  In this paper we review our recent work on the theoretical approach to
quantum Loschmidt echoes, i.e. various properties of the so called echo
dynamics -- the composition of forward and backward time evolutions generated
by two slightly different Hamiltonians, such as the state autocorrelation
function (fidelity) and the purity of a reduced density matrix traced over a
subsystem (purity fidelity). Our main theoretical result is a linear response
formalism, expressing the fidelity and purity fidelity in terms of integrated
time autocorrelation function of the generator of the perturbation.
Surprisingly, this relation predicts that the decay of fidelity is the slower
the faster the decay of correlations. In particular for a static
(time-independent) perturbation, and 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, as the
perturbation and classical limits do not commute. The theoretical results are
demonstrated numerically for two models, the quantized kicked top and the
multi-level Jaynes Cummings model. Our method can for example be applied to the
stability analysis of quantum computation and quantum information processing.
