What was the fallacy in EPRs chain of arguments?

Question

Let's say, there is an entangled system of two electrons with opposite spins; The joint system is in a state of eigenvectors for z-Spin ( $S_z$) with both particles far away from each other:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}(| +\rangle | -\rangle - |- \rangle | +\rangle)$$

When $S_z$ is measured on A and the result is $s_z$ the measurement of $S_z$ on B will give -$s_z$ with certainty. On the other hand, when $S_x$ is measured on A and the result is $s_x$ the measurement of $S_x$ on B will give -$s_x$ with certainty.

To put an example:

We measure $S_z$ on A and get +. Then a measurement of $S_z$ on B will give -.

Would we get the same outcome on B if $S_x$ was measured prior on A or even nothing has been measured yet on A? For my feeling it would be rather ridiculous, if the outcome of a measurement of B would depend on a prior measurement of A when the measurements are separated space-like. Moreover, whether A or B was first measured, depends on the observer due to relativity. So the only solution is, that both the outcomes for $S_z$ and $S_x$ are somehow pre-determined.

EPR claim in their 1935 paper, that QM must be incomplete, because we can simultaneously attribute definite values to non commuting observables (like in my example):

Previously we proved 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. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.

Today we know, that EPR were wrong, but what was the fallacy in their argumentation? Personally, I find the arguments "logically", but of course I'm wrong...where is the problem? I get a headache every time I think about this problem.

Answer 1

As @TobiasFunke and @WillO already said above, they made an unwarranted assumption. I will show you the exact wording of that assumption. First, they seemingly successfully demonstrated that the outcomes of measurements on spatially separated systems must actually be predetermined. At first glance, the ability to predict the outcome of Bob's measurement with certainty based on the result of distant Alice's similar measurement appears ironclad. But even they realized there was an implicit assumption in that conclusion. From EPR 1935:

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

Ah, so they realized that it was not possible to perform counterfactual observations on a pair of entangled particles. Thus only 2 measurement choices need be considered on 2 particles. But... they didn't know of Bell's Theorem, which arrived nearly 30 years later. But Bell showed there was much more to their assumption, which in fact is eminently reasonable. Though reasonable, it is wrong. It turns out that entangled expectation values are a function of a future context, and that context need not be local.

EPR's argument seems to work simply because they picked a special case where there are so-called "perfect" correlations: they were considering dual measurement of either momentum (P) and/or position (Q). But with Bell, spin was being considered. That gave a much larger set of measurement settings to choose from. And they are not "either/or" types of measurements. When you switch to measuring spin at angles where there is NOT perfect correlation, their assumption is shown to be invalid. Obviously, had Einstein, Podolsky and Rosen known of Bell's Theorem, they would have seen the problem in their "reasonable" assumption.

Answer 2

One invents a mathematical model of physical systems, and then tests them against experiment. This is the long history of physics.

In the EPR case, they made 2 assumptions in their analysis:

1. Physics is local. ie. Physical systems only influence other physical systems that are "close by."

2. Any parameter of their theory existed before an actual measurement took place. This is called realism.

The experiments have shown (and was the subject of the 2022 Noble Prizes) that local realism is violated.

The issue has nothing to do with their logic. It's just that nature doesn't work the way they thought it would.

Answer 3

The mistake is to assume there is a classical joint probability distribution for the outcomes of the various possible measurements.

Answer 4

You write, “Today we know, that EPR were wrong”, but this statement is certainly not correct. There is no conceivable way to prove that the mathematical quantum wavefunction is a complete description of physical reality, so we actually don’t know that they were wrong about this conclusion. Indeed, many of the currently-researched ideas in quantum foundations (Bohmian mechanics, retrocausal/all-at-once accounts, spontaneous collapse theories, etc.) all agree with the EPR conclusion about the limitations of the wavefunction. Indeed, the only formal interpretation which would disagree with the EPR conclusion is an Everettian viewpoint, where the universal wavefunction was the fundamental reality in question.

Still, certain steps of the reasoning in the EPR paper are now considered suspect. They took it for granted that the act making a measurement in one place couldn’t affect the state of reality at some distant space-like-separated location. And buried inside this assumption is the assumption that it is even possible to talk about a “spacetime-based reality”, concerning what is actually happening in some particular region of space-time. (This is the notion that there are indeed “spacetime-based parameters”, as assumed by all of classical physics.) Many modern physicists would say that at least one of these two assumptions has been shown to be incorrect.

