A Clarification regarding argument in EPR paper

[RespectableCheese]

TL;DR

The argument presented in the EPR paper seems flawed/confusing, curious about the reasoning.

Greetings all. I read through the original EPR paper recently and ran into some confusion regarding the central argument. As I understand it, the authors assert the following two definitions:

Assumption 1: A physical theory is called complete if every element in physical reality has a corresponding element in the physical theory.

Assumption 2: If a physical quantity can be predicted with certainty, then its corresponding element exists in physical reality.

They then go on to make the following assertion:

Proposition 1: It cannot be the case that both (1) The quantum theory is a complete physical theory and (2) The eigenvalues corresponding to two non-commuting observables have simultaneous physical reality.

They then go on to show how in principle an entangled system could in theory be constructed such that by measuring either one of two non-commuting observables on one of the entangled system's subsystems, a definite value for that observable's eigenvalue could be yielded at the un-measured system. To preserve the property of locality for that system, it would have to be the case that the observables' eigenvalues at the un-measured subsystem, while initially assumed to be indefinite, were actually well-defined and predictable all along. Therefore in this case the eigenvalues of non-commuting values do in fact have simultaneous reality, and so, by the law of disjunction elimination and the truth of proposition 1, it follows that the quantum theory is in-complete.

This conclusion clearly follows if proposition 1 is assumed true, however I am having some difficulty in figuring out how that proposition is justified from just the assumptions given. Their justification is given verbatim as follows:

"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.

It seems like it would be more elegant to say that, in the constructed example with the entangled system, it is possible according to the quantum theory to predict with certainty and simultaneity eigenvalues for non-commuting observables, and that since this is empirically impossible, the theory itself must be flawed in some manner.

As I understand it Einstein later distanced himself from this paper and clarified that his main issue was with the non-locality that was implied by entangled quantum states. So perhaps it's not fruitful to pick this paper apart, but I thought it might be worth bringing up.

The paper is also attached below for convenience. Thanks.

 

[mad mathematician]

What does constitue as "physical reality"?

Do we assume materialism in the background?

 

[PeroK]

"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.

It seems like it would be more elegant to say that, in the constructed example with the entangled system, it is possible according to the quantum theory to predict with certainty and simultaneity eigenvalues for non-commuting observables, and that since this is empirically impossible, the theory itself must be flawed in some manner.

As I understand it Einstein later distanced himself from this paper and clarified that his main issue was with the non-locality that was implied by entangled quantum states. So perhaps it's not fruitful to pick this paper apart, but I thought it might be worth bringing up.

The paper is also attached below for convenience. Thanks.

I would tend to agree with your analysis.

 

[Morbert]

They then go on to show how in principle an entangled system could in theory be constructed such that by measuring either one of two non-commuting observables on one of the entangled system's subsystems, a definite value for that observable's eigenvalue could be yielded at the un-measured system. To preserve the property of locality for that system, it would have to be the case that the observables' eigenvalues at the un-measured subsystem, while initially assumed to be indefinite, were actually well-defined and predictable all along. Therefore in this case the eigenvalues of non-commuting values do in fact have simultaneous reality, and so, by the law of disjunction elimination and the truth of proposition 1, it follows that the quantum theory is in-complete.

[...]

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.

They can't be predicted simultaneously, but the observer can choose to measure either of the two non-commuting quantities, and hence learn the corresponding quantity of the distant, undisturbed system. How is the observer free to learn about either quantity of the undisturbed system?

 

[martinbn]

I don't understand your confusion. The EPR argument goes like this: you can predict with 100% certainty the value of an observable without disturbing the system, therefore (according to EPR) this value should be already there. But the theory doesn't have that, so it (the theory) must be incomplete (in the EPR sense).

 

[PeroK]

I don't understand your confusion. The EPR argument goes like this: you can predict with 100% certainty the value of an observable without disturbing the system, therefore (according to EPR) this value should be already there. But the theory doesn't have that, so it (the theory) must be incomplete (in the EPR sense).

I thought that part of the problem was that there are many ways in which you can try to get round the HUP, but it's only when you specify the details of your experiment that the HUP asserts itself, sometimes in quite subtle ways.

For example, you could try to set up an experiment where you know the spin of an electron about two coordinate axes at the same time. See section 6.4.3 here.

https://physics.mq.edu.au/~jcresser/Phys304/Handouts/QuantumPhysicsNotes.pdf

But by suitable fiddling with the beam, the magnetic field strengths and so on it should
be possible in principle, at least from the point of view of classical physics, to minimize this effect,
or at least determine exactly how much precession occurs, and take account of it. But in practice,
it turns out that all these attempts fail.

The same might be true of measurements of position and momentum for a pair of particles If you take a too-classical approach, then you can imagine an experiment where one particle's position and one particle's momentum are measured simultaneously to a jointly higher precision than allowed by the HUP, throwing each particle into an impossible state. Again, however, it's in the practical details of the experiment where the HUP asserts itself. Or, to put it in more modern terms: the attempted preparation of an impossible state cannot be made to work.

