Clarification regarding argument in EPR paper

[RespectableCheese]

This is not their argument! Their argument is
1) every value that can be predicted with certainty should be in the theory, if the theory is complete
2) here is an example of something that can be predicted with certainty but is not in the theory
Therefore the theory is not complete.

No, you second point is backwards.

"... [the theory] would contain these values; these would then be predictable. This not being the case, we are left with the alternatives stated."

This chain of reasoning is immediately preceded by:

"More generally, it is shown in quantum mechanics that, if the operators corresponding to two physical quantities, say A and B, do not commute, that is, if AB ≠ BA, then the precise knowledge of one of them precludes such a knowledge of the other. Furthermore, any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the first."

(The values they are referring to in the first quoted segment are the eigenvalues of non-commuting observables).

 

[PeroK]

2) here is an example of something that can be predicted with certainty but is not in the theory
Therefore the theory is not complete.

The argument seems quite weak in another way. It's like making a big deal out of a probability of 1 (or 0) and saying it's not really a probability any more. That it's only really a probability when it's strictly between 0 and 1.

 

[martinbn]

The argument seems quite weak in another way. It's like making a big deal out of a probability of 1 (or 0) and saying it's not really a probability any more. That it's only really a probability when it's strictly between 0 and 1.

Yes, for them if the probability for measuring and finding a value is 1, then the value must have been so before the measurement.

 

[RespectableCheese]

This is not their argument! Their argument is
1) every value that can be predicted with certainty should be in the theory, if the theory is complete
2) here is an example of something that can be predicted with certainty but is not in the theory
Therefore the theory is not complete.

I'm not sure I understand this - what is the "something" that actually can be predicted with certainty but is not "in the theory"?

 

[Morbert]

That's exactly what I mean, that's the issue with their argument. It would have to be the other way around for their logic to hold together.

Their argument doesn't rest on the two quantities A and B of a remote system being simultaneously predictable. Their argument instead rests on the freedom to to learn, with certainty, either the value of A or the value of B of a remote system without ever disturbing it.

 

[martinbn]

I'm not sure I understand this - what is the "something" that actually can be predicted with certainty but is not "in the theory"?

Simultaneous values for momentum and position of particle B are not in the theory. Using particle A they can measure and then predict the corresponding value for B. But B is far away, so it has not been disturbed. Which means that both values must be real.

 

[RespectableCheese]

Simultaneous values for momentum and position of particle B are not in the theory. Using particle A they can measure and then predict the corresponding value for B. But B is far away, so it has not been disturbed. Which means that both values must be real.

Oh, I see what you're saying - that's referring to the latter part of their argument. Given that both values are physically real, this line of reasoning:

"From this follows that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. For if both of them had simultaneous reality—and thus definite values—these values would enter into the complete description, according to the condition of completeness. If then the wave function provided such a complete description of reality, it would contain these values; these would then be predictable. This not being the case, we are left with the alternatives stated."

Is used to conclude that the theory is not complete. (Because then (2) is false, so (1) must be true by disjunctive elimination). But it's this argument where I see the flaw, as I described earlier.

 

[DrChinese]

"For if both of them had simultaneous reality - and thus definite values - these values would enter into the complete description, according to the description of completeness. If the wave function provided such as complete description of reality, it would contain these values; these would then be predictable. This not being the case, we are left with the alternatives stated."

I don't see how this argument follows, given the known empirical reality that the eigenvalues of non-commuting observables can not be predicted simultaneously with absolute certainty. For the predictability of a physical quantity is, from assumption 2, only a sufficient but not necessary condition for those elements existing in physical reality, and so the fact alone that they are not predictable proves nothing. An additional implicit assumption would have to be that if a quantity exists in a physical theory, then it is predictable.

You are quite correct, there is something missing here. They actually make an additional assumption at the end of the paper:

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but no both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depends upon the process of measurement carried out on the first system, which does not disturb the second system in any way."

