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 $$\epsilon $$ϵ in its neighborhood. We say that such nearby states are $$\epsilon $$ϵ-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 $$\epsilon $$ϵ-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 $$\epsilon $$ϵ, and provide a constructive method for finding $$\epsilon $$ϵ-discordant states.