  Particle number fluctuations, no matter how small, are present in
experimental set-ups. One should rigorously take these fluctuations into
account, especially, for entanglement detection. In this context, we generalize
the spin squeezing inequalities introduced by T\'oth et al. in Phys. Rev. Lett.
99, 250405 (2007). These new inequalities are fulfilled by all separable states
even when the number of particle is not constant, and may present quantum
fluctuations. These inequalities are useful for detecting entanglement in
many-body systems when the super-selection rule does not apply, or when only a
subspace of the total systems Hilbert space is considered. We also define
general dichotomic observables for which we obtain a coordinate independent
form of the generalized spin squeezing inequalities. We give an example where
our generalized coordinate independent spin squeezing inequalities present a
clear advantage over the original ones.
