  Quantum degeneracy is the central many-body feature of ideal quantum gases
stemming from quantum mechanics. In this work we address its relationship to
the most fundamental form of non-classicality in many-body system, i.e.
many-body entanglement. We aim at establishing a quantitative link between
quantum degeneracy and entanglement in spinful ideal gases, using entanglement
witness criteria based on the variance of the collective spin of the spin
ensemble. We show that spin-1/2 ideal Bose gases do not possess entanglement
which can be revealed from such entanglement criteria. On the contrary, ideal
spin-1/2 Fermi gases exhibit spin entanglement revealed by the collective-spin
variances upon entering quantum degeneracy, due to the formation of highly
non-local spin singlets. We map out the regime of detectable spin entanglement
for Fermi gases in free space as well as in a parabolic trap, and probe the
robustness of spin entanglement to thermal effects and spin imbalance. Spin
entanglement in degenerate Fermi gases is amenable to experimental observation
using state-of-the-art spin detection techniques in ultracold atoms.
