Proof of Bell's inequality

The experimental situation is the following:

The state ψ is arbitrary. The measurement device consists of two parts M1, M2, with three possible input values (control parameters) for each of the two devices a1, a2 ∈ {v1,v2,v3}. We measure two results y1, y2 ∈ {1,-1}, one for each device Mi.

Application of realism for the description of the measurement results

We measure the expectation value of the product f(y) = y1 ⋅ y2. Following the definition of realism, a realistic explanation for the measurement results has the following form:

E(f, a1, a2) = ∫ y1(x, a1, a2) ⋅ y2(x, a1, a2) ρ(x) dx

Application of strong Einstein causality

Assume the two measurement devices Mi are located far away from each other, and the measurements (that means, the input of the a1 and the fixation of the y1) happens in short enough time (say, less than a minute) at approximately the same time, for example on Earth and Mars, so that light signal need several minutes to reach the other device. (The velocities of Mars and Earth are small enough, so that relativity of simultaneity does not matter, but, for definiteness, we can use the system of rest of the Sun to define "the same time".)

Thus, a signal from M1 after the choice of a1 by the first experimenter reaches M2 only after y2 has been fixed, and reverse. That means, that, in case of strong Einstein causality, that the function y2(x,a1, a2) cannot depend on a1, and, as well, y1(x,a1, a2) cannot depend on a2. Therefore, the formula reduces to

E(f, a1, a2) = ∫ y1(x, a1) ⋅ y2(x, a2) ρ(x)dx

This is already the main formula (2) used by Bell to prove his inequality. Everything philosophically or methodologically important or possibly problematic has already happened. The problematic assumptions — realism and strong Einstein causality — are already used. What remains is quite elementary math.

Derivation of the inequality

It should be noted here, that there are many different variants of the Bell inequalities, with different advantages and disadvantages. For practical tests, one needs variants which allow for uncertainties of the measurements or preparations — in exchange for increasing complexity. We, therefore, consider only the variant we have used in our game.

If nobody cheats in the game, it follows that we have the following property: If a1=a2=a, then

E(f,a, a) = ∫ y1(x,a) ⋅ y2(x,a) ρ(x)dx = 1,

and this is possible only if always y1=y2. Therefore, the two functions y1(x,a) and y2(x,a) should be equal. This defines a common function y(x,a) ∈ {-1,1} so that

E(f,a1, a2) = ∫ y(x,a1) ⋅ y(x,a2) ρ(x)dx.

For such functions with values in {-1,1} we have always

y(x, v1) ⋅ y(x, v2) + y(x, v2) ⋅ y(x, v3) - y(x, v3) ⋅ y(x, v1) ≤ 1,

as can be easily seen: Each of the three products can be only ±1. To violate the inequatlity, all three product terms would have to give the value +1. But this is impossible, because it would follow that y(x,v1)=y(x,v2), y(x,v2)=y(x,v3), but y(x,v3)=-y(x,v1), which is impossible.

It remains to integrate the inequality to obtain

E(f, v1, v2) + E(f, v2, v3) - E(f, v3, v1) ≤ 1,

which is the variant of Bell's inequality we need in our game.