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

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=0.5(ln 3/ln 2-1)approximate to 0.292 481 25 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 root 3(m+1)/(3(m)+3) x2(-(1+gamma)m)equivalent to root 3(m+1)/(3(m)+3)x6(-m/2) (note that kappa=root 3), compared to the previous 2x2(-3m/2). This implies that with parameters realistic for current experiments, nuclear magnetic resonance (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)->infinity.