [39]. The purpose of our investigation is to expand our knowledge regarding how intrinsic decoherence and inhomogeneities impact quantum correlations in hybrid systems under the influence of the DM interaction and external magnetic fields.

The primary objective of this research is to investigate the temporal variations of skew information correlations and logarithmic negativity in a hybrid-spin Heisenberg XXZ model that is subjected to DMI and an inhomogeneous external magnetic field. We will examine how coupling strengths, anisotropic intensity, external magnetic fields intensities, initial state mixture parameter, and DM interaction parameter affect skew information correlations and entanglement. To accomplish this, we will utilize two measures to characterize skew information correlations within the system : local quantum uncertainty $(\mathcal{U})$ and uncertainty-induced non-locality $(\mathcal{U}_\mathcal{C})$. $\mathcal{U}$ is a measure of uncertainty in a quantum state that is determined through the measurement of a single local observable and has been extensively studied (see references [51,52,53]). $(\mathcal{U}_\mathcal{C})$ is another measure of non-classical correlations that is defined as the maximal skew information [53, 54]. Furthermore, we will use logarithmic negativity $(\mathcal{L}_\mathcal{N})$ [55] to quantify entanglement.

This manuscript is structured as follows. In Sect. 2, we offer a comprehensive overview of three quantifiers employed in this manuscript: logarithmic negativity, uncertainty-induced non-locality, and local quantum uncertainty. Section 3, delves into the specifics of the qubit–qutrit system under investigation, including its diagonalization and the impact of intrinsic decoherence. Moving on to Sect. 4, we examine how the aforementioned quantifiers behave in the hybrid system being studied and provide a discussion of our discoveries. Finally, in Sect. 5, we summarize the key findings from our research.

2 Quantifiers of quantum correlations

The three quantifiers used in this study are defined in this section. These are logarithmic negativity $(\mathcal{L}_\mathcal{N})$, local quantum uncertainty $(\mathcal{U})$, and uncertainty-induced non-locality $(\mathcal{U}_\mathcal{C})$ as indicators of quantum entanglement and skew information correlations, respectively.

2.1 Logarithmic negativity

The logarithmic negativity $(\mathcal{L}_\mathcal{N})$ is an important indicator of entanglement for quantum states [55, 56], which can be easily computed. It is given for a state ${\hat{D}}$ acting on a Hilbert space $H={H}^{A} \otimes {H}^{B}$, as follows:

$$\begin{aligned} \mathcal{L}_ \mathcal{N}({\hat{D}})=\log _2|{\hat{D}}^{T_2}|_1. \end{aligned}$$

(1)

Here, ${\hat{D}}^{T_2}$ denotes the partially transposed bipartite density operator ${\hat{D}}$ with respect to the second party B [57, 58]. The trace norm of an operator represented by $|.|_1$ in equation (1) is given by:

$$\begin{aligned} |\kappa |_1=Tr\left( \sqrt{\kappa ^{\dagger }\kappa }\right) . \end{aligned}$$

(2)

$\mathcal{L}_\mathcal{N}({\hat{D}})$ is an entanglement monotone under deterministic LOCC operations and an additive quantity. The calculation of $\mathcal{L}_\mathcal{N}({\hat{D}})$ using the absolute eigenvalues ${\nu _i }$ of the partial transpose density operator $({{\hat{D}}})^{T_{2}}$ is given by:

$$\begin{aligned} \mathcal{L}_\mathcal{N}({\hat{D}})= \log _{2} \left( \sum _{i} |\nu _i| \right) . \end{aligned}$$

(3)

A value of $\mathcal{L}_\mathcal{N}({\hat{D}})=0$ indicates that the state ${\hat{D}}$ is entirely separable, while a value of $\mathcal{L}_\mathcal{N}({\hat{D}})=1$ indicates that the density matrix ${\hat{D}}$ has maximal entanglement.

2.2 Uncertainty-induced non-locality

We will employ uncertainty-induced quantum non-locality ($\mathcal{U}_\mathcal{C}$) as a genuine non-local correlation quantifier, which is one of the non-classical correlation quantifiers associated with the Wigner–Yanase skew information (WYSI) concept ${\mathcal {I}}({\hat{D}}, \Gamma _A \otimes {{\mathbb {I}}_B})$ [59, 60]. The quantifier ($\mathcal{U}_\mathcal{C}$), which was introduced by Wu et al. in [61], is used to detect the amount of non-classical correlations in multipartite systems [60]. For any bipartite state ${\hat{D}}$, the quantifier $\mathcal{U}_\mathcal{C}$ is defined as follows [61]:

$$\begin{aligned} \mathcal {U}_\mathcal{C}({\hat{D}})=\max _{\Gamma ^{C}}{\mathcal {I}}({\hat{D}}, \Gamma ^{C}), \end{aligned}$$

(4)

where the quantity

$$\begin{aligned} {\mathcal {I}}({\hat{D}}, \Gamma )=-\frac{1}{2}Tr\left[ \sqrt{{\hat{D}}}, \Gamma \right] ^{2}, \end{aligned}$$

(5)

is used to quantify the uncertainty produced when the observable $\Gamma$ acts on the density matrix ${\hat{D}}$. The maximization in (4) is applied to all local maximally informative commuting observables $\Gamma ^{C}=\Gamma _{A}^{C}\otimes {\mathbb {I}}_{B}$, where $\Gamma ^{C}_{A}$ is a Hermitian operator acting on qubit A with a distinct spectrum and ${\mathbb {I}}_{B}$ is the identity matrix associated with the second qubit. The form of $\mathcal{U}_\mathcal{C}$ for each ($2\otimes d$)-dimensional quantum system ${\hat{D}}$ is precisely defined as [61].

$$\begin{aligned} {\mathcal {U}}_\mathcal{C}({\hat{D}})=\left\{ \begin{array}{cc} 1-\Lambda _{min}({\mathcal {W}}), &{} \quad \pmb {r}=\pmb {0} \\ 1-\frac{1}{|\pmb {r}|^{2}}\,\pmb {r}\, {\mathcal {W}}\,\pmb {r}^{T}, &{} \quad \pmb {r} \ne \pmb {0} \end{array}\right. \end{aligned}$$

(6)

with $\pmb {r}^{T}$ denotes the transpose of the Bloch vector $\pmb {r}$, and $\Lambda _{min}({\mathcal {W}})$ represents the smaller eigenvalue of the symmetric matrix ${\mathcal {W}}_{3\times 3}$ whose elements are as follows

$$\begin{aligned} \Big ({\mathcal {W}}\Big )_{ij} \equiv \textrm{Tr} \left\{ \sqrt{{\hat{D}}} \left( \sigma _{Ai}\otimes {\mathbb {I}}_{B}\right) \sqrt{{\hat{D}}} \left( \sigma _{Aj}\otimes {\mathbb {I}}_{B}\right) \right\} \end{aligned}$$

(7)

with $\sigma _{Ai} (i={x},{y},{z})$ being the 1/2-spin operators on 1st subsystem (qubit).

