Introduction

The realisation of quantum phenomena, including entanglement and other quantum correlations, in real qubit systems is considered the only component of quantum information science. Quantum correlated two-qubit states are crucial for quantum cryptography1, the proposed realisation of quantum computers2, quantum teleportation3 and quantum memory4,5. With the determination to build quantum computers, many real systems have also been proposed to realise qubits. These qubit systems are based on superconducting circuits6,7 (with which the teleporting two-qubit entanglement over 19 qubits3 was demonstrated), trapped ions8, silicon9, nitrogen vacancy (NV) centers in a diamond lattice10.

In particular, NV-centers spin systems have emerged as one of the real qubit systems for a scalable quantum processor architecture11,12. Due to their long coherence times even at room temperature and their compatibility with optically detected magnetic resonance techniques, they have the potential to implement qubit quantum computers at room temperature13,14. However, NV center qubits show particular resistance to environmental noise when embedded in a well-designed quantum architecture15,16. Moreover, nitrogen-vacancy-center systems have been used to realize more quantum information resources. These include the study of quantum memory-assisted entropic uncertainty and the two-qubit nonlocality of nitrogen-vacancy centers inside open nanocavities17, the study of coherence of NV electronic spin ensembles18, and the study of non-Markovian dynamics of quantum correlation between two entangled two-qubit nitrogen-vacancy centers inside open photonic-crystal cavities19. Furthermore, the quantum correlation dynamics of separated nitrogen-vacancy centers was investigated using both quantum discord and entanglement of formation20. The studying of quantum correlations between two nitrogen-vacancy centers still requires further investigation, especially under the influence of zero-field splitting, external magnetic fields, external electric control fields and dipole-dipole coupling. The ability of these systems to realise multiple quantum information resources is high, and existing semiconductor technologies continue to position nitrogen-vacancy-centers as good real qubit systems21,22,23.

Furthermore, the realisation of quantum Fisher information (QFI) is a crucial, significant tool for the implementation of quantum information resources, such as the quantum acceleration time of perturbed open systems24, the improvement of quantum teleportation25, the creation of entanglement criteria26, the exploration of quantum phase transitions27 and further estimates of quantum metrology28,29,30. Furthermore, QFI is an important link between entanglement, quantum metrology and other quantum correlations31,32. However, the local quantum Fisher information (LQFI) or “interferometric power” was introduced to characterise other quantum correlations beyond entanglement34,35,36. It is shown that QFI is statistically underlying Wigner-Yanase (WY) skew information, which quantifies the amount of information in non-commuting observables associated with conserved operators37. The minimization (local quantum uncertainty (LQU)38) and maximization (uncertainty-induced nonlocality39) of this WY skew information were used as two important quantifiers quantify other quantum correlations beyond entanglement. Moreover, entanglement is not the only form of quantum correlation. In fact, certain separable (disentangled) states can still exhibit features of quantum non-locality40. Following the realization that quantum discord provides a more general framework for quantifying quantum correlations than entanglement alone41, several alternative measures have been proposed. Among these are quantifiers based on Wigner–Yanase skew information and quantum Fisher information, which aim to capture quantum correlations beyond entanglement. However, the LQU and LQFI have been used to realize related correlations42 in several proposed qubit systems43,44,45.

The motivations for this work are: First, as mentioned above, nitrogen-vacancy center systems are important proposed systems for building quantum computers. Second,the experimental estimation of the quantum Fisher information resource46. Finally, previous studies on quantum information resources in nitrogen-vacancy center systems are limited and there is a need to investigate how increasing zero-field splitting, external magnetic fields, external electric control fields, dipole-dipole coupling and decoherence affect the generated correlations between two NV centers. Therefore, this study aims to investigate the decoherence of the quantum correlation dynamics of two qubit Nitrogen-vacancy (NV) centers through LQU, LQFI, and concurrence. Using an intrinsic decoherence model under the effects of increasing zero-field splitting, external magnetic fields, external electric control fields, dipole-dipole coupling, and decoherence.

