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

Modified Landauer's principle: How much can the Maxwell's demon gain by using general system-environment quantum state?