I have seen various explanations, including pse1 pse2 pse3 pse4 pse5

and so far none of them have made sense to me.

It is a counter-intuitive probability theory argument, like the Monty Haul question. i have found that when I look carefully at that sort of problem after awhile it makes sense. So for example with Monty Haul, you start out with a 1/3 chance for each of 3 choices. You choose one of them. Monty is obligated to show you one of the other choices that fails. There's always at least one of them that fails so he can do that. When he does, it doesn't improve the chance that your choice is correct. So the remaining choice is 2/3 likely to be correct. If Monty chose one of the other possibilities at random and if it was the correct one then he just told you so, then you'd do just as well by choosing either of the remaining choices, unless you had already lost.

I haven't seen any description of this problem that actually describes the probability argument coherently. I could go to old questions and ask in comments for a real answer. Or I could answer an old question, saying that I didn't actually have an answer but none of the existing answers did what I wanted either. It looks to me like a new question is appropriate, unless an existing question that I have overlooked inspired a good answer.

Here is how the problem looks to me. A good answer does not depend on reading how I misunderstood it, maybe you can make a good explanation without needing to understand why I didn't understand others.

QM predicts probability distributions of outcomes, given probability distributions of inputs. When there's no known way to measure inputs as more than probability distributions, this is the best we can do.

Some people want to think that there might be deterministic equations that describe what really happens on a subatomic level, but we can't yet collect the data that would reveal them. Others want to think that it's all just probabilities and there's nothing else but probabilities. At first sight with no way to measure anything else, we can't tell whether there IS anything else -- and it doesn't much matter until we can measure it.

However, Bell's Theorem and various similar things say that deterministic models must fail if they accept "locality". The central concept appears to go like this:

Suppose there are two particles a and b that are separated by some distance. They can't immediately affect each other, not until light can get from one to the other. Any attempt at explaining their behavior will have a limit on how much correlation there can be between them. But QM shows that in some particular circumstances their behavior has MORE correlation than any reasonable explanation can possibly allow. Therefore there can be no explanation. All we can do is apply QM to get correct probability distributions.

What I'm looking for is a descriptive probability story. Perhaps with game show contestants locked into their soundproof boxes where they can't communicate with each other. What is it that happens that can't fit a reasonable story, that's analogous to the events that QM describes that similarly can't fit a reasonable story.

  • $\begingroup$ The story you are asking for is in my answer. $\endgroup$
    – WillO
    Aug 18, 2022 at 23:46
  • $\begingroup$ @WillO Woo! You are telling me that Bell's Theorem is not about a counter-intuitive probability theory result. Bell's theorem is about experimental evidence that demonstrates the world cannot make sense. $\endgroup$
    – J Thomas
    Aug 19, 2022 at 2:22
  • $\begingroup$ If two particles are perfectly correlated, sent different directions and tested later, they will match quantum predictions. Perfect correlations includes speed, trajectory, frequency/oscillation, and timing/coherency. $\endgroup$ Aug 19, 2022 at 2:23
  • $\begingroup$ When statisticians discover a result that is statistically impossible, the instinct is first to look for censored data, and second to look at whether what is actually being measured is not what is theoretically claimed to be measured. QM could be correctly predicting that the observations are biased, or it could be correctly measuring something that is not what theorists think is being measured. Or maybe reality just does not make sense. Or something else I haven't thought of. $\endgroup$
    – J Thomas
    Aug 19, 2022 at 4:24
  • $\begingroup$ @JThomas: I have no idea what you're talking about. Bell's theorem is a statement that certain statistical patterns cannot be consistent with classical probability theory. Quantum mechanics predicts that some of those statistical patterns will occur, and experimental evidence confirms those predictions. The story in my answer above is another example of a statistical pattern different from what occurs in quantum mechanics, but having the same flavor and violating Bell's theorem in exactly the same way. I thought that was the kind of story you were looking for. $\endgroup$
    – WillO
    Aug 19, 2022 at 5:36

5 Answers 5


The type of correlations taking place in a Bell experiment are like this.

Let us assume that the hidden variable theory which should explain the quantum phenomenology is realistic. In other words, all observables are always defined and there is a common probability space where I can describe all them.

Within this framework, the data I obtain from quantum measurements permit to assert what follows.

The value of an observable A on the right side of the system is correlated with which observable B (belonging to a definite set) I decide to measure on the left side of the system.

I can make a choice of B randomly and I can make that choice in a region of spacetime which is causally separated from the region of spacetime where the outcome of A is recorded.

