  The Landauer principle states that decrease in entropy of a system,
inevitably leads to a dissipation of heat to the environment. This statement is
usually established by considering the system to be in contact with an
environment that is initially in a thermal state with the system-environment
initial state being in a product state. Here we show that a modified Landauer
principle, with correction terms, still holds even if the system and
environment are initially correlated and the environment is in an athermal
state. Furthermore, we consider a case where the system is in contact with a
large athermal environment, such that the system dynamics allow Born-Markov
approximations, and we derive the finite-time modified Landauer's bound for the
same.
