Is This Video on Bell's Theorem Wrong?

This is really a comment rather than an answer, but it's not going to fit in the comment box. I haven't yet watched the video, so I am taking your quotes at face value.

I will say that the Veritasium channel has in my experience had really excellent research, and their clickbaity "everyone is wrong about X" videos tend to be correct explanations about differences between popular science and actual research that people tend to slowly realize over a year or so in graduate school. That kind of communication is hard, because you have to (a) explain what a person qualified for graduate school might understand, then (b) explain how that understanding might be slowly modified over a year or two of study of related topics. The fraction of this journey that a person will make during a forty-minute video depends a lot on the viewer's starting point and their individual acumen. The starting point that "the video" as an entire object is "right" or "wrong" is itself a misunderstanding of what's being communicated.

To your questions:

However, as I understand it, Quantum Mechanics is a local theory, or at least it isn't necessarily non-local.

I don't have a lot of confidence that everyone involved in a discussion will come to the table using all of these technical terms the same way.

When most people say "quantum mechanics," they mean the model based on the Schrödinger eigenfunction equation,

$$ \left( \frac{\hat p^2}{2m} + \hat V(x) \right)\psi(x,t) = \hat E\psi(x,t), $$

where the momentum and energy operators $\hat p = -i\hbar\partial_x$ and $\hat E=+i\hbar\partial_t$ mean that a plane-wave state $\psi = e^{i(kx-\omega t)}$ has momentum eigenvalue $\hbar k$ and energy eigenvalue $\hbar \omega$.

Note that the Schrödinger equation is explicitly nonrelativistic. The relativistic energy for a free particle obeys $E^2 = (pc)^2 + (mc^2)^2$, which approaches the Newtonian $E = mc^2 + (pc)^2 / 2mc^2$ only in the low-momentum limit $pc\ll mc^2$. (The constant offset $mc^2$ turns out to be irrelevant here.) So at some level, tension between the Schrödinger equation and relativity is to be expected. Consider that Dirac's development of a relativistic wave equation immediately predicted both electron spin and antimatter electrons.

Contrary to your understanding, solutions to the Schrödinger equation are explicitly nonlocal functions. Functions of complex numbers which solve differential equations tend to be analytic, which means (among other things) that you can't change the behavior of a function in one place without changing it in some way everywhere. Students in introductory quantum courses will solve lots of "boundary value problems," where you demand that the wavefunction or its derivatives have particular values at one place, and combine those constraints with your potential function to solve for the wavefunction everywhere.

In the context of the video, it also seems to imply that reality must be non-local. I thought it was not locally real.

I think there's also a problem with your technical terms here. I think the correct replacement for "it" in your second sentence would be "reality is not locally real," which sounds unnecessarily bizarre.

In any case, we don't know what reality "is." We know what this or that model of reality is, and whether those models describe phenomena that we can observe. It is often the case that our best models of some microscopic phenomenon have behaviors or internal features which are quite different from the intuition that we developed from observing macroscopic phenomena. But a statement like "our model which is X is good, therefore reality is X" is fraught. To quote a cliché, the map is not the territory.

However, I thought the main point of Bell's theorem was that it does rule out local hidden variable theories.

Careful here. Bell's theorem doesn't rule out anything. Bell observes that, if two particles carry their own state information away with them to separate detectors, you get a particular statistical distribution of the correlation between their measurements. If the state information is shared or "entangled" between both particles, even as they separate, you'll get a different distribution of correlations. The theorem is that the quantum-mechanical superposition gives the largest possible correlations.

It is experiments which have observed, consistently, that it's possible to create ensembles of particle pairs whose measurement correlations follow the distribution for maximally-entangled particles. In low-noise experiments, it's possible to exclude a large class of models where the state information travels with the particles. The Nobel Prize for entanglement went to the experimentalists Aspect, Clauser, and Zeilinger. (If Bell were alive he would certainly have been included; he would have turned 98 this year.)

This is also stated on the Wikipedia page of Bell's theorem.

