Decoherence and the Loschmidt echo

Environment--induced decoherence causes entropy increase. It can be quantified using, e.g., the purity $\varsigma={\rm Tr}\rho^2$. When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo $\bar M(t)$. It is given by the average squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of $\varsigma(t)$ and $\bar M(t)$. In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of $\varsigma$ and $\bar M$ has a regime where it is dominated by the classical Lyapunov exponents