Fidelity Decay Saturation Level for Initial Eigenstates
  We show that the fidelity decay between an initial eigenstate state evolved under a unitary chaotic operator and the same eigenstate evolved under a perturbed operator saturates well before the 1/N limit, where $N$ is the size of the Hilbert space, expected for a generic initial state. We provide a theoretical argument and numerical evidence that, for intermediate perturbation strengths, the saturation level depends quadratically on the perturbation strength. 
Yaakov S. Weinstein, Joseph V. Emerson, Seth Lloyd, David G. Cory