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

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. This is the most general quantum mechanically allowed operation in the Maxwell demon's arsenal, and, in particular, includes non-completely positive but physically realizable maps on the system. The correction terms provide an advantage: they reduce the work required by the Maxwell's demon to erase its memory. The modified principle also incorporates the possibility of arbitrary charge flows, including the usual heat flow, between system and environment. Furthermore, we consider a case where the system is in contact with a large initially-decoupled athermal environment, and we derive the finite-time modified Landauer's bound for the same.