The only way out seems to assume that my (random) choice as well as the outcome of A were predetermined by a common cause.

This is the loophole of superdeterminism (a sort of cosmic conspiracy), a metaphisical position not very fruitful in my view.

Rejecting the cosmic conspiracy, there are only two non-mutually exclusive possibilities for a (hidden variable) theory capable to explain the quantum phenomenology. The theory is non-local or it is non-realistic.

Quantum mechanics opts for the second way. But, for instance, the Bohmian mechanics prefers the first option.

I stress that in spite of the presence of the aforementioned non local correlations, it is not possible to transmit information through them. So no causal paradoxes can be produced in this way.

For instance, looking at the statistics of the outcomes in only one side, it is not possible to desume which observable is (will be, was) measured in the other side and also the outcomes of the measurements of that observable.

We would obtain the same statistics in one side even if no observable is (will be, was) measured in the other side.

This is a theorem of quantum mechanics.

The correlation appears just when comparing the statistics of outcomes of both sides.

This comparison can be made a posteriori and by communicating the outcomes through subluminal transmission of information. No violation of causality takes place.

The experimental knowledge of these phenomena is nowadays so consolidated that they are exploited in technology: quantum distribution of cryptographic keys through entangled states of photons.

  • $\begingroup$ I'm not sure I get that. I'll say it in my own words and see if I have it right: QM predicts that in some circumstances, you can choose which measurement to take in one place, and your choice affects the probability of the outcome of somebody else's experiment somewhere else, before light from you could reach them. The implication is that the world does not make sense, and QM does not make sense, and therefore no theory which makes sense can possibly be correct. Is that what you're saying? $\endgroup$
    – J Thomas
    Aug 17, 2022 at 21:47
  • 1
    $\begingroup$ Yes I am saying that. But not that the world does not make sense. For instance we cannot transmit information between the to sides of the system through these correlations. If we could, then yes the world would not have much sense, in my view, because of causal paradoxes one could produce. $\endgroup$ Aug 17, 2022 at 21:55
  • $\begingroup$ If we could transmit information between the sides quickly enough to cause paradoxes, that would be a world that made even LESS sense. But it sounds like you are asserting that no causal theory can be correct, but still QM which asserts statistical results based on observed previous statistics, is correct without making sense. $\endgroup$
    – J Thomas
    Aug 19, 2022 at 4:39
  • $\begingroup$ Looking at the statistics of the outcomes at only one side, it is not possible to desume which observable is (will be, was) measured at the other side. The correlation appears just when comparing the statistics of outcomes of both sides. This comparison can be made a posteriori and by communicating the outcomes through subluminal transmission of information. No violation of causality takes place. $\endgroup$ Aug 19, 2022 at 7:19
  • $\begingroup$ Yes, our world is quite strange, but not self contradictory and certainly much more subtle than we expected. I would like to stress that the knowledge of these phenomena is nowadays so consolidated that they are exploited in technology: quantum distribution of cryptographic keys through entangled states of photons. $\endgroup$ Aug 19, 2022 at 7:25

Suppose there are two particles a and b that are separated by some distance. Neither can have any effect on the other before light has traveled the distance between them. So any deterministic theory must say that their behavior will be uncorrelated. They will be statistically independent.

This is incorrect. The two particles are allowed to interact in the past, but then they are left to increase their mutual distance. Then any theory, deterministic or not, allows for some correlations.

But QM sometimes shows they will in fact be correlated. So any theory which expects them to be uncorrelated, which every reasonable deterministic theory must agree to, is wrong.

Bell has shown that according to standard rules of quantum theory, the particles that come from one interacting system in the past, may have stronger correlations than is consistent with local hidden variable models (under some assumptions). This is formulated in terms of certain inequalities. Quantum theory breaks those inequalities, while those hidden variable models obey them.

The conclusion from this is that any alternative to quantum theory that would be consistent with Bell's theorem has to be either non-local, or non-realist(no hidden variables can ever fully describe the system).

