Dynamics of two-qubit quantum nonlocality in a Heisenberg chain model with the intrinsic decoherence

This paper investigates the dynamics of two-spin nonlocality generation in a Heisenberg XXX chain with Dzyaloshinskii-Moriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions. We analyze the two-spin nonlocality dynamics by using uncertainty-induced nonlocality, maximal Bell function, and log-negativity. We demonstrate that a separable two-spin Heisenberg XXX chain state, induced by two-spin antiferromagnetic interaction as well as x-component of DM and KSEA interactions, could evolve to maximal two-spin nonlocality state. The ability of preserving the maximal uncertainty-induced nonlocality can be enhanced by increasing the coupling strength of the spin-spin interaction coupling. The hierarchy principle is maintained for the two-spin Bell nonlocality and log-negativity entanglement. The two-spin log-negativity dynamics exhibits the phenomena of sudden death and birth. The sudden-death phenomenon is due to the intrinsic decoherence, which also causes a reduction in the two-spin nonlocality. While the sudden-birth phenomenon is due to two-spin antiferromagnetic interaction as well as x-component of DM and KSEA interactions. The two-spin uncertainty-induced nonlocality is more robust, against the intrinsic decoherence, than the other types of the nonlocality. The results indicate that by boosting the two-spin antiferromagnetic interaction, the produced nonlocality (resulting from the DM and KSEA x-component interactions) can be shielded from the intrinsic decoherence effect.


Introduction
The center of several quantum studies is the examination of the capacity of quantum qubit systems to produce two-qubit quantum information resources (including, coherence, nonlocality, and its variants: Bell nonlocality, entanglement, quantum discord, and other quantum correlations) (Horodecki et al. 2009;Vedral 2002;Wei et al. 2005;Ashhab et al. 2007; Mohamed et al. 2019a, b). Two-qubit quantum information resources have several potential applications in quantum information (Datta and Vidal 2007;Benabdallah et al. 2020;Nielsen and Chuang 2000;. Bell nonlocality and entanglement are the first introduced types of the quantum nonlocality. They verify the hierarchy principle (Costa et al. 2016;Qureshi et al. 2018;Abdel et al. 2020), according to which all quantum states that exhibit nonlocality identified by the violation of the maximum Bell function are necessary but not sufficiently entangled. After proving that some separable quantum states with quantum discord can be utilized to perform quantum information processing (Ollivier and Zurek 2001), several quantum information resources, including but not limited to entanglement, have been proposed to be implemented. These resources include quantum discord and skew information (Wigner and Yanase 1963;Girolami et al. 2013;Wu et al. 2014). The minimization and maximization of the skew information are used to define local quantum uncertainty and uncertainty-induced non-locality, respectively, which are used to measure other quantum coherence beyond the entanglement (Girolami et al. 2013;Wu et al. 2014;Slaoui et al. 2018Slaoui et al. , 2019. A significant amount of effort has been spent on studying and characterizing the coherence, and nonlocality (Hammar et al. 1999;Eggert et al. 1994) using the Heisenberg spin chain models due to the prospective uses of two-qubit quantum information resources (HSCMs). The spin-spin interactions are described by the HSCM, which is also essential in explaining the magnetic and thermodynamic properties of many-particle systems (Lee and Johnson 2004;Kohno 2009;Chen et al. 2010). In addition, numerous additional quantum systems, such as quantum dots (Burkard et al. 1999) and nuclear spins (Kane 1998), are also described by HSCM. Previously, the two-qubit quantum information resources have been investigated in several Heisenberg spin chain models (Fouokeng et al. 2022;Houça et al. 2022). The effects of Dzyaloshinskii-Moriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions on the two-spin nonlocality in Heisenberg XYZ chain models were studied at thermal equilibrium (Yurischev 2020;Fedorova and Yurischev 2022). It is discovered through these research that adding the DM and KSEA z-component interactions to the Heisenberg spin chain can improve the generation of the two-qubit quantum information resources.
