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 that property mentioned above -- they all have $\epsilon$-entangled states at any small distance in their neighborhood. Furthermore, the entanglement they have is proven 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.