2.3 Local quantum uncertainty

We employ local quantum uncertainty (LQU) as the second indicator of skew information correlations. LQU is a non-classical correlation quantifier related to the Wigner–Yanase skew information (WYSI) notion ${\mathcal {I}}({\hat{D}}, \Gamma _A \otimes {\mathbb {I}}_B)$ [59]. We consider a bipartite system with a density matrix ${\hat{D}}$, and LQU with respect to the first subsystem is defined as [62]:

$$\begin{aligned} {\mathcal {U}}({\hat{D}}) \equiv \min _{\Gamma _A} {\mathcal {I}}({\hat{D}}, \Gamma _A \otimes {\mathbb {I}}_B), \end{aligned}$$

(8)

with $\Gamma _A$ acting as the local observable on subsystem A and ${\mathbb {I}}_B$ acting as the identity operator on subsystem B and

$$\begin{aligned} {\mathcal {I}}({\hat{D}}, \Gamma _A \otimes {\mathbb {I}}_B)=-\frac{1}{2}\textrm{Tr}([\sqrt{{\hat{D}}}, \Gamma _A \otimes {\mathbb {I}}_B]^{2}). \end{aligned}$$

(9)

If there exists a local operator $\Gamma _A$ for which ${\mathcal {I}}({\hat{D}}, \Gamma _A \otimes {\mathbb {I}}_B) =0$, then the bipartite quantum system represented by the state ${\hat{D}}$ does not exhibit any quantum correlation. To compute $\mathcal{U}$ analytically, we minimize Eq. (8) over all observables that act on the first part of the bipartite system. Specifically, for a $2 \otimes d$-dimensional quantum system, the formula for $\mathcal{U}$ with respect to subsystem A is provided by [60]:

$$\begin{aligned} \mathcal{U}\left( {{{\hat{D}}}} \right) = 1 - \max \left( {{\Lambda _{1}},{\Lambda _{2}},{\Lambda _{3}}} \right) , \end{aligned}$$

(10)

where $\Lambda _{i=1,2,3}$ represent the eigenvalues of the symmetric matrix ${\mathcal {W}}_{3\times 3}$ Eq. (7).

3 (1/2; 1) mixed-spin XXZ model under intrinsic decoherence(ID) and with DM interaction

This section examines the Heisenberg qubit–qutrit XXZ chain that is subjected to an inhomogeneous magnetic field and a DM interaction. The following is the Hamiltonian that characterizes such a hybrid system.

$$\begin{aligned} H_{{XXZ}} & = J\left( {s_{{1x}} S_{{2x}} + s_{{1y}} S_{{2y}} + \delta ~s_{{1z}} S_{{2z}} } \right) + D_{z} \left( {s_{{1x}} S_{{2y}} - s_{{1y}} S_{{2x}} } \right) \\ & + (B + b)S_{{2z}} + (B - b)s_{{1z}} , \\ \end{aligned}$$

(11)

where $S_{2\mu }$ and $s_{1\mu }$ $(\mu = x; y; z)$ stand for the spatial components of the spin-1 (qutrit) and spin-1/2 (qubit) operators, respectively. The qutrit operators take the following form

$$\begin{aligned} S_{x}=\frac{\hbar }{\sqrt{2}}\left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{}1 \\ 0 &{} 1 &{} 0 \end{array} \right) , \quad S_{y}=\frac{\hbar }{\sqrt{2}}\left( \begin{array}{ccc} 0 &{} -i &{} 0 \\ i &{} 0 &{}-i \\ 0 &{} i &{} 0 \end{array} \right) , \quad S_{z}=\hbar \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{}0 \\ 0 &{} 0 &{} -1 \end{array} \right) . \end{aligned}$$

(12)

The parameter $\delta$ represents the XXZ exchange anisotropy of this exchange interaction, and the parameter $D_{z}$ corresponds to the intensity of the DMI oriented along the z-direction. The coupling constant J designates the Heisenberg exchange interaction between spin-1/2 and spin-1 particles. If $J > 0$, this is equivalent to the anti-ferromagnetic interaction between sites of the model; while if $J < 0$, this is the ferromagnetic interaction. Moreover, a static external magnetic field is indicated by the parameter B. And b represents the magnetic field’s degree of inhomogeneity. We notice that we are working in units so that the parameters: b, $D_z$, B, $\delta$ and J are dimensionless. Here we specifies that the basis vectors $\left| 0 \right\rangle _1$ and $\left| 1 \right\rangle _1$ form the eigenbasis of z-component of spin-$\frac{1}{2}$ operator $s_{1z}$ while $\left| {{\textbf {0}}} \right\rangle _2, \left| {{\textbf {1}}} \right\rangle _2$ and $\left| {{\textbf {2}}} \right\rangle _2$ form the eigenbasis of spin-1 operators $S_{2z}$ along the z-axis. In the computational basis, the Hamiltonian (11) has the following matrix representation.

$$\begin{aligned} H_{XXZ}=\left( \begin{array}{cccccc} \frac{b+J\delta +3B}{2}&{}0&{}0&{}0&{}0&{}0\\ 0&{}\frac{-b+B}{2}&{}0&{}\frac{(J+iD_z)}{\sqrt{2}}&{}0&{}0\\ 0&{}0&{}\frac{-3b-J\delta -B}{2}&{}0&{}\frac{(J+iD_z)}{\sqrt{2}}&{}0\\ 0&{}\frac{(J-iD_z)}{\sqrt{2}}&{}0&{}\frac{3b-J\delta +B}{2}&{}0&{}0\\ 0&{}0&{}\frac{(J-iD_z)}{\sqrt{2}}&{}0&{}\frac{b-B}{2}&{}0\\ 0&{}0&{}0&{}0&{}0&{}\frac{-b+J\delta -3B}{2}\\ \end{array} \right) . \end{aligned}$$

(13)

Using extremely basic calculations (by setting $\hbar =1$), the eigenvalues of the Hamiltonian $H_{XXZ}$ (11) are determined as

$$\begin{aligned} {\mathcal {R}}_{1,2}=\frac{1}{2}\left( \mp b\mp 3B+J\delta \right) , {\mathcal {R}}_{3,4}=\frac{1}{4}\left( 2b+2B-J\delta \mp \varepsilon \right) , {\mathcal {R}}_{5,6}=\frac{1}{4}\left( -2b-2B-J\delta \mp \vartheta \right) \end{aligned}$$