The amazing thing about Wikipedia — the thing that makes it trustworthy enough that people treat it incorrectly as an oracle — is that when it's wrong, you can click "edit" and fix it. The next sentence you quote says that this interpretation is present in "all sorts of physics textbooks and papers and whatnot." You found one.

I could not find a source on Bell stating that [it's really quite remarkable how many people make that error].

Bell discusses this at length in his book "Speakable and unspeakable in quantum mechanics" (1987).

Lastly, I was wondering if they could be using the definitions of locality, realism, and hidden-variables differently or ambiguously. Are these statements not wrong in terms of the definitions used by physicists?

If you're not sure that everyone is using the same terms, the end of the question is an inefficient place to raise that concern.

Bell's 1964 paper is much easier to find now than it used to be. (When I went down my Bell's Theorem rabbit hole as a grad student twenty years ago, I had to use secret library jujitsu to find a copy; the journal that published it failed after about a year.) If you have questions about how things are defined, going backwards to original sources is generally the way to go.

I think I have been explicit above about how I understand the distinction between a "local" state variable, associated with an individual particle, versus a "nonlocal" state variable, associated collectively with the pair.

The question of "realism" has to do with whether the result of the measurement exists before it is measured. I have a bowl of fruit in my kitchen that has some apples in it. I don't know how many are left; it is either two, three, or four. But I don't believe that the number of actual apples in my actual bowl is ambiguous, nor that the wavefunction of the bowl will collapse into e.g. the three-apple state when I go to look at it, nor that I can keep the number of apples ambiguous by pulling one off the top while carefully not looking at the bottom of the bowl. All of these counterintuitive statements are things people do say about quantum-mechanical states. These ideas are particularly associated with the "Copenhagen interpretation" of quantum mechanics, in which a special process called a "measurement" produces an instantaneous global change in a wavefunction.

The idea of a "hidden variable" (which originated somewhere along the line from EPR to Bohm to Bell) is that, while an observable like spin certainly exhibits all of the Copenhagen non-realism, perhaps the spin is actually related to some other state information which isn't experimentally accessible.

I did wind up watching the video while I was writing this. I had apparently watched the first half earlier and stopped, so I didn't go back to the beginning, but your questions are all about the ending. The video's statements at the end about what has or hasn't been proved are all consistent with my understanding, but the point they are making is pretty subtle, and I don't think your question summarizes it correctly. As I wrote at the top of this answer, I think it's fine to sit with the idea that your previous understanding is different from the actual state of things, even if you can't yet summarize the actual state of things to your own satisfaction.

In the precise sense of ‘elements of reality’ used by EPR, Bell’s theorem, combined with experimental violations of Bell inequalities, decisively rule out local completions of quantum mechanics. However, to understand exactly what this means requires being very careful with language.

The idea of "locality" is a slippery one in quantum mechanics. There are at least two different meanings.

1. The first meaning is that a local theory is one in which measurement outcomes are determined by variables local to each wing of the experiment and a shared past state.
2. The second meaning is that a local theory is one in which it is impossible to make a local measurement at a point $x$ and learn something about the state of the Universe at a point $y$ where light has not had time to travel from $y$ to $x$. (In more technical terms: observable operators outside the light cone commute).

Bell’s theorem assumes local causality in the sense of point 1, not locality in the sense of no superluminal signaling.

In particular, the starting point of Bell's theorem is to assume that reality can be described by a model with the properties of point 1 above. There is also an assumption that two experimenters can choose settings on their devices independently of each other that I'll return to later.

Then, the experimenters measure various properties of the particles, such as the particle's spin around different axes. Each experimenter chooses the parameters for each measurement (such as the axis they use to measure polarization); they generally do not choose the same parameters. After many trials, the experimenters compare their results.

Bell's theorem asserts that under these conditions (made suitably precise), there is an upper bound on certain combinations of correlations (Bell inequalities) that the experimenters will find. In the common case of spin-1/2 particles, this violation is often presented in terms of the angle dependence of the two-point correlation function between the polarization directions measured by the experimenters.

Under these same conditions, quantum mechanics predicts an amount of correlation that is larger than this upper bound. Therefore, quantum mechanics cannot be a local theory in the sense of point 1 above.

