  We address the "major open problem" of evaluating how much increased
efficiency in estimation is possible using non-separable, as opposed to
separable, measurements of N copies of m-level quantum systems. First, we study
the six cases m = 2, N = 2,...,7 by computing the the 3 x 3 Fisher information
matrices for the corresponding optimal measurements recently devised by Vidal
et al (quant-ph/9812068) for N = 2,...,7. We obtain simple polynomial
expressions for the ("Gill-Massar") traces of the products of the inverse of
the quantum Helstrom information matrix and these Fisher information matrices.
The six traces all have minima of 2 N -1 in the pure state limit, while for
separable measurements (quant-ph/9902063), the traces can equal N, but not
exceed it. Then, the result of an analysis for m = 3, N = 2 leads us to
conjecture that for optimal measurements for all m and N, the "Gill-Massar
trace" achieves a minimum of (2N-1)(m-1) in the pure state limit.