Understanding the impact of intrinsic decoherence (Milburn 1991) on dynamics and the capacity of quantum two-qubit real systems to produce two-qubit quantum information resources play a crucial role in controlling quantum features (Mohamed et al. 2022;Liao 2022;Mohamed et al. 2021;Benabdallah et al. 2022). The intrinsic decoherence model was used to investigate dynamics of two-qubit quantum correlation in a Heisenberg chain model (Guo et al. 2014(Guo et al. , 2015 and the dynamical behaviors of entropic uncertainty relation in a two-qutrit system in the presence of external magnetic field and DM interaction (Guo et al. 2018).
Only the Heisenberg spin chain models for two-qubit X-matrix states have been examined in previous studies of the two-qubit quantum information resources. For the two-spin Heisenberg chain with x-components of DM and KSEA interactions, in particular, the dynamics of two-qubit nonlocality, of uncertainty-induced nonlocality, maximal Bell function, as well as logarithmic negativity, with the non-X-state/general state, remain unexplored.
Therefore, in the current study, using different quantifiers of information, as: uncertaintyinduced nonlocality, the maximal Bell function, and logarithmic negativity, we will examine the dynamics of variously produced two-spin nonlocality in a Heisenberg chain with intrinsic decoherence and DM and KSEA x-component-interactions.
The manuscript is structured as follows: The physical model of the Heisenberg XXX chain with DM and KSEA x-component interactions in the presence of intrinsic decoherence is addressed in Section (2). Section(3) introduces the two-spin quantum information quantifiers. We present and analyze the outcomes of the numerical simulations in Section (4). We end up in section (5) by a conclusion.

The intrinsic decoherence two-spin model
Here, we investigate a two-spin Heisenberg XXX-model under the influence of the x-component interactions with the DM and KSEA. The Milburn equation (Milburn 1991) describes the system's dynamics when there is intrinsic decoherence: M(t) designs the time-dependent two-spin density matrix. represents the decoherence rate. The Hamiltonian Ĥ of the two 1 2 -spins isotropic Heisenberg XXX chain with DM and KSEA x-direction interactions, is (Houça et al. 2022;Yurischev 2020): The first term designs the two-spin interaction with the exchange two-spin coupling F (the chain is antiferromagnetic for F > 0 and ferromagnetic for F < 0 ). K represents the KSEA interaction coupling. D designs DM interaction coupling strength. The quantum simulation of the Heisenberg spin models has been performed using a two transmon-qubit circuit quantum electrodynamics (Salathé et al. 2015). Experimentally, the DM interaction can be realized by a system of qubit-spin pyrochlores R 2 V 2 O 7 (Xiang et al. 2011) and it arises from spin-orbit coupling (Moriya 1960). Also, the KSEA interactions have been confirmed experimentally in Ba 2 CuGe 2 O 7 (Zheludev et al. 1998). Here, with the case of the intrinsic decoherence, we consider the time-dependent state M (t) of only two spins ( N = 2 ) from the Heisenberg chain model (that contains N spins). Based on matrix-product-state technique,different quantum nonlocality has been investigated for the N-spin Heisenberg chain state only when the system is initially at thermal equilibrium (Maziero et al. 2010;Liu et al. 2011;Jafari and Akbari 2020), which is controlled by X-state. This technique is used to investigate entanglement with phase decoherence, however it is limited to a simple quantum Heisenberg XY chain (Li and Xu 2003). It is very difficult to establish the timedependent state for the Heisenberg chain model when N > 3 because, in this scenario, the intrinsic decoherence model solution requires 2 N eigenvectors and 2 N eigenvalues of the N-spin Hamiltonian. Therefore, here we study the two-qubit quantum nonlocality dynamics of only N = 2 . In this case, we deal with a general time-dependent state (not with X-state).
For the case of N = 3 , different quantum information resources of time-dependent Heisenberg chain X-state induced by the intrinsic decoherence model have been investigated (Guo and Song 2009;Mohamed 2013).
