  We show that for an m-qubit quantum system, there is a ball of radius
asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at
the identity matrix, of separable (unentangled) positive semidefinite matrices,
for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much
smaller in magnitude than the best previously known exponent, from our earlier
work, of 1/2. For normalized m-qubit states, we get a separable ball of radius
sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + \gamma)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times
6^{-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 with
standard pseudopure-state preparation techniques can access only unentangled
states if 36 qubits or fewer are used (compared to 23 qubits via our earlier
results). We also obtain an improved exponent for m-partite systems of fixed
local dimension d_0, although approaching our earlier exponent as d_0
approaches infinity.

Quantum Physics