Quantum correlations and coherence in a two-qubit anisotropic $XY$ under magnetic field

Abstract: We study thermal quantum correlations and coherence in Heisenberg $XY$ model with anisotropic interactions under a uniform magnetic field $ B $. Using concurrence $C$, local quantum uncertainty (LQU), Bell-Clauser-Horne-Shimony-Holt (CHSH) nonlocality $ \mathbb{B}$, and coherence $C_l$ as quantifiers, we analyze how magnetic anisotropy $ δ_m $, coupling anisotropy $ δ_c $, Dzyaloshinskii-Moriya (DM) interaction $ D $, temperature $ T $, and magnetic field $ B $ modulate quantum resources. At low temperatures and relevant magnetic fields, the entanglement is maximized, but exhibits sudden death for $ δ_m = 0 $, which turns into a smooth decay as $ δ_m $ increases, highlighting its stabilizing role. LQU shows that stronger anisotropy suppresses quantum correlations, while $ \mathbb{B} $ induces a non-monotonic response peaking at a critical field $ B_c $. Bell-CHSH nonlocality violations ($ \mathbb{B} > 2 $) persist below $ B_c $, but thermal noise ($ T \geq 1 $) suppresses them. Coherence $ C_l $ is most robust to thermal fluctuations, especially for high ( δ_m ), which also dampens abrupt quantum phase transitions. The DM interaction is essential for entanglement generation, with $ D $ and anisotropy synergistically enhancing correlation resilience. We identify a hierarchy of thermal degradation: nonlocality ($ \mathbb{B} $) vanishes first, followed by entanglement ($ C $), then general quantum correlations (LQU), while coherence $ C_l $ persists the longest. These results demonstrate tunable control of quantum resources via anisotropy and external parameters, providing insights for the design of robust spin-based quantum technologies.

• The paper provides a closed-form analytical evaluation of entanglement, discord-like correlations, nonlocality, and coherence in the thermal state of a two-qubit anisotropic XY model.
• It demonstrates that magnetic and coupling anisotropies, along with Dzyaloshinskii-Moriya interaction, serve as tunable mechanisms for stabilizing quantum resources.
• The paper establishes a thermal hierarchy where nonlocality is most fragile and coherence remains robust over a wide parameter range.

Quantum Correlations and Coherence in Anisotropic Two-Qubit $XY$ Systems under Magnetic Field

Overview

The work analyzes the thermal quantum correlations and quantum coherence in a two-qubit anisotropic $XY$ Heisenberg model in the presence of both uniform magnetic field and Dzyaloshinskii-Moriya (DM) interaction. The study provides a systematic and quantitative assessment of concurrence, local quantum uncertainty (LQU), Bell-CHSH nonlocality ($\mathbb{B}$), and $l_1$-norm quantum coherence ($C_l$). It investigates how these resources are influenced by magnetic anisotropy ($\delta_m$), coupling anisotropy ($\delta_c$), DM interaction strength ($D$), temperature ($T$), and external field strength ($B$). The results elaborate on mechanisms for stabilizing quantum resources and detail a thermal hierarchy in resource fragility.

System, Hamiltonian, and Analytical Solution

The system is modeled by an anisotropic two-qubit $XY$0 Heisenberg Hamiltonian under a tunable magnetic field: $XY$1 with parameters recast in terms of mean and difference (anisotropies), giving dimensionless control over the interaction features. Analytical diagonalization yields eigenenergies and eigenstates, facilitating closed-form characterization of the canonical (thermal) density matrix for arbitrary $XY$2.

Key to the approach is explicit construction of the thermal state in the computational basis, resulting in an X-structured density matrix. This structure enables efficient and exact evaluation of all the quantum correlation and coherence measures employed.

Quantification of Quantum Resources

Entanglement (Concurrence)

Concurrence serves as the principal bipartite entanglement quantifier. For the thermal state, the explicit formula is given and its behavior is analyzed via numerical simulations. The results exhibit:

• Maximal entanglement at low $XY$3 and within optimal $XY$4 ranges.
• At $XY$5, 'entanglement sudden death' occurs for increasing $XY$6 or $XY$7.
• Increasing $XY$8 softens abrupt entanglement loss, producing smoother decay and wider parameter regions with nonvanishing concurrence.
• The DM interaction strength $XY$9 is identified as essential for the generation and stabilization of entanglement; $\mathbb{B}$0 is strictly zero when $\mathbb{B}$1.

