  Recent investigations of the quantum properties of an antiferromagnet in the
spin wave approximation have identified the eigenstates as two-mode squeezed
sublattice states. The uniform squeezed vacuum and one-magnon states were shown
to display a massive sublattice entanglement. Here we expand this investigation
and study the squeezing properties of all sublattice Fock states throughout the
magnetic Brillouin zone. We derive the full statistics of the sublattice magnon
number with wave number $\vec k$ in the ground state and show that magnons are
created in pairs with opposite wavevectors, hence, resulting in entanglement of
both modes. To quantify the degree of entanglement we apply the
Duan-Giedke-Cirac-Zoller inequality and show that it can be violated for all
modes. The degree of entanglement decrease towards the corners of the Brillouin
zone. We relate the entanglement to measurable correlations of components of
the N\'eel and the magnetization vectors, thus, allowing to experimentally test
the quantum nature of the squeezed vacuum. The distinct $\vec k$-space
structure of the probabilites shows that the squeezed vacuum has a nonuniform
shape that is revealed through the $\vec k$-dependent correlators for the
magnetization and the N\'eel vectors.

Mesoscale and Nanoscale Physics;Materials Science;Quantum Physics