Three-qubit entanglement witnesses with the full spanning properties

We construct three-qubit entanglement witnesses with the full spanning property, by which we mean that all the partial transposes have the spanning properties. These witnesses determine faces for separable states whose interior lies in the interior of positive partial transposes (PPT) states, and so they detect sets of PPT entangled states with nonempty interiors. These witnesses also detect fully bi-separable states as well as all kinds of bi-separable pure product states. Our construction depends on the observation that multi-partite entanglement witnesses correspond to positive multi-linear maps. (paper)