# Probability and correlation function, interpretation of a result

My question is originated from the paper

• J.F.Clauser,M.A.Horne,A.Shimony and R.Holt,"Proposed experiment to test local hidden-variable theories",Phys.Rev.Let.,23(15),1969

which is a physics paper. But my question is only about the mathematical treatment made in it.

Let us have $A(a)=\pm 1$ and $B(b)= \pm 1$ be the results of the measure $A$ depending on $a$ and $B$ depending on $b$. $A$ and $B$ can be interpreted as the emergence ($+1$) and the non emergence ($-1$) of a photon through a filter.

Let also $A(\infty)$ and $B(\infty)$ be the results of $A$ and $B$ when the filters are removed (the result $A(\infty)$ and $B(\infty)$ are thus necessarily $+1$).

Let $P(a,b)$ be the emergence correlation function between $A$ and $B$. Let the rate of coincidence detection be $R(a,b)$ and the probability that $A(a)=\pm 1$ and $B(b)=\pm 1$ be $w[A(a)_{\pm},B(b)_{\pm}]$ (which I will rather note $w[\pm,\pm]$ in order to be concise).

In the following, $R_0 = R(\infty,\infty)$, $R_1(a) = R(a,\infty)$ and $R_2(b)=R(\infty,b)$.

• Question 1

In the article, they express $P(a,b)$ as $$P(a,b) = w[+,+]+w[-,-]-w[+,-]-w[-,+]$$ which I find logic but that I can't explain... How could this evident formula be explained ? I'd say

$P=1$ means that all results are perfectly correlated and that $P=-1$ means that they are all opposite. Hence, if we add the probability for the two results to be equal and that we substract the probability for the two results to be opposite, we get the correlation function

But this explanation doesn't seem very good to me...

• Question 2

They then say

$w[A(a)_+,B(\infty)_+](=w[+,\infty])=w[+,+]+w[+,-]$ and of similar formulas for $w[\infty,+]$ and $w[\infty,\infty]$ we obtain $$P(a,b) = \frac{4R(a,b)}{R_0} - \frac{2R_1(a)}{R_0} -\frac{2R_2(b)}{R_0} +1$$

I don't understand how to get to it. I've first expressed $w[\infty,+]$ and $w[\infty,\infty]$ as this \begin{align} w[\infty,+]&=w[+,+]+w[-,+] \\ w[\infty,\infty] &= w[+,+]+w[-,-]+w[+,-]+w[-,+] (=1?)\tag{1} \end{align}

and $R(a,b),R_1(a)$ and $R_2(b)$ as this

\begin{align} R(a,b)&=R_0.\left(w[+,+]+w[-,-]\right) \tag{2}\\ R_1(a)&=R_0.\left(w[+,+]+w[+,-]\right) \tag{3}\\ R_2(b)&=R_0.\left(w[+,+]+w[-,+]\right) \tag{4} \end{align}

Since I can't find the same result as theirs, I imagine (1) and/or (2-4) are not the good expression... But I don't see why.

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