# Bell's Theorem - a paradox in reality

In my previous post about Quantum Mechanics, I mentioned Bell’s Theorem in passing. However, this is one of the most fundamental aspects of Quantum Theory and understanding it was a fully mind-blowing moment for me, at least as mind blowing as the double-slit experiment. So I thought I’d come in here and give some more background and an explanation for what I said in the other thread, that the experiments validate Bell’s Theorem that “hidden local variables” can’t account for the statistics we see with entangled particles.

A bit of historical background: when Quantum Mechanics was first coming to the attention of theoretical physics, it had a lot of pushback, and for precisely that reason: that it seemed to imply some sort of non-local causality. Einstein himself objected to it, and played a central role in formulating the EPR Paradox paper in1935:

It wasn’t until 1964 (unfortunately years after Einstein’s death) that someone devised an experiment that could prove which side of that debate was right - and it would bbe years after that until those experiments would be actually performed. John Stewart Bell noticed that, using the correlations predicted by Quantum Mechanics for the spin of photons, you could devise a set of experiments which would be impossible to reproduce using classical assumptions. This was Bell’s Theorem.

First, about measuring spin with singlet-state entangled photons, and setting up our toy experiment

In very basic terms, when two photons (or electrons) are entangled in a singlet-spin-state way, then when you measure one photon’s spin as up, the other one is always down. However, note that you can measure spin at any angle, but you can only get a positive or negative result for that angle.

To help visualise this, what I’m going to do is describe to you a completely non-quantum classical experimental setup that should fully mirror the Quantum tests at a conceptual level.

Imagine you have a bag of millions balls. Each ball has two halves - red and blue - and you can pull the ball apart and see, on each inner side, an angle represented. And, just like with the electrons that have always opposite spin, the angle in these balls is always opposite as well - 180° from each other. An illustration:

And just like with the electrons, with these balls we’ll invent a measuring apparatus - a detector that can be rotated to any angle. Each detector can only give you a “positive” or “negative” result - “positive” if it’s in the coloured area of the detector, negative otherwise. When the two detectors are set at the same angle, the have opposite results 100% of the time. When one is set to 180° of the other one, they have the same results 100% of the time. However, other angles in between only have a certain percent chance to match, based on the angle. In this example, if you rotated the Red detector to 45° but kept the blue one at 0, you’d see a double-negative result:

(you may have to scroll to see the full image) In order to fully match the quantum experiment, when they’re running the experiment with the balls, the experimenters aren’t allowed to look at the angle, they can only use the measurement device to get a negative or positive. And we can only measure each one once.

Now, if you’re an experimenter with these balls, you wouldn’t be able to take that double-negative result and know which exact angle either ball is at, BUT we know that, because the balls are entangled, the measurement of the red ball tells us something about the blue ball, and vice versa – we know that IF we had instead measured the blue ball at 45deg, we would get a positive, since red gave negative. And we know that IF we had instead measured the red one at 0deg, we would have a positive there too, since the blue gave us a negative.

So now that we know how the balls work and how the detectors work, we can start…

Running the actual experiment

We’re going to run an experiment on our physical balls, but ALSO simultaneously we’re going to have some physicists run the same experiment on their photons, and we’ll compare results after.

The experiment works like this: First, I set up my blue detector at 0° and I set up my red detector at 22.5°, and I send as many balls as I can stand to wait for through to be measured, and I count how many of them come out positive on both red and blue. Then, I rotate my detectors: blue detector at 22.5°, red detector at 45°, and again I send as many as I can through and count how many come out positive on both. Then, I rotate my detectors one more time, this time blue is at 0° and red is at 45°, and once again I send as many as I can through and count how many come out as ‘positive’ on both.

And of course I ask my physicist friends to do the same with their photons - they’ve got a blue and a red spin detector to

What we find with our blue and red balls is:

Setup 1, 0° and 22.5°, out of 10,000 balls sent through, 625 have come back with both detectors positive (6.25%)
Setup 2, 22.5° and 45°, also has 625 out of 10,000 balls come back with both detectors positive (6.25%)
Setup 3, 0° and 45°, has 1250 out of 10,000 balls come back with both detectors positive (12.5%)

Nothing out of the ordinary there!

However, oddly, our physicist friends got a … different result

Setup 1, 0° and 22.5°, out of 10,000 photon-pairs sent through, 732 have come back with both detectors positive (7.32%)
Setup 2, 22.5° and 45°, also has 732 out of 10,000 photon-pairs come back with both detectors positive (7.32%)
Setup 3, 0° and 45°, has 2500 out of 10,000 photon-pairscome back with both detectors positive (25%)

The paradox is in there, but we’ll have to do some work to get it out

We need to look a bit more closely at the results the physicists got, because there’s a paradox in those results that isn’t entirely obvious at first glance. This paradox is at the heart of Quantum Mechanics.

