I introduce a sufficient condition for multi-particle entanglement which is also necessary and sufficient for sub shot-noise sensitivity in the estimation of a collective rotation angle. The connection is provided by the Fisher information, which is a simple measure of statistical distance among quantum states. I show how to produce this "useful" entanglement with dilute Bose-Einstein condensates and discuss applications on Mach-Zehnder interferometry with phase sensitivity at the Heisenberg limit.