  We examine the dynamics of entanglement entropy of all parts in an open
system consisting of a two-level dimer interacting with an environment of
oscillators. The dimer-environment interaction is almost energy conserving. We
find the precise link between decoherence and production of entanglement
entropy. We show that not all environment oscillators carry significant
entanglement entropy and we identify the oscillator frequency regions which
contribute to the production of entanglement entropy. Our results hold for
arbitrary strengths of the dimer-environment interaction, and they are
mathematically rigorous.
