  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 auto-correlation function of the generator of perturbation.
Surprisingly, this relation predicts the slower decay of fidelity the faster
decay of correlations is. 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 time scale 1/delta as
opposed to mixing dynamics where the fidelity is found to decay exponentially
on a time-scale 1/delta^2, where delta is a strength of perturbation. A
detailed discussion of a semi-classical regime of small effective values of
Planck constant is given where classical correlation functions can be used to
predict quantum fidelity decay. Note that the correct and intuitively expected
classical stability behavior is recovered in the classical limit hbar->0, as
the two limits delta->0 and hbar->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 a celebrated example of a
quantized kicked top.
