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We examine the elements of the balance equation of entropy in open quantumevolutions and their response as we go from a Markovian to a non-Markoviansituation. In particular, we look at the heat current and entropy productionrate in the non-Markovian reduced evolution, as well as a Markovian limit ofthe same, experienced by one of two interacting systems immersed in a Markovianbath. The analysis naturally leads us to define a heat current deficit and anentropy production rate deficit, which are differences between the global andlocal versions of the corresponding quantities. The investigation leads, incertain cases, to a complementarity of the time-integrated heat current deficitand the relative entropy of entanglement between the two systems.

Heat current and entropy production rate in local non-Markovian quantum dynamics of global Markovian evolution