Optimal ensemble length of mixed separable states

The optimal (pure state) ensemble length of a separable state A is the minimum number of (pure) 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 bounds are sharp. A consequence of our results is that for all two-particle systems, except possibly those with one 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 probabilty 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system of p qubits with probability 1 it is greater than 2(2p)/(1+2p), i.e., almost the square of the rank. [S0022-2488(00)02510-X].