In the two-spin basis: , the twospin Hamiltonian eigenstates �E i ⟩(i = 1, 2, 3, 4) and the eigenvalues E i of the Eq. (2) are given by with and the corresponding eigenvalues are, By utilizing the eigenvalues E k ( k = 1, 2, 3, 4 ) and the two-spin Hamiltonian eigenstates �E k ⟩ , the time evolution of the two-spin density matrix of Eq. (1) is (Milburn 1991), The terms of the interaction X mn (t) and the intrinsic decoherence Y mn (t) can be written as For an initial pure state as: �11⟩ , in the eigenvectors representation, the initial two-spin density matrix M (0) is given by In the following, we investigate the ability of the two-spin antiferromagnetic, Dzyaloshinskii-Moriya, and Kaplan-Shekhtman-Entin-Wohlman-Aharony interactions to generate quantum nonlocality via maximum Bell function, uncertainty-induced nonlocality, and logarithmic-negativity using an initial pure state (that does not have quantum nonlocality).

Quantum nonlocality quantifiers
• Uncertainty-induced nonlocality (UIN): Here, we measure the two-spin quantum nonlocality using UIN. It is defined by the maximization of the skew information quan- The two-spin Bell-nonlocality is realized when the Bell inequality is violated, i.e., the two-spin state possesses Bell-nonlocality when the maximum Bell function B max (t) satisfies the inequality: ( B max (t) ≥ 2 ) (Banaszek and Wódkiewicz 1998;Horodecki et al. 1995). In this work, we redefine the maximal Bell function as: B(t) = B max (t) − 1 , i.e., the two-spin Bell-nonlocality exists when the redefined maximal Bell function fulfills B(t) ≥ 1 . The analytical expression of the maximal Bell function B(t) is given by √ 1 + 2 − 1, k (k = 1, 2) represent the two largest eigenvalues of the matrix K = T † T , which depend on the correlation matrix T = [t ij ] (Horodecki et al. 1995), with t ij = Tr{M(t) (1) i (2) j } , i, j = 1, 2, 3.
• Logarithmic negativity The logarithmic-negativity (Vidal and Werner 2002) of the two-spin density matrix M(t) can be used to determine the entanglement generated due to the two-spin anti-ferromagnetic, Dzyaloshinskii-Moriya, and Kaplan-Shekhtman-Entin-Wohlman-Aharony interactions. The closed form of the logarithmic negativity is given by T (Vidal and Werner 2002) is the two-spin density matrix negativity, defined by the absolute sum of the negative eigenvalues of the partial transpose matrix (M(t)) T A∕B of the two-spin density M (t) with respect to subsystem A/B. The elements of the matrix (M(t)) T A are given by, For the separable states, the log-negativity disappears at N(t) = 0 . It possesses nonzero values for partially 0 < N(t) < 1 and maximally ( N(t) = 1 ) two-spin entangled states. The negativity is monotonically related to the log-negativity that quantifies an upper bound to the quantum entanglement. They are utilized to quantify the entanglement between two-qubit states.

Two-spin nonlocality dynamics
Uncertainty-induced nonlocality, maximal Bell function, and logarithmic negativity are utilized to analyze the dynamics of different types of nonlocality for the two-spin system. Our aim to discus the numerical results of the two-spin nonlocality dynamics in particular we are interested to: Dzyaloshinskii-Moriya interaction coupling D effect (illustrated in Fig. 1), anti-ferromagnetic interaction coupling F effect (shown in Fig. 2), as well as the KSEA interaction coupling K (shown in Fig. 2). While in Figs.3 and 4, the intrinsic decoherence is investigated for different anti-ferromagnetic, DM and KSEA interaction coupli ngs.