Experiments have been done and shown that in reality, the observed correlation violates the upper bound, and is also completely consistent with quantum mechanics. This means two things. First, quantum mechanics passes a non-trivial experimental test, so we gain confidence in it. Second, even if quantum mechanics is wrong, a local theory in the sense of point 1 above cannot be a correct description of nature.

In other words, Bell’s theorem rules out theories satisfying local causality, not merely “local hidden variables” in the naïve sense. Bell objected to the phrase “local realism” because realism is not an independent assumption: given local causality and the existence of definite outcomes, realism follows rather than being separately postulated.

The only way out of this conclusion, without getting rid of standard causal structure, is to violate measurement independence, a view known as superdeterminism, which abandons the assumption that experimenters’ choices are statistically independent of the system being measured. (Loosely speaking: it means that if experimenter 1 chooses to measure the spin along some axis, then there is something in the Universe that forces experimenter 2 to measure the spin along a particular, correlated axis; if you allow for this "something," then you can produce whatever correlation between experimental results you like, and in particular the one that comes out of quantum mechanics).

However, quantum mechanics (and in particular quantum field theory) is local in the sense of point 2 above. There is no local measurement that can be done that tells you what is happening in a part of the Universe where light has not had time to reach you. The fact that quantum mechanics is able to thread the needle between points 1 and 2 is remarkable.

No, I would not say that the video is wrong, but I do agree that this particular quote is confusing:

We do often teach Bell's theorem to physics students, and it's taught as something that rules out local hidden variables. That's just not true.

At first glance, this seems to imply that local hidden variables are not ruled out by Bell's theorem (and the experimental verification of quantum-mechanical predictions). Given that the video goes to great length in explaining how QM is non-local, I do not believe that this is the intended meaning. I am guessing the speaker wanted to say that Bell's theorem does not just rule out local hidden variables, but rather locality itself. It's not the hidden variables that introduce the problem. This point is made clearer elsewhere in the video, e.g. starting at 35:06.

So, to clarify: Local hidden variables are disproven by Bell's theorem, but not because of the hidden variables.

If you are interested in a hidden variable theory that is compatible with QM (and remedies many of its conceptual and onthological difficulties), I can recommend reading up on the de Broglie–Bohm theory, for example Jean Bricmont's Making Sense of Quantum Mechanics gives a nice overview including a detailed discussion of Bell experiments in that theory. It also helps differentiate claims derived from QM between the serious (e.g. "reality is non-local") and the bogous (e.g. "particle trajectories cannot exist" or "the world is demonstrably non-deterministic").

I was reluctant about writing an answer for this question, with so many good answers already here, but there are some points that I would like to mention, and I think it is missing in this discussion.

First, Bell's Theorem is a no-go theorem. It means that it proves that a specific situation is physically impossible, using a proof by contradiction. The situation in question is the existence of Local Hidden Variables (LHV) explaining the correlations obtained in a Bell test that are produced by measuring over a shared quantum system in an entangled state. If you set the quantum state and the measurements properly, it is possible to show a mathematical contradiction with the assumption of the LHV and what is predicted by Born rule.

The theorem doesn't mean that all quantum correlations are non-local. We could find a lot of quantum correlations that are local. For example, if in the Bell test, you choose a product state or a separable state, it would be local, since entanglement is a necessary condition for non-locality (but not sufficient, since some entangled states are local either). However, since there are some correlations which is non-local, it rules out the possibility that someone could find one day a local model "reproducing quantum theory", or more precisely reproducing all possible quantum correlations in a Bell test.

Another important thing about Bell theorem is its theoretical meaning. I disagree with rob that the theorem just raise a question that was answered years later by the experimentalists. To get sense of it, think about this situation: Imagine that I decide to invent a new model for electromagnetism, alternatively of Maxwell equations and I decide to name it egocentrically Ruffolo equations. Now, imagine that Ruffolo equations have very nice features, but it predicts that a point charge would have a field that is no spherically symmetric, contradicting Coloumb's Law and what Maxwell equations predicts.

