Separable balls around the maximally mixed multipartite quantum states

We show that for an m-partite quantum system, there is a ball of radius 2^-(m/2-1) in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an {ε} below which mixtures of {ε} of any density matrix with 1-ε of the maximally mixed state will be separable. The {ε} thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which nuclear magnetic resonance with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 23 qubits (compared to 13 qubits via earlier results). A ball of radius 1 is obtained for multipartite states separable over the reals.