Shannon information theory in heat engines

Open quantum system techniques help us to understand how density matrix evolve in subsystems. This is only useful to evaluate time evolution of physical quantitues, which are linear in density matrix. Shannon entropy is not so, in fact it is nonlinear in density matrix. We introduce a technique, called extended Keldysh formalism, to understand time evolution of nonlinear quantities in density matrix. As result we show that Shannon entropy can evolve using much larger class of Kubo-Martin-Schwinger correlators, therefore evaluation of Shannon entropy based on standard open quantum techniques, such as Lindblade equations etc., are misleading and inconsistent with quantum nature of entropy. We address inconsistencies and also address new results.