Outgrowth of quasi pure states in isolated coupled-harmonic-oscillator

Isolated coupled-harmonic-oscillator here is the system of two distinguishable particles coupled with a harmonic oscillator interaction potential. Each particle stays in a mixed state due to entanglement. However, in center-of-mass reference frame, we obtain quasi wavefunction of the first particle expressing quasi pure state by replacing the second coordinate in the total wavefunction. We discuss the similar systems with the first particle and the potential being same and the second mass changing from micro to macro one. Measured by fidelity and coherence, the quasi pure state approaches to the pure state of a usual harmonic oscillator with same mass and similar potential. It conversely shows that the latter purely superposed state in position representation and its coherence originate from those of the first particle, which are related with some neglected macro object and the interaction between them. The current results provide a possible clue to new insights into quantum states.