Quantum steering and entanglement for coupled systems: exact results

Using the Wigner function in phase space, we study quantum steering and entanglement between two coupled harmonic oscillators. We derive expressions for purity and quantum steering in both directions and identify several important selection rules. Our results extend the work reported in {\color{blue} [Phys. Rev. E 97, 042203 (2018)]} focused on the weak coupling regime, revealing significant deviations in the ultra-strong coupling regime. In particular, Makarov's prediction of a separable ground state contrasts with our exact calculations, highlighting the limitations of his approach under strong coupling conditions. We show that quantum steering between excited oscillators is completely absent even in the ultra-strong coupling regime. Similarly, resonant oscillators have no steering, and ground states cannot steer any receiver state. We find that quantum steering becomes notably more pronounced as the system approaches resonance and within specific ranges of ultra-strong coupling. This behavior is marked by a clear asymmetry, where steering is present in only one direction, highlighting the delicate balance of interaction strengths that govern the emergence of quantum correlations. These results advance our understanding of how excitation levels and coupling strengths influence quantum steering and entanglement in coupled harmonic oscillators.