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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 P: It cannot be the case that both (A) The quantum theory is a complete physical theory and (B) 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 P, it follows that the quantum theory is in-complete.

This conclusion clearly follows if proposition P 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 found here for your convenience. Thanks.

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  • $\begingroup$I've lightly edited your question to stop it from have three different uses of "1", however, I cannot fix your paper link.$\endgroup$
    – naturallyInconsistent
    CommentedMar 5 at 2:37
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    $\begingroup$Related$\endgroup$
    – Tobias Fünke
    CommentedMar 5 at 6:09

2 Answers 2

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I like to use spins as an example when thinking about EPR. We have our entangled particles, and send one to Alice and one to Bob. Alice has the choice to measure the spin component $S_x$ or the spin component $S_z$. These don't commute as operators in quantum mechanics. Once she has chosen to measure, say, $S_x$, then she gets a value, say spin up. At this point, she knows with certainty that if Bob is to measure $S_x$, then he will get spin down (I didn't tell you exactly how I defined the entangled state but this is one way to set things up).

Now because of locality, Einstein et al would argue that the state of the electron at Bob cannot be affected by Alice's measurement. This is a sneaky step that is later refuted by experiments confirming that Nature violates Bell's inequalities. So everything after this is wrong.

Therefore, Einstein et al say, because we can definitely say what Bob will measure for $S_x$, and because Alice's measurement can't possibly affect Bob's electron, it must be the case that it was possible to predict the value of $S_x$ all along, even before Alice made the measurement. You can repeat the argument and conclude $S_z$ must also be predictable in a complete theory of Nature. The fact that quantum mechanics does not predict the outcomes of Bob's measurements of $S_x$ or $S_z$ before Alice makes her measurements, means that quantum mechanics cannot be complete.

As I stated above, the error here is in imposing a form of locality that is too strong. Bell carefully analyzed the case where there is an underlying hidden variable theory that is local (technically called "local realism"), and found that it had to obey an inequality. Quantum mechanics predicts that this inequality is violated, because in some sense it is non-local (even though it is also consistent with special relativity and causality). But the true test is that experiments were done, and showed that Nature violates the predicted inequalities, meaning that quantum mechanics can explain the data, while a theory of local realism cannot.

I think the error (or at least an error) in your reasoning is when you say "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." I would be very careful with the words "empirical reality." Quantum mechanics claims that eigenvalues of non-commuting observables cannot be predicted simultaneously with absolute certainty. But this is not empirical reality. Empirical reality corresponds to what we actually measure, and shouldn't refer to things like "predictions" or "operators." An empirical reality in our example is that every time Alice measures $S_x$ up, then Bob measures $S_x$ down, and every time Alice measures $S_z$ up, then Bob measures $S_z$ down. Einstein et al would then claim this empirical reality implies that quantum mechanics can't be complete, based on the reasoning involving locality I explained above. To restate it: quantum mechanics predicts that you cannot predict both $S_x$ and $S_z$ with certainty because they are non-commuting, but empirically there are situations where Alice's results are perfectly anticorrelated with Bob's results for $S_x$ and $S_z$, and Einstein et al would argue that because of locality, that must mean that there is some way for a truly complete physical theory to predict $S_x$ and $S_z$. However, experimental tests of Bell's inequality show that Nature disagrees.

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  • $\begingroup$(+1) Let's say $S_x$ is a polarizer (in the photon version of EPR) oriented facing a specific direction. The second polarizer is then carried 1 light year away for measurement. But since light paths have bent by this distance, the correct placement of which direction the second should face must be experimentally determined, correct?$\endgroup$
    – James
    CommentedMar 5 at 8:37
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    $\begingroup$@James Before the experiment happens, you suppose that Alice and Bob have some way of aligning their polarizers so that they are using the same $x$ and $z$ directions. The details of how they do this is left as an exercise to the reader :D In practice these experiments are done on Earth where it's a lot easier to guarantee to polarizers are aligned.$\endgroup$
    – Andrew
    CommentedMar 5 at 12:08
  • $\begingroup$thank you. I suspect that this alignment cannot be done. E.g. use an equilateral triangle shaped polarizer (minimal number of vertices for a surface) as the first polarizer. Then emit 3 photons from each end of the polarizer. Emit a 4th photon to double check whether it lands properly at the expected location. Mark the location of all 3 photons after 1 light year and make a bigger/smaller triangle and call it the 2nd polarizer at these vertices. Now check the 4th photon. It is not constrained by any laws of physics to be coplanar as the other three, right? Most likely it will not be...$\endgroup$
    – James
    CommentedMar 5 at 14:44
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    $\begingroup$@James These boil down to technical challenges in designing the experiment. I don't see a fundamental reason that would prevent the alignment from being done, even if it is difficult. For example, you could take a source with a known polarization, and move it far away from both polarizers so that the rays from the source are essentially parallel when they reach Alice and Bob, and have Alice and Bob calibrate their polarizers to the known light source. But anyway this is kind of irrelevant because the experiments have been done, so this problem is solved in the lab where it matters.$\endgroup$
    – Andrew
    CommentedMar 5 at 15:29
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    $\begingroup$Alain Aspect is one of the main experimentalists in this field and if you are interested in how the alignment works in real experiments then I would suggest looking up his papers.$\endgroup$
    – Andrew
    CommentedMar 5 at 15:30
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It is definitely the case that you are wrong (subtly), but it took a while for me to narrow down where the mistake is (still having an uncomfortably large range of possibilities) when the nice other answer appeared.

