  Exchange interactions in spin systems can give rise to quantum entanglement
in the ground and thermal states of the systems. In this paper, we consider a
spin tetramer, with spins of magnitude 1/2, in which the spins interact via
nearest-neighbour, diagonal and four-spin interactions of strength J1, J2 and K
respectively. The ground and thermal state entanglement properties of the
tetramer are calculated analytically in the various limiting cases. Signatures
of quantum phase transition (QPT) in terms of appropriate entanglement measures
are obtained. The magnetic properties of a S = 1/2 AFM polyoxovanadate
compound, V12, are well explained by spin tetramers with only nearest-neighbour
exchange interactions. Treating the magnetic susceptibility chi as an
entanglement witness (EW), an estimate of the lower bound of the critical
entanglement temperature, T_c, above which the multipartite entanglement
disappears in the experimental compound, is determined. A second EW based on
energy provides an estimate of the entanglement temperature T_E, below which
bipartite entanglement is certainly present in the system, is determined. The
cases of the symmetric trimer and the tetrahedron are also considered.