(14)

with $\varepsilon =\sqrt{8(D_{z})^{2}+8(J)^{2}+(J\delta -4b)^{2}}$ and $\vartheta =\sqrt{8(D_{z})^{2}+8(J)^{2}+(J\delta +4b)^{2}}$. The corresponding orthonormal eigenvectors can be expressed as

$$\begin{aligned} \begin{array}{cc} \left| {U}_{1}\right\rangle =\left| 1,{{\textbf {2}}} \right\rangle , &{} \left| {U}_{2}\right\rangle =\left| 0,{{\textbf {0}}}\right\rangle \\ \left| {U}_{3}\right\rangle =\frac{1}{\sqrt{1+\frac{1}{8}|\eta _{+}|^{2}}}\left( -\frac{\eta _{+}}{2\sqrt{2}}i\left| 0,{{\textbf {1}}} \right\rangle +\left| 1,{{\textbf {0}}} \right\rangle \right) , &{} \left| {U}_{4}\right\rangle =\frac{1}{\sqrt{1+\frac{1}{8}|\eta _{-}|^{2}}}\left( -\frac{\eta _{-}}{2\sqrt{2}}i\left| 0,{{\textbf {1}}} \right\rangle +\left| 1,{{\textbf {0}}} \right\rangle \right) \\ \left| {U}_{5}\right\rangle =\frac{1}{\sqrt{1+\frac{1}{8}|\upsilon _{+}|^{2}}}\left( -\frac{\upsilon _{+}}{2\sqrt{2}}i\left| 0,{{\textbf {2}}} \right\rangle +\left| 1,{{\textbf {1}}} \right\rangle \right) , &{}\left| {U}_{6}\right\rangle =\frac{1}{\sqrt{1+\frac{1}{8}|\upsilon _{-}|^{2}}}\left( -\frac{\upsilon _{-}}{2\sqrt{2}}i\left| 0,{{\textbf {2}}} \right\rangle +\left| 1,{{\textbf {1}}} \right\rangle \right) , \end{array} \end{aligned}$$

(15)

where $\eta _{\pm }=\frac{(4b-J\delta )\pm \varepsilon }{(D_{z}+iJ)}$ and $\upsilon _{\pm }=\frac{4b+J\delta \pm \vartheta }{(D_{z}+iJ)}$.

Introducing the Milburn decoherence model [26], which postulates that quantum systems undergo a sequence of similar unitary transformations instead of a unitary evolution, we incorporate the influence of ID. The evolution is governed by the following equation [26].

$$\begin{aligned} \frac{d{\hat{D}}(t)}{dt} = \frac{1}{\gamma }\Bigl [e^{-i\gamma H_{XXZ}}{\hat{D}}(t) e^{i\gamma H_{XXZ}}-{\hat{D}}(t)\Bigr ], \end{aligned}$$

(16)

where ${\hat{D}}(t)$ is the evolved state that relates to the Hamiltonian $H_{XXZ}$ and $\gamma$ is the ID rate. In the limit of $\gamma ^{-1} \rightarrow \infty$, Eq. (16) is reduced to the typical von Neumann equation characterizing the Heisenberg qubit–qutrit XXZ chain evolution $(\frac{d{\hat{D}}(t)}{dt}=-i [H_{XXZ},{\hat{D}}(t)])$. Milburn introduced a novel term to the Schrödinger equation, enabling the spontaneous loss of quantum coherence during the quantum system’s evolution, which does not require the involvement of the reservoir or the conventional energy dissipation that typically occurs with decay. The equation can be obtained using the following derivation

$$\begin{aligned} \frac{d{\hat{D}}(t)}{dt}=-i [H_{XXZ},{\hat{D}}(t)]-\frac{\gamma }{2}[H_{XXZ},[H_{XXZ},{\hat{D}}(t)]]. \end{aligned}$$

(17)

The presence of the term $\frac{\gamma }{2}[H_{XXZ},[H_{XXZ},{\hat{D}}(t)]]$ signifies the non-unitary evolution resulting from the ID in the hybrid-spin Heisenberg XXZ model we are investigating. The solution of Eq. (17) is obtained using the Kraus operators $M_{{\hat{f}}}$, [26, 63,64,65]

$$\begin{aligned} {\hat{D}}(t)= \sum ^{\infty }_{{{\hat{f}}}=0} M_{{\hat{f}}}(t) {\hat{D}}(0)M^{\dagger } _{{\hat{f}}}(t). \end{aligned}$$

(18)

Here, ${\hat{D}}(0)$ denotes the density matrix of the hybrid system at the initial time $t=0$ and $M_{{\hat{f}}}(t)$ are given by

$$\begin{aligned} M_{{\hat{f}}}(t) = \sqrt{\frac{(\gamma t)^{{{\hat{f}}}}}{{{\hat{f}}}!}} {H_{XXZ}}^{{\hat{f}}} \exp \left( -i{H_{XXZ}}t\right) \exp \left( -\frac{\gamma t}{2}{H_{XXZ}}^{2}\right) \quad \text {with} \quad \sum ^{\infty }_{{{\hat{f}}}=0} M_{{\hat{f}}}(t) M^{\dagger }_{{\hat{f}}}(t) ={\mathbb {I}}. \end{aligned}$$

The expression for the evolved density matrix of the qubit–qutrit Heisenberg XXZ model, taking into account the effects of intrinsic decoherence, can be presented in the following form

$$\begin{aligned} {\hat{D}}(t) = \sum _{{{\hat{l}}},{{\hat{m}}}} \exp \left[ -\frac{\gamma t}{2}({{\mathcal {R}}}_{{\hat{l}}}-{{\mathcal {R}}}_{{{\hat{m}}}})^{2}- i({{\mathcal {R}}}_{{{\hat{l}}}}-{{\mathcal {R}}}_{{{\hat{m}}}})t\right] \langle {U}_{{{\hat{l}}}}\vert {\hat{D}}(0)\vert {U}_{{{\hat{m}}}}\rangle \vert {U}_{{{\hat{l}}}}\rangle \langle {U}_{{{\hat{m}}}}\vert . \end{aligned}$$

(19)

Here, ${{\mathcal {R}}}_{{{\hat{l}}},{{\hat{m}}}}$ and ${|{{U}_{{{\hat{l}}},{{\hat{m}}}}}\rangle }$ are, respectively, the eigenvalues and related eigenstates of the Hamiltonian $H_{XXZ}$ defined in equation (11). Equation (19) describes how quantum coherence is degraded as the system evolves in the basis ${|{{U}_{{{\hat{l}}},{{\hat{m}}}}}\rangle }$ corresponding to the system.

The main focus of this research is to examine the influence of intrinsic decoherence on the entanglement and skew information correlations in a hybrid spin (1/2; 1) Heisenberg XXZ chain system that is subject to DMI. We will use Local quantum uncertainty, Uncertainty-induced non-locality, and $\mathcal{L}_\mathcal{N}$ to assess the impact of intrinsic decoherence on the quantum resources of the qubit–qutrit hybrid system. To analyze this, we will use a particular set of qubit–qutrit states, denoted by ${\hat{D}}(0)$, that are characterized by a single parameter. These states consist of a qubit system that interacts locally with a qutrit system and are used to initialize the quantum qubit–qutrit model. The considered initial state ${\hat{D}}(0)$ can be defined in the computational basis as

$$\begin{aligned} {\hat{D}}(0)&= \frac{r}{2}\left( \left| 0{{\textbf {0}}}\right\rangle \left\langle 0{{\textbf {0}}}\right| +\left| 0{{\textbf {1}}}\right\rangle \left\langle 0{{\textbf {1}}}\right| +\left| 1{{\textbf {2}}}\right\rangle \left\langle 1{{\textbf {2}}}\right| +\left| 1{{\textbf {1}}}\right\rangle \left\langle 1{{\textbf {1}}}\right| +\left| 1{{\textbf {2}}}\right\rangle \left\langle 0{{\textbf {0}}}\right| +\left| 0{{\textbf {0}}}\right\rangle \left\langle 1{{\textbf {2}}}\right| \right) \nonumber \\ {}&+\frac{1-2r}{2}\left( \left| 0{{\textbf {2}}}\right\rangle \left\langle 0{{\textbf {2}}}\right| +\left| 0{{\textbf {2}}}\right\rangle \left\langle 1{{\textbf {0}}}\right| +\left| 1{{\textbf {0}}}\right\rangle \left\langle 0{{\textbf {2}}}\right| +\left| 1{{\textbf {0}}}\right\rangle \left\langle 1{{\textbf {0}}}\right| \right) . \end{aligned}$$