This work is organised as follows: In Sect. "Physical nitrogen-vacancy-centers model of the system" we present the physical model of the NV center system in the presence of intrinsic decoherence. Section "Quantum information quantifiers" revisits the definitions of quantum information quantifiers in the context of LQU, LQFI, and concurrence are discussed in Sect. "LQU, LQFI, and concurrence dynamics". Finally, our conclusions are presented in Sect. “Conclusion”.

Physical nitrogen-vacancy-centers model of the system

Our system considers two qubit nitrogen-vacancy (NV)centres (A and B) in diamond with isolated electronic spins in the presence of zero-field splitting, an external electric control field and the spin-NV dipole-dipole interaction47. Moreover, an external magnetic field \(\textbf{B} = (0, 0, B_z)\) applied along the z direction induces a Zeeman interaction that causes a splitting of the spin states of the NV centres. Furthermore, we assume that the interactions between the two spin NV centres are described by the dipole-dipole coupling \(\Gamma\)48,49, which depends on the relative orientation and distance r between the NV centres. In addition, an external electric field \(E_x\) is applied in the x direction50. The total Hamiltonian of the considered system can be expressed as47,48,49,50,51:

$$\begin{aligned} \hat{H}= & \sum _{k=A,B} \{ D_k \sigma _z^{k} + \chi B_z \sigma _z^{k} + \Gamma \frac{1}{r^3} [\sigma _x^{k} \sigma _x^{3-k} \nonumber \\ - & 3 (\vec {{\textbf {r}}} \cdot \hat{\sigma }^{k})(\vec {{\textbf {r}}} \cdot \hat{\sigma }^{3-k})]\}+ E_x(\sigma _x^{A} + \sigma _x^{B}), \end{aligned}$$
(1)

where \(D_k\) represents the zero-field splitting strengths, and \(\chi\) represents the electron gyromagnetic ratio. \(\sigma _z^{(1)}\) and \(\sigma _z\) are the Pauli-z spin-1/2 operator describing the two-qubit NV centers. \(\quad \vec {{\textbf {r}}}=(r_{x},r_{y},r_{z})\) is the two NV centers’ unit vector. In our work, we take \(\vec {{\textbf {r}}}=(0,1,0)\). In the two-NV-center-qubit space states: \(\{ |1_{A}1_{B}\rangle , |1_{A}0_{B}\rangle ,|0_{A}1_{B}\rangle ,|0_{A}0_{B}\rangle \}\), the total Hamiltonian has the following form:

$$\begin{aligned} \hat{H}= & \left( \begin{array}{cccc} DB, & 0 & 2E_{x} & \frac{8}{r^3}\Gamma \\ 0 & D_{A}-D_{B} & \frac{-4}{r^3}\Gamma & 2E_{x} \\ 2E_{x} & \frac{-4}{r^3}\Gamma & D_{B}-D_{A} & 0 \\ \frac{8}{r^3}\Gamma & 2E_{x} & 0 & -DB \\ \end{array} \right) , \end{aligned}$$

where \(DB=D_{A}+D_{B}+2\chi B_{z}\).

There are several approaches to modeling decoherence. One such approach is intrinsic decoherence, which provides an explanation for the deterioration of quantum coherence. It is based on the hypothesis that closed quantum systems do not evolve unitarily according to the Schrödinger equation, but instead follow more generalized dynamics that incorporate intrinsic decoherence (pure dephasing)52. In contrast, other approaches are based on the Lindblad master equation, which accounts for irreversible effects such as dissipation and decoherence (or pure dephasing)53,54. These models describe the interaction between the system and its surrounding reservoir, naturally leading to decoherence as information is irretrievably lost during the system’s evolution. Due to that the decoherence of the NV center spin is dominated by pure dephasing55,56, the time evolution of the decoherence of quantum local Fisher and uncertainty information in the generated two-NV-center-qubit system states, described by the density matrix \(\hat{M}(t)\), will be explored by using Milburn intrinsic decoherence model52,

$$\begin{aligned} \frac{d}{d t} \hat{M}(t)=-i[\hat{H},\hat{M}]-\frac{1}{2\gamma }[\hat{H},[\hat{H},\hat{M}]], \end{aligned}$$
(2)

where \(\gamma\) is the intrinsic NV-center decoherence coupling.