Consider setup 1: of all the photons that went to the red detector, 732 were positive at 22.5°, but their sister-photon was positive at 0°, which means the red ones would have been negative if they were measured at 0°
setup 2 as well: 732 photons sent to the red detector that were positive at 45° but would have been negative if they were measured at 22.5°
setup 3: 2500 photons to red, all positive at 45°, all would have been negative if measured at 0°

These 2500 photons have a problem: we didn’t measure them at 22.5°, but we know that IF WE HAD, they would have been either positive or negative if we did.
We know they’re positive at 45°, we know they’re negative at 0°.
Imagine if we split this 2500 into two groups: those that would have been positive at 22.5°, or those that would have been negative at 22.5°.
We can deduce that the half that would have been positive at 22.5° would have passed in setup 1.
We can deduce that the half that would have been negative at 22.5° would have passed in setup 2.
Which means EVERY photon that passed in setup 3 should theoretically either pass in setup 1 or setup 2.
Which means the sum of the passes in setup 1 and setup 2 should be AT LEAST equal to the passes in setup 3.

But they’re not. They’re not even close. 732 + 732 is 1464, nowhere near the 2500.

But this is the result that QM predicts, and this is the results that experiments prove.

So those are the numbers, and they don’t add up, but what does that mean?

The reason why this is important is, no matter what way you try to arrange my classical balls, there’s no way to get an experimental result like this. There’s no way for the balls to have a fixed angle of spin as they separate from each other, that produces these numbers. No matter how you fix the angle of spin in the classical balls, the number of balls that pass setup 1 and setup 2 will always add up to setup 3 perfectly.

And reality seems to match what Quantum Mechanics predicts in this case, not our classical ideas of fixed properties that get measured later. That’s why “Bell’s theorem proves that quantum physics is incompatible with local hidden-variable theories.” The spin of these entangled photon cannot be fixed at the time of entanglement, and also produce the measured results we get. The numbers don’t add up for that to be the case.

But spooky action at a distance, randomness, etc. go away under either Many Worlds or Superdeterminism.

Many Worlds still has the phenomenon that spooky action at a distance is talking about, it just has a novel way of describing how that can happen without getting rid of locality.

Superdeterminism gets rid of spooky action at a distance entirely, and in it’s place posits that the entire universe has conspired to trick you into thinking that spooky action at a distance happens.

Well, if you keep locality you get rid of spooky action at a distance by definition, because spooky action at is a distance just is nonlocality.

As to superdeterminism, it strikes me as nuts, but I’m not a physicist.

Sure, it’s just not ‘spooky’ anymore (well, maybe it’s still a bit spooky). But you still have measurements of indeterminate properties separated in space that are perfectly correlated, which is exactly what Einstein was calling ‘spooky action at a distance’. It’s just that neither he nor anybody else had conceived of Many Worlds at that point, or many other interpretations that were later to come.

Here’s a good, relatively short paper on spooky action at a distance and MWI: Quantum Nonlocality Does Not Exist

Note that in the linked paper Tipler writes:

Bold by me.

And that’s just it for me, though again I’m no physicist. Postulating collapse is the root of all quantum mysteries, and yet collapse seems entirely at variance with any kind of science, a piece of magic or a kind of deus ex machina. Get rid of collapse and you get rid of a whole bunch of problems. Accepting Many Worlds is straightforward from this but that doesn’t trouble me at all. Maybe I’m just weird like that.

It’s straight forward because you live in a world where someone invented the idea for you XD. It wasn’t even thought of, as far as I’m aware, for many decades after QM came out.

“Collapse” naturally is what you need to happen to the wave function in a single-world universe. You’re right that it’s a bit of magic that they have to add on top of the equations, but when the thought doesn’t even occur to you that the other possibilities in the wave function are real because you’ve taken single-world universe for granted because all you’ve ever seen is a single world, then you can see why they came up with collapse, why it was the default for so long, and even why it’s still possibly more popular than many worlds among actual physicists (though probably not by very much).

I seem to recall from my readings of the history of QM that even the founders had qualms about wave function collapse for the reasons listed above, but found no alternative to it.

Here is Hugh Everett’s 1957 paper on the relative state.

Einstein died in 1955, two years before this came out. I often wonder what he would have made of MWI. He certainly didn’t like the standard QM he helped invent/discover.

This is one of the great jokes of science history in my opinion. Einstein was absolutely a next level thinker. Relativity was some deep meta level thinking, and he was so cockily sure of it, and he was goddamn right.

He contributed to the early development of QM but could never get on board with the rest of it. And as far as our experiments have shown since then, he was wrong.

Well, he wasn’t really wrong, though, if Many Worlds is right. MWI gives him what he sought — locality, realism, and determinism. There are just all these … many worlds … that come associated with it. That’s why I oft wonder what he would have thought of it.

Superdeterminism presaerves locality, realism and determinism too, but at a price much steeper it seems to me than MWI. Sabine Hossenfelder, a big SD supporter, insists it isn’t quantum mechanics at all, not an interpretation of QM but a deeper theory from which QM is derived.

