Entanglement Detection by Approximate Entanglement Witnesses

The problem of determining whether a given quantum state is separable is known to be computationally difficult. We develop an approach to this problem based on approximations of convex polytopes in high dimensions. By showing that a convex polytope constructed from a polynomial number of hyperplanes approximates the Euclidean ball arbitrarily well in high dimensions, we find evidence that a polynomial-sized set of approximate entanglement witnesses is potentially sufficient to determine the entanglement of a state with high probability.