Largest separable balls around the maximally mixed bipartite quantum state

For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral $l_p$ norms for $1<~p<~\infty ,$ of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\mathrm{tr}{\rho }^2$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by nuclear magnetic resonance quantum computation implementations or other experimental situations.