Bell’s inequality – Bells’s inequality

We now need a general criterion in order to test any desired theory that assumes the independence of Alice and Bob’s measurements. The general criterion we get from Bell’s inequality, and similar inequalities based on the same key ideas.
Alice and Bob each choose two different angles (α1, α2), as well as (β1, β2). On the board, we see two angle pouches, which show the angles chosen, α1 or β1.

Next, we will test the validity of the product ansatz for the correlation function. We assume that the product ansatz is correct, and check whether there is a combination of correlation functions that contradicts this assumption. We begin with the statement that the mathematical expression shown is never greater than two. The difference illustrated here is likewise never greater than two. As an example, we select the extreme case. The sum is 2 + 0 equals 2, and is thus likewise not greater than 2. This also applies in general.

We use the values from the angle pouches in this inequality. Finally, we multiply out the brackets. After averaging over many measurements, the respective correlation functions in the four angle combinations arise. This inequality is a variant of the well-known Bell’s inequality – the CHSH inequality. It was deduced based on the assumption that the measurements of Alice do not influence Bob’s results, and vice versa.