(20)

The mixture parameter r is limited to the interval $0\leqslant r\leqslant \frac{1}{2}$, which ensures the positivity condition of ${\hat{D}}(0)$. It is worth noting that the initial state ${\hat{D}}(0)$ is entangled for all values of r within this range, except for the special case of $r=\frac{1}{3}$. In the standard basis $\{{|{i{{\textbf {j}}}}\rangle }\}_{i=0,1;{{\textbf {j}}}={{\textbf {0}}},{{\textbf {1}}},{{\textbf {2}}}}$, the initial state (20) can be expressed as a matrix as

$$\hat{D}(0) = \left[ {\begin{array}{*{20}c} {\frac{r}{2}} & 0 & 0 & 0 & 0 & {\frac{r}{2}} \\ 0 & {\frac{r}{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{{1 - 2r}}{2}} & {\frac{{1 - 2r}}{2}} & 0 & 0 \\ 0 & 0 & {\frac{{1 - 2r}}{2}} & {\frac{{1 - 2r}}{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{r}{2}} & 0 \\ {\frac{r}{2}} & 0 & 0 & 0 & 0 & {\frac{r}{2}} \\ \end{array} } \right].$$

(21)

After considering Eq. (19), one can express the time-varying density matrix that is utilized to evaluate the development of quantum correlations in the spin (1/2; 1) Heisenberg XXZ chain model with a qubit–qutrit hybrid as shown below:

$$\begin{aligned} {\hat{D}}(t)= {\begin{bmatrix} {\hat{D}}_{11}(t)&{}0&{}0&{}0&{}0&{}{\hat{D}}_{16}(t)\\ 0&{}{\hat{D}}_{22}(t)&{}{\hat{D}}_{23}(t)&{}{\hat{D}}_{24}(t)&{}{\hat{D}}_{25}(t)&{}0\\ 0&{}{\hat{D}}_{32}(t)&{}{\hat{D}}_{33}(t)&{}{\hat{D}}_{34}(t)&{}{\hat{D}}_{35}(t)&{}0\\ 0&{}{\hat{D}}_{42}(t)&{}{\hat{D}}_{43}(t)&{}{\hat{D}}_{44}(t)&{}{\hat{D}}_{45}(t)&{}0\\ 0&{}{\hat{D}}_{52}(t)&{}{\hat{D}}_{53}(t)&{}{\hat{D}}_{54}(t)&{}{\hat{D}}_{55}(t)&{}0\\ {\hat{D}}_{61}(t)&{}0&{}0&{}0&{}0&{}{\hat{D}}_{66}(t)\\ \end{bmatrix}}. \end{aligned}$$

(22)

In the appendix, we shall delve into the matrix elements of ${\hat{D}}(t)$ in detail. It is important to note that the X form of ${\hat{D}}(t)$ does not remain constant during its temporal evolution. This stands in contrast to other bipartite systems, such as the qubit–qubit system, where previous research has shown that the system’s initial state structure is preserved throughout its temporal dynamics.

Our examination will focus on the dynamics of three quantifiers: $\mathcal{L}_\mathcal{N}$, $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$. Each of the three quantifiers, $\mathcal{L}_\mathcal{N}$, $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$, serve to characterize different types of quantum correlations. The $\mathcal{L}_\mathcal{N}$ quantifies the degree of entanglement by taking the logarithm of the negativity, which is calculated as the absolute sum of the negative eigenvalues resulting from the partial transpose of the density matrix of the system. The UIN ($\mathcal{U}_\mathcal{C}$) is determined by the maximum skew information attainable between a specific bipartite quantum state and a locally commuting observable. It serves to measure the degree of non-locality present within the specified quantum state. The LQU ($\mathcal{U}$), is characterized as the minimal skew information and seeks to measure the smallest amount of quantum uncertainty that arises in a quantum state when a single local observable is measured.

To elucidate the distinctions and resemblances in the behavior of the three aforementioned quantifiers, we conduct a comparative analysis, examining their responses in relation to the intrinsic decoherence rate, parameters of the hybrid qubit–qutrit Heisenberg XXZ system, the strength of DM interaction and the mixing parameter of the initial state.

4 Results and discussion

In this section, we aim to delve into the impact of various parameters on the dynamics of quantum correlations in a qubit–qutrit Heisenberg XXZ model. Our investigation will primarily focus on examining how different parameters, including coupling strengths denoted as J, anisotropic intensity represented by $\delta$, magnetic field intensities denoted as B and b, mixture parameter indicated as r, and DM interaction denoted as $D_z$, influence skew information correlations and entanglement. The ultimate objective is to identify the optimal values for these quantities that can enhance the skew information correlations within the system under consideration. To quantify quantum entanglement, we will employ the notation $\mathcal{L}_\mathcal{N}$, while both $\mathcal{U}$ and $\mathcal{U}_\mathcal{C}$ will be used to quantify the skew information correlations.Our study aims to achieve a better understanding of how different parameters of the Heisenberg XXZ hybrid-spin model interact, and how they affect quantum correlations within the system. To begin our analysis, we will investigate the effects of intrinsic decoherence on quantum correlations. We will consider a scenario where the system is initially configured in an entangled state with $r=0.5$, and the coupling strengths $J=4>0$, indicating anti-parallel spin orientation in anti-ferromagnetic materials. Figure 1 displays the results of our analysis.

Fig. 1

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) and a comparison of the three measures’ results (d) in terms of t and $\gamma$.

Fig. 1 illustrates the presence of three distinct scenarios for each quantifier, which arise due to variations in the intrinsic decoherence (ID) rate, $\gamma$. These scenarios exhibit different characteristics depending on the specific value of $\gamma$. Notably, as the intrinsic decoherence rate, $\gamma$, surpasses a critical threshold denoted as $\gamma _e$, the system undergoes a transition. The precise value of this critical rate, $\gamma _e$, is contingent upon the system configuration. At this critical point, the damping effect becomes dominant, rendering oscillations impossible.

• First, when intrinsic decoherence is not present ($\gamma =0$), we note that the correlations display a uniform oscillatory pattern, leading to a regular sequence of oscillations of skew information correlations and bipartite entanglement within the examined system.

• Second, when $\gamma <\gamma _e$, we witness a phenomenon of damped oscillations. In this case, the amplitude experiences a gradual decline until it settles into a stable state that deviates from zero. The quantifiers that measure correlations in the system, namely $\mathcal{U}_\mathcal{C}({\hat{D}})$ (Fig. 1a) and $\mathcal{U}({\hat{D}})$ (Fig. 1b), ultimately stabilize at approximately 0.75, while $\mathcal{L}_\mathcal{N}({\hat{D}})$ (Fig. 1c) dwindles to zero. The extent of damping decreases with decreasing values of $\gamma$, and the intrinsic decoherence effect causes a dissipation of quantum resources within the qubit–qutrit hybrid system.

