# How is superdeterminism a loophole to Bell's theorem?

I read wiki about superdeterminism and Bell's theorem and the lecture on free will, and I struggle to understand what free will has to do with the Bell's theorem.

From my understanding, for Bell's inequality to be true, one requires local realism and that the hidden variables associated with Alice, Bob and detector ($$a$$, $$b$$ and $$\lambda$$) to be uncorrelated, from which one can derive the CHSH inequality, which is almost the same thing as Bell's inequality. Therefore, if the Bell's inequality is violated we have to either give up local realism or the absence of correlation between $$a$$, $$b$$ and $$\lambda$$. Gerard 't Hooft in his Cellular Automaton interpretation of quantum mechanics on p. 43 derives the conditional probability function for $$\lambda$$, given $$a$$ and $$b$$ (in certain interpretation of the variables): $$P( \lambda|a, b)=\frac{1}{2}|\sin(4\lambda-2a-2b)|.\tag{1}$$
From the said above, it would seem, that whenever one wants to build a classical theory that matches quantum measurements, they would have to make sure the conditional probability (1) holds no matter what. To me, this is the gist of the problem, since the chaotic behavior of the external noise usually kills any correlations between spatially separated systems over time, given the noise actually influences the hidden variables. And if the noise does not influence $$a$$, $$b$$ and $$\lambda$$, then they are constants of motion and the outcomes of the experiment are going to be always the same! I cannot even think of what kind of system would have such bizarre property that the individual variables are influenced by noise, but the mutual correlation would not be destroyed.

Of course, one could hypothetically solve the equations of motion backwards and actually obtain the configuration of noise that would produce the desired correlation. But then, the predictive power of such theory would be close to zero, since even though the noise preserved the correlation till now (for which we performed the fitting), it is not guarantied to preserve it in the future.

Where does "free will" comes into play here? There seem to be no point where it would be invoked into the argument.

• Free Will has nothing to do with it and Don’t give up on local realism. Think correlation instead of entanglement. May 3, 2021 at 0:47
• I am not giving up local realism. I just wonder what kind of theory would (A) have local hidden variables influenced by noise and (B) have the correlation between the three variables not influenced by noise. Free choice seems to be pretty much irrelevant to this problem May 3, 2021 at 2:06
• The "free will" argument is a bit moot. The point is; if the entire world is a 3D movie, and every single atom in the universe is behaving not according to an internal set of rules, but as a predetermined "frame" of this movie - there is no violation. May 3, 2021 at 7:26

You never get to observe the probability distribution directly. You only make individual measurements and build up statistics by combining these measurements. If we assume that the measurements you make are independent trials, then sampling many times from the experiment will give you a sense of the underlying distribution.

But, if for some reason, the measurements were correlated, it could be that you are not really sampling from the "true" distribution. For example, if you are a puppet being controlled by some higher being engaging in a conspiracy, you could simply be carrying out a sequence of measurements that were carefully selected to give results that look as though they were random samples from the quantum mechanical probability distribution, but in fact were deterministic.

My personal feeling is that it is not worth thinking for very long about superdeterminism.

• Wait, you don't mean that superdeterminists imply there is some other conscious being making choices for us, right? I got an impression that they believe that everything is predetermined and decided by the "laws of nature". In other words, the theory generates the weird correlations from within (I find hard to imagine such theory, but I thought this is what superdeterminism is about). Am I wrong in my understanding? May 3, 2021 at 2:10
• @Pavlo.B. That is my understanding too. The point of super-determinism is that if the observer and the system are correlated, then the observers might never make a choice that would cause the (deterministic) system to contradict QM. Bell's theorem assumes the choices to be independent. Imagine that you simply choose a classical particle universes in which no experiment that contradicts QM ever occurs. No one ever asks the right question. The main counter to this is that that seems very unlikely. The counter counter is - based on what assumptions? May 3, 2021 at 8:45
• @Pavlo.B. Well, I think the idea is that something in the initial conditions of the Universe arranged for all measurements performed to be consistent with quantum mechanics. Whatever the "something" is doesn't need to be a conscious entity. But I have to be honest that I do not think of superdeterminism as a serious proposal -- it's a logical possibility, but not one that provides any scientific value. It's inconceivable to me that the initial conditions of the Universe just by chance encoded every measurement in the Universe and conspired to make them consistent with quantum mechanics. May 3, 2021 at 11:07
• If superdetermism were true, I would find it quite likely there was something intentionally arranging a conspiracy. May 3, 2021 at 11:08
• I haven't seen an argument for superdeterminism that doesn't, at its root, imply some sort of conscious or unconscious conspiracy. Violations of bell's inequality imply that the universe would have to conspire with itself to trick us into making it look like there are no local hidden variables to account for those measurements, while also having local hidden variables that account for those measurements. There's no clean way around that without a "conspiracy" of some sort, that I can see.
– TKoL
Oct 14, 2022 at 14:51

