Quantum mechanics is a weird thing. It says that we can never really know all there is about reality. If we measure a particle, we can’t know its exact momentum and position at the same time. If we know one, there has to be an uncertainty about another. There is a certain “fuzziness” to reality – an uncertainty that is inherent in the Universe.

But it’s interesting to ask – do particles *actually* have a well-defined position and momentum? Is it just us who just lack the ability to observe both of these at once? Or is this fuzziness actually inherent in nature?

In order to figure out if particles actually have well-defined properties and it is us who just can’t measure them simultaneously, a nifty theorem was developed by John Bell in 1964. This experiment involves observing the properties of entangled particles. It can show us what type of Universe we live in – one where reality is fuzzy itself, or one where our understanding of reality is fuzzy.

In order to understand this theorem, let’s set up our own simplified experiment. To do so, we will need quantum entangled coins and a friend on the other side of the Universe. Where you get your entangled coins is up to you. Let’s get started.

Let’s say that you and your friend have two sets of entangled coins – a penny, a dime, and a quarter. Whenever you throw your penny and it lands as heads, when your friend also throws her penny, it will land as heads, regardless of where in the Universe she is.

If our particles, coins in this case, actually have an inherent property within them, a “headness” or “tailness” so to speak, we would only just think that these coins could be heads or tails until we flip them. The idea that these coins have an inherent “headness” or “tailness” is called the hidden variable theory. According to the hidden variable theory, these entangled coins decided whether or not they would be heads or tails when they were in contact with one anther. This “headness” or “tailness” remains hidden from us until we choose to measure it.

On the other hand, perhaps the hidden variable theory is wrong. In this case, the coins are both heads and tails until we measure them. At the point of measurement, the coins choose which side they will pick. This is the prediction of quantum mechanics. But remember – our coins are entangled and always must give the same result – no matter where they are and even if they are thrown simultaneously! If these particles communicate their headness or tailness, that means their message would travel faster than the speed of light!

This makes the lovers of relativity very unhappy – nothing, not even information, should be able to travel faster than the speed of light. So which is right – relativity or the communicating particles of quantum mechanics?

Our coin experiment will be able to help. You and your friend both pick a coin at random and throw it, and record whether or not you get the same result. Sometimes, you’ll both choose the same coin and both get the same result. But what if you choose different coins? What is the probability that, if you are both throwing coins at random, you get the same result?

Let’s say the penny is heads, the dime is tails, and the quarter is tails. We could write this as HTT. There are only a few possibilities. They all are:

HHH

HHT

HTH

THH

TTH

THT

HTT

TTT

According to quantum theory, the probability that you and your friend both throw an entangled coin at random and get the same result is 1/2. (In order to get this result, you’ll need to know some basic geometry and know about particle spins. I won’t go into it here, but if you are willing to step away from our coin example, here is a good explanation of where this number comes from.)

If the hidden variables theory is correct, these coins decide on their headness or tailness without your knowledge before they are sent across the Universe from one another. Now we can compute the likelihood, according to the hidden variable theory, that both you and your friend get the same result if you choose a coin at random and throw it.

Let’s take the situation where TTT or HHH. In this case, no matter what coin you and your friend pick, you will get the same result. Therefore, the probability is 1.

For the other cases, the math is the same. We will look at one of these: the situation where the penny is tails, the dime is heads, and the quarter is tails (THT). You both pick a coin at random and throw it and measure how many times you get each result. The various scenarios are:

You throw the penny and your friend throws the penny – same

You throw the penny and your friend throws the dime – different

You throw the penny and your friend throws the quarter – same

You throw the dime and your friend throws the penny – different

You throw the dime and your friend throws the dime – same

You throw the dime and your friend throws the quarter – different

You throw the quarter and your friend throws the penny – same

You throw the quarter and your friend throws the dime – different

You throw the quarter and your friend throws the quarter – same

Therefore, you get the same result 5/9 times, or more than half the time. Since there are two cases where you always get the same side (HHH or TTT), this means that more than half of the time, you will get the same side as your friend.

This is the prediction of the hidden variables theory. Fundamentally, this does not agree with quantum mechanics, which predicts 1/2! (What we’ve shown here is in regards to local hidden variables, or those that cannot travel faster than the speed of light.)

So which one is right? We actually have been able to do this experiment in the real world several times (albeit, without entangled coins) in several different ways, even using light from distant quasars and with video games. And the verdict? The results agree with quantum mechanics – particles decide their properties only when we observe them, and somehow communicate at vast distances.

That’s our reality, and it’s pretty weird.

Now don’t throw relativity out the window quite yet. This information that is transmitted between the two coins is ‘random information’. In all likelihood, we still can’t send any useful information faster than the speed of light. So no, we can’t ring up Alpha Centauri and expect an answer tomorrow. Sorry.

Bell’s theorem is a great topic of conversation in metaphysics to talk about over a drink with your physics and philosophy-inclined friends. As a tribute to Bell’s Theorem, the British Journal for the Philosophy of Science recently released a virtual journal in tribute to Bell’s Theorem and all the weird and wonderful predictions it makes. It contains enough reading about Bell’s Theorem to keep you busy for a while, and give you plenty of philosophy to haunt your dreams.

Until next time, stay fuzzy, my friends. Make sure to check on links that appear on the post, They are pretty informative.