And they explicitly assume:

"No reasonable definition of reality could be expected to permit this".

 

[DrChinese]

The paper is also attached below for convenience.

Your link did not work for me. If anyone needs it:

Can Quantum-Mechanical Description of Physical Reality Be Considered Complete ?

 

[RespectableCheese]

You are quite correct, there is something missing here. They actually make an additional assumption at the end of the paper:

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but no both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depends upon the process of measurement carried out on the first system, which does not disturb the second system in any way."

And they explicitly assume:

"No reasonable definition of reality could be expected to permit this".

I don't think that really addresses the concern - in that excerpt the authors refute a potential objection to the criterion of reality they had assumed (that is, that an element of a system is physically real if it can be predicted with absolute certainty without disturbing the system). The latter half of their argument relies on the hypothetical case of the entangled system, in which either eigenvalue for the two non-commuting observables can be predicted with absolute certainty and without disturbing the second, distant system. The potential objection they address here posits that for two values to be considered simultaneously real by the given criterion, they must be simultaneously predictable. Because in the contrived example, the observer can only make one measurement of the two observables, they are not simultaneously predictable. This would imply that the reality of one or the other eigenvalue is then determined at the time of measurement, the measured eigenvalue gaining the status of phsyical reality at the expense of the unmeasured. The authors then dismissed this as being in principle unsatisfactory as a general criterion of physical reality. There is no additional assumption made; they describe a potential objection and reject it.

 

[DrChinese]

I don't think that really addresses the concern - in that excerpt the authors refute a potential objection to the criterion of reality they had assumed (that is, that an element of a system is physically real if it can be predicted with absolute certainty without disturbing the system). The latter half of their argument relies on the hypothetical case of the entangled system, in which either eigenvalue for the two non-commuting observables can be predicted with absolute certainty and without disturbing the second, distant system. The potential objection they address here posits that for two values to be considered simultaneously real by the given criterion, they must be simultaneously predictable. Because in the contrived example, the observer can only make one measurement of the two observables, they are not simultaneously predictable. This would imply that the reality of one or the other eigenvalue is then determined at the time of measurement, the measured eigenvalue gaining the status of phsyical reality at the expense of the unmeasured. The authors then dismissed this as being in principle unsatisfactory as a general criterion of physical reality. There is no additional assumption made; they describe a potential objection and reject it.

True enough. But that exact objection (I call it an assumption) is precisely what Bell exploited!

Bell said: let’s assume both can’t be predicted, but merely have values. And in fact all (but at least 3) potential measurement settings have definite preexisting values. That leads to Bell Inequalities, and we know where those lead.

In other words: if they (at least 3) have values, what are they? There aren’t such values, independent of the experimenter’s choice of measurement basis. So pick out of this what you choose to be the critical assumption, and you’re home.

 

[PeterDonis]

The authors then dismissed this as being in principle unsatisfactory as a general criterion of physical reality.

And the problem is, as Bell first deduced theoretically, and as has now been confirmed experimentally, that there is no model that would be satisfactory to EPR in terms of capturing their notion of physical reality, that also agrees with experiments.

In other words, we now know, experimentally, that however "reality" works is not reasonable according to EPR.

 

[RespectableCheese]

True enough. But that exact objection (I call it an assumption) is precisely what Bell exploited!

Bell said: let’s assume both can’t be predicted, but merely have values. And in fact all (but at least 3) potential measurement settings have definite preexisting values. That leads to Bell Inequalities, and we know where those lead.

In other words: if they (at least 3) have values, what are they? There aren’t such values, independent of the experimenter’s choice of measurement basis. So pick out of this what you choose to be the critical assumption, and you’re home.

I mean that's a true enough statement about Bell's theorem but I don't see what it has to do with this paper.

 

[RespectableCheese]

And the problem is, as Bell first deduced theoretically, and as has now been confirmed experimentally, that there is no model that would be satisfactory to EPR in terms of capturing their notion of physical reality, that also agrees with experiments.

