Stability of quantum motion and correlation decay

We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time autocorrelation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster the decay of correlations. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay, of fidelity is predicted on a timescale proportional to 1/delta as opposed to mixing dynamics where the fidelity is found to decay exponentially on a timescale proportional to 1/delta(2), where delta is the strength of perturbation. A detailed discussion of a semiclassical regime of small effective values of Planck constant h is given where classical correlation functions can be used to predict quantum fidelity decays Note that the correct and intuitively expected classical stability behaviour is recovered in the classical limit h --> 0, as the two limits delta --> 0 and h --> 0 do not commute. In addition, we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on the celebrated example of a quantized kicked top.