A Clarification regarding argument in EPR paper

[martinbn]

But if particle B's spin about the z axis "exists in physical reality", what about particle B's spin about, say, an axis tilted by 60 degrees from z? EPR don't discuss that at all. They don't even appear to have considered it.

Because they didn't consider spin. That was Bohm's later version.

 

[PeterDonis]

Because they didn't consider spin. That was Bohm's later version.

That's true, EPR's original paper discussed two particles that are entangled in configuration space (position and momentum). But EPR also don't discuss measurements of, for example, momentum in different directions.

Measurements of momentum along orthogonal axes, at least, commute, so EPR might not have thought there was an issue in that regard. But considering the spin case, as Bohm did, makes it clear that there is.

 

[RespectableCheese]

No, that's not the issue. The issue is with an element that, according to their criterion, does "exist in physical reality", because we can predict it with certainty, and what that implies. Again, in the case under discussion, we have two particles whose spins are entangled, so if we measure particle A's spin about, say, the $z$ axis, we can now predict with certainly particle B's spin about the $z$ axis, so the latter must "exist in physical reality".

But if particle B's spin about the $z$ axis "exists in physical reality", what about particle B's spin about, say, an axis tilted by 60 degrees from $z$? EPR don't discuss that at all. They don't even appear to have considered it.

But Bell showed that, once you say that particle B's spin about the $z$ axis "exists in physical reality", that has implications for particle B's spin about other axes--and those implications require that the Bell inequalities are obeyed. But we know, experimetally, that the Bell inequalities are violated.

The only way to avoid the implications of Bell inequality violations is to make "exists in physical reality" basically mean nothing over and above its definition: yes, if we measure particle A's $z$ spin, we can predict with certainty particle B's $z$ spin. But that's all it can mean.

And that's not the position EPR were arguing for. They were arguing for the position that "exists in physical reality" means more than just being able to predict something with certainty--being able to predict something with certainty was just an indicator to them. They were arguing that QM must be incomplete because it doesn't include these extra things that "exist in physical reality". But any such "more complete" model of reality would have to obey the Bell inequalities--and reality itself doesn't.

It seems like you might be circling back to my initial point - it's not their criterion of reality alone that's the issue, it's how it fits into their broader argument regarding complete theories and entanglement.

Can you elaborate on how the existence of a particle's spin "in physical reality" has implications about the particle's spin about other axes? Did Bell use his own well-defined criterion of physical reality? I'm familiar with the idea behind Bell's inequalities (I think I read the GHZ version and not Bell's) but I'm not sure what you're referring to here. What attribute is it precisely that both EPR and Bell thought existed beyond the sufficient criterion given (that of absolute predictability) that elements existing in physical reality possess?

 

[RespectableCheese]

My main gripe with the EPR argument, just to re-summarize, is that it seems to contain redundancies. They describe a theoretical scenario in which it is possible, under the assumption of causal locality, to predict with absolute certainty and without disturbing the target system, the values of two non-commuting observables (1). From this they use their criterion of reality to infer that those values must have simultaneous physical reality (2). If QM is a complete theory then by their definition of completeness these values should be described by or contained in the theory (3). Then (and here is where the strict logic breaks down unless additional assumptions are made), these values should be predictable (from the theory?). This not being the case, the assumption of completeness must be abandoned. Unless predictability from a given theory and predictability in the more general sense are distinguishable, it seems like just result (1) would be sufficient to point out the inconsistency in the existing framework. I suppose they probably wanted to explicitly negate the proposition "QM is a complete theory", given what they considered sound philosophical assumptions derived from first principles. Maybe that makes a better headline (in 1935)?

 

[PeterDonis]

Did Bell use his own well-defined criterion of physical reality?

Mathematically, he made the required conditions for proving his theorem quite clear. There are two key ones:

First, the joint probability for two measurements has to factorize so that the probability of each result only depends on the measurement settings for that measuring device, not on the settings of the other device (this assumption is what is usually referred to as his "locality" assumption).

Second, it has to be meaningful to ask what the result would have been of a measurement that was not actually made, if it had been made (this assumption is usually called "counterfactual definiteness" in the literature).

Most discussions of Bell's Theorem focus on the locality assumption, since it's the one that QM obviously violates (the joint probability is a function of the angle between the two spin measurements, which obviously has to depend on both measurement settings). But EPR's criterion of "an element of physical reality" IMO is more closely related to the counterfactual definiteness assumption: if a particle's $z$ spin has a definite value, its spin about other axes should have definite values too, even if we don't measure them. Otherwise it's really hard to see how the spin about any axis could be "an element of physical reality", since whether or not the particle's spin is "real" shouldn't depend on which axis we choose for measuring it. At least, that seems to be the viewpoint that EPR were arguing from.

 

[RespectableCheese]

Mathematically, he made the required conditions for proving his theorem quite clear. There are two key ones:

First, the joint probability for two measurements has to factorize so that the probability of each result only depends on the measurement settings for that measuring device, not on the settings of the other device (this assumption is what is usually referred to as his "locality" assumption).

