Question on the logical structure of the EPR argument and Bell's inequalities

Question

Recently I have read a lot online about the EPR argument and Bell's inequalities and its implications. When comparing what people write there online with the actual research articles of Einstein and Bell, it seems to me - put drastically - that people online talk about something but not about Bell and Einstein. Let me explain, what I mean.

Online most of the times the logical structure of the argument is roughly given in the following way (see, e.g., here in the first few paragraphs):

Assuming locality and pre-existing properties, Bell's inequalities follow. Quantum mechanics' predictions (and experiments) are in contradiction to the inequality. Hence, one can abandon either locality or pre-existing properties.

Reading the EPR paper (available here) however, the structure seems to be quite different. They consider the following criterion of an element of the physical reality (and a criterion is not an assumption!): If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. This, I think, is very reasonable. One can even very well argue that this is an analytical statement because of the phrase 'without in any way disturbing a system'. So one does not have to assume the correctness of the criterion as it is true anyhow!

It is employed in the following situation (Bohm's version of the EPR set-up):

Think of the singlet state of two spin-1/2 particles with total spin zero. If one measures the spin in the x direction of particle A, one can predict with certainty the spin in x direction of particle B, no matter how far the particles are separated (think of a space-like separation). For the sake of the argument it is not needed that we consider different directions for the two particles.

Now assuming locality, they must conclude by their criterion that there really exist pre-existing properties. That is Einstein.

So, the correct reasoning should be (also in Bell's understanding as he pointed out repeatedly) that locality implies the pre-existing properties. The consequence of Bell's inequalities then is that one cannot abandon either locality or pre-existing properties but must abandon locality.

So, who's right? Why are there two different conclusions? Are Einstein and Bell missing an essential point? I often read that counterfactual definiteness is tacitly assumed. But that isn't an assumption right? It is simply a property of the singlet state if one measures the same spin direction for both particles which is sufficient here.

It would be very much appreciated if it could be pointed out in Einstein's and Bell's original papers where the mistakes or debatable passages are (if there are any), since I think they are a very good basis for this discussion. Thanks.

Answer 1

Have a look at this table. Note that, in particular, true counterfactual definiteness is rare among the interpretations.

Counterfactual definiteness (which is what is usually called realism in the context of Bell's theorem) says that we can meaningfully talk about every result of every measurement, regardless of whether it is performed or not. That we can predict a single measurement with definiteness is not sufficient for counterfactual definiteness, it must be the case for all. Bell's derivation of his theorem assumes locality and that the probabilities to measure the particle at certain angles "exist", i.e. can be used in statistical calculations as if they are real, definite probabilities, and reality violates Bell's inequality, so at the very least it is either non-local or has no counterfactual definiteness. Note that "no counterfactual definiteness" does not prohibit that some things may be counterfactually determined.

The EPR argument shows then merely that it is not sufficient to give up CD to preserve locality - see again the table of interpretation and note that there are indeed interpretations which are neither local nor have CD. That the definiteness of some measurements can be achieved by entanglement even for spacelike separated measurements is as per the EPR argument the OP relates is the other hurdle a interpretation must take to call itself local. The Copenhagen interpretation throws its hands up and just says "Fine, non-locality it is, then" concerning this "collapse" of the wave-function, but Many Worlds and others (think they) find a way around.

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[Gets out the soapbox]
As I do not grow tired to remark, none of this has any physical implications. Whether the world is local, collapsing and non-CD or non-local, non-collapsing and CD has no bearing on the fact that all interpretations must predict the same things and are empirically indistingushable. You may like one better than the others, but they are nothing more than soothing balm for our minds that feel incapable of dealing with the unadorned quantum world. "Shut up and calculate" remains the mantra of the working physicist.

Answer 2

This disagreement may just be a disagreement over what "locality" means. Arguably, relativistic QFT is local (after all, it doesn't violate Lorentz symmetry in any detectable way, and doesn't permit faster-than-light signaling). If it is, then you obviously don't have to abandon locality.

I think there is a definition of "locality" for which your analysis is correct, but it's tricky to state what it is. It might be something like this: spacelike separated choices between options A and B are independent in the sense that you can't rule out any of the outcomes (A,A), (A,B), (B,A), (B,B). What is a "choice"? I don't know, but Bell's theorem requires some notion of localized choice on the experimenter's part: if the initial conditions determine the measurement axes and the particles can observe those initial conditions, then the argument doesn't go through. The choice needn't be "free will"; it can just be some form of true randomness.