 

[martinbn]

I thought that part of the problem was that there are many ways in which you can try to get round the HUP, but it's only when you specify the details of your experiment that the HUP asserts itself, sometimes in quite subtle ways.

For example, you could try to set up an experiment where you know the spin of an electron about two coordinate axes at the same time. See section 6.4.3 here.

https://physics.mq.edu.au/~jcresser/Phys304/Handouts/QuantumPhysicsNotes.pdf

But by suitable fiddling with the beam, the magnetic field strengths and so on it should
be possible in principle, at least from the point of view of classical physics, to minimize this effect,
or at least determine exactly how much precession occurs, and take account of it. But in practice,
it turns out that all these attempts fail.

The same might be true of measurements of position and momentum for a pair of particles If you take a too-classical approach, then you can imagine an experiment where one particle's position and one particle's momentum are measured simultaneously to a jointly higher precision than allowed by the HUP, throwing each particle into an impossible state. Again, however, it's in the practical details of the experiment where the HUP asserts itself. Or, to put it in more modern terms: the attempted preparation of an impossible state cannot be made to work.

But they are not talking about measurement of two conjugate variables, just predicting the values. And their calculation is not problematic. It is the interpretations of their argument that disagree with each other.

 

[PeroK]

But they are not talking about measurement of two conjugate variable

Of course they are. The paper says "Suppose now the quanitity A is measured and is is found that it has the value $a_k$."

 

[martinbn]

Of course they are. The paper says "Suppose now the quanitity A is measured and is is found that it has the value $a_k$."

There they are just explaining the reduction of the state postulate.

 

[PeroK]

There they are just explaining the reduction of the state postulate.

The only way to be certain of a measureable is to measure it and send the state to an eigenstate/function. That's what EPR are doing. See the paragraph after equation (18):

Thus, by measuring either A or B we are in a position to predict with certainty ...

The whole argument is based on complementary measurments on an entangled system.

 

[martinbn]

The only way to be certain of a measureable is to measure it and send the state to an eigenstate/function. That's what EPR are doing. See the paragraph after equation (18):

Thus, by measuring either A or B we are in a position to predict with certainty ...

The whole argument is based on complementary measurments on an entangled system.

I don't disagree with that, may be I didn't phrase it well. But their concern is the predicted values, not the measured ones.

 

[PeroK]

I don't disagree with that, may be I didn't phrase it well. But their concern is the predicted values, not the measured ones.

That's interesting. They terminate their argument at a point where there appears to be no contradiction with QM. I must confess that I thought the argument was more sophisticated than that.

 

[Demystifier]

It seems like it would be more elegant to say that, in the constructed example with the entangled system, it is possible according to the quantum theory to predict with certainty and simultaneity eigenvalues for non-commuting observables, and that since this is empirically impossible, the theory itself must be flawed in some manner.

The paper was written in 1935, at that time it was not so clear that such thing is empirically impossible. Even today, just because nobody knows how to do it in the laboratory doesn't imply that it's absolutely impossible. They wanted to prove a theorem, so they didn't want an argument that depends on successes of experimental physics. They wanted an argument that depends only on theoretical assumptions.

 

[martinbn]

That's interesting. They terminate their argument at a point where there appears to be no contradiction with QM. I must confess that I thought the argument was more sophisticated than that.

I think (might be wrong), that they don't claim that there is a contradiction, but that there are values that should be there but are not in the theory, hence it is incomplete.

 

[RespectableCheese]

I don't understand your confusion. The EPR argument goes like this: you can predict with 100% certainty the value of an observable without disturbing the system, therefore (according to EPR) this value should be already there. But the theory doesn't have that, so it (the theory) must be incomplete (in the EPR sense).

I'm not sure that's exactly how their reasoning goes; they seem to be trying to say that, under the assumption of locality, the eigenvalues of two non-commuting observables can have simultaneous reality at a system, but that this cannot be the case given our apparent inability to empirically measure such values simultaneously. The strict logic of their argument doesn't seem to hold together though if you follow it rigorously. Technically the fact that you can make those predictions with absolute certainty actually does follow directly from the established theory (as demonstrated by their constructed example), but it doesn't seem to line up with empirical reality.

 

[PeroK]

I think (might be wrong), that they don't claim that there is a contradiction, but that there are values that should be there but are not in the theory, hence it is incomplete.

That's what I meant to say. It's interesting how much water has flowed under the bridge since 1935 and yet this paper remains one of the most famous. It's quite underwhelming and reinforces my view that the arguments against QM were always much weaker than they were given credit for.

 

[RespectableCheese]

The paper was written in 1935, at that time it was not so clear that such thing is empirically impossible. Even today, just because nobody knows how to do it in the laboratory doesn't imply that it's absolutely impossible. They wanted to prove a theorem, so they didn't want an argument that depends on successes of experimental physics. They wanted an argument that depends only on theoretical assumptions.

