Extrapolated quantum states, void states and a huge novel class of distillable entangled states

A nice and interesting property of any pure tensor product state is that each such state has distillable entangled states at an arbitrarily small distance epsilon in its neighborhood. We say that such nearby states are epsilon-entangled, and we call the tensor product state in that case, a "boundary separable state," as there is entanglement at any distance from this "boundary." Here we find a huge class of separable states that also share the property mentioned above-they all have epsilon entangled states at any small distance in their neighborhood. Furthermore, the entanglement they have is proved to be distillable. We then extend this result to the discordant/classical cut and show that all classical states (correlated and uncorrelated) have discordant states at distance epsilon, and provide a constructive method for finding epsilon-discordant states.