  We study the complexity for an open quantum system. Our system is a harmonic
oscillator coupled to a one-dimensional massless scalar field, which acts as
the bath. Specifically, we consider the reduced density matrix by tracing out
the bath degrees of freedom for both regular and inverted oscillator and
computed the complexity of purification (COP) and complexity by using the
operator-state mapping. We found that when the oscillator is regular the COP
saturates quickly for both underdamped and overdamped oscillators.
Interestingly, when the oscillator is underdamped, we discover a kink like
behaviour for the saturation value of COP with varying damping coefficient. For
the inverted oscillator, we found a linear growth of COP with time for all
values of bath-system interaction. However, when the interaction is increased
the slope of the linear growth decreases, implying that the unstable nature of
the system can be regulated by the bath.
