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I am working through my quantum optics textbook by Grynberg, Aspect and Fabre, and this concept has tripped me up a little.
(13) is an inseparable state, whereas (15) IS separable - but they are the same state, just written using different bases. I can't get my head around how changing our basis has somehow changed what I thought to be inherent to the state.
If this is a stupid question and I need to go back in the textbook, let me know! :)
Any state, no matter how entangled in the original basis, can be written as a product state in another basis, and vice versa. The catch is that the basis transformation has to be global, not local. Entanglement between subsystems $A$ and $B$ cannot be changed by changes of basis which affect $A$ and $B$ alone ($U=U_A\otimes U_B$), but they can be changed by global changes of basis ($U=U_{AB}$).
A change of basis can be any unitary operator and for any given initial state $\lvert\psi\rangle$ and any final state $\lvert\phi\rangle$, there is a unitary which maps $\lvert\psi\rangle$ to $\lvert\phi\rangle$, so any state can be fully disentangled by some change of basis.
The original entangled degrees of freedom are still entangled after the basis change, but the new basis is written in terms of new (perhaps complicated nonlocal) unentangled degrees of freedom. Those degrees of freedom were still unentangled in the original basis. The basis change doesn't change anything physical: it just changes which degrees of freedom we explicitly keep track of.
