Rings a Bell

A couple weeks ago, it was announced that this year's Nobel Prize in Physics was going to a group of scientists who experimentally tested Bell's Theorem. As with many concepts in quantum mechanics, this can be a bit tricky to understand, so I wanted to build it up piece by piece (as much for myself as for you).

When Quantum Mechanics was first developed, many people (including Einstein) were disturbed by the implication that, not only did interactions have random results, but that entangled particles could communicate those results instantaneously, breaking the speed of light. One explanation that was proposed to avoid this problem was the "Hidden Variable Hypothesis", which claimed the final states of the entangled particles were actually determined by some unknown property present at their joint creation, and therefore no information needed to be exchanged. John Bell came up with a way to test for the presence of a hidden variable, which I'll outline below.

First, suppose we have some collection of measurements. Each measurement has three properties: A, B, and C, which can either be +1 or -1. If each property is assigned randomly, we can think about the probability of two properties being the same, e.g. P(A = B). Now we can write

If you're not sure about this, you can try a couple sets of values, but the key is that we only have 2 choices, +1 and -1, for 3 properties. Now these properties are exactly the type of hidden variables that we're suggesting may exist. If we can come up with an experiment that can measure yes/no for three different properties, then we can simply count the outcomes and check if this inequality holds.

Looking again at that old post I linked to, we can imagine the following experiment: We produce entangled particles with opposite spin, and send them in opposite directions. Each goes into a Stern-Gerlach box and gets measured as spin-up or spin-down on the box's axis. However, we vary the angle of each box between 3 possibilities: 0°, 120°, and 240°. Our three properties are "spin-up along n-degree axis", and negation represents spin-down.

With this setup, we can think of the hidden variables as a set of rules for how the particles respond to each of the three angles. When a pair is produced, each is assigned one of these rule sets, e.g.

Since the particles have opposite spin, comparing the two detectors means the equalities in the equation above become not-equal. Now we can look at all the possible combinations of A, B, C between the two detectors, as well as each set of rules the pairs could be assigned, and find the probability that the two measurements are opposite:

In the inequality above, each term contributes 1/3, which gives us a total of 1 and satisfies the relation.

Quantum mechanics, though, predicts a different result. When one of the entangled particles is measured to be spin-up along a particular axis, we immediately know the other one is spin-down along that same axis. Knowing the second particle's orientation, we can find the probabilities of measuring opposite spin along each of the possible axes:

Adding up the terms again, according to Quantum Mechanics we only get 3/4, violating Bell's Inequality! That means we have an experimental method to test for hidden variables. Unfortunately, dealing with entangled particles is a delicate process, and the experiment needs to be repeated several times to accurately measure the statistical distributions, which is why it has taken more than 50 years to confirm this result. A well-deserved Nobel Prize for these scientists!