After calculating the eigenvalues \(E_{k}\) (\(k=1, 2, 3, 4\)) and the eigenstates \(|E_{k}\rangle\) of the two-NV-center-qubits Hamiltonian in Eq. (1), the density matrix \(\hat{M}(t)\) in Eq. (2) can be computed by

$$\begin{aligned} \hat{M}(t)= \!\! \sum ^{4}_{m, n=1} \!\! X_{mn}(t) \,Y_{mn}(t) \,\langle E_{m}|\hat{M}(0)|E_{n}\rangle \, |E_{m}\rangle \langle E_{n}|, \end{aligned}$$
(3)

which depends on the following unitary interaction \(X_{mn}(t)\) and intrinsic NV-center decoherence \(Y_{mn}(t)\) terms:

$$\begin{aligned} X_{mn}(t)= & e^{-iD_{E}t}, \nonumber \\ Y_{mn}(t)= & e^{ -\frac{\gamma }{2} D_{E}^{2}t}, \end{aligned}$$
(4)

which depend on the difference between the two NV centers’energies \(D_{E}=(E_{m}-E_{n})\). To investigate the ability of the two-qubit system of Nitrogen-vacancy (NV) centers’interactions to generate two-qubit correlations, we consider that the system starting in an excited-state triplet, where the two electron spins are aligned upwards. This initial state is represented as \(\hat{M}(0)=|1_{A}1_{B}\rangle \langle 1_{A}1_{B}|\). Therefore, Eq.(3) is used to numerically calculate and explore the dynamics of a two-NV-center-qubit system through their generated quantum correlations LQFI, LQU, and concurrence under the influence of intrinsic interactions and the couplings of gyromagnetic factor, external magnetic field \(B_{z}\), external control electric field, and dipole-dipole interaction, and zero-field splitting strengths.

Quantum information quantifiers

Here, the behaviour of the two-NV-center-qubits states will be investigated by the LQFI and local quantum uncertainty (LQU), and their association with entanglement measured by concurrence having the following quantifiers.

  • LQFI Here, we use LQFI to quantify another type in the generated two-NV-center-qubits’ quantum correlation beyond entanglement. After calculating the eigenvalues \(\pi _{k}\) (\(k=1, 2, 3, 4\)) and the eigenstates \(|\Pi _{k}\rangle\) of the two-NV-center-qubits state having the representation matrix: \(M(t)=\sum _{m}\pi _{m}|\Pi _{m}\rangle \langle \Pi _{m}|\) with \(\pi _{m}\ge 0\) and \(\sum _{m}\pi _{m}=1\), the quantum Fisher information associated with the local evolution generated by \(I_{a}\otimes H_{b}\) can be written as

    $$\begin{aligned} F(\rho _{b},H_{b})= & 4\text {Tr}\{\rho _{ab} H^{2}_{b}\} \nonumber \\ & - \sum _{m,n} \frac{ 8\pi _{m}\pi _{n} }{\pi _{m}+\pi _{n}} |\langle \Pi _{m}|I_{a}\otimes H_{b}|\Pi _{n}\rangle |^{2}. \end{aligned}$$
    (5)

