  We investigate the quantum entanglement of systems of coupled harmonic
oscillators on the basis of thermo-field dynamics (TFD). For coupled harmonic
oscillators at equilibrium, the extended entanglement entropy is derived using
the TFD method, and it is demonstrated to be controlled by temperature and
coupling parameters. For non-equilibrium systems, in addition to temperature
and coupling parameters, the time dependence of the extended entanglement
entropy is calculated in accordance with the dissipative von Neumann equation,
and the dissipative dynamics of the systems of coupled harmonic oscillators is
discussed. Consequently, based on TFD, the physical parametrization of the
entanglement entropies is confirmed in both the equilibrium and non-equilibrium
cases of harmonic oscillator systems by means of the laws of thermodynamics.
