Bell Nonlocality and EPR Steering Decay in Dephasing Hyperfine Spins

This work presents a comprehensive study of quantum correlations and their degradation under environmental dephasing within the atomic hydrogen system. By analyzing the magnetic coupling between the electron and proton spins in the $1s$ hyperfine state, we elucidate how coherent spin interactions generate entangled states and govern their temporal evolution. The investigation focuses on three key measures of quantum correlations—Bell nonlocality, Einstein–Podolsky–Rosen (EPR) steering, and quantum purity—each reflecting a different level within the hierarchy of nonclassical correlations. Analytical formulations and numerical simulations reveal that, in the absence of decay, all quantities remain steady, indicating the preservation of coherence. When dephasing is introduced, each measure decays exponentially toward a stationary lower bound, with Bell nonlocality identified as the most fragile, followed by steering and purity. A three-dimensional analysis of Werner states under dephasing further establishes the critical purity thresholds required for Bell inequality violations. The results highlight the interdependence between magnetic coupling, decoherence, and initial entanglement, providing a unified framework for understanding correlation dynamics in open quantum systems. These findings have direct implications for the development of noise-resilient quantum information protocols and spin-based quantum technologies, where preserving nonlocal correlations is essential for reliable quantum operations.