From Fig. 1a, we find that for the case where the Dzyaloshinskii-Moriya interaction coupling strength is very weak (D = 0.05) and the intrinsic decoherence is absent, the relatively weak spin-spin and Kaplan-Shekhtman-Entin-Wohlman-Aharony interactions, F = K = 0.5 , have high ability to generate regular nonlocality. The two-spin uncertainty-induced nonlocality, the maximal Bell function, and the logarithmic negativity grow and oscillate with -period indicating that the two-spin nonlocality states can be produced by the two-spin and KSEA interactions. The generated two-spin nonlocality have the same oscillatory dynamics. The two spins have maximally correlated states at t = 1 2 (2n + 1) (n = 0, 1, 2, ...) , where U max = 1 , N max = 1 and B max = 2 √ 2 − 1 ≈ 1.8284. Fig. 1b shows the effect of the increase of the Dzyaloshinskii-Moriya interaction coupling strength, D = 2 . UIN, MBF, and logarithmic-negativity two-spin quantum information are increased, quickly, due to the increase of the Dzyaloshinskii-Moriya interaction coupling strength. The regular dynamics of the two-spin quantum information functions disappears. The amplitudes of the maximal Bell function, and the log-negativity decrease while of uncertainty-induced nonlocality increases.
For high Dzyaloshinskii-Moriya interaction coupling strength D = 7 , see Fig. 1c, we note that the generated two-spin uncertainty-induced nonlocality, Bell-function, and the log-negativity effects are enhanced. The increase of the amplitudes and frequencies of the quantum quantifiers enhance the probability of maximally correlated states. Previous investigations reveal that the generated skew information and Bell function quantum correlation Fig. 1 Dynamics of the two-spin nonlocality of the Bell nonlocality B(t), uncertainty-induced nonlocality U(t) as well as the logarithmic negativity N(t) for the spin-spin and KSEA interaction coupling which are relatively weak F = K = 0.5 in the absence of the decoherence. Different DM interaction coupling strengths are considered: D = 0.05 in (a), D = 2 in (b), and D = 7 (in this case, we take t ∈ [0, 2 ] ) in (c) of a two-qubit Heisenberg XYZ chain are both greatly improved by the DM interaction . Fig. 2 illustrates how the amount, regularity, and speed of the dynamics of the produced UIN, MBF, and log-negativity two-spin quantum information resources are affected by an increase in the two spin antiferromagnetic interaction coupling. By comparing the results of Figs.1b and 2a, we note that the increase of the spin-spin interaction coupling ( F = 2 ) enhances the amplitudes and frequencies of the quantum quantifiers U(t), B(t) and N(t). The generated two-spin Heisenberg XXX states tend to have maximal UIN, MBF, and log-negativity nonlocality. UIN stabilizes to its maximal value U max = 1 , i.e., the generated two-qubit Heisenberg XXX has high ability to preserve maximal UIN nonlocality. Fig. 2b shows that the ability of preserving the maximal UIN nonlocality can be enhanced by increasing the coupling strength of the spin-spin interaction coupling ( F = 8 ). In this case, the two-spin uncertainty-induced nonlocality is stabilized for a long time. For the relatively strong Dzyaloshinskii-Moriya interaction coupling, the increase of the two spin antiferromagnetic interaction coupling improves the generation of the maxmal two-spin uncertainty-induced nonlocality, Bell-function, and logarithmic-negativity nonlocality. The results demonstrate that the two spin antiferromagnetic interaction has a great capacity to produce a two-qubit Heisenberg XXX chain state with maximal nonlocality as a result of the increase of the Dzyaloshinskii-Moriya coupling. The time windows for the Bell's inequality violation get larger for the two-spin Heisenberg XXX chain. UIN amplitudes increase and outpace logarithmic-negativity amplitudes. Fig. 3 shows the ability of strong two-spin antiferromagnetic as well as the DM and KSEA x-component interactions to generate maximal two-qubit Heisenberg XXX nonlocality. By comparing the results of Figs.1b and 3a, we find that the amplitudes and frequencies of the two-spin Bell-function, uncertainty-induced nonlocality, and the log-negativity quantifiers are enhanced, as the time evolves, due the increase of the KSEA interaction. This mean that after considering relatively strong Kaplan-Shekhtman-Entin-Wohlman-Aharony interaction ( K = 2 ), the generated two-spin Heisenberg XXX chain states tend to have maximal UIN, MBF, and log-negativity nonlocality.
By comparing the results of Figs.2b and 3b, we observe that the strong two-spin antiferromagnetic interaction enhances the ability of the Kaplan-Shekhtman-Entin-Wohlman-Aharony x-component interaction to increase the two-spin nonlocality. -period oscillations of the two-spin nonlocality have large amplitudes and high frequencies.
