Entanglement and extreme spin squeezing of unpolarized states

We present criteria to detect the depth of entanglement in macroscopic ensembles of spin- j particles using the variance and second moments of the collective spin components. The class of states detected goes beyond traditional spin-squeezed states by including Dicke states and other unpolarized states. The criteria derived are easy to evaluate numerically even for systems of very many particles and outperform past approaches, especially in practical situations where noise is present. We also derive analytic lower bounds based on the linearization of our criteria, which make it possible to define spin-squeezing parameters for Dicke states. In addition, we obtain spin squeezing parameters also from the condition derived in (Sørensen and Mølmer 2001 Phys. Rev. Lett. https://doi.org/10.1103/physrevlett.86.4431 86 https://doi.org/10.1103/physrevlett.86.4431 ). We also extend our results to systems with fluctuating number of particles.