  We examine the elements of the balance equation of entropy in open quantum
evolutions, and their response as we go over 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, and 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, being differences between the
global and local versions of the corresponding quantities. The investigation
brings us, in certain cases, to a complementarity of the timeintegrated heat
current deficit with the relative entropy of entanglement between the two
systems.