In Fig. 4, the effect of the intrinsic decoherence for small rate ( = 0.05 ) is investigated for DM and KSEA interaction couplings D = K = 0.5 and different values of the spin-spin couplings F = 0.5 in (a) and F = 5 in (b). Fig. 4a illustrates that the amplitudes and the frequencies of the generated Bell-function, uncertainty-induced nonlocality, and log-negativity nonlocality decrease, as the time evolves, due the intrinsic decoherence. For particular time windows, the entanglement log-negativity disappears suddenly and then arises suddenly, i.e., the phenomenon of the sudden death and birth in the two-spin lognegativity dynamics (Yu and Eberly 2009) occurs. The sudden-death phenomenon is due to the intrinsic decoherence, which also causes a reduction in the two-spin nonlocality. While the sudden-birth phenomenon is due to two-spin antiferromagnetic interaction as well as x-component of DM and KSEA interactions. The uncertainty-induced nonlocality is more robustness than the other measures of the nonlocality. In several time intervals, the violation of Bell's inequality does not occur and the two-spin state possesses partial entanglement log-negativity (s.t., 0 < N(t) < 1 ) in the same time intervals. This is in line with the hierarchy principle (Costa et al. 2016;Qureshi et al. 2018;Abdel et al. 2020). The Bell nonlocality implies entanglement log-negativity nonlocality, which is necessary and is not sufficiently. In Fig. 4b, we observe that the grow of the two-spin antiferromagnetic interaction can protect the generated nonlocality (due to the DM and KSEA x-component interactions) against the intrinsic decoherence. The phenomenon of the entanglement sudden death and birth disappears. Fig. 5 shows the effect of the intrinsic decoherence for = 0.05 on the generated nonlocality, due to strong DM and KSEA x-component interactions (for x-component interactions D = K = 0.5 ). We notice that the effect of the intrinsic decoherence is accelerated as the DM and KSEA interaction coupling strengths rise. After a brief period of time, the created negativity entanglement and Bell nonlocality are completely destroyed. When the DM and KSEA x-component interactions are strengthened, uncertainty-induced nonlocality is acceleraterd and becomes more strong and stable. While the entanglement and Bell nonlocality are extremely vulnerable to the intrinsic decoherence.

Conclusion
We have examined the two-spin nonlocality dynamics resulting from the two-spin antiferromagnetic, Dzyaloshinskii-Moriya, and Kaplan-Shekhtman-Entin-Wohlman-Aharony x-component interactions in the Heisenberg XXX model. By adjusting the intensity of the interactions between the two-spin antiferromagnetic, Dzyaloshinskii-Moriya, Kaplan-Shekhtman-Entin-Wohlman-Aharony, as well as the intrinsic-decoherence rate, we can control the produced nonlocality of the two-spin Heisenberg XXX chain state. The capacity to create two-spin Heisenberg XXX states with two-qubit nonlocality is highly enhanced by the DM and KSEA x-component interactions. The generated two-qubit Heisenberg XXX chain has a high capacity to maintain maximal UIN nonlocality. The two-spin antiferromagnetic interactions have the ability to enhance the amplitudes and frequencies of the two-spin nonlocality. The hierarchy principle is shown to be respected for the produced Fig. 5 The two-spin nonlocality generation dynamics of Fig. 4a are investigated under strong DM and KSEA interaction couplings two-spin Bell nonlocality and log-negativity entanglement. Due to interactions between the DM and KSEA x-components in the Heisenberg XXX chain model, an increase in intrinsic decoherence causes a decrease in the Bell-function, uncertainty-induced nonlocality, and log-negativity. The phenomenon of sudden death and birth occurs in the dynamics of the log-negativity. Compared to the other Bell function nonlocality, the two-spin uncertaintyinduced nonlocality is more resilient against the intrinsic decoherence. By strengthening the two-spin antiferromagnetic interaction, the generated nonlocality may be shielded from the intrinsic decoherence. While an an enhancement of the DM and KSEA x-component interaction couplings speeds up the intrinsic decoherence effect.