Quantum Discord-like Correlations (LQU)

LQU is employed as an operational discord-type quantifier indicating quantum correlations in both entangled and separable regimes. The analytical structure allows its computation through the largest eigenvalue of a matrix constructed from the square root of the density operator. The study reveals:

• At ($\mathbb{B}$2, $\mathbb{B}$3), LQU is insensitive to anisotropy.
• For finite $\mathbb{B}$4 or $\mathbb{B}$5, LQU generally decreases with $\mathbb{B}$6, demonstrating suppression of local quantum correlations by magnetic anisotropy.
• The temperature dependence is nonmonotonic in certain $\mathbb{B}$7 regimes, with critical behavior near field-induced transitions, reflecting the interplay of competing interactions.

Nonlocality (Bell-CHSH Observable)

Nonlocality is evaluated via maximal Bell-CHSH violation $\mathbb{B}$8 using the Horodecki criterion, which depends on the correlation tensor elements of the X-state. Main findings include:

• $\mathbb{B}$9 (violation) is robust only for low $l_1$0 and within critical field regions.
• Nonlocality is the most fragile resource under both thermal and magnetic perturbations.
• Both anisotropy and coupling parameters can enhance or suppress the critical field $l_1$1 at which $l_1$2 transitions below the classical threshold.

Quantum Coherence ( $l_1$3-norm )

The $l_1$4-norm quantifies off-diagonal coherence in the reference basis. The study provides:

• Coherence persists over the widest parameter regime, outlasting entanglement and LQU as $l_1$5 increases.
• $l_1$6 has a significant stabilizing effect on coherence, shifting critical fields and suppressing abrupt transitions.
• Coupling anisotropy $l_1$7 enables revival of coherence at reduced magnetic fields, especially at larger $l_1$8.

Complementary measures such as relative entropy of coherence are discussed, and the bounds for finite-dimensional systems are explicitly referenced, establishing a benchmark for maximal coherence.

Thermal Hierarchy of Quantum Resources

A core contribution is the explicit identification and demonstration of the order in which quantum resources degrade with increasing $l_1$9:

1. Nonlocality ($C_l$0): Vanishes at the lowest $C_l$1, highly sensitive to thermal fluctuations.
2. Entanglement (Concurrence): Lost at higher $C_l$2 than nonlocality but prior to LQU and coherence.
3. Discord-like correlations (LQU): Persist after entanglement, vanishing near the ultimate decoherence threshold.
4. Coherence ($C_l$3): Most robust, remaining nonzero over the largest swath of parameter space.

This sequence empirically substantiates a theorized hierarchy of quantum correlations and provides a quantitative mapping within a concrete model.

Synergistic Effects of Model Parameters

• Magnetic Anisotropy ($C_l$4): Smooths phase transitions, stabilizes and enhances all quantum resources, particularly at higher values.
• Coupling Anisotropy ($C_l$5): Facilitates coherence revival, tunes the critical field for quantum transitions.
• DM Interaction ($C_l$6): Necessitates entanglement; increasing $C_l$7 promotes both entanglement and discord-type correlations, in cooperation with anisotropy.

The interplay of these parameters is shown to provide practical 'knobs' for resource engineering in spin-based quantum technologies.

Implications and Future Directions

These results carry practical relevance for design and operation of quantum devices, such as solid-state qubits, where environmental decoherence and thermal effects are inescapable. The hierarchy identified here implies that certain protocols may rely on coherence or discord-like correlations even when entanglement and nonlocality are lost—quadruple resource quantification is essential for realistic benchmarking. The explicit control afforded by anisotropy and DM interactions enables adaptive strategies for resource preservation and optimization.

Further theoretical development is suggested in several directions:

• Extension to multi-qubit and many-body spin chains to examine how the hierarchy persists or is modified in larger Hilbert spaces.
• Inclusion of non-Markovian environmental dynamics to scrutinize robustness under nontrivial decoherence models.
• Experimental validation in solid-state spin platforms exploiting DM and anisotropic exchange interactions.

Conclusion

The paper establishes a comprehensive, quantitatively resolved account of the interplay and durability of different quantum resources in an anisotropic $C_l$8 two-qubit Heisenberg model under magnetic field, with an essential role played by the DM interaction. The study reveals a clear and explicit hierarchy in the resilience of quantum nonlocality, entanglement, discord-like correlations, and coherence to thermal and external-field-induced decoherence. Magnetic and coupling anisotropies, together with DM interaction, are shown to offer unique and synergistic means for stabilizing quantum resources, holding concrete implications for control protocols in quantum information processing and quantum technology development.

Reference: "Quantum correlations and coherence in a two-qubit anisotropic $C_l$9 under magnetic field" (2606.07051)