We present the 1964 paper of John Stewart Bell titled On the Einstein Podolsky Rosen paradox.

An Experiment:

A nucleus with zero angular momentum emits electrons and positrons. Thus, if you fix any direction $\vec{a}$ and you measure the spin of the electron and positron in that same direction $\vec{a}$, they must be opposite of each other

To measure the spin of an electron/positron, you first fix a direction $\vec a$, and then you measure the spin in that direction. When you do that, you obtain values $+1$ or $-1$. Let

$$ \begin{align*} e_{\vec{a}}&= \text{the value of the spin of the electron when measured in the direction } \vec{a},\\ p_{\vec{a}}&= \text{the value of the spin of the positron } p \text{ when measured in the direction } \vec{a}.\end{align*} $$

$e_{\vec{a}}$ and $p_{\vec{a}}$ are not classical random variables! They are observables! We will follow Bell's proof to obtain a contradiction assuming $e_{\vec{a}}$ and $p_{\vec{a}}$ are random variables. Einstein-Podolsky-Rosen (1935) proposed that they are random variables (hidden variables) since locality can not explain instantaneous transmision of information.

Assumptions about the random variables $e_a$ and $p_a$:

(1) $e_{\vec{a}}$ and $p_{\vec{a}}$ take values in $\{-1,+1\}$; (2) $e_{\vec{a}}=+1 \Leftrightarrow e_{-\vec{a}}=-1$; (3) $p_{\vec{a}}=+1 \Leftrightarrow p_{-\vec{a}}=-1$; (4) $e_{\vec{a}}=\pm 1 \Leftrightarrow p_{\vec{a}}=\mp 1$; (5) One cannot make two measurements simultaneously on the same particle, e.g. nether $e_{\vec{a}} e_{\vec{b}}$ nor $p_{\vec{a}}p_{\vec{b}}$ is allowed.

Bell’s Theorem 1:

Assume that $e$ and $p$ are random variables satisfying the assumptions above. Let $\vec{a}, \vec{b}, \vec{c}$ be three directions. Then the expectation $E$ satisfies

$$ \left|E\left(e_{\vec{a}} p_{\vec{b}}\right)-E\left(e_{\vec{a}} p_{\vec{c}}\right)\right| \leq 1+E\left(e_{\vec{b}} p_{\vec{c}}\right). $$

Bell’s Theorem 2:

Consider an entangled electron/ positron pair with opposite spins $\{\pm 1\}$ when measured in any direction. Assume that the state of the system is the singlet state $=\frac{1}{\sqrt{2}} \left( | 10\rangle-| 01\rangle\right)$. Then

$$ E\left(e_{\vec{a}} p_{\vec{b}}\right)=-\vec{a} \cdot \vec{b}. $$

Concluding the contradiction from Bell’s Theorems 1 and 2:

Now make the following choices

$\vec a = (0,0,1), \quad \vec b =(1, 0, 0), \quad \vec c = \left(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right)$