 In 1964, an Irish physicist, John Bell, published a mathematical paper proposing a way to test for hidden variables. His work is called Bell's Theorem, or Bell's Inequalities. It was based on entangled electrons and Stern-Gerlich apparatus spin detectors. But we'll use the more easily managed particles, photons, and polarized lenses. Bell's idea was to assume Einstein's hidden variables hypothesis is true and then show how it leads to a contradiction. This would prove that the hidden variables hypothesis is false. The best way to understand Bell's theorem is to use Venn diagrams from basic set theory. Here's a simple Venn diagram example. Consider the set of all people in a town, say, Paris, Illinois, who go out on a particular rainy day wearing a hat. Some of these people are also wearing gloves. This would be a subset of the whole. Now we count the number of people with hats, and we count the number of people with hats and gloves. If the number of people with hats and gloves is greater than the number of people with hats, you have a contradiction, a violation of the basic assumption. The assumption that they are counting people in the same town on the same day must be false. For example, this violation could happen if the count for hats was indeed taken in Paris, Illinois, but the count for hats and gloves was taken in Paris, France. Bell's thought experiment involves sending photons through polarized filters. If a photon passes through a filter, it is referred to as past. If it's blocked, it is referred to as failed. The probability that a photon will pass or fail depends entirely on the angle between its polarization state and the filter's polarized state. Here we have three tests, A, B, and C. Test A sends vertically polarized photons into a vertically polarized filter. Test B sends vertically polarized photons into a filter polarized at an angle theta. And Test C sends vertically polarized photons into a filter polarized at an angle 2 theta. Now the object of the exercise is to examine the role of Einstein's entangled particle hidden variables hypothesis. So we'll use quantum entangled photons along with the assumption that interacting with one of them does not change the state of the other. So all tests start out with vertically polarized entangled photons. The thought experiment used tests in three particular combinations. One was to run a photon through test A followed by running its entangled photon through test B. The second was to run a photon through test B followed by running its entangled photon through test C. And the third was to run a photon through test A followed by running its entangled photon through test C. What Bell was looking for are the number passing test A followed by failing test B called A not B. The number passing test B followed by failing test C called B not C. And the number passing test A followed by failing test C called A not C. Now consider the three sets. Set A of all the tests that passed test A. Set B of all the tests that passed test B. And set C of all the tests that passed test C. Notice where they overlap and where they don't. Here's the subset A not B and B not C. When we combine them you can see that A not C is a subset. From set theory we know that the number in A not B plus the number in B not C must be greater or equal to the number in A not C. This is the famous Bell inequality. Remember that our assumption is that the states of the entangled particles depend only on their original hidden variables and cannot change just because there was a measurement taken on the other particle. Being a thought experiment we cannot actually run the tests and count the results, but we can use the quantum state probabilities to compute the results for these three numbers. For an angle of 45 degrees we get 0.75 is greater than or equal to 1. Clearly not true. This is called a Bell violation. It tells us that the assumption that states are determined by hidden variables must be false. The problem is that complex thought experiments like this are filled with assumptions and loopholes and in the 1960s there was no known way to build an entangled photon generator. If we could create and manage such photons in large enough numbers we could flood volumes and see entanglement behavior directly. As of now this is not possible, but today we can produce entangled photons at will and see the states of entangled particles change.