We show that the quantum Fisher information provides a sufficient condition to recognize multi-particle entanglement in a N qubit state. The same criterion gives a necessary and sufficient condition for sub shot-noise phase sensitivity in the estimation of a collective rotation angle θ. The analysis therefore singles out the class of entangled states which are {\it useful} to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the non-linear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.
@article{arxiv.0711.4840,
title = {Entanglement, Non-linear Dynamics, and the Heisenberg Limit},
author = {L. Pezze' and A. Smerzi},
journal= {arXiv preprint arXiv:0711.4840},
year = {2015}
}