• When the ID rate, $\gamma$, surpasses the critical threshold $\gamma _e$, it becomes evident that excessive damping takes place. This excessive damping leads to the absence of oscillations or rapid decay of correlations, ultimately reaching a steady state. In this steady state, the quantity $\mathcal{L}_\mathcal{N}({\hat{D}})$ attains a value of zero, while $\mathcal{U}_\mathcal{C}({\hat{D}})$ and $\mathcal{U}({\hat{D}})$ approximately reach a value of 0.75.

Figure 1 highlights that even in separable hybrid systems, a significant quantity of skew information correlations persists under intrinsic decoherence. Moreover, the steady-state value achieved by the skew information quantifiers remains unchanged when the intrinsic decoherence rate $\gamma$ is varied, except for $\mathcal{L}_\mathcal{N}$, which completely vanishes at a later time t. In Fig. 1d, it is observed that at the onset ($t=0$), both $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$ initially capture a more considerable amount of quantum correlations, approximately 0.87, in contrast to $\mathcal{L}_\mathcal{N}$. However, with the passage of time, $\mathcal{U}_\mathcal{C}$ captures more quantum correlations than $\mathcal{U}$, ultimately converging to a stable state of roughly 0.75. In contrast, $\mathcal{L}_\mathcal{N}$ completely vanishes at large time t.

Next, Fig. 2 displays the time-dependent behaviors of the three measures, namely $\mathcal{U}_\mathcal{C}$, $\mathcal{U}$, and $\mathcal{L}_\mathcal{N}$, for various values of the parameter J while keeping the anisotropy parameter $\delta$ fixed at 0.5. These figures provide a more crucial findings of the relationship between the parameter J and the quantum correlation quantifiers employed in this investigation.

Fig. 2

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) in terms of t and J.

It is worth mentioning that the oscillations’ frequency and damping time of the system are closely related to the coupling constant J. When the value of J decreases, the amplitudes of the oscillations increase, and the damping time of the correlations becomes longer. Besides, the steady-state value of the system also rises as J decreases. Our analysis reveals that $\mathcal{U}_\mathcal{C}$ (Fig. 2a) and $\mathcal{U}$ (Fig. 2b) are better at capturing quantum correlations than $\mathcal{L}_\mathcal{N}$ (Fig. 2c), and it is worth noting that while $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$ remain stable, $\mathcal{L}_\mathcal{N}$ tends towards zero. This implies that even in the absence of quantum entanglement, the quantum correlations captured by $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$ persist in the system and can still be utilized for some quantum tasks. Furthermore, we observe that for each value of J, the frozen states attained by $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$ are almost identical. To provide a more detailed comparison between the three quantifiers, we focus on the case where $J=2$. At the initial time $t=0$, both $\mathcal{U}_\mathcal{C}$ (Fig. 2a) and $\mathcal{U}$ (Fig. 2b) are initially equal to 0.87, while $\mathcal{L}_\mathcal{N}$ is nearly equal to 0.58. After a long time, the quantum correlations stabilize at 0.75, while $\mathcal{L}_\mathcal{N}$ approaches zero, indicating the absence of quantum entanglement. Even after $t > 40$, when entanglement no longer exists, significant quantum correlations still remain in the system. The most significant outcome from our analysis is that the hybrid-spin chain model exhibits greater resilience to intrinsic decoherence when the coupling strength J is weak. This finding is highly relevant to quantum computing and information processing.

In the following, we shall explore in Fig. 3, the relationship between quantum correlation quantifiers, the coupling constant J, and the anisotropic intensity $\delta$. It is evident that any alterations in the quantum correlation quantifiers resulting from an increase in $\delta$ are symmetrically distributed around $\delta =0$.

Fig. 3

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) in terms of J and $\delta$.

It is important to note that the curve of quantum correlations’ variation decreases as the absolute value of parameter $\delta$ increases, with the presence of some fluctuation. This behavior could be attributed to the coupling strength of the term $s_{1z}S_{2z}$ in the Hamiltonian of the system, which is proportional to the $\delta$ parameter. Additionally, the fluctuations in the quantum correlation quantifiers are influenced by the value of parameter J, with smaller values resulting in lower quantifier fluctuations. Another notable observation is that the highest values of correlations for each quantifier occur around $\delta =0$, possibly due to the same reason previously stated; i.e., the coupling strength of the $s_{1z}S_{2z}$ term in the Hamiltonian becomes weak along the z-direction when The parameter $\delta$ is approximately close to zero. To gain a more comprehensive comprehension of the impact of the anisotropy parameter $\delta$ on quantum correlations, we performed an extensive examination of the dynamic variations in the measurements $\mathcal{U}_\mathcal{C}$, $\mathcal{U}$, and $\mathcal{L}_\mathcal{N}$. This analysis encompassed a comparison between two distinct $\delta$ values, while keeping the parameter J constant. The results of our investigation are visually depicted in (Fig. 4).

Fig. 4

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) and a comparison of the three measures’ results (d) in terms of t and $\delta$.

At the initial time $t=0$, the quantum correlations captured in Fig. 4a and b for both $\delta$ values are indistinguishable, as shown by the overlapping curves. However, as time progresses, a slight difference arises in the amount of quantum correlations captured between $\delta =0.2$ and $\delta =0.7$. This observation suggests that the amplitude of quantum correlations is slightly affected by this small variation of the parameter $\delta$. Figure 4c presents an interesting finding where the time required for the bipartite entanglement to completely vanish increases with decreasing anisotropy intensity $\delta$, while the steady-state value remains unaffected by $\delta$. Moreover, Fig. 4d shows that $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$ capture more quantum correlations than $\mathcal{L}_\mathcal{N}$, with the former two functions converging to a stationary value of 0.75, while the latter function tends towards zero. Although $\mathcal{U}_\mathcal{C}$ has a slightly higher initial value than $\mathcal{U}$, both eventually stabilize at the same non-zero steady-state value. It is noteworthy that the plot representing $\mathcal{L}_\mathcal{N}$ exhibits death and revival phenomena before reaching a stationary state, during which $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$ overlap.

Next, we illustrate in (Fig. 5)a, b and c the time evolution of $\mathcal{U}_\mathcal{C}$, $\mathcal{U}$ and $\mathcal{L}_\mathcal{N}$ for different values of the DM interaction intensity $D_z$.

Fig. 5

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) and a comparison of the three measures’ results (d) in terms of t and $D_z$.

