  The optimal (pure state) ensemble length of a separable state, A, is the
minimum number of (pure) product states needed in convex combination to
construct A. We study the set of all separable states with optimal (pure state)
ensemble length equal to k or fewer. Lower bounds on k are found below which
these sets have measure 0 in the set of separable states. In the bipartite case
and the multiparticle case where one of the particles has significantly more
quantum numbers than the rest, the lower bound for non-pure state ensembles is
sharp. A consequence of our results is that for all two particle systems,
except possibly those with a qubit or those with a nine dimensional Hilbert
space, and for all systems with more than two particles the optimal pure state
ensemble length for a randomly picked separable state is with probability 1
greater than the state's rank. In bipartite systems with probability 1 it is
greater than 1/4 the rank raised to the 3/2 power and in a system with p qubits
with probability 1 it is greater than (2^2p)/(1+2p), which is almost the square
of the rank.