This is the relevant paragraph from the paper:

"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 empirical impossibility of measuring non-commuting observables simultaneously follows from the theory, true. I suppose a better way of phrasing it is that it is possible to contrive a scenario in the theory (the authors' constructed example with the entangled state) that makes contradictory predictions compared to other aspects of the theory (the impossibility of having simultaneous perfect knowledge of non-commuting eigenvalues).

 

[martinbn]

I'm not sure that's exactly how their reasoning goes; they seem to be trying to say that, under the assumption of locality, the eigenvalues of two non-commuting observables can have simultaneous reality at a system, but that this cannot be the case given our apparent inability to empirically measure such values simultaneously.

I don't think that they say that. They say that the theory as it is doesn't have simultaneous values of noncommuting observables. I don't think it has to do with what we can measure and what not.

The strict logic of their argument doesn't seem to hold together though if you follow it rigorously.

Where does it break?

Technically the fact that you can make those predictions with absolute certainty actually does follow directly from the established theory (as demonstrated by their constructed example), but it doesn't seem to line up with empirical reality.

 

[martinbn]

That's what I meant to say. It's interesting how much water has flowed under the bridge since 1935 and yet this paper remains one of the most famous. It's quite underwhelming and reinforces my view that the arguments against QM were always much weaker than they were given credit for.

I think this is because today entanglement is understood (at least mathematically). Then it was one of the first time they've seen it.

 

[PeroK]

The empirical impossibility of measuring non-commuting observables simultaneously follows from the theory, true.

Actually, it's much subtler than that. You would be better to read Cresser's section on spin measurements than the EPR paper. It's great that you are looking at original papers, but that was an anti-QM paper and I can't see how the argument even stacks up.

PS There's been so much progress since this paper that almost anything it says may be out of date and have been superseded by theoretical developments in the past 90 years.

 

[RespectableCheese]

I don't think that they say that. They say that the theory as it is doesn't have simultaneous values of noncommuting observables. I don't think it has to do with what we can measure and what not.

Where does it break?

I'm making that inference from these two statements:

"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..."

"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."

If a quantity is "predictable" with certainty, that seems to imply that you can first claim that you will observe some particular value, given your knowledge of the state of the physical system at a point in time, and then be certain to actually observe just that value after measuring the state.

The logic seems to break down in the second quoted paragraph - there's no reason to infer that if a complete physical theory contains definite values, they are therefore predictable (by the authors' definition of a complete theory). If the values exist in a theory, and the theory is complete, then they exist in physical reality, but this doesn't imply that they be predictable with certainty, because absolute predictability is a sufficient but not necessary criterion for physical existence, as the authors state explicitly.

This paper frustrates me because they seem to have the right idea but they fumbled the details in a really baffling way.

 

[PeroK]

This paper frustrates me because they seem to have the right idea but they fumbled the details in a really baffling way.

If they had had the right idea then the the experiments to test Bell's theorem would have proved QM wrong. That's where these ideas were eventually tested experimentally.

 

[martinbn]

The logic seems to break down in the second quoted paragraph - there's no reason to infer that if a complete physical theory contains definite values, they are therefore predictable (by the authors' definition of a complete theory). If the values exist in a theory, and the theory is complete, then they exist in physical reality, but this doesn't imply that they be predictable with certainty, because absolute predictability is a sufficient but not necessary criterion for physical existence, as the authors state explicitly.

No, you have it backwards. They don't say that values that exist should be predictable. They say that values which are predictable with certainty should exist.

 

[RespectableCheese]

Actually, it's much subtler than that. You would be better to read Cresser's section on spin measurements than the EPR paper. It's great that you are looking at original papers, but that was an anti-QM paper and I can't see how the argument even stacks up.

Oh yes, I'm aware that this is basically just a historical artifact, clearly we've progressed since then. The statement about non-commuting observables is a consequence of the uncertainty principle, which was established at that time.

Do you mean you think the EPR argument is flawed as well? I guess Einstein was already fairly old at this point; kind of amusing to see how even brilliant scientists aren't immune to getting out of touch. It did get people talking about entanglement though.

 

[RespectableCheese]

If they had had the right idea then the the experiments to test Bell's theorem would have proved QM wrong. That's where these ideas were eventually tested experimentally.

Yes, that is correct. I mean they had the right idea in noting the tension between QM and locality; they assume locality and derive contradictory results.

 

[RespectableCheese]

No, you have it backwards. They don't say that values that exist should be predictable. They say that values which are predictable with certainty should exist.

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.

 

[Demystifier]

I think this is because today entanglement is understood (at least mathematically). Then it was one of the first time they've seen it.

In one paper I argue that Einstein used a version of EPR argument already in 1930.
https://arxiv.org/abs/1203.1139

 

[martinbn]

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.

No! Why?

 

[RespectableCheese]

No! Why?

"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."

This logic can only follow in two cases:

1) Physical reality of a quantity implies predictability (not the case, based on their definition)
2) A value's existence in a complete theory implies predictability (not stated)

 

[martinbn]

"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."

This logic can only follow in two cases:

1) Physical reality of a quantity implies predictability (not the case, based on their definition)
2) A value's existence in a complete theory implies predictability (not stated)

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.