  • $\begingroup$ I'm fine with no hidden variables ever fully describing the system. No finite set of axioms can define the real numbers, after all. I'd be very happy with a set of hidden variables that describe only 99.99% of the system. Your statement of it looks more cautious than others I've seen. They said that NO local hidden variable model can fit the reality, although some can work in limited contexts. Thank you for explaining that some correlation is allowed. $\endgroup$
    – J Thomas
    Aug 18, 2022 at 0:55
  • $\begingroup$ Authors of "no-go" theorems don't typically deal with such possibilities as "not possible to have hidden variable theory (HVT) that reproduces QT 100%, but possible to have one that reproduces 99.99%". They seek nice results such as "under these assumptions, no HVT can reproduce QT in all cases". If we allow for imperfections, things are too broad and "no-go" theorem about HVT would be very hard to derive and unattractive. $\endgroup$ Aug 18, 2022 at 2:03
  • $\begingroup$ If we had an HVT that reproduced QT 99.99%, then it would be worth doing tests to see which comes closer to experimental results in the other 0.01%. $\endgroup$
    – J Thomas
    Aug 18, 2022 at 2:32

The easiest way to understand the issue is to think about an extreme case where Bell's Theorem is violated even more than it is in quantum mechanics.

You and I each have a particle that we can look at under a microscope. The microscope eyepiece is small, so you can only look through it using one eye at a time.

So imagine this scenario:

Every day, you and I independently make choices about whether to look through our left eye or our right eye.

On days when either one of us (or both of us) use our left eyes, we always see that our particles are opposite colors.

On days when both of us use our right eyes, we always see that our particles are the same color.

Note that unless something very non-local is going on, my particle cannot in any sense "know" whether you are using your left eye or your right eye. Therefore if I use my right eye, my particle cannot "know" whether it's supposed to be the same as yours or the opposite of yours. Nevertheless, it always conforms to the rules.

That is exactly the issue that arises in quantum mechanics --- the statistics we see in both theory and experiment are a bit less dramatic than the statistics in my imaginary experiment, but they cannot be explained with local hidden variables for exactly the same reason that my imaginary experiment can't be explained with local hidden variables.

Your story about Frank, Betty and Alice completely ignores the main issue, because in your story, the correlations between the measurements depend only on what's in the envelopes and not on decisions that different people made separately in separate places about how they were going to open those envelopes.

Your story, in other words, does not violate Bell's theorem. Quantum mechanics does violate Bell's theorem (as does my imaginary experiment), so your story cannot be relevant to understanding this aspect of quantum mechanics.

  • $\begingroup$ In my example, which horse race has its probability field collapse is determined entirely by which envelope Betty chooses to open. The probability for the one she doesn't open is still 0.5, but the probability for the one she does open is 1. If in many trials she only opened the envelope when one particular horse won, that would be very strange. She couldn't know ahead of time whether that horse won, so how could she know whether to open it or not? Are you sure that's what's happening with QM, or is it only that the probabilities shift because of what the observer knows? $\endgroup$
    – J Thomas
    Aug 18, 2022 at 14:03
  • $\begingroup$ @JThomas : Your example does not violate Bell's theorem, so it does not illustrate any of the issues you're trying to grapple with. Even if Alice opened her envelope only when one particular horse won, it would be odd but would still not violate Bell's theorem --- there would still be a joint probability distribution for all the relevant events. $\endgroup$
    – WillO
    Aug 18, 2022 at 15:00
  • $\begingroup$ If that isn't enough to violate Bell's theorem, I'm having trouble imagining anything that would. Can you point to an example of a meaningful story that would violate bell's theorem? $\endgroup$
    – J Thomas
    Aug 18, 2022 at 17:21
  • $\begingroup$ Yes. The story in the post you are commenting on violates bell's theorem. $\endgroup$
    – WillO
    Aug 18, 2022 at 17:23

There is a wonderful explanation of the Bell inequalities on-line somewhere and when I remember where I will edit this response, but the argument goes more or less as follows:

You are walking door-to-door asking survey questions about home appliance ownership ("do you own a vacuum cleaner?" "do you own a dishwasher?" etc.) But there are some questions you are not allowed to ask, like "do you own a stove?" for example.

But you can ask questions like, "do you own either x or y?" or "do you own both x and y?" or "do you own x and not y?" and in principle, you can put together a logic tree that allows you to deduce the answers to the questions you were not allowed to ask, and thereby construct a table which portrays the data like this:

% of households that contain a stove: 89%

% of households that contain a dishwasher: 14%

% of households that contain a vacuum cleaner: 75% and so on.

Then you do a consistency check where you do the probability sums of the form (% of households with stoves + % of households without stoves) each of which sums must add up to 100%, and then you discover to your amazement that despite the airtight logic behind the percentage table they do not, and you quit your job in disgust and puzzlement.

Of course, in the real ("classical") world of door-to-door surveys this would never happen but in the (un)real world of quantum measurement statistics, it does, and Bell figured out how to express that mathematically.