I think general relativity was some deep meta-level thinking, not so much special relativity. SR had a lot of underpinnings before Einstein took the next step. Not so much GR, which took a big conceptual breakthrough.

I’d say probably that Newtons’ posit of the concept of mass and bodies falling toward one another were bigger conceptual breakthroughs, because in thinking this stuff up he really had no shoulders to stand on, unlike Einstein. It is interesting though that Newton worried about gravitational attraction — how to explain a force propagating across space. It took Einstein, in GR, to figure out that there was no force as such, but the curvature of spacetime. I’m sure Newton would have been wowed!

I was recently reading that Newton, in a vague way, anticipated quantum mechanics. He wondered why windows, though transparent, partly also reflect light — we can see our reflections, vaguely, in windows. He couldn’t come up with an explanation. QM does explain this.

When you propose a fictional analogy to an actual process but your analogy suggests a different result than the actual process produces - doesn’t that tell you right away that something is wrong with your analogy?

In your red/blue ball analogy - you suggest that 6.25% of the 10,000 tests should come back positive. But then you say the real experiment produces 7.32%. So what was wrong with your analogy?

Discover that and you will see that by doubling the passage gap - you should get about 4 times the amount of passage - and they do.

That’s actually the entire point of the analogy. That’s the entire points of bell’s theorem. Bell’s theorem says that quantum mechanics gives a different prediction than any theory of local hidden variables, and that local hidden variables cannot account for the actual results we see.

So yes, something is wrong with the analogy. The analogy is meant to illustrate local hidden variables, and the thing that’s wrong with it is that local hidden variables cannot account for the statistics we see.

Produce for me a single set of balls that, when I run them through each test, give me the expected percentages. It can be as small or big of a sample as you like, and it doesn’t have to perfectly match the statistics, just approximately.

For full clarity, I’m looking for these statistics:

Setup 1, 0° and 22.5°, out of 10,000 photon-pairs sent through, 732 have come back with both detectors positive (7.32%)
Setup 2, 22.5° and 45°, also has 732 out of 10,000 photon-pairs come back with both detectors positive (7.32%)
Setup 3, 0° and 45°, has 2500 out of 10,000 photon-pairscome back with both detectors positive (25%)

Just so we’re clear, I of course don’t mean literal balls. You can send me over a csv, where each row represents a blue/red ball pair, and the value in each cell is an angle.

blue,red
20,200
0,180
90,270
95,275

You’ll of course notice in the above list that they’re always at 180degrees from each other. Please make sure your dataset obeys that relationship.

Ever heard of a “strawman”?

When you propose that “the truth is –this– therefore –that–” – and “that” is found to be not true - then your premise must not be true. And that is irrelevant to whether it has anything to do with QM.

Your initial argument was “if my strawman proves to not reflect reality - then reality is broken.”

And now your response is “well think of a better analogy/strawman”. I might or might not be able to think of a better analogy - that isn’t the point. It is a simple science tenant that if your theory does not reflect the reality of the experiment - then your theory is incorrect (although the contra-proposal is not true).

Your argument depends on Bell’s hypothesis being correct to begin with - and clearly it isn’t.

The reason it isn’t seems too easy to discern - and that tells me that this is just something peasants are just supposed to believe (religions do that - even the newer ones) - because I don’t believe the actual highly intelligent physicists from 100 years ago couldn’t figure this one out.

And it seems a similar argument to another proposed theory involving aether - “If there is aether - it must work this way. But since it doesn’t work that way - it is proof that aether doesn’t exist”.

That argument came from the same people who appear to be so arrogant as to assume they know too much about their strawman premise and then draw a conclusion that is nothing but a diversion from finding the truth. - Or more likely they just don’t want the world of peasants to discover the truth.

The error in Bell’s hypothesis involves the proposed perfection of the filters and detectors. Imagine your same analogy but with the straight edges of the filtering more blurred such that out of 10,000 tests on a single filter - 6,000 photons passed instead of the proposed 5,000. Then rethink what to expect when using two skewed filters and a detector. The math isn’t as easy - but the expectation is more accurate.

Um … no.

There is no “strawman” here. Go look up the meaning.

The experiment was a test of the hypothesis of local hidden variables in QM. If there were local hidden variables, you’d get result x. If local hidden variables are ruled out, you get result y. The test yielded result y — no local hidden variables. It’s that simple.

In your analogy - I have a little trouble discerning whether you mean to take the intersection of the red and blue (to the right) or the neither-set (to the left) as being a positive detection. For one you could use a 55% passage (is better than 60%) but if you mean the other - take a 45% passage.

I am just guessing at those percentages. I know that no real filter/detector is going to be at exactly 50% and I can see that with more realistic specifications in the analogy - you would get very close to what the real experiment demonstrates.

Can you quote the part of my post you’re asking about here? Happy to clarify anything.