  Landauer's principle provides a perspective on the physical meaning of
information as well as on the minimum working cost of information processing.
Whereas most studies have related the decrease in entropy during a
computationally irreversible process to a lower bound of dissipated heat,
recent efforts have also provided another lower bound associated with the
thermodynamic fluctuation of heat. The coexistence of the two conceptually
independent bounds has stimulated comparative studies of their close
relationship or tightness; however, these studies were concerned with finite
quantum systems that allowed the revival of erased information because of a
finite recurrence time. We broaden these comparative studies further to open
quantum systems with infinite recurrence times. By examining their dependence
on the initial state, we find the independence of the thermodynamic bound from
the initial coherence, whereas the entropic bound depends on both the initial
coherence and population. A crucial role is indicated by the purity of the
initial state: the entropic bound is tighter when the initial condition is
sufficiently mixed, whereas the thermodynamic bound is tighter when the initial
state is close to a pure state. These trends are consistent with previous
results obtained for finite systems.
