It is my understanding that you can use phonons to make a gaussian packet, which would behave like a quantum particle. I also believe that you can make two such packets and entangle them, that is create a two phonon state in which two phonons move away from each other and that you can write as:
$|\Psi\rangle=|\psi^{(1)}_a\rangle |\psi^{(2)}_b\rangle+|\psi^{(1)}_b\rangle |\psi^{(2)}_a\rangle$
where the numbers identify which packet (left or right) and the letter the state (some different quantum number). Also, I read that the atoms in the solid are not entangled (see here). Now, assume the solid is very large, and then measure the state of both phonons almost simultaneously.
Then, my question is: it looks like the phonons experience quantum non-locality, but this non-locality is only apparent, because the atoms interact locally and are themselves not entangled. Then, where this apparent non-locality come from?
