This will be in words that most people modestly math-literate can understand, is my goal. It’s a bit long, but bear with me, I think it’s worth it for anybody with any interest in QM.

In 1935, Einstein and friends published a paper called “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” in which they laid out the EPR Paradox. Long story short, the paper made the argument that, given the results of the quantum experiments on entangled particles done so far, there are only two possibilities:

(1) ‘Spooky action at a distance’ – faster-than-light influence between two particles. If this is true, this means that if I do something to a photon over here which is entangled with a photon over near you, what I do to my photon can affect your photon in some way, and that affect, in violation of Relativity, can propagate through space at faster-than-light speeds, which would imply that physics is non-local (something all physicists shudder at the thought of).

(2) There are some variables, as yet undiscovered and unaccounted for in the Quantum equations, present within each entangled particle that determined the results of the experiments. This was the possibility that Einstein and friends favored.

Now, I don’t personally believe (and I know that most people with a phd in physics also don’t believe) that this dichotomy is necessarily a good/true one – that assuming one is false doesn’t imply the other is true. There are some assumptions ingrained into the logic producing this dichotomy that were, at the time, unexamined assumptions, assumptions taken to be so self-evident they weren’t even discussed for decades. That has to be said before we go on, because one of those possibilities is going to be disproven, and I don’t want you going around saying ‘Quantum Mechanics proved the other one is true!’ because it didn’t. 9 times out of 10 when you hear someone saying Quantum Mechanics proved something on the internet, they’re wrong.

So, Bell saw the paper, and the clever guy came up with a way of disproving 2. In other words, disproving the idea that there is a preexisting fact about the photons, prior to measurement, that would determine the result of the measurement. This is what I hope to explain to you – how he disproved that.

First, a bit about entanglement and the base experiment: if I entangle two photons in a particular way, and send one towards, let’s say, Paris and the other towards London, here’s what happens: If I measure the one in Paris with a polarized filter set at 0°, then that photon can either pass through the filter or be absorbed by the filter (each one happens 50% of the time), and whichever happens to the one in Paris, the opposite will ALWAYS happen to the one in London if measured also by a 0° polarized filter. So, every time it passes through that filter in Paris, the other is absorbed in London, and every time it’s absorbed in Paris, the other passes through in London. This experiment was already done, it was already known, it’s accepted, you can take it as a fact: if you entangle two photons in that way, and measure them by a polarized filter at the same angle, they will ALWAYS have the opposite result. Fact.

But what happens when you measure them at different angles? Quantum Mechanics predicts that if the photon passed through (I’ll use the term ‘transmitted’ from here on out) - if the photon was transmitted by a filter in Paris, the probability that the entangled photon in London will be transmitted by a filter that’s at angle X° to the filter in Paris is (sin(X°))^2. So, just as an example, if the one in Paris is measured at 0° and is transmitted, and the one in London is measured at 30°, to calculate the probability that the one in London is transmitted you would first take the sin of 30°, which is .5, and square it, which is .25. 1/4 of the time that the photon in Paris is transmitted by a 0° filter, the photon in London is also transmitted by a 30° filter. Recall earlier that I also said that it’s transmitted by the first filter 50% of the time and absorbed 50% of the time – so, taking the two results in conjunction, if I have a filter at 0° in Paris and 30° in London, 1/2 of the time it will be transmitted by the filter in Paris, and 1/4 of the times that it’s transmitted in Paris it’s also transmitted in London, that means that 1/8 of the time it’s transmitted in both Paris and London. In the example I used Paris at 0° and London at 30°, but the same math would apply in exactly the same way for Paris at 2° and London at 32° – the X in the (sin(X°))^2 is the difference between the two angles – London° - Paris°.

If you are following up to now, you should know how to calculate the probability that the photon in London will be transmitted at a given angle given that the photon in Paris was transmitted at a different given angle.

Now, here’s where we get into talking about Bell’s Theorem. This is the real juicy stuff. What we do now is we look at the experimental results, and we Assume that Einstein was right, that (2) was right, that whether or not a photon passes through a given filter is determined by a preexisting variable within the photon itself. Using this assumption, the results…well, you’ll see.

First, I set up a filter in Paris at 0° and one in London at 30°, and we send entangled photons through one at a time as above. Let’s say we run this experiment 1,000,000 times – we send 1,000,000 pairs of entangled photons through the filters. 1/2 of the time, the one in Paris is transmitted, and of those times that it’s transmitted, 1/4 of the time the one in London is also transmitted, so 1/8 of the time during the entire experiment, each photon is transmitted. Just as above – exact same experiment I already described, no new math here.