    where \(\pi _{m}+\pi _{n}>0\). Therefore, the generated two-NV-center-qubits states’ quantum correlation is defined as the minimum QFI over all the local Hamiltonian: \(H_{b}= \vec {r}.\vec {\sigma }\), where \(|\vec {r}| = 1\) and \(\vec {\sigma }= {\sigma ^{x},\sigma ^{y},\sigma ^{z}}\) is the Pauli vector. This minimization has been performed analytically35, and the local quantum Fisher information associated with the generated two-NV-center-qubits states having the density matrix \(\hat{M}(t)\) is given by the following expression35,36

    $$\begin{aligned} L(t)=1-W_{\max }, \end{aligned}$$

    which depends on the highest eigenvalue \(X_{\max }\) of the symmetric matrix \(W=[w_{ij}]\) depending on the elements \(Z_{mn}^{i}=\langle \Pi _{m}|I\otimes \sigma ^{i}|\Pi _{n}\rangle\), the symmetric matrix elements \(x_{ij}\) are given by

    $$\begin{aligned} w_{ij}=\sum _{\pi _{m}+\pi _{n}\ne 0}\frac{2\pi _{m}\pi _{n}}{\pi _{m}+\pi _{n}}Z_{mn}^{i}(Z_{nm}^{j})^{\dagger }. \end{aligned}$$

    For a maximally correlated two-NV-center-qubits state, the LQFI is \(L(t)=1\). Otherwise, the LQFI is bounded by the inequality \(0<L(t)<1\), indicating that the two-NV-center-qubits states have partial LQFI correlation.

  • Wigner–Yanase (WY) information quantifier local quantum uncertainty (LQU)37,38,39 was defined as the minimum of Wigner–Yanase (WY) information quantity37, and it was introduced to realize another quantum correlation37,38,39. In the first, we calculate the largest eigenvalue \(L_{max}(L_{AB})\) of the \(3\text {x}3\)-matrix \(\Lambda =[l_{ij}]\) having the following elements:

    $$\begin{aligned} l_{ij}=\text {T}r\mathbf {\big \{}\sqrt{M(t)}(\sigma _{i}\otimes I)\sqrt{M(t)}(\sigma _{j}\otimes I)\mathbf {\big \}}. \end{aligned}$$

    Then the LQU associated the two-NV-center-qubits states having the density matrix \(\hat{M}(t)\)is computed by using the following closed expression38:

    $$\begin{aligned} U(t)= & 1-L_{max}(L_{AB}), \end{aligned}$$
    (6)

    The LQU associated with the two NV-center qubit states oscillates between zero-value and one, i.e., \(0 \le LU(t) \le 1\). If \(U(t)=1\), the generated two NV-center qubit have maximal LQU correlation, otherwise indicating that the two NV-center qubit states have partial LQU correlation.

  • Concurrence Here, we use the concurrence57 to investigate the generated two-NV-center-qubits entanglement resulting from:

    $$\begin{aligned} C(t)=\max \{0, e_{1} -e_{2}-e_{3}-e_{4}\}, \end{aligned}$$
    (7)

    with \(e_{i}>e_{i+1}\), and \(e_{i}\) represent the square roots of the eigenvalues for the matrix: \(R=M(t)(\sigma ^{y}\otimes \sigma ^{y}) M^{*}(t)(\sigma ^{y}\otimes \sigma ^{y})\). For \(C(t)=0\), the two-NV-center-qubits state is a disentangled state. A maximally entangled state with two spins is when \(C(t)=1\). Otherwise (\(0<C(t)<1\)), the two NV-center qubits have a partially entangled state.

LQU, LQFI, and concurrence dynamics

In this section, we examine the effects of increasing various physical parameters on the dynamical analysis, such as the external magnetic field, zero-field splitting strengths, external control electric field, and intrinsic decoherence, on the dynamical analysis of the generated two NV-center qubit correlation dynamics. The simulation results are presented in the following figures, using appropriately scaled dimensionless physical parameters. This analysis employs quantum tools including LQFI, LQU, and concurrence.