Tables of Possibilities

In order to help you see that EPR's argument is valid and unsound, as opposed to them making an elementary logical error, it will help to visualise in tabular form what is being discussed. In the absence of any argument at all, we should have

QM is completeQM is incomplete
Non-commutating observables simultaneously pinned downpossiblepossible
Non-commutating observables cannot simultaneously pinned downpossiblepossible

Your question is worded in such a way that I cannot even ascertain if you understood that EPR had two distinct sections. You have stated the either-or conclusion of the first section, i.e. at the end of the first section, the table looks like

QM is completeQM is incomplete
simultaneousimpossiblep
not simultaneouspp

Having established the above possibility table, the main argument in the second section is to show that the table should instead read

QM is completeQM is incomplete
simultaneouspirrelevant to second section
not simultaneousimpossibleirrelevant to section section

So that by combining the two sections, EPR arrives at the table

QM is completeQM is incomplete
simultaneousimpossiblep
not simultaneousimpossiblep

Note that this table makes it clear that the conclusion is logically valid, since the cases where QM is complete are all refuted.

Your mistake: Attacking whether non-commuting observables can have simultaneous reality or not is irrelevant to the EPR argument; they left open the possibility that both QM is incomplete and non-commutating observables cannot have simultaneous reality.

The language used in the paper is so old and convoluted that it would be helpful to somewhat modernise them. You wrote of two assumptions. There is a subtlety here that I think you are missing. Those are not assumptions in and of themselves. Instead, the EPR paper is defining:

EPR definition of complete: every element of the physical reality must have a counterpart in the physical theory

It helps to consider what this is saying in much more familiar terms. Consider the spinless Coulombic Hydrogen atom. The set $\{\mathcal H,L^2,L_z\}$ serve as the complete set of commuting operators for this fictitious system, and so when you label the energy eigenstates $\left|\psi\right>$, you label them as $\left|n,\ell,m\right>$. Thus, in modern terms, EPR's definition means that every simultaneously 100% predictable eigenvalue should then simultaneously appear in the state labelling set.

EPR definition of reality: 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.

In modern terms, this just means that 100% predictable stuff must be describable by a quantum state purely living in the specific-eigenvalue's associated eigenspace of each observable's operator being scrutinised.

Of course, it is definitely an assumption that these two definitions are relevant! And not nonsense in and of themselves!

It took a few decades before Bell could come along and point out that EPR made some other hidden assumptions. It should be not at all surprising that Einstein would be interested in assuming that his other baby, relativity and thus locality, is true; it might surprise the reader that Einstein was already onto this problem back at the 1927 Solvay conference where everybody else totally failed to understand what he was on about; which in turn is not too surprising, because Einstein had something like a 20-year head start on everybody else thinking about the tension between (overly strong) locality and quantum theory.

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  • $\begingroup$I think you've completely misunderstood the two assumptions - the first definition of a complete theory is more general than QM, and relies on an understanding of scientific theories as abstractions that map or mirror physical reality. The second definition is similarly more general: they're just saying that any element that can be predicted in the manner described exists in "physical reality" in the most general possible sense.$\endgroup$
    – bishop-fish
    CommentedMar 5 at 21:22
  • $\begingroup$Of course they are meant to be more general than QM, but without expressing them in terms that fit in a modern QM setting, it is difficult to see what the words could even mean. So, no, I think you are quibbling something that I have already fully understood.$\endgroup$
    – naturallyInconsistent
    CommentedMar 6 at 0:18
  • $\begingroup$Regarding completeness, you claim the modern version of their definition would imply that any eigenvalue that is absolutely predictable would appear in a labeling set (in the theory). I guess this would be true if you combine it with their second definition to mean that those eigenvalues therefore exist in physical reality, and then must have some correspondents in the theory (namely, the labels). So with respect to those physical elements that would serve as an instance of a theory that satisfies their criteria; but I'm not sure what your point is otherwise?$\endgroup$
    – bishop-fish
    CommentedMar 6 at 5:24
  • $\begingroup$(cont.) Your second point about the criterion of reality doesn't seem to correct; the only strict implication following therefrom is that %100 predictable elements exist in physical reality, and not as part of some specific theoretical framework (in your description, the the mathematical framework of eigenvalues and operators). Your main point (following "Your mistake:") is just wrong though; it is absolutely relevant whether the eigenvalues of non-commuting observables can have simultaneous physical reality; if this is true, it is implied by the disjunctive proposition that QM is incomplete.$\endgroup$
    – bishop-fish
    CommentedMar 6 at 5:31
  • $\begingroup$Re: Your comment starting with completeness, yes, you are correct; my point is to highlight that the modernised recasting of what EPR is saying, would be to end up with the monstrosity that is $\left|x,p\right>$, which is manifestly laughable to anybody who understands QM. Note, however, that Einstein himself repeatedly asserted that he understood and agreed with HUP and had never intended to attack HUP, even in the 1927 Solvay conference. My point with bringing these up is precisely to highlight the serious tension that is being presented.$\endgroup$
    – naturallyInconsistent
    CommentedMar 6 at 5:39

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