Mathematically, the only type of correlations that can be made in a local Einstein-Bell realistic world are of the form

$$C(\mathbf{a}, \mathbf{b}) = \int_{\lambda \in \Lambda} A(\mathbf{a}, \lambda) B(\mathbf{b}, \lambda)\ d\mu_\Lambda$$

where we integrate with respect to a probability measure $$\mu_\Lambda$$ on the state of objective system states $$\Lambda$$, where we are extracting the spin components in the directions of $$\mathbf{a}$$ and $$\mathbf{b}$$ respectively for the two particles. However, to match the predictions of quantum theory, we must have

$$C(\mathbf{a}, \mathbf{b}) = -\mathbf{a} \cdot \mathbf{b}$$

This cannot be written as an integral of the form given, because a product of two scalar functions of single vectors cannot equal a single scalar function of two vectors generally. Namely, $$\mathbf{a} \cdot \mathbf{b}$$ does not factor as

$$\mathbf{a} \cdot \mathbf{b} = f(\mathbf{a})\ g(\mathbf{b})$$

for individual functions $$f: \mathrm{Vec}(\mathbb{R}, 3) \rightarrow \mathbb{R}$$ and $$g: \mathrm{Vec}(\mathbb{R}, 3) \rightarrow \mathbb{R}$$. You need to mix vector components together in a way that this type of multiplication cannot do.

But this only is a problem so long as there's no mixing factor in the integral: One way to have one is to have $$A$$ and $$B$$, the probabilities that measurements at the two different particles will yield a result given their objective states, depend on both $$\mathbf{a}$$ and $$\mathbf{b}$$. This is non-local realism, because now the local measurement represented by $$A$$ must somehow yank distant information from the one represented by $$B$$, and conversely.

The other way, though, is to say that $$\lambda$$ is correlated to $$\mathbf{a}$$ and $$\mathbf{b}$$ themselves. Either this means merely setting the instruments alone changes the particles, or that the measurement settings themselves are somehow influenced, whether by the particles or by some other event in the past. This last approach is the superdeterministic approach.

• Thank you for the response! It is quite to the point, but I have some questions concerning an event in the past influencing the settings. If such thing happened and the settings were correlated, wouldn't the external noise "decorrelate" the settings back? For instance, the settings could explicitly rely on signal from parts of the universe that were not available at the moment when the whole experiment started. These parts of the universe would be not available at the moment when the whole thing was set up, so the detector cannot know which setting $\lambda$ it is supposed to choose. May 3, 2021 at 2:31
• @Pavlo B. If the device relies on receipt of a signal from a distant area of the universe, then that just means the correlating event has to be so far back in time that both device and distant signal source are in its future light cone - e.g. if they are separated by 100 million light years, then the correlating event must have been at least 50 million years ago (not considering the cosmic expansion). Ultimately, all events share a "common causal ancestor" of this type if the Universe is considered to have started in a Big Bang where all parts were coincident. May 3, 2021 at 2:48
• So... it assumed that the "event in the past" is the very starting state of the Universe? Basically, the initial state of the universe set all these hidden variables up, and they are not decorrelated by interactions with each other? But then again, I struggle to imagine what kind of theory could have such local hidden variables that do not get decorrelated from the past by chaos. May 3, 2021 at 3:14
• I mean, in theory you can have a "common cause" that correlated all hidden variables in a necessary way. But isn't this the same in practice as just saying that the initial state of the universe was super-duper fine-tuned? Which is, of course, almost always possible, but as a theory practically useless for predictions (unless one postulates that the universe was fine-tuned for arbitrary number of experiments, in which case, if the universe was built on a discrete lattice, such fine-tuning quite likely does not exist due to being too restrictive) May 3, 2021 at 3:25
• @Pavlo.B. You've discovered the exact reason the vast majority of physicists don't take superdeterminism seriously. It's a "theory" in the same sense that "the moon vanishes when you're not looking at it" is a theory. May 3, 2021 at 20:26

I've read this article by Sabine Hossenfleder (review by Gerald Hooft!), and it is a nice recent review of the Bell's theorem, superdeterminism and arguments for it and against it, as well as examples of some superdeterministic models. The "free will" was indeed claimed to be "red herring", and it was effectively stated that the whole business of superdeterminism is about how to make the hidden variable of the source $$\lambda$$ dependent on the on the hidden variables of Alice and Bob $$a$$ and $$b$$ (or the other way around).

Unfortunately, the problem of the hidden variables being influenced by noise but the correlations not being influenced by noise was not raised at all. I am not sure if it is because there is no good response, or because the author is not aware about the problem

When you talk about “noise” in the superdeterminists view (at least in my view) you are talking about something with a very large but finite number of external factors. A random number generator is never truly random, just meant to capture as much entropy from its local system as possible. It is impossible to design something that does anything random, no matter how much noise you throw at it.

The word “random” is the true conundrum to me, and the implication is that “spontaneous” processes are where the truly interesting problems are found. Why does an electron spontaneously emit a photon when it chooses to? What is alpha decay? I’ve heard people explain it as caused by quantum tunneling and the inherent randomness of the cosmos, but that doesn’t make any sense to me. There must be some tiny machine in there, not dice. Those are the hidden variables we should be looking for. Spontaneous emission seems the most approachable thing to study.

And I personally don’t understand why one dismisses this as undermining science or the independence of the observer. Perhaps it cannot be known, but I am convinced that it is a mechanism fundamental to the universe. If there were 2 identical universes that electron would emit that photon at the exact same instant.

Since nothing can be truly random, everything in the universe is necessarily correlated.

• "Since nothing can be truly random" -- why not? Feb 22, 2022 at 14:10