Why does entanglement lie at the basis of quantum mechanics?

Question

In this question I read:

I am writing this because of the mostly hostile reception given to my question about entanglement. It was made clear that it was pure nonsense to want to examine what a world without entanglement would be as QM foundations were entirely dependent on it.

I'm not so sure I understand this. Of course, QM is a non-local theory. All values of the position of a particle are involved in the wavefunction corresponding to the particle. But this is the case also when entanglement is not involved. Or can we say that all positions are entangled? When a measurement is made, and a small range of the values of the particle's position is measured, the other values can't be measured anymore (shortly after the measurement). Is this the same as making a measurement on the spin of one of two particles that are lightyears away from each other? This is only the case if the two particles were in contact with each other somewhere in the past. A single particle is always in contact with itself, so does entanglement (for single particles) refer to this? Can we say that all positions are entangled? Or is this non-locality?

If I measure the position of a single particle, then upon measurement the wavefunction has "collapsed" to a smaller range of positions. Just so, the two spins of separated particles "collapse" upon spin-measurement of one particle. So why we say that spins are entangled and positions not.

Can't we say that the wavefunction is a linear combination of position "eigenstates" (or Dirac delta distributions) with corresponding eigenvalues, the values of position?

Answer 1

The claim is not that "everything is entangled", the claim is that you cannot remove the notion of entanglement from quantum mechanics and still have something that deserves to be called "quantum mechanics".

The possibility of entanglement is a direct consequence of the basic structure of quantum mechanics, in particular the idea that states can be linearly combined and so the space of states of composite systems is the tensor product of the individual spaces of states instead of the direct product from classical mechanics (see this question and its answers for more discussion of this point). I also discuss in this answer that combining systems, entanglement and density matrices are all interlinked - you can't have one without the other.

Entanglement is therefore not some phenomenon tacked onto quantum theory that you could remove at will, or that would only be relevant in some exotic situations. As soon as you have more than one system, your potential states include entangled states, and you can't just ignore that, so it is as ill-defined to say "quantum mechanics without entanglement" as it would be to prompt people to think about "Newtonian mechanics without Newton's laws".