Fig. 1
figure 1

The dynamics of LQFI, LQU, and concurrence for two NV-center qubits are illustrated under the following parameter settings: gyromagnetic factor \(\chi =0.5\), inter-qubit distance \(r=1\), external magnetic field \(B_z=0.5\), external control electric field \(E_x=2\), and dipole-dipole interaction strength \(\Gamma =0.4\) for different zero-field splitting strengths \(D_{B}=D_{A}-0.3\) and \(D_{A}=0.8\) in (a), and \(D_{A}=2.5\) in (b). The intrinsic decoherence is taken with a small coupling: \(1/\gamma = 5\times 10^{-5}\).

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Figure 1 illustrates the ability of interactions within a two-qubit system of NV centers to generate two-qubit correlations in terms of LQFI, LQU, and concurrence. For small zero-field splitting strengths, specifically \(D_{B}=D_{A}-0.3\) (this value of 0.3 is only chosen to ensure that \(D_{k}\) differs) with \(D_{A}=0.8\), Fig. 1a illustrates that the generated dynamics of LQFI, LQU, and concurrence-entanglement associated with two-NV-center qubits, starting from the initial state \(\hat{M}(0)=|11\rangle \langle 11|\), exhibit no quantum correlations: \(L(0)=U(0)=C(0)=0\). Moreover, the generated two-NV-center-qubits states exhibit oscillatory generations in their LQFI, LQU, and concurrence-entanglement dynamics, all occurring at the same frequency. At particular times, we find that the generated LQFI, LQU, and concurrence-entanglement associated with two-NV-center qubits can approximately reach a maximally entangled state. Besides, for a weak intrinsic decoherence coupling, where \(1/\gamma = 5\times 10^{-5}\), the LQFI and LQU quantifiers of the generated quantum local Fisher and uncertainty information correlations of two-NV-center qubits have the same oscillatory generations, beyond the intervals characterized by strong concurrence-entanglement. Having LQFI and LQU quantifiers the same oscillatory generations means that the minimizations of the Wigner–Yanase skew and the Fisher information realize a special type of two-qubit correlations, which will be called “Wigner–Yanase-Fisher correlation”. This concepts of Wigner–Yanase–Fisher correlation arises from the relationship between the quantum Fisher information \(F(\rho , H)\) and skew information \(I(\rho , H)\) satisfying the following inequality relation58: \(I(\rho , H) \le F(\rho , H) \le 2I(\rho , H)\). From this inequality, it follows that: \(LQU \le LQFI \le 2LQU\). However, as the time increases, the generated two-NV-center-qubits Wigner–Yanase-Fisher correlation and concurrence-entanglement exhibit the same oscillatory generations, which is due the unitary interaction term: \(\cos (V_{m}-V_{n})t-i \sin (V_{m}-V_{n})t\). Moreover, at specific moments, the generated two NV-center qubits exhibit partial concurrence-entanglement without quantum local Fisher and uncertainty information correlations. Figure 1b demonstrates that increasing the zero-field splitting strength (\(D_{B} = D_{A} - 0.3\) and \(D_{A} = 2.5\)) reduces the amplitudes of oscillatory generations linked to the correlations of the generated two NV-center qubits. Consequently, this enhances the Wigner–Yanase-Fisher correlation intervals, which are characterized by weak concurrence-entanglement. As the iterations of the two NV centers evolve, the increasing coupling of the zero-field splitting term in the Hamiltonian enhances transitions between low-energy states of the two NV center qubits, resulting in faster and larger oscillations in generated correlations’ LQFI, LQU, and concurrence. The results confirm that the amplitudes of the concurrence-entanglement are greater than those of the generated LQFI, LQU, and Wigner–Yanase-Fisher correlation, (\(U(t) \le L(t) \le C(t)\)).

Fig. 2
figure 2

The dynamics of LQFI, LQU, and concurrence for two NV-center qubits of Fig.(1a) are shown but for different external magnetic field couplings: \(B_z=2\) in (a), \(B_z=7\) in (b).

