Separability gap and large deviation entanglement criterion

For a given Hamiltonian $H$ on a multipartite quantum system, one is interested in finding the energy $E_0$ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value $\lambda_{\rm min}^{\otimes}$ of $H$ among all product states. For several concrete model Hamiltonians, we investigate the difference $\lambda_{\rm min}^{\otimes}-E_0$, called separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable among a given state is sufficiently distant from the average value, the state is almost surely entangled.