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Consider experiments involving entangled spins, say two-spin 1/2 particles in the singlet state:

$$\left|\psi\right> =\frac{1}{\sqrt{2}}\left[\left|\uparrow\downarrow\right> - \left|\downarrow\uparrow\right>\right]$$

where Alice is going to measure one spin in the $z$-direction and Bob the other spin in the same $z$-direction. Alice and Bob are space-like separated at the time they perform their measurements. There then seems to be a (benign) locality paradox. Suppose that in some frame, Alice performs her measurement first. She then knows what Bob will find, but how can the information of Bob's result already exist before Bob makes his measurement?

The problem here is that we know that there are no local hidden variables underlying quantum mechanics. John Bell demonstrated this by showing that a local hidden variables theory gives rise to certain inequalities (now know as the Bell inequalities) for certain correlations of spin measurements of entangled spins, while quantum mechanics implies that these inequalities can be violated in certain cases (the same singlet state we're considering here, can show correlations that do not satisfy the Bell inequalities). later the violations of these inequalities was verified in an experiment conducted by Alan Aspect.

So, the non-existence of local hidden variables means that the information that Alice has about what Bob will find cannot also be present locally at Bob's place. Now, in principle, there is no problem in principle with information about this popping up non-locally at Alice's place. One cannot exploit this to allow for faster than light communications. Also we can note that non-local hidden variable theories can be compatible with quantum mechanics. But this does go against the spirit of locality. It then seems that there is an inherent benign non-local aspect to quantum theory.

However, in the MWI, there is no non-locality at all. After Alice measures her spin, there are two copies of her and both copies are equally for for Bob, so it's not true that in the frame where Alice has made her measurement and Bob hasn't, that the information about what Bob will find already exists in Alice's place.

Now, the whole point of the MWI is to treat observers and measurement devices in a quantum mechanical way too instead of in an ad-hoc classical way and thereby to get rid of the collapse of the wavefunction. But if we do treat observers in a classical way and stick to a single universe theory, then we do get a non-locality problem, albeit a benign one.

Then having resolved the source of the apparent non-locality in this case as a classical spanner thrown into the quantum world, we can ask if this is always the case. E.g. we can read here that Lev Vaidman claims that this is also true for the famous Aharonov-Bohm effect, but also that this is in dispute. Vaidman's argument is rather technical, but he argues that ultimately the whole non-locality issue there is artifact of using classical potentials.

Is there an example of a non-local aspect of quantum mechanics that cannot explained away as an artifact of a classical concept?

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    $\begingroup$ For me, the problem lies in the concept of locality. I do not think we can have a satisfactory definition of locality without a clear picture of spacetime structure. I believe that the later is closely related with the structure of quantum state space so that the concepts of entanglement, spacetime, locality & causality are somehow 'entangled'. Without a 'correct' definition of locality, the non-locality is meaningless. $\endgroup$ – XXDD Jun 27 '17 at 15:54

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