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In Fig. 2, we illustrate the dynamics of the two NV center qubits’ LQFI, LQU, and concurrence, as shown in Fig. 1(a), but the external magnetic field couplings are increased, while keeping the other couplings fixed. Moreover, applying an external magnetic field, affecting transitions between states of the two NV center qubits, and their generated quantum correlations. In the case of Fig. 2a) with \(B_z=2\), the two-NV-center-qubits system exhibits quick and more oscillations. Increasing the external magnetic field reduces the amplitudes of oscillatory generations linked to the correlations. Consequently, this enhances the Wigner–Yanase-Fisher correlation intervals, which are characterized by weak concurrence-entanglement and returning the two-NV-center-qubits system to its initial uncorrelated state. Moreover, with the strong applied external magnetic field considered in Fig. 2(b), we observe faster oscillatory generations with small amplitudes, and increased intervals of Wigner–Yanase-Fisher correlations. This is because the applied external magnetic field enhances state transitions of the two NV center qubits, thereby limiting quantum information exchange between them.

Fig. 3
figure 3

The dynamics of LQFI, LQU, and concurrence for two NV-center qubits of Fig.(1a) are shown but when increasing the external control electric field coupling to be \(E_x=4\) in (a), and \(E_x=8\) in (b).

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Figure 3 visualizes the dynamics of two-NV-center-qubits LQFI, LQU, and concurrence dynamics of Fig.(1a) are shown but when the external control electric field is taken with large couplings. In Fig. 3(a), with a large coupling of \(E_x=4\), the state transitions of the two NV centers, supported by the Hamiltonian term: \(2E_{x}[|1_{A}1_{B}\rangle \langle 0_{A}1_{B}| + |0_{A}1_{B}\rangle \langle 1_{A}1_{B}| + |1_{A}0_{B}\rangle \langle 0_{A}0_{B}| + |0_{A}0_{B}\rangle \langle 1_{A}0_{B}|]\), enhance the oscillatory generation of quantum correlations by increasing their amplitudes and reducing their frequencies. In this case of \(E_x=4\), the interactions between the two NV centers exhibit a high capacity to generate strong, quasi-stable two-qubit LQFI and concurrence correlations. Additionally, the intervals where LQFI and LQU display different amplitudes with the same oscillatory generations are increased, while the intervals of the Wigner–Yanase-Fisher correlation are reduced. However, the results of Fig. 3a,b show that the relatively strong external control electric field with \(E_x=8\) reduce the frequency of the generated oscillatory generations. The generated two-NV-center-qubits states exhibits a relatively strong and stable concurrence correlation with small oscillations through a large time intervals in which the difference between amplitudes of the LQFI and LQU are very large. The Wigner–Yanase-Fisher correlation intervals are reduced.

Fig. 4
figure 4

The dynamics of LQFI, LQU, and concurrence for two NV-center qubits of Fig.(1a) are shown but for different dipole-dipole couplings: \(\Gamma =0.8\) in (a), and \(\Gamma =1.5\) in (b).