As the value of $D_z$ increases, we observe that the frequency of oscillations for the three quantifiers increases, but the fluctuations decrease, and the amounts of quantum correlations stabilize at non-zero values. The entanglement disappears completely as $D_z$ increases. Despite having smaller amplitudes between their lower and upper bounds, higher values of $\mathcal{U}_\mathcal{C}$ (Fig. 5a) and $\mathcal{U}$ (Fig. 5b) are observed compared to $\mathcal{L}_\mathcal{N}$. The steady-state values for all three quantifiers remain the same regardless of the $D_z$ value, except for $D_z=0$, where the steady-state value is higher. This indicates that the presence of a DM interaction along the z-direction makes the system less resistant to intrinsic decoherence effects. Figure 5c illustrates the emergence and disappearance of entanglement, with the rate of these phenomena decreasing as $D_z$ increases, ultimately resulting in complete separability of the system. This phenomenon is more clearly observed by extending the time interval in Fig. 5c. Increasing the strength of the DM interaction, represented by the $D_z$ value, reduces or even suppresses the quantum correlation fluctuations within the system. This suppression of fluctuations can be observed in the comparative plot in Fig. 5d, where we can see that fluctuations within the system are suppressed after a certain period of time, while quantum correlations continue to decline before stabilizing.

Now we show in Fig. 6, the effects of the external homogeneous magnetic field on the behaviors of the three quantum correlation measures.

Fig. 6

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) and a comparison of the three measures’ results (d) in terms of t and B.

Upon analyzing the results, we observe that the initial quantum correlations are equal among the different magnetic field (B) values for each quantifier. The intrinsic decoherence process leads to a reduction in the quantum correlations of the system. When the Heisenberg chain is subjected to a high-intensity magnetic field, the amplitudes of the quantifiers become very small. As B increases, the fluctuations diminish rapidly, and the quantum correlations stabilize at non-zero values, while entanglement completely disappears. Although $\mathcal{U}_\mathcal{C}$ (Fig. 6a) and $\mathcal{U}$ (Fig. 6b) have higher values compared to $\mathcal{L}_\mathcal{N}$ (Fig. 6c), the ranges between their lower and upper bounds are narrower. It is seen also that each quantifier reach the same steady states regardless of the value of B. In Fig. 6c, we can observe the birth and death of entanglement, but as B increases, the system becomes fully separable, resulting in a decrease in the occurrence rate of these events. Therefore, we can conclude that in the hybrid-spin chain model, if the magnetic field is extremely strong, the quantum correlations initially remain at a low level and eventually settle at a significant steady state. Moreover, irrespective of the value of B, each quantifier attain identical steady state. To address the question of how to enhance the value of the steady state, we can refer to Fig. 7, which illustrates the impact of the inhomogeneous degree of the magnetic field b on the behaviors of the three different quantifiers utilized in this study.

Fig. 7

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) in terms of t and b.

For selected values of the parameter b, both skew information quantifiers depicted respectively in Fig. 7a and b initially capture an equal amount of quantum correlations. However, as b increases, the quantum correlations decrease, and the damping effect intensifies. Modifying the parameter b leads to changes in the steady-state values of the quantifiers. Moreover, the steady-state value increases with the increase in the inhomogeneous magnetic field. When b is fixed, both skew information quantifiers reach the same steady states. Figure 7c portrays the birth and death of entanglement. However, as b increases, the occurrence rate of these events decreases, and the system becomes completely separable. Therefore, adjusting the value of the inhomogeneous magnetic field b can help regulate the steady-state value.

Finally, we depict in Fig. 8 the behavior of $\mathcal{U}_\mathcal{C}$, $\mathcal{U}$, and $\mathcal{L}_\mathcal{N}$ over time for various mixing parameter values. It is evident that the three quantifiers $\mathcal{U}_\mathcal{C}$, $\mathcal{U}$, and $\mathcal{L}_\mathcal{N}$ have the maximum amplitudes when $r = 0$.

Fig. 8

$\mathcal{U}_\mathcal{C}$ (a), $\mathcal{U}$ (b),$\mathcal{L}_\mathcal{N}$ (c) and a comparison of the three measures’ results (d) in terms of t and r.

By increasing the mixing parameter, the amplitudes of the three quantifiers diminish. When the parameter $r=\frac{1}{3}$ specifically, the amplitudes become extremely small, resulting in the lowest quantum correlation due to the initial separability of the system. The damping time of the system extends as the mixing parameter increases, eventually eliminating fluctuations. Even though entanglement vanishes, the quantum correlation values persist at non-zero levels. Among the quantifiers $\mathcal{U}_\mathcal{C}$, $\mathcal{U}$, and $\mathcal{L}_\mathcal{N}$, $\mathcal{U}_\mathcal{C}$ (Fig. 8a) and $\mathcal{U}$ (Fig. 8b) yield higher values compared to $\mathcal{L}_\mathcal{N}$ (Fig. 8c), and all three quantifiers reach the same steady states regardless of the value of r. The impact of the mixing parameter on the quantifiers $\mathcal{U}_\mathcal{C}$, $\mathcal{U}$, and $\mathcal{L}_\mathcal{N}$ is substantial, with $\mathcal{L}_\mathcal{N}$ being the most affected as its value remains non-zero ($\mathcal{L}_\mathcal{N}(r=0)=0.0444136$) even when the mixing parameter is zero. This suggests that selecting a maximally entangled initial state could enhance quantum entanglement and enhance the system’s resilience against inherent decoherence. Figure 8d displays the plots of the three quantifiers when $r=0.01$. It is evident that $\mathcal{U}_\mathcal{C}$ and $\mathcal{U}$ capture a greater amount of quantum correlations compared to $\mathcal{L}_\mathcal{N}$, despite initially having the same value. Although entanglement diminishes, the quantum correlation values remain stable at non-zero levels. The Fig. 8d clearly illustrates the significant influence of the initial state and the mixing value on the damping time and the quantity of quantum correlations. Hence, selecting an appropriate initial state becomes crucial to prevent the loss of entanglement in the qubit–qutrit system. Exploring alternative maximally entangled initial states should be considered to enhance quantum entanglement and quantum correlations and strength the system’s resistance against intrinsic decoherence.

5 Concluding remarks

Our research focused on the impact of intrinsic decoherence on various correlations in Heisenberg qubit–qutrit XXZ chains, with the additional effects of DM interaction in the presence of homogeneous and inhomogeneous magnetic fields. We employed the Milburn model to obtain the evolved state and used $\mathcal{U}$, $\mathcal{U}_\mathcal{C}$, and $\mathcal{L}_\mathcal{N}$ to measure the skew information correlations and entanglement within the hybrid system. Our results demonstrate that the amount of bipartite entanglement and skew information correlations are significantly influenced by the decoherence parameter, magnetic field strength, system parameters, and DMI magnitude. Specifically, increasing rates of intrinsic decoherence $\gamma$ negatively impact the quantum correlations exhibited by the model. We also discovered that tuning different system parameters, such as the initial state mixing parameter, magnetic field strength, and DMI strength $D_z$, can adjust the robustness of quantum correlations. Moreover, we found that non-classical correlations in the hybrid-spin chain model exhibit greater resistance to intrinsic decoherence when the coupling strength J is weak, and when the strengths of the magnetic field and DMI ($D_z$) are reduced or eliminated. These findings have substantial implications for the fields of quantum computing and information processing, as they offer valuable insights into the quantum properties of the hybrid-spin chain model, which holds great promise for practical quantum technology applications.

