  The SU(2) and SU(3) Lie algebras lend themselves naturally to studies of two-
and three-well Bose-Einstein condensates, with the group operators being
expressed in terms of bosonic annihilation and creation operators at each site.
The success of these representations has led to the purities associated with
these algebras to be promoted as a measure of entanglement between the atomic
modes in each well. In this report, we show that these purities do not provide
an unambiguous measure, but instead give results which depend on the quantum
statistical states of each atomic mode. Using the example of totally uncoupled
modes which have never interacted, we quantify these purities for different
states and show that completely separable states can give values which have
been claimed to indicate the presence of entanglement.