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Figure 4 examines how increasing the dipole-dipole coupling significantly affects the dynamics of generated LQFI, local quantum uncertainty, and concurrence-entanglement due to interactions between two NV centers under the same conditions as Fig. 1(a). Due to the Hamiltonian dipole-dipole coupling term: \(\frac{8}{r^3}\Gamma [|1_{A}1_{B}\rangle \langle 0_{A}0_{B}|+|0_{A}0_{B}\rangle \langle 1_{A}0_{B}|] -\frac{4}{r^3}\Gamma [|1_{A}0_{B}\rangle \langle 0_{A}1_{B}| + |0_{A}1_{B}\rangle \langle 1_{A}0_{B}| ]\) modifies the distribution and coupling between the two NV-center qubits, an increase in dipole-dipole coupling enhances significantly the oscillatory generation of quantum correlations by increasing their amplitudes and their frequencies. By comparing the results of Fig. 1(a) (\(\Gamma =0.4\)) with those depicted in the dipole-dipole coupling regime (\(\Gamma =0.8\)) of Fig. 4(a), we find that the amplitudes and frequencies of the generated LQFI, local quantum uncertainty, and concurrence-entanglement increase rapidly with more irregular oscillations. This indicates that increasing the dipole-dipole coupling accelerates the fluctuations of quantum information resources between two NV-center qubits, enhancing the generated LQFI, local quantum uncertainty, and concurrence-entanglement. However, as the intervals characterized by strong generated concurrence-entanglement increase, the intervals of Wigner–Yanase-Fisher correlation are reduced. In the relatively strong dipole-dipole coupling regime (\(\Gamma =1.5\), Fig. 4b), the results confirm that increasing the dipole-dipole coupling accelerates the fluctuations of quantum information resources between two NV-center qubits, enhancing the generated LQFI, local quantum uncertainty, and concurrence-entanglement. Moreover, the generated two-NV-center-qubits states exhibit a relatively strong concurrence correlation through several time intervals in which the difference between amplitudes of the LQFI and LQU are very large.

Moreover, we can deduce that dipole-dipole coupling is a significant tool in facilitating interactions between two NV centers to generate strong quantum information resources.

Fig. 5
figure 5

The dynamics of LQFI, LQU, and concurrence for two NV-center qubits of Fig.1a are shown when the intrinsic decoherence is taken with a large coupling \(1/\gamma = 5\times 10^{-3}\) in (a), and \(1/\gamma = 5\times 10^{-2}\) in (b).

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Figures 5 and 6 examine how increasing the intrinsic decoherence coupling significantly affects the oscillatory generation of different quantum correlations due to interactions between two NV centers under the same circumstances considered in the previous figures. These figures highlight how the intrinsic decoherence coupling suppresses quantum correlations over time.

Fig. 6
figure 6

Two-NV-center-qubits LQFI, LQU, and concurrence dynamics of Figs.2a, 3a, 4a are shown but when the intrinsic decoherence is taken with large coupling: \(1/\gamma = 5\times 10^{-3}\), respectively.

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Figure 5 shows the dynamics of two-NV-center-qubits LQFI, LQU, and concurrence of Fig. (1a) but when the intrinsic decoherence is taken with a large coupling \(1/\gamma = 5\times 10^{-3}\). We find that the exponential decoherence term, \(e^{ - \frac{1}{\gamma } (E_{m} - E_{n})^{2} t}\) depending on the intrinsic NV-center decoherence coupling \(1/\gamma\) leads to the degradation of the LQFI, local quantum uncertainty, and concurrence correlations associated with the generated two-NV-center qubits.

Moreover, we observe that the considered correlation quantifiers indicate the decoherence of the QFI, local quantum uncertainty, and concurrence-entanglement as time increases, by reducing the amplitudes and frequencies of their oscillatory generations. However, the generated concurrence correlation of two NV-center qubits is more robust than that of LQFI and local quantum uncertainty. Additionally, despite the two NV-center qubits exhibiting partial concurrence-entanglement, the intervals where LQFI and LQU display different amplitudes as the same oscillatory generations are increased, while the intervals of the Wigner–Yanase-Fisher correlation are reduced. To show a more pronounced effect of decoherence, the two-NV-center-qubits LQFI, LQU, and concurrence dynamics of Fig. 1a are shown but when the intrinsic decoherence is taken with a large coupling \(1/\gamma = 5\times 10^{-2}\) in Fig. 5b. We find that increasing the decoherence coupling \(1/\gamma\) accelerates the degradation of the NV-center qubits’ LQFI, local quantum uncertainty, and concurrence correlations, leading them to reach their damped oscillatory generations more rapidly, with small quasi-dependent amplitudes. As time evolves, the two NV-center qubits reach their time quasi-dependent stationary states in terms of LQFI, local quantum uncertainty, and concurrence correlations.