Appendix A

$$\begin{aligned} {\hat{D}}_{11}(t)&={\hat{D}}_{66}(t)=\frac{r}{2},\end{aligned}$$

(23)

$$\begin{aligned} {\hat{D}}_{22}(t)&=\frac{\frac{\zeta _-^2 \left( 8 (1-2 r) \left( D_z^2+J^2\right) +\zeta _-^2 r\right) }{\digamma _-^2}-\frac{2 \zeta _+^2 \left( \zeta _+ r (\delta J-4 b)+4 (r-1) \left( D_z^2+J^2\right) \right) }{\digamma _+^2}}{2 \left( D_z^2+J^2\right) {}^2}+\frac{32 (3 r-1) \left( 1+e^{i \alpha t}\right) e^{-\frac{1}{8} \alpha t (\alpha \gamma +4 i)}}{\digamma _- \digamma _+},\end{aligned}$$

(24)

$$\begin{aligned} {\hat{D}}_{33}(t)&=\frac{1}{2} \left( \frac{\frac{\xi _-^2 \left( 8 r \left( D_z^2+J^2\right) +\xi _-^2 (1-2 r)\right) }{\varsigma _-^2}+\frac{\xi _+^2 \left( 8 r \left( D_z^2+J^2\right) +\xi _+^2 (1-2 r)\right) }{\varsigma _+^2}}{\left( D_z^2+J^2\right) {}^2}-\frac{64 (3 r-1) \left( 1+e^{i \beta t}\right) e^{-\frac{1}{8} \beta t (\beta \gamma +4 i)}}{\varsigma _- \varsigma _+}\right) ,\end{aligned}$$

(25)

$$\begin{aligned} {\hat{D}}_{44}(t)&= 4 \left( \frac{\frac{8 b \zeta _+ r-8 (r-1) D_z^2-8 J^2 (r-1)-2 \delta \zeta _+ J r}{\digamma _+^2}+\frac{(8-16 r) \left( D_z^2+J^2\right) +\zeta _-^2 r}{\digamma _-^2}}{D_z^2+J^2}-\frac{8 (3 r-1) \left( 1+e^{i \alpha t}\right) e^{-\frac{1}{8} \alpha t (\alpha \gamma +4 i)}}{\digamma _- \digamma _+}\right) ,\end{aligned}$$

(26)

$$\begin{aligned} {\hat{D}}_{55}(t)&= 4 \left( \frac{\frac{8 r \left( D_z^2+J^2\right) +\xi _+^2 (1-2 r)}{\varsigma _+^2}-\frac{2 (2 r-1) \left( 4 b \xi _-+\delta J \left( \xi _--\beta \right) \right) }{\varsigma _-^2}}{D_z^2+J^2}+\frac{8 (3 r-1) \left( 1+e^{i \beta t}\right) e^{-\frac{1}{8} \beta t (\beta \gamma +4 i)}}{\varsigma _- \varsigma _+}\right) ,\end{aligned}$$

(27)

$$\begin{aligned} {\hat{D}}_{16}(t)&={\hat{D}}_{61}(t)= \frac{1}{2} r e^{-\frac{1}{2} t (b+3 B) (b \gamma +3 B \gamma +2 i)},\end{aligned}$$

(28)

$$\begin{aligned} {\hat{D}}_{23}(t)&= \frac{i \sqrt{2} (1-2 r)}{\left( D_z-i J\right) {}^2 \left( D_z+i J\right) }\left( -\frac{\zeta _- \xi _-^2 e^{-\frac{1}{32} t \tau _+ \left( \gamma \tau _++8 i\right) }}{\varsigma _- \digamma _-}+\frac{\zeta _+ \xi _+^2 e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _-+8 i\right) }}{\varsigma _+ \digamma _+}+\frac{\zeta _+ \xi _-^2 e^{-\frac{1}{32} t y _- \left( \gamma y _--8 i\right) }}{\varsigma _- \digamma _+}\right. \nonumber \\ {}&\left. -\frac{\zeta _- \xi _+^2 e^{-\frac{1}{32} t y _+ \left( \gamma y _++8 i\right) }}{\varsigma _+ \digamma _-}\right) ,\end{aligned}$$

(29)

$$\begin{aligned} {\hat{D}}_{24}(t)&= \frac{i \sqrt{2}}{\left( D_z-i J\right) {}^2 \left( D_z+i J\right) } \left( -\frac{2 \zeta _+ \left( -4 b \zeta _+ r+4 (r-1) D_z^2+4 J^2 (r-1)+\delta \zeta _+ J r\right) }{\digamma _+^2}\right. \nonumber \\ {}&\left. +\frac{8 (3 r-1) \left( D_z^2+J^2\right) e^{-\frac{1}{8} \alpha t (\alpha \gamma +4 i)} \left( \zeta _--\zeta _+ e^{i \alpha t}\right) }{\digamma _- \digamma _+}-\frac{\zeta _- \left( (8-16 r) \left( D_z^2+J^2\right) +\zeta _-^2 r\right) }{\digamma _-^2}\right) ,\end{aligned}$$

(30)

$$\begin{aligned} {\hat{D}}_{25}(t)&= \frac{4 (1-2 r) \left( -\frac{\zeta _+ \xi _+ e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _-+8 i\right) }}{\varsigma _+ \digamma _+}+\frac{\zeta _- \xi _- e^{-\frac{1}{32} t \tau _+ \left( \tau _++8 i\right) }}{\varsigma _- \digamma _-}-\frac{\zeta _+ \xi _- e^{-\frac{1}{32} t y _- \left( \gamma y _--8 i\right) }}{\varsigma _- \digamma _+}+\frac{\zeta _- \xi _+ e^{-\frac{1}{32} t y _+ \left( \gamma y _++8 i\right) }}{\varsigma _+ \digamma _-}\right) }{\left( D_z-i J\right) {}^2},\end{aligned}$$

(31)

$$\begin{aligned} {\hat{D}}_{32}(t)&= -\frac{i \sqrt{2} (1-2 r)}{\left( D_z-i J\right) \left( D_z+i J\right) {}^2} \left( -\frac{\zeta _- \xi _-^2 e^{-\frac{1}{32} t \tau _+ \left( \gamma \tau _+-8 i\right) }}{\varsigma _- \digamma _-}+\frac{\zeta _+ \xi _+^2 e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _--8 i\right) }}{\varsigma _+ \digamma _+}+\frac{\zeta _+ \xi _-^2 e^{-\frac{1}{32} t y _- \left( \gamma y _-+8 i\right) }}{\varsigma _- \digamma _+}\right. \nonumber \\&\left. -\frac{\zeta _- \xi _+^2 e^{-\frac{1}{32} t y _+ \left( \gamma y _+-8 i\right) }}{\varsigma _+ \digamma _-}\right) ,\end{aligned}$$

(32)

$$\begin{aligned} {\hat{D}}_{34}(t)&=\frac{4 (1-2 r) \left( \frac{\xi _-^2 e^{-\frac{1}{32} t \tau _+ \left( \gamma \tau _+-8 i\right) }}{\varsigma _- \digamma _-}+\frac{\xi _+^2 e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _--8 i\right) }}{\varsigma _+ \digamma _+}+\frac{\xi _-^2 e^{-\frac{1}{32} t y _- \left( \gamma y _-+8 i\right) }}{\varsigma _- \digamma _+}+\frac{\xi _+^2 e^{-\frac{1}{32} t y _+ \left( \gamma y _+-8 i\right) }}{\varsigma _+ \digamma _-}\right) }{D_z^2+J^2},\end{aligned}$$

