Quantum steering and entanglement in two coupled harmonic oscillators

We study quantum steering and entanglement in two coupled harmonic oscillators using Wigner function techniques in phase space. We derive expressions for purity and quantum steering in the x- and y-directions. Our results extend previous studies from the weak coupling regime to the ultra-strong coupling regime, revealing significant deviations in behavior. In particular, we show that quantum steering between excited oscillators is completely absent even in the ultra-strong coupling regime. Similarly, resonant oscillators show no steering, and ground states cannot steer any receiver state. We find that quantum steering increases as the system approaches resonance and in certain ultra-strong coupling regimes. This behavior shows a clear asymmetry, with steering occurring in only one direction. It highlights the delicate balance of interaction strengths that shapes quantum correlations. These results improve our understanding of how excitation levels and coupling strengths affect quantum steering and entanglement in coupled harmonic oscillators.