Figure 6(a-c) examines how increasing the external magnetic field to \(B_z=2\), the external control electric field to \(E_z=4\), and the dipole-dipole coupling to \(\Gamma =0.8\) significantly affects, respectively, the degradation of the NV-center qubits’ LQFI, local quantum uncertainty, and concurrence correlations as shown in Fig. 5(a). From Fig. 6a, we observe that increasing the external magnetic field to \(B_z=2\), which affects transitions between states of the two NV center qubits, accelerates the degradation of the NV-center qubits’ LQFI, local quantum uncertainty, and concurrence correlations. The two NV-center qubits reach their time quasi-dependent stationary states in terms of LQFI, local quantum uncertainty, and concurrence correlations, rapidly. The decoherence of the LQFI, local quantum uncertainty, and concurrence correlations shown in Figs. 5a and 6b confirm that increasing the control electric field to \(E_z=4\) enhances the intrinsic decoherence effect. This additional resource for suppressing decoherence significantly affects the oscillatory generation of quantum correlations. The intervals of appearance of the time quasi-dependent stationary states of the two NV-center qubits, in terms of LQFI, local quantum uncertainty, and concurrence correlations are visible. Moreover, we observe that the concurrence associated with the generated two NV-center qubits states drops suddenly to zero at a specific time for a short interval, exhibiting sudden-death and sudden-birth phenomena in the generated NV-center qubits entanglement. During this death interval, the NV-center qubits states exhibit partial LQFI and local quantum uncertainty correlations. Also, from Fig. 6c, we find that the dipole-dipole coupling to \(\Gamma =0.8\) enhances the intrinsic decoherence effect. After a particular time, the NV-center qubits’ LQFI, local quantum uncertainty, and concurrence correlations exhibit stationary, regular oscillatory behavior, with the same oscillation patterns but different amplitudes.

Conclusion

In this study, we analyzed the two-qubit quantum correlations of LQU, LQFI, and concurrence in the two NV-center qubits, considering the effects of increasing the zero-field splitting, external magnetic field, the external control electric field, and dipole-dipole coupling. Under the influence of these physical parameters, the two generated NV-center qubits exhibit a general non-X state. This contrasts with previous studies on NV centers, where the dynamics of quantum correlations were analyzed without incorporating such parameters19,20,59, thereby ensuring that the resulting two NV-center qubits maintained an X-state structure. It is found that the interactions between the two NV-center qubits, characterized by small couplings for a gyromagnetic factor, an external magnetic field, an external control electric field, and a dipole-dipole interaction, exhibit a high capability to generate LQFI, LQU, and concurrence correlations, even when intrinsic decoherence is present with small coupling. The generated LQFI, LQU, and concurrence exhibit oscillatory generations with the same frequency. Most of the time, LQFI and LQU exhibit the same Wigner–Yanase-Fisher correlation, except during strong entanglement intervals. The results confirm that this high capability to generate LQFI, LQU, and concurrence correlations with the same oscillatory generations can be weakened by increasing external magnetic fields and zero-field splitting fields. Conversely, increasing external control electric fields and dipole-dipole couplings enhance this high capability of the interactions to generate LQFI, LQU, and concurrence correlations.

For a large intrinsic NV-center decoherence coupling, the dynamics of LQFI, LQU, and concurrence have been investigated when the interactions between the two NV-center qubits are characterized by small couplings as in the previous case. Moreover, the exponential decoherence term, which depends on the intrinsic NV centre-decoherence coupling, is shown to lead to a degradation of the LQFI, the local quantum uncertainty and the concurrence correlations associated with the generated two-NV-center qubits. Finally, we examined how increasing the external magnetic field, external control electric field, and dipole-dipole coupling significantly impacts these correlation degradations.