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