(33)

$$\begin{aligned} {\hat{D}}_{35}(t)&= -\frac{i \sqrt{2}}{\left( D_z-i J\right) {}^2 \left( D_z+i J\right) }\left( \frac{\xi _+^3 (2 r-1)-8 \xi _+ r \left( D_z^2+J^2\right) }{\varsigma _+^2}\right. \nonumber \\ {}&\left. -\frac{2 \xi _- \left( b^2 (16-32 r)+4 b (2 r-1) (\beta -2 \delta J)-4 (r-1) D_z^2+J \left( J \left( \delta ^2-2 \left( \delta ^2+2\right) r+4\right) +\beta \delta (2 r-1)\right) \right) }{\varsigma _-^2}\right. \nonumber \\ {}&\left. -\frac{8 (3 r-1) \left( D_z^2+J^2\right) e^{-\frac{1}{8} \beta t (\beta \gamma +4 i)} \left( \xi _-+\xi _+ e^{i \beta t}\right) }{\varsigma _- \varsigma _+}\right) ,\end{aligned}$$

(34)

$$\begin{aligned} {\hat{D}}_{42}(t)&=-\frac{i \sqrt{2}}{\left( D_z-i J\right) \left( D_z+i J\right) {}^2} \left( -\frac{2 \zeta _+ \left( \zeta _+ r (\delta J-4 b)+4 (r-1) \left( D_z^2+J^2\right) \right) }{\digamma _+^2}\right. \nonumber \\ {}&\left. +\frac{8 (3 r-1) \left( D_z^2+J^2\right) e^{-\frac{1}{8} \alpha t (\alpha \gamma +4 i)} \left( -\zeta _++\zeta _- e^{i \alpha t}\right) }{\digamma _- \digamma _+}-\frac{\zeta _- \left( (8-16 r) \left( D_z^2+J^2\right) +\zeta _-^2 r\right) }{\digamma _-^2}\right) , \end{aligned}$$

(35)

$$\begin{aligned} {\hat{D}}_{43}(t)&= \frac{4 (1-2 r) \left( \frac{\xi _-^2 e^{-\frac{1}{32} t \tau _+ \left( \gamma \tau _++8 i\right) }}{\varsigma _- \digamma _-}+\frac{\xi _+^2 e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _-+8 i\right) }}{\varsigma _+ \digamma _+}+\frac{\xi _-^2 e^{-\frac{1}{32} t y _- \left( \gamma y _--8 i\right) }}{\varsigma _- \digamma _+}+\frac{\xi _+^2 e^{-\frac{1}{32} t y _+ \left( \gamma y _++8 i\right) }}{\varsigma _+ \digamma _-}\right) }{D_z^2+J^2},\end{aligned}$$

(36)

$$\begin{aligned} {\hat{D}}_{45}(t)&= \frac{8 i \sqrt{2} (1-2 r) \left( \frac{\xi _- e^{-\frac{1}{32} t \tau _+ \left( \gamma \tau _++8 i\right) }}{\varsigma _- \digamma _-}+\frac{\xi _+ e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _-+8 i\right) }}{\varsigma _+ \digamma _+}+\frac{\xi _- e^{-\frac{1}{32} t y _- \left( \gamma y _--8 i\right) }}{\varsigma _- \digamma _+}+\frac{\xi _+ e^{-\frac{1}{32} t y _+ \left( \gamma y _++8 i\right) }}{\varsigma _+ \digamma _-}\right) }{D_z-i J},\end{aligned}$$

(37)

$$\begin{aligned} {\hat{D}}_{52}(t)&= \frac{4 (1-2 r) \left( \frac{\zeta _- \xi _- e^{-\frac{1}{32} t \tau _+ \left( \gamma \tau _+-8 i\right) }}{\varsigma _- \digamma _-}-\frac{\zeta _+ \xi _+ e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _--8 i\right) }}{\varsigma _+ \digamma _+}-\frac{\zeta _+ \xi _- e^{-\frac{1}{32} t y _- \left( \gamma y _-+8 i\right) }}{\varsigma _- \digamma _+}+\frac{\zeta _- \xi _+ e^{-\frac{1}{32} t y _+ \left( \gamma y _+-8 i\right) }}{\varsigma _+ \digamma _-}\right) }{\left( D_z+i J\right) {}^2},\end{aligned}$$

(38)

$$\begin{aligned} {\hat{D}}_{53}(t)&= \frac{i \sqrt{2}}{\left( D_z-i J\right) \left( D_z+i J\right) {}^2}\left( -\frac{8 (3 r-1) \left( D_z^2+J^2\right) e^{-\frac{1}{8} \beta t (\beta \gamma +4 i)} \left( \xi _++\xi _- e^{i \beta t}\right) }{\varsigma _- \varsigma _+}\right. \nonumber \\&\left. -\frac{2 \xi _- \left( -4 b (2 r-1) (4 b-\beta +2 \delta J)-4 (r-1) D_z^2+J \left( J \left( \delta ^2-2 \left( \delta ^2+2\right) r+4\right) +\beta \delta (2 r-1)\right) \right) }{\varsigma _-^2}\right. \nonumber \\ {}&\left. +\frac{\xi _+^3 (2 r-1)-8 \xi _+ r \left( D_z^2+J^2\right) }{\varsigma _+^2}\right) ,\end{aligned}$$

(39)

$$\begin{aligned} {\hat{D}}_{54}(t)&=-\frac{8 i \sqrt{2} (1-2 r) \left( \frac{\xi _- e^{-\frac{1}{32} t \tau _+ \left( \gamma \tau _+-8 i\right) }}{\varsigma _- \digamma _-}+\frac{\xi _+ e^{-\frac{1}{32} t \tau _- \left( \gamma \tau _--8 i\right) }}{\varsigma _+ \digamma _+}+\frac{\xi _- e^{-\frac{1}{32} t y _- \left( \gamma y _-+8 i\right) }}{\varsigma _- \digamma _+}+\frac{\xi _+ e^{-\frac{1}{32} t y _+ \left( \gamma y _+-8 i\right) }}{\varsigma _+ \digamma _-}\right) }{D_z+i J}, \end{aligned}$$

(40)

where $\alpha =\sqrt{(\delta J-4 b)^2+8 D_z^2+8 J^2}$, $\beta =\sqrt{(4 b+\delta J)^2+8 D_z^2+8 J^2}$, $\zeta _{\pm }=\alpha {\pm }(4 b-\delta J)$, $\xi _{\pm }=4 b+\delta J{\pm }\beta$, $\varsigma _{\pm }= 8+\left| \frac{\xi _{\pm }}{i J+D_z}\right| ^2$, $\digamma _{\pm }= 8+\left| \frac{\zeta _{\pm }}{i J+D_z}\right| ^2$, $y _{\pm }={\pm }(4 b+4 B)+\beta +\alpha$, $\tau _{\pm }={\pm }(\alpha -\beta )+4 b+4 B$.