Decoherence and the Loschmidt Echo

Decoherence causes entropy increase that can be quantified using, e.g., the purity $\varsigma =\mathrm{T}\mathrm{r}{\rho }^2$. When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo $M(t)$. It is given by the squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of $\varsigma (t)$ and the average $\overline{M}(t)$. 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 $\overline{M}$ has a regime where it is dominated by the Lyapunov exponents.