Now, if our local hidden-variables theory is to be a success, it must predict the same matching probability as quantum mechanics. So, what does quantum mechanics say when spins that are emitted in a singlet state are detected by spin-filters mutually inclined at 120º? Well, according to the mathematics of quantum mechanics, the probability of a match is not ⅔ – it is ¾. For the quantum enthusiast, I show this here, but you can skip it without interrupting the flow of the website.

Maybe we could devise a different local, hidden-variables theory where the matching agrees with the probability of ¾ predicted by quantum mechanics?  But then we could go on forever, of course, trying out a new theory, seeing it fail and trying yet another. Fortunately, we don’t need to do that, because what Bell did in his 1964 paper (see step 1.6) was to show that no local, hidden-variables theory can reproduce the correlations between Alice’s and Bob’s results predicted by quantum mechanics.

He did this by assuming the most general local, hidden-variables theory and then showing that, in such a theory, a particular combination of measurement outcomes from the two particles must be less than a given number. This inequality was later named a Bell Inequality (there are different types, depending upon the chosen combinations of measurements). He went on to show that the same combination for the outcomes of the same measurements as predicted by quantum mechanics (rather than the local, hidden-variables theory) was in fact greater than the constraining number; such a situation would be called a violation of the Bell Inequality.