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We show that for an m-qubit quantum system, there is a ball of radiusasymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered atthe identity matrix, of separable (unentangled) positive semidefinite matrices,for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is muchsmaller in magnitude than the best previously known exponent, from our earlierwork, of 1/2. For normalized m-qubit states, we get a separable ball of radiussqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + \gamma)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times6^{-m/2} (note that \kappa = \sqrt{3}), compared to the previous 2 * 2^{-3m/2}.This implies that with parameters realistic for current experiments, NMR withstandard pseudopure-state preparation techniques can access only unentangledstates if 36 qubits or fewer are used (compared to 23 qubits via our earlierresults). We also obtain an improved exponent for m-partite systems of fixedlocal dimension d_0, although approaching our earlier exponent as d_0approaches infinity.Comment: 30 pp doublespaced, latex/revtex, v2 added discussion of Szarek's  upper bound, and reference to work of Vidal, v3 fixed some errors (no effect  on results), v4 involves major changes leading to an improved constant, same  exponent, and adds references to and discussion of Szarek's work showing that  exponent is essentially optimal for qubit case, and Hildebrand's alternative  derivation for qubit case. To appear in PR

Better bound on the exponent of the radius of the multipartite separable ball