# How can a basis change make a state suddenly separable?

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! :)

• The edition of Grynberg Aspect & Fabre Introduction to Quantum Optics that I am looking at has 700 pages and equation numbers of the form (V.W.XY) none of the form (XY). Without further background from what exactly you are reading (edition/page number) it is hard to handle this question. Commented Jan 17, 2022 at 14:13
• Sorry, I should have clarified, the picture is not from the textbook, it's a supplementary set of notes. I mentioned the textbook to give an idea of what level I am at and what information I'm using most. Commented Jan 17, 2022 at 15:13
• A local matrix operation can make it separable if it is irreversible, like a measurement process, for example the matrix whose colums are all the corresponding endstate. Commented May 29, 2022 at 9:34

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 ($$U=U_{AB}$$).