In other words, we now know, experimentally, that however "reality" works is not reasonable according to EPR.

I think their initial criterion of reality (that a quantity's being predictable with absolute certainty without disturbing the system of the corresponding element entails that element's physical reality) could still hold even given Bell's results, which imply that local hidden variable theories are not tenable; in their example of the entangled state, it's possible for the distant system to obtain a definite real value when the measurement is made by abandoning the condition of local realism.

 

[PeterDonis]

in their example of the entangled state, it's possible for the distant system to obtain a definite real value when the measurement is made by abandoning the condition of local realism.

Only for the case where both measurements are made in the same direction (about the same spin axis). But that case isn't the one that violates the Bell inequalities. The cases that violate those inequalities are cases where the two spin measurements on the two entangled particles are not made about the same axis. A "definite real value" for the spin doesn't just commit the particle to a particular result for a spin measurement about one axis; it commits the particle to particular results for any spin measurement. EPR didn't even address that issue, but it's crucial: a "definite real value" actually means a lot more than EPR appear to have realized.

 

[RespectableCheese]

Only for the case where both measurements are made in the same direction (about the same spin axis). But that case isn't the one that violates the Bell inequalities. The cases that violate those inequalities are cases where the two spin measurements on the two entangled particles are not made about the same axis. A "definite real value" for the spin doesn't just commit the particle to a particular result for a spin measurement about one axis; it commits the particle to particular results for any spin measurement. EPR didn't even address that issue, but it's crucial: a "definite real value" actually means a lot more than EPR appear to have realized.

I think this already pretty tangential, but I'm not sure what you're getting at; their criterion of reality is that a value being predictable with certainty is sufficient for the corresponding element to exist in physical reality. So I don't think that situation violates their criterion, i.e. by demonstrating a case where a value that can be predicted with absolute certainty does not have real phsyical existence.

 

[PeterDonis]

a value being predictable with certainty is sufficient for the corresponding element to exist in physical reality

Yes, but what does "exist in physical reality" mean?

In the case of spin, if a particle's spin about a certain axis "exists in physical reality" (because it's entangled with another particle whose spin has just been measured about that axis), that has implications for its spin about other axes as well. EPR didn't consider those implications. But Bell did.

 

[RespectableCheese]

Yes, but what does "exist in physical reality" mean?

In the case of spin, if a particle's spin about a certain axis "exists in physical reality" (because it's entangled with another particle whose spin has just been measured about that axis), that has implications for its spin about other axes as well. EPR didn't consider those implications. But Bell did.

I think they just mean that it has the ontological status of existing in the objective physical world? Maybe that sounds like a circular definition. They were reasoning metaphysically, that is, prior to any particular scientific theory or empirical observation, with their definition of completeness and criterion for physical reality.

What exactly is it about the Bell states that makes their criterion inconsistent/untenable? From a purely logical standpoint you'd have to provide an example of a case where you can predict with absolute certainty a particular quantity for some element that does not exist in physical reality, in the most general meaning of that term. This just seems impossible a priori. Otherwise what are you predicting? You would have some well-defined physical quantity, say mass, for which a specific value can be predicted with certainty for a physical system, but for which no actual physical element exists in that system to which that quantity is attached. As far as reasoning from first principles goes this seems pretty unbreakable as a sufficient criterion for something existing in the objective physical world.

 

[PeterDonis]

From a purely logical standpoint you'd have to provide an example of a case where you can predict with absolute certainty a particular quantity for some element that does not exist in physical reality

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.

 

[PeroK]

This just seems impossible a priori.

QM does, in fact, predict the impossible! Bell's inequality was a limit on what was possible. The mathematics of QM predicted that QM could do better. And, experiments showed that nature can, in fact, do the impossible. Precisely as QM predicted.

 

[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.