  We consider a quantum linear oscillator coupled at an arbitrary strength to a
bath at an arbitrary temperature. We find an exact closed expression for the
oscillator density operator. This state is non-canonical but can be shown to be
equivalent to that of an uncoupled linear oscillator at an effective
temperature T_{eff} with an effective mass and an effective spring constant. We
derive an effective Clausius inequality delta Q_{eff} =< T_{eff} dS, where
delta Q_{eff} is the heat exchanged between the effective (weakly coupled)
oscillator and the bath, and S represents a thermal entropy of the effective
oscillator, being identical to the von-Neumann entropy of the coupled
oscillator. Using this inequality (for a cyclic process in terms of a variation
of the coupling strength) we confirm the validity of the second law. For a
fixed coupling strength this inequality can also be tested for a process in
terms of a variation of either the oscillator mass or its spring constant. Then
it is never violated. The properly defined Clausius inequality is thus more
robust than assumed previously.