  • $\begingroup$ In an actual statistical study like this, your conclusions about the population that the samples were drawn from always have some imprecision and you can possibly get results with ranges that appear to have impossible overlaps. But the samples themselves have to add up. If the samples themselves add up to something impossible, then either some of the people lied, or some of the data has been lost. $\endgroup$
    – J Thomas
    Aug 19, 2022 at 4:34
  • $\begingroup$ @JThomas, or your system statistics were being altered by Bell without your knowledge, which was the point of the article in the first place. $\endgroup$ Aug 19, 2022 at 5:04
  • $\begingroup$ We can also always suppose that our system statistics are being altered by invisible fairies. $\endgroup$
    – J Thomas
    Aug 19, 2022 at 6:13
  • $\begingroup$ @JThomas, you are misunderstanding my point. $\endgroup$ Aug 19, 2022 at 6:38
  • $\begingroup$ I agree. I am missing your point. I'm starting to think that explanations in english just can't work, and I will have to do the math. Not that there's anything wrong with you or the other explainers, it just doesn't work. Maybe explaining Bell's theorem is kind of like coming up with a local realist theory for QM. If it's true, but it doesn't make sense, you're not going to find an explanation that makes sense $\endgroup$
    – J Thomas
    Aug 19, 2022 at 12:37

Bell's theorem is about hidden variable theories, in which the correlations between supposedly random outcomes when measuring widely-separated particles arise because the random decision was made before they separated, and merely hidden.

If Alice can do either two sorts of measurements with outcomes $a_0$ and $a_1$, which have definite $\pm 1$ values defined before they separate, and Bob can do either of two sorts of measurements with outcomes $b_0$ and $b_1$, likewise well defined $\pm 1$ values, then we can calculate the number $a_0b_0+a_0b_1+a_1b_0-a_1b_1$ which must be $\pm 2$ for any set of choices for $a_0$, $a_1$, $b_0$, $b_1$. So if we run lots of experiments and count the pairs we observe, the average result has to be between $-2$ and $+2$. Quantum mechanics offers an example where the number turns out to be $2\sqrt{2}$, and the bound is exceeded.

There is, in fact, another interpretation of QM that is both realist and local, as well as being fully deterministic. This is the Everett Interpretation - often called the Many Worlds Interpretation, although that's a slightly misleading name.

According to Everett, what happens is that whenever one quantum system observes another, their wavefunctions become correlated. They are a superposition of joint states, one with one outcome and an observer observing that outcome, another with the other outcome and an observer observing the other outcome. The two components don't interact, they can't see one another, it is as if they were in separate worlds.

So in the experiment, the particles are initially correlated with each other, they split and the observers separately become correlated with their own particle and hence each other, and then when the observers return home to compare notes, only the compatible combinations can see one another (are in the same 'world'). The correlations arise and can exceed Bell's bound because the other outcome is still hanging around in the bit of the universe you can't see.

It's deterministic: all the outcomes happen, every time. Nothing is random. It's local: no effects travel faster than light. It's realist: the universe objectively exists and has a definite state at all times. It's time-symmetric: there are no irreversible, information-destroying processes. It's complete: it just uses plain quantum mechanics; there is no need to posit the mysterious, non-linear "wavefunction collapse" process, the mechanism and trigger for which has always been rife with handwaving and wildly speculative theories. It applies at all scales: we don't have separation between quantum physics at atomic sizes and classical physics at macroscopic sizes. It's (relatively) sensible!

There are a few issues. There is a dispute over whether it truly explains the probability laws we see (not that any of the other do, either), and there are issues over how the preferred measurement basis is determined, and a few other technical issues that I think are answerable, but I admit require some handwaving. And of course, there is the mind-boggling concept of the outcomes of every decision in the history of the universe all still existing, but in a place we can never see, detect, or prove one way or the other. I personally think those are minor shortcomings compared to non-realism and non-locality, which would normally kill any other physical theory, but opinions differ. Some physicists like it for its elegance and simplicity, but it's not the most popular of the Interpretations.

  • $\begingroup$ "So if we run lots of experiments and count the pairs we observe, the average result has to be between −2 and +2. Quantum mechanics offers an example where the number turns out to be 22–√, and the bound is exceeded." No individual result can be bigger than 2. But the average result is 2*sqrt(2), which is bigger than 2. I don't understand. $\endgroup$
    – J Thomas
    Aug 18, 2022 at 13:55

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