David Wyde

Bell Tests and Coin Flips



I previously wrote a paper on an interpretation of quantum mechanics. It makes the same predictions as existing explanations while sounding a lot like classical physics.

My goal with this post is to give a less formal explanation of that theory, with a few new ideas.


I’m going to start with a thought experiment. Its setting is classical physics, but the ideas should help illustrate some important aspects of quantum mechanics.

I’ll then go into more detail on photon polarization, and how it may be possible to explain certain parts of quantum mechanics in a straightforward fashion.

Coin Flips

I’m going to design a machine for flipping coins, with a few unusual features.

First, I need two coin flippers. Each flipper can be thought of as a mechanical thumb. I can place a coin on a platform, then push a button, and the machine will flip the coin. The coin may land on either side; I don’t know which, but, if I knew all of the variables down to a microscopic level, it seems like I might be able to use the laws of physics to predict which way it will land.

I’ll have a way of controlling the angle of the platform on which a coin is placed. Some people think it’s necessary to be able to set the angle randomly, rather than with a manual dial, or else I might have a loophole in the integrity of the experiment. That concern is hopefully going to seem unnecessary when I’m done, but people spend a lot of time worrying about it.

The two coin flippers should be made as similar to each other as possible. In theory, the same coin should have the same result if it were able to be exactly copied and then flipped separately by both flippers.

I also need to make special coins. I have a coin processor at the start of each trial. It takes a standard US quarter as input, then melts it down and creates two new coins. Each new coin has one side marked heads and one side marked tails. The new coins will be flipped, one at each flipper. A pair of coins has the property that, if the angles of the flippers are the same, either both coins will land on heads or both coins will land on tails.

I can try to model what will happen if I set the two flippers to different angles before flipping each pair of coins. It turns out that, if I call the difference between the angles of the flippers θ, the chance that the coins agree should vary with cos2(θ). You might imagine that there is some function that determines the correlation, and in this particular setup, that’s what it is. It means that even slight differences in the angles of the flippers can lead to a result with one heads and one tails.

To run the experiment, I first make the coin flippers very far apart from each other. Then, I create a series of pairs of coins. For each pair, I send one coin to one flipper and the other coin to the other flipper. I try to disturb the coins as little as possible in transit, to avoid them getting knocked out of sync in an effect called decoherence. Then, I flip them.

For each pair of coins, I vary the angles of the flippers as unpredictably as I can. After a number of trials, I compare the rate of heads-heads and tails-tails pairs against the predicted value.

Bell Tests

There is a very similar experiment in quantum mechanics called a Bell test. In one version, it uses photons instead of coins and polarizers instead of flippers.

Bell tests are usually interpreted to rule out certain kinds of local theories. That means people think, if the flippers are far enough apart, something must happen faster than the speed of light in order for them to agree at a rate of cos2(θ). That would be in conflict with relativity, and is generally regarded as a surprising and deep result.

Given the way they were created as a pair, it seems possible that the coins have some sort of shared state. One might then wonder by what physical mechanism the coins always agree on the same result when they are flipped at the same angle, even when they are far away from each other.

An alternative view is superdeterminism, which says that the angle of one flipper and the result of heads or tails at the other flipper are caused by a common event in their past. Many people seem to think that this means there is no free will, and reject the idea based on that, rather than on whether they think that kind of correlation makes any sense on its own. A related view to superdeterminism is retrocausality, the idea is that the future causes the past.

It seems that these coin flips are quite tricky to explain.

A Simpler Explanation

A simpler explanation is that the states of the two coins in a pair are synchronized, rather than shared. Each coin is completely separate from the other, but they will still agree if sent to identical flippers.

In that case, the main thing left to explain is why the correlation is cos2(θ) when the angle varies. This is where the analogy between the classical and quantum versions starts to break down: for coins, it isn’t clear why it would be that value; however, for Bell tests, that is what is observed.

An Interpretation of Quantum Mechanics

The paper I linked to at the beginning of this post goes into more detail on an interpretation of quantum mechanics that I’m going to describe now.

The idea is to start with Bohmian mechanics, then try to remove its nonlocality.

Bohmian mechanics explains the double-slit experiment by each particle having an associated wave. The wave goes through both slits, while the particle goes through only one slit. The particle is drawn where the interference of the wave is constructive, and not to where it is destructive. That is the cause of the interference pattern in the double-slit experiment.

I’ll use the explanation for the wave-particle duality and the double-slit experiment from Bohmian mechanics. Entanglement involves nonlocality in normal Bohmian mechanics, but I will instead use the coin experiment’s synchronized states explanation for entanglement and thus Bell tests. The resulting theory is local and deterministic.

A common criticism of Bohmian mechanics is that it requires extra math. I think that’s a small price to pay for being able to say the precise position of a particle, rather than just a probability distribution of where the particle is.

Imagine I asked you to guess a number chosen uniformly at random from 1 to 100, then told you its square. You could say it has a 50% chance of it being even, or you could do some extra math and tell me the exact number.

Malus’s Law

There are several ways in which cos2(θ) comes up that are related to the topics in this article.

One example is Malus’s law. It says that if you put one linear polarizer after another, with an angle of θ between them, the intensity of light that passes through both is proportional to cos2(θ).

It can normally be assumed that the light involved when applying Malus’s law isn’t polarized in any particular way. That is, it’s a collection of photons with varying polarizations, distributed roughly uniformly at random.

Photons are quanta, so a photon must either completely pass through or completely be blocked at a polarizing filter. The intensity of the light will be linearly related to the number of photons that pass through. So, one would expect each photon to have a chance of cos2(θ) of passing through both filters.

In this article’s interpretation, there is no randomness. Each photon has some internal state that causes it to pass through a polarizing filter with a probability that depends on cos2(θ). Given that internal state and the angle of polarization, whether the photon passes through is deterministic: the probability results from a lack of knowledge about the state and polarization.

There’s an experiment in which one polarizing filter is at 0° and another is at 90° behind it. In that case, almost no light gets through. Then, a third filter is inserted between them at 45°, and some light gets through. All of the photons entering the second filter should be polarized the same way; that some but not all of them get through can be explained by varying internal state.


It may be possible to think of a photon as like a particle or a wave, given the assumption that it has an associated wave as in Bohmian mechanics. One advantage of thinking of a photon as like a wave is that it provides another possible explanation for the cos2(θ) value.

The energy in a wave is proportional to the amplitude squared. A linear filter blocks the part of a wave that is moving in one dimension, while allowing the rest to pass through. The proportion of the energy of a wave moving in the direction that passes through the filter is cos2(θ), since the amplitude depends on cos(θ). Part of the wave passing through may be less relevant to polarization because photons are quanta, but there is some conceptual overlap.

It may also be worth noting that the Born rule, in simple cases, says that the probability of making a measurement depends on the amplitude of the wave function, squared. That seems like it could be related to the above points if the wave function is seen as physically real, and possibly to the explanation for the double-slit experiment in Bohmian mechanics.


It seems that some results in quantum mechanics, such as Bell tests and double-slit experiment, can be explained in an intuitive way.

It may be possible to have an interpretation of quantum mechanics that preserves locality, particles having a defined state at all times, one result per measurement, and other ideas from classical physics.