Are all classically impossible quantum possibilities entangled?

Question

Any entangled state represents a quantum possibility that is classically impossible.
Is the converse true?
That is, are all states that are quantum mechanically possible but classically impossible entangled in some way?
If so, can you give a proof, or a reference to a proof?
If not, can you give a counterexample?

Answer 1

I don't know if this is what OP is asking(v1), but consider a single qubit, whose states can be visualized via the Bloch sphere. The classical states (which form a bit) are the north and the south pole. The pure quantum states are the surface of the 2-sphere $S^2=\partial(B^3)$, and the mixed quantum states are the interior 3-ball $B^3$. Since there is just a single qubit, it is not entangled.

Answer 2

"The things that trouble me most are not the true facts I don't know, but the false facts I do know"--Ancient Proverb
I was very surprised by how simple (and low dimensional) the first answer was.
I think the issue revolves around the definition of what is classically possible or not.
Gleason's theorem only holds in three dimensions or more, not two or less.
You can map Hilbert space to phase space for one particle, not two or more.
If you can simulate something with local hidden variables, it is classically possible.
I think you can simulate a single qubit with fewer than eight real numbers ie with local hidden variables.
This makes me doubt that any single qubit operation is classically impossible, let alone the mere existence of a qubit.
On the other hand you can prove or test Bell's theorem with two or more entangled photons or electrons or whatever.
I accept this as a demonstrable classical impossibility.
Therefore my reaction is that the mere existence of a qubit while perhaps classically impossible in some sense, cannot be proven to be classically impossible in the same way as the Bell correlations can.