  A new concept of eventually entanglement breaking divisible (eEB-divisible)
dynamics is introduced. A dynamical map is eEB-divisible if any propagator
becomes entanglement breaking in finite time. It turns out that any completely
positive dynamical semigroup with a unique faithful stationary state enjoys
this property. Interestingly, it is shown that eEB-divisibility is quite
general and holds for a pretty large class of quantum evolutions.
