Entanglement, Nonlinear Dynamics, and the Heisenberg Limit

We show that quantum Fisher information provides a sufficient condition to recognize multiparticle entanglement in an 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 theta. The analysis therefore singles out the class of entangled states which are useful to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the nonlinear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.