Size of the separable neighborhood of the maximally mixed bipartite quantum state

For finite-dimensional bipartite quantum systems, we find the size of the largest balls, in spectral $l_p$ norms for $1 \le p \le \infty$, of separable (unentangled) matrices around the identity matrix. We discuss corollaries and applications to density matrices and bulk quantum information processing such as high-temperature nuclear magnetic resonance.