Second, it has to be meaningful to ask what the result would have been of a measurement that was not actually made, if it had been made (this assumption is usually called "counterfactual definiteness" in the literature).

Most discussions of Bell's Theorem focus on the locality assumption, since it's the one that QM obviously violates (the joint probability is a function of the angle between the two spin measurements, which obviously has to depend on both measurement settings). But EPR's criterion of "an element of physical reality" IMO is more closely related to the counterfactual definiteness assumption: if a particle's $z$ spin has a definite value, its spin about other axes should have definite values too, even if we don't measure them. Otherwise it's really hard to see how the spin about any axis could be "an element of physical reality", since whether or not the particle's spin is "real" shouldn't depend on which axis we choose for measuring it. At least, that seems to be the viewpoint that EPR were arguing from.

Right, I think I understand now. The thought experiments in the Bell and EPR cases are not identical but are pretty similar conceptually I think - in the EPR experiment, the systems I and II can be represented by two distinct wavefunctions w1(x1, x2) = a(x1)*p(x2) or w2(x1, x2) = b(x1)*q(x2), where x1 and x2 are the local variables for the respective systems. By measuring either A or B at system I, one will yield an eigenvalue in the eigenspectrum of a(x1) or b(x1), respectively, while the second system "collapses" into a correlated eigenvalue in the spectra of p(x2) or q(x2). Under the assumption of locality, these second wavefunctions must exist prior to the measurement at the first system being made, and so system II after the measurement can be described by two wavefunctions simultaneously, which EPR show can be the eigenfunctions of non-commuting observables (and from there proceed the difficulties).

So this is similar to the scenario you describe; because you should be able to measure the spin along any axis, all such spin eigenvalues must exist simultaneously prior to measurement (under the assumption of locality). It's not enough to say that you can predict the spin at the undisturbed system give a measurement result at the first, all such potential outcomes must somehow already exist there. Of course if you do away with locality that is not an issue.

But that wasn't really the point of my question; I was nitpicking the internal logic of the EPR paper, which doesn't seem to hold together in and of itself.

 

[PeterDonis]

if you do away with locality that is not an issue.

In a sense this is true, but I'm not sure it's the sense you intended.

"Do away with locality", if you leave counterfactual definiteness in place, gets you Bohmian mechanics. Which certainly works--as in, it makes correct predictions. And Bell himself said that it makes the nonlocality so explicit that you can't ignore it.

Note, however, that in Bohmian mechanics, all the potential outcomes do already exist! That is, if you knew the full state of the system, which includes, not just the wave function, but the exact positions of every particle, you would know the results of all possible measurements on every particle in advance. Even if it was entangled with another particle--because all the information needed to keep the two particles' results consistent with the entangled state get transmitted non-locally, faster than light, through the quantum potential.

I believe Bell once commented of Bohmian mechanics that he could not imagine a solution (meaning, to the question EPR asked) that Einstein would have liked less. The above is why.

 

[RespectableCheese]

In a sense this is true, but I'm not sure it's the sense you intended.

"Do away with locality", if you leave counterfactual definiteness in place, gets you Bohmian mechanics. Which certainly works--as in, it makes correct predictions. And Bell himself said that it makes the nonlocality so explicit that you can't ignore it.

Note, however, that in Bohmian mechanics, all the potential outcomes do already exist! That is, if you knew the full state of the system, which includes, not just the wave function, but the exact positions of every particle, you would know the results of all possible measurements on every particle in advance. Even if it was entangled with another particle--because all the information needed to keep the two particles' results consistent with the entangled state get transmitted non-locally, faster than light, through the quantum potential.

I believe Bell once commented of Bohmian mechanics that he could not imagine a solution (meaning, to the question EPR asked) that Einstein would have liked less. The above is why.

I'm not familiar with the intricacies of Bohm's theory, but I don't think doing away with the requirement of strict local causality necessarily entails that particular interpretation. If you admit inherent stochasticity it could be the case that the potential observables are totally indeterminate until a measurement is made, and then a single value snaps into existence, even across space like intervals (the Copenhagen interpretation basically).

 

[PeterDonis]

I don't think doing away with the requirement of strict local causality necessarily entails that particular interpretation.

Perhaps not, but that interpretation does have the very interesting property I described, that it does do away with local causality but also has all measurement results completely determined in advance. In other words, there is no "inherent stochasticity" in reality in this interpretation, and the fact that we have to use probabilities is due to our ignorance of initial conditions, just as in classical physics.

If you admit inherent stochasticity it could be the case that the potential observables are totally indeterminate until a measurement is made, and then a single value snaps into existence, even across space like intervals (the Copenhagen interpretation basically).

But this, by itself, still doesn't explain how Bell inequality violations are enforced for cases where, for example, spin measurements about different axes are made on two entangled particles. It's not enough for "a single value to snap into existence" for such cases because knowing the measurement result for one particle does not enforce a single value for the measurement result on the other particle; the "inherent stochasticity" itself has to have nonlocal structure to it.