Entanglement depth for symmetric states

Entanglement depth characterizes the minimal number of particles in a system that are mutually entangled. For symmetric states, there is a dichotomy for entanglement depth: An N-particle symmetric state is either fully separable or fully entangled - the entanglement depth is either 1 or N. We show that this dichotomy property for entangled symmetric states is even stable under nonsymmetric noise. We propose an experimentally accessible method to detect entanglement depth in atomic ensembles based on a bound on the particle number population of Dicke states, and demonstrate that the entanglement depth of some Dicke states, for example the twin Fock state, is very stable even under a large arbitrary noise. Our observation can be applied to atomic Bose-Einstein condensates to infer that these systems can be highly entangled with the entanglement depth that is on the order of the system size (i.e., several thousands of atoms).