  The destruction of entanglement of open quantum systems by decoherence is
investigated in the asymptotic long-time limit. Starting from a general and
analytically solvable decoherence model which does not involve any
weak-coupling or Markovian assumption it is shown that two fundamentally
different classes of entangled states can be distinguished. Quantum states of
the first class are fragile against decoherence so that they can be
disentangled asymptotically even if coherences between pointer states are still
present. Quantum states of the second type are robust against decoherence.
Asymptotically they can be disentangled only if also decoherence is perfect. A
simple criterion for identifying these two classes on the basis of two-qubit
entanglement is presented.
