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 κ2^-γm in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent ${γ}=0.5(ln{{0.2em}{0ex}}3∕ln{{0.2em}{0ex}}2{-}1){≈}0.292{{0.2em}{0ex}}481{{0.2em}{0ex}}25$ 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 $√{{3}ᵐ⁺¹∕({3}ᵐ+3)}×{}{2}^{{-}(1+{γ})m}{≡}√{{3}ᵐ⁺¹∕({3}ᵐ+3)}×{}{6}^{{-}m∕2}$ (note that ${κ}=√3$), compared to the previous $2×{}{2}^{{-}3m∕2}$. 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 d₀, although approaching our earlier exponent as d₀→∞.