We can say that the right thing to do in this situation is to do the experiment. To test Ruffolo equations against Maxwell equations, one should try to build a point charge experimentally, which is impossible to do exactly. So we should charge a small object and try to observe the electric field of it as far as we could, so it could be approximately a point, and we should do it in many directions we could, to testify if it is at least approximately spherically symmetric or not. But we don't need to spend this money and time. Maxwell equations and Ruffolo equations are not in the same stage. The former are being experimentally tested for almost two centuries. We trust in Maxwell equations, so any new model should at least reproduce what it already predicts.

The Bell theorem is quite the same situation that I described above. The quantum postulates, whose nobody question about its precision and correctness, already conflicts with LHV. The difference between local models and the Ruffolo equations is that one could argue, at least in principle, that Bell made some additional assumptions that could not fit in an experimental setup. To address the concern about these additional assumptions, experimentalists started to build loophole-free Bell tests.

There are some claims about Bell's personal point of view and also the point of view of Adam Becker, who is the physicist in the video. In the very nice article "Hidden variables and the two theorems of John Bell" by David Mermin, we find his last words

To those for whom nonlocality is anathema, Bell's Theorem finally spells the death of the hidden-variables program. But not for Bell. None of the no-hidden-variables theorems persuaded him that hidden variables were impossible. What Bell's Theorem did suggest to Bell was the need to reexamine our understanding of Lorentz invariance, as he argues in his delightful essay on how to teach special relativity and in Dennis Weaire's transcription of Bell's lecture on the Fitzgerald contraction. What is proved by impossibility proofs, " Bell declared, "is lack of imagination."

We don't need to take Bell's personal point of view as mandatory about his theorem. Curiously, Mermin left a last footnote in this article:

Although I gladly give John Bell the last word, I will take the last footnote to insist that he is unreasonably dismissive of the importance of his own impossibility proofs. One could make a complementary criticism 'of much of contemporary theoretical physics: What is proved by possibility proofs is an excess of imagination. Either criticism undervalues the importance of defining limits to what speculative theories can or cannot be expected to accomplish.

If you want to have a perspective about how the community thinks about Bell inequality violations, take the poll "A Snapshot of Foundational Attitudes Toward Quantum Mechanics" made by Maximilian Schlosshauer, Johannes Kofler and Anton Zeilinger, in the conference “Quantum Physics and the Nature of Reality,” held in July 2011 at the International Academy Traunkirchen, Austria. The question "What is the message of the observed violations of Bell’s inequalities?" led to the following answers:

About this result, the authors says:

The Bell inequalities are a wonderful example of how we can have a rigorous theoretical result tested by numerous experiments, and yet disagree about the implications. The results of our poll clearly support this observation.

In conclusion, Veritasium video is a good introduction, but it is partial about advocating one perspective of the theorem. It is partially caused by Adam Becker personal point of view about it. In literature, we could find other perspectives about its meaning.

I mean this entirely respectfully, and I don't mean it as a dodge of your question, but IMO this is just too big and complicated of a topic to fully resolve in one YouTube video aimed at a popular audience, or in one Physics Stack Exchange question page. Entire books have been written on this subject. In order to fully address it, you need to spend at least a few pages just carefully defining your terms and your top-line philosophical assumptions about the basic nature of reality and of scientific theories. As rob said, being either "right" or "wrong" is too much to ask of a YouTube video on this topic. (Well, okay - it may not be possible for it to be "right" within the space of 45 minutes, but I guess it could definitely be wrong within that span of time.)

Just to illustrate the challenges: all of the quotes in your question occur between the 33- and the 40-minute mark. But then, in the last five minutes, Derek goes on to introduce an entire new conceptualization - the many-worlds interpretation - which he seems to imply is fully local. (David Deutsch, in particular, argues strongly that the MWI proves a completely local formulation of QM, albeit one that lacks counterfactual definiteness - another huge aspect of the debate that the video barely has time to touch on.)

Taken on its face, this seems to suggest that the last five minutes of the video obviate all of your quotations. But then again, Derek says "Maybe there is a way out" (right after the dramatic cymbal crash) - so it isn't entirely clear what he is trying to argue.

