Abstarct: In this thesis, we obtain the dynamics of the problem of two coupled harmonic oscillators using the Langevin quantum equation approach. We then calculate the covariance matrix using these equations and we provide the term for non-equilibrium entropy production rate for quantum oscillators, which allows us to give a simple interpretation of the irreversible start. Then we obtain the entropy production rate and the quantum system correlations, mutual information, via covariant matrix, numerically. Finally, we apply our results to the analysis for a nano-mechanical resonator (optomechanics).