Entanglement and extreme spin squeezing for a fluctuating number of indistinguishable particles

We extend the criteria for k-particle entanglement from the spin-squeezing parameter presented in Srensen and Mlmer Phys. Rev. Lett. 86, 4431 (2001) to systems with a fluctuating number of particles. We also discuss how other spin-squeezing inequalities can be generalized to this situation. Further, we show how, by employing additional degrees of freedom, it is possible to extend the bounds to the case when the individual particles cannot be addressed. As a by-product, this allows us to show that in spin-squeezing experiments with cold gases the particles are typically distinguishable in practice. Our results justify the application of the Srensen-Mlmer bounds in recent experiments on spin squeezing in Bose-Einstein condensates.