Hidden Variable - How to calculate the straight line

[Edgardo]

There are plots that compare the expectation value E(a,b) of
(i) a hidden variable theory
(ii) quantum mechanics

For example here:
1. Talk given by Alain Aspect, video at 2m28s
2. PF Thread: The Unfair Sampling Assumption & Bell Tests

For the hidden variable theory the expectation value E(a,b) or E(theta) looks like a line. My question: How do you get this line?

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Let me further explain:

From what I understand the hidden variable is constructed as follows:
E(a,b) is given by:

$E(\textbf{a},\textbf{b}) = \int \rho(\lambda) A(\lambda, \textbf{a}) B(\lambda,\textbf{b}) d\lambda$

(Let's call this expression HV-Expectation)

where the vectors a and b are the directions of the polarizers.
(See Aspect talk at 0m54s)


Aspect asks us at 2m28s to construct a hidden variable and comes up with:

$A(\lambda, \textbf{a}) = sign( \{cos(2(\theta_a-\lambda )\}$
$B(\lambda, \textbf{b}) = sign( \{cos(2(\theta_b-\lambda )\}$
(Let's call them HV-equations)

where $\theta_a$ and $\theta_b$ are the angles of the vectors a and b with respect to some axis.

Now, I suppose that $\lambda$ is the angle of a vector $\vec{\lambda}$. This vector $\vec{\lambda}$ represents the hidden variable (sort of like a classical polarization). At least that's what I understood after reading Bell's original paper, see Eq (4).
(I also don't understand why Bell introduces the vector a').

Anyways, Aspect then explains that we get the line after plugging the HV-equations into HV-Expectation.

How is the calculation done?

 

[Edgardo]

I have meanwhile found a nice explanation in the book Quantum theory: concepts and methods by Asher Peres, see page 161-162.

The hidden variable expectation value is:

$E(\theta) = -1 + 2\theta/\pi$

where $\theta$ is the angle between the vectors $a$ and $b$. This is indeed a line.