Bell inequality violation versus entanglement in presence of local   decoherence
  We analyze the effect of local decoherence of two qubits on their entanglement and the Bell inequality violation. Decoherence is described by Kraus operators, which take into account dephasing and energy relaxation at an arbitrary temperature.   We show that in the experiments with superconducting phase qubits the survival time for entanglement should be much longer than for the Bell inequality violation. 
A. G. Kofman and A. N. Korotkov