Evolution of entanglement under echo dynamics

Echo dynamics and fidelity are often used to discuss stability in quantum information processing and quantum chaos. Yet fidelity yields no information about entanglement, the characteristic property of quantum mechanics. We study the evolution of entanglement in echo dynamics. We find qualitatively different behavior between integrable and chaotic systems on one hand and between random and coherent initial states for integrable systems on the other. For the latter the evolution of entanglement is given by a classical time scale. Analytic results are illustrated numerically in a Jaynes Cummings model.