I think that trying to decipher this video's "true" argument is not the right way to go. It's supposed to be a fun and engaging introduction to these ideas to a lay audience, not to win over experts with an airtight rigorous argument.

I'm not going to comment on the video directly but only on the quotes you gave. There are two quotes you gave about Bell's theorem:.

We need to be teaching Bell's theorem in a different way. We do often teach Bell's theorem to physics students, and it's taught as something that rules out local hidden variables. That's just not true.

and

but this is one of the most misunderstood experiments in all of physics. - You'll find in all sorts of physics textbooks and papers and whatnot, that what Bell's theorem proves is that it rules out local hidden variables or local realism. John Bell said that was an error, you know, he, he said like, it's really quite remarkable how many people make that error.

These claims are true but, as I will describe later, they are also almost entirely irrelevant. Bell's theorem constrains correlations between systems satisfying the following constraints.

(1) When you measure a system you get one result: this is commonly called realism.

(2) Systems only interact with one another locally by signals travelling at or below the speed of light in a vacuum.

(3) The measuring instrument and measured system aren't engaged in a conspiracy to produce particular kinds of correlations.

The violation of Bell's theorem is often taken as evidence that local hidden variable theories are false. But techically you could have a theory that is local and realistic in which the measuring instrument and measured system conspire with one another.

You will notice that I haven't mentioned quantum theory so far. One weird thing about this controversy and many other controversies about quantum theory is that many of the people engaged in it completely ignore what the equations of motion of quantum theory say about reality in general and the experiment they are discussing in particular. What do the equations actually say?

In classical physics the evolution of a measurable quantity, such as the $x$ position of a particle is described by a function $x(t)$ such that if you measure $x$ at time $t$ you get the result $x(t)$.

In quantum physics the evolution of a measurable quantity $x$ is described by an observable $\hat{x}(t)$ whose value at each time $t$ is a linear operator whose eigenvalues are the possible results of measuring $x$. In general the results of an experiment depend on what happens to all of the possible values of $x$: quantum interference. For an example see

https://arxiv.org/abs/math/9911150

But when information is copied out of a quantum system regardless of whether it is done by an observer or a collision with an air molecule or whatever, interference is suppressed, this is called decoherence

https://arxiv.org/abs/1911.06282

For objects you see in everyday life, information is copied out on scales of space and time much smaller than those over which the system changes significantly and interference is very effectively suppressed. In general information is copied out of a large system multiple times. For systems like electrons in atoms the effect is a lot weaker and so interference is a significant effect for an electron in an atom.

Decoherence doesn't eliminate the other possible states it just implies they are dynamically isolated from the result you see so there are multiple versions of the measurement result. This is often called the many worlds interpretation:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

Many people dislike this idea and will go to any lengths to avoid it. This includes proposing wildly different alternatives to quantum theory such as collapse and pilot wave theories and calling them interpretations of quantum theory despite the fact that they make different predictions:

https://arxiv.org/abs/2310.14969

https://arxiv.org/abs/2411.10782

What does quantum theory say about Bell correlations? The relevant variants of quantum theory for making predictions in situations where the speed of light is a relevant constraint are quantum field theories and they are local, as can be seen in many books on quantum field theory such as "Quantum field theory for the gifted amateur" by Lancaster and Blundell and "The conceptual framework of quantum field theory" by Anthony Duncan. Events at a particular point in spacetime can only change events at another point if that point is not spacelike separated from it. Almost all experimentally tested predictions of quantum theory come from relativistic quantum field theories and so almost all experimentally tested predictions of quantum theory are incompatible with non-locality. So the quote:

So quantum mechanics is non-local, but it doesn't lead to the sort of catastrophic paradoxes you might expect from relativity

is false unless we're willing to completely throw out theories that have been thoroughly tested by experiment. There is currently no replacement that makes the same predictions:

https://arxiv.org/abs/2205.00568

Quantum theory violates the first constraint of Bell's theorem because observables don't have a single value after a measurement. Rather, a system's observables carry information about the other systems it has interacted with in a form that can't be accessed by measurements on that system alone: locally inaccessible information. That information gives rise to Bell correlations:

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

https://arxiv.org/abs/1109.6223