Probing quantum entanglement from magnetic-sublevels populations: beyond spin squeezing inequalities

Spin squeezing inequalities (SSI) represent a major tool to probe quantum entanglement among collection of few-level atoms, and are based on collective spin measurements their fluctuations. Yet, for atomic ensembles spin-$j$ atoms ultracold spinor gases, many experiments can image the populations in all Zeeman sublevels $s=-j, -j+1, \dots, j$, potentially revealing finer features not captured by SSI. Here we present systematic approach which exploits Zeeman-sublevel population order construct novel criteria, illustrate our ground states spin-1 spin-2 Bose-Einstein condensates. Beyond these specific examples, allows one infer, manner, optimal permutationally-invariant witness any given set an ensemble $d$-level systems.