At this point, I’d like to add a touch of clarity that I haven’t seen before in another explanation – I’ll explicitly talk about each photon having a boolean variable for whether it passes through each filter – we’ll create a separate variable for each filter. A is the first variable - A is true if and only if the photon did (or would have) been transmitted by a filter at 0°. B likewise for 30°, and C for 60°.

So, again, let’s look at the above experiment that we just did. We only need, for the sake of Bell’s theorem to make sense, to look at the instances of the photons going through both filters - and furthermore, we only need to look at the London photons in particular.

Now, looking at the 1/8 of the times that both photons were transmitted, what can we say about the London photons? They were transmitted by a 30° filter, so we can say that B is true. Futhermore, they were entangled with a photon in Paris that was transmitted by a 0° filter, and you’ll recall from above that that means that if the London photon was measured by a 0° filter, it would have been absorbed – every time they’re measured at the same angle, they have opposite result – so that means A is false.

Using Einsteins assumptions and the results of this experiment, we can say that for 1/8 of photons, A is false, B is true, and (C can either be true or false).

Now we do our next experiment: in Paris, we now set the filter at 30°, and in London at 60°. Again, 1,000,000 trials. 1/2 of the photons are transmitted in Paris, and of the times that the one in Paris is transmitted, 1/4 of the time the one in London is also transmitted. So 1/8 of the time, they’re both transmitted.

Using the exact same logic as before, we can say that for 1/8 of photons, (A can be true or false), B is false and C is true.

Now our final experiment: in Paris, we set the filter at 0°, and in London at 60°. 1,000,000 trials. 1/2 of the photons are transmitted in Paris, and of the times it’s transmitted in Paris, 3/4 of the time it’s also transmitted in London. So, 3/8 of the time it’s transmitted in both Paris and London.

Again, exact same logic as before, for 3/8 of photons, A is false, (B can be true or false), and C is true.

Now, I’ll lay out the important details again for absolute clarity:

Using EPR assumptions, we can interpret the results of these experiments as so:

[1] for 1/8 of photons, A is false, B is true, and (C can either be true or false).

[2] for 1/8 of photons, (A can be true or false), B is false and C is true.

[3] for 3/8 of photons, A is false, (B can be true or false), and C is true.

Now, you have to look really closely to see the problem. It’s not immediately obvious to most people what the problem with this is – it sure as hell wasn’t immediately obvious to me.

In each of the experiments, one variable is unmeasured. In the last experiment, B was unmeasured. B corresponds, again, to the question ‘Will this photon pass through a filter at 30°?’ Einstein’s assumption is that, for any given photon, this variable has a preexisting real value – we didn’t measure it, but that obviously doesn’t mean it doesn’t exist. So, for some of those photons in group [3], B is true, and for some of them, B is false. Right? And it has to have some value – mutually exclusive, mutually exhaustive. Law of non-contradiction and law of excluded middle.

So we can assume using Einstein’s assumptions that, of the ones in group [3], some have B as true and some have B as false. Let’s look at the ones that are true: the ones for which B is true, we already know that A is false and C is true. Interestingly, that makes them photons that would fit into group [1], right? For a photon to be in group [1], A has to be false and B has to be true, and C can be anything. We know that the A value is false, and we’re talking about those photons for which the B value is true. So, for all photons in group [3] who’s B value is True, they are also group [1] photons.

The remaining group [3] photons, the ones for which B is false, on the other hand, fit into group [2] for the same reasons. To fit into group [2], C has to be true and B has to be false – A can be anything. So, for all photons in group [3] who’s B value is False, they are also group [2] photons.

3/8 of photons are in group [3]. Some of those photons are also members of group [1], and the remaining are also members of group [2]. So, for 3/8 of photons, they all have to fit in either group [1] or group [2].

But group [1] only comprises 1/8 of photons, and group [2] only comprises 1/8 of photons as well. That means that, of all photons, only 2/8 can fit into either group [1] or group [2]. So how can what I just said be possible? How can we say of the 3/8 of photons that are in group [3], that they’re also either part of group [1] or group [2] depending on their B variable? It’s probabilistically impossible. But it’s also necessary given Einstein’s assumptions.

Einstein’s assumptions are wrong. QED.