But it’s important to stress that even if EPR were wrong about the assumption that one measurement can’t affect a distant space-like separated region, they might very well still turn out to be right about the incompleteness of the wavefunction. It’s commonly and lazily claimed that Bell’s Theorem ‘proves’ there are no hidden variables, but that’s not true in the slightest. It only proves that any hidden variables would have to be subject to certain nonlocal or retrocausal influences, which is a far cry from the conclusion that they couldn’t possibly exist.

Answer 5

There are a couple of possibilities. One, as already mentioned by other answers, is that measurements can indeed influence distant events by faster-than-light backwards-in-time means with an unknown mechanism. No known force or mechanism in physics travelled faster than light, so the collapse of the wavefunction propagates through space via unknown physics.

Einstein thought true action-at-a-distance was a philosophical nonsense, and retrocausality led to such inconsistencies and ambiguities that the major principles underpinning modern physics would have to be abandoned. He had argued the absolute necessity of special relativity - with spectacular success - by relying on precisely those sorts of principles. It would have gone deeply against the grain to abandon all that.

The other possibility is that the collapse does not happen, and all the possibilities remain in superposition. However, the observer is also a quantum system, and when it interacts with a particle enters a correlated superposition as well. Thus, the question of the 'reality' of the physical quantities P or Q is only applicable to each orthogonal component of the observer superposition. Because the components are orthogonal, they do not interact, and cannot perceive one another. Thus, each sees only one definite outcome. No "collapse" propagates from such an event; there are no faster-than-light rips in the universe. Instead, the correlation is explained because when two widely-separated observers become correlated to particles which are correlated with one another, the observers become correlated with each other too. Thus, when they return home (at sublight speeds), the observer who saw 'up' can only perceive the partner who saw 'down', and the observer who saw 'left' can only perceive the partner who saw 'right'. But an observer who saw 'up' is not orthogonal to partners who saw 'left' and 'right', and so when they interact enters a superposition of observer states, between them seeing both.

Einstein understood that to be consistent with relativistic causality the exact correlations between experimental outcomes could only originate locally, at a single location somewhere that the worldlines of the two experiments met. He assumed it had to be in the past, because like everyone else, he believed that at the place and time when the measurements were made the matter was decided. But in fact they can also occur in the future, after the experiment has happened, and the two experimenters return home to compare notes.

This interpretation is linear, local, deterministic, realist (in the sense of considering quantities to be 'real' if they can be predicted with 100% reliability), considers the wavefunction to be ontologically real and complete, and requires no extra axioms, assumptions or mechanisms that were not already included in quantum mechanics. In fact, the only thing needed was to throw away an assumption - the collapse. I don't know for certain if Einstein would have liked it had he lived long enough to see it - it does multiply unobservable entities beyond necessity - but it does everything he said he wanted from a theory.

Answer 6

A definition of physical reality is given in EPR paper:

The elements of the physical reality cannot be determined by a priori philosophical considerations, but must be found by an appeal to results of experiments and measurements. A comprehensive definition of reality is, however, unnecessary for our purpose. We shall be satisfied with the following criterion, which we regard as reasonable. 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. It seems to us that this criterion, while far from exhausting all possible ways of recognizing physical reality, at least provides us with one such way, whenever the conditions set down in it occur. Regarded not as a necessary, but merely as a sufficient, condition of reality, this criterion is in agreement with classical as well as quantum-mechanical ideas of reality.

In substance this means that EPR are assuming that real physical quantities have only a single value at any given time.

You write:

Let's say, there is an entangled system of two electrons with opposite spins; The joint system is in a state of eigenvectors for z-Spin ($S_z$) with both particles far away from each other

...

Would we get the same outcome on B if $𝑆_𝑥$ was measured prior on A or even nothing has been measured yet on A? For my feeling it would be rather ridiculous, if the outcome of a measurement of B would depend on a prior measurement of A when the measurements are separated space-like. Moreover, whether A or B was first measured, depends on the observer due to relativity. So the only solution is, that both the outcomes for $𝑆_z$ and $𝑆_𝑥$ are somehow pre-determined.

You are talking about the outcome of a measurement so you too are assuming physical quantities are single valued.

Both the EPR paper and your discussion assume that physical quantities are single valued. In both cases this conclusion about the way the world works doesn't come from quantum theory. It is somewhat odd to try to work out the implications of a theory without looking at what the theory itself claims but that is the procedure followed by EPR and you and many others.

Quantum theory in the Heisenberg picture describes physical quantities in terms of evolving observables represented at each time by Hermitian operators, not by a number nor by a stochastic variable. If you work out how the observables change over time during the experiment, then systems become dependent on one another only as a result of local interactions:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223

The correlations arise after the results are compared. The information that gives rise to the correlations is conveyed by local interactions in the form of locally inaccessible quantum information in decoherent channels.