Measure the two spins of a singlet along directions $\mathbf{a}$ and $\mathbf{b}$ at angle $\theta$. Each result alone is a coin toss, but the mean of their product is $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot\mathbf{b}=-\cos\theta$: always opposite at $\theta=0$, unrelated at $90^\circ$, always equal at $180^\circ$. The probability that the two agree is $\sin^{2}(\theta/2)$. It follows from letting $A$’s result fix $B$’s state. A simple local model, in which each pair carries a random direction and each spin answers by the side of it its axis lies on, gives a straight line instead of the cosine. They agree at 0, $90^\circ$ and $180^\circ$, and nowhere else; the next lesson turns that gap into Bell’s test.

Two dancers mirror each other perfectly when they face the same way. Turn one of them and the mirroring fades — not by a fixed amount per degree, but along a smooth curve, strong for small angles. The cosine is that fading.

The mean product of the two results against the angle $\theta$ between the axes, for $\theta$ from 0 to $\pi$: $-\cos\theta$ for the singlet, and the straight line $-1+2\theta/\pi$ of a local model with random directions. They meet at 0, and ; in between the quantum curve is the stronger correlation.

Spin $A$ measured along $\mathbf{a}$ and spin $B$ along $\mathbf{b}$, at angle $\theta$, each giving $\pm1$:

• Each result alone is $\pm1$ with probability $\tfrac12$: $\langle\mathbf{a}\cdot\vec{\sigma}\rangle=0$ on either side.

Take $A$’s result first: each value comes with probability one half. A result of $+1$ along $\mathbf{a}$ leaves $B$ down along $\mathbf{a}$, which is up along $\mathbf{b}$ with probability ; a result of leaves up along , the mirror case. Averaging the product over the four cases gives the difference of and of the half angle, which is .

Spin $A$ alone is completely mixed.

As along $z$, so along $\mathbf{a}$: the singlet has the same form along every axis.

$\ket{\downarrow_{\mathbf{a}}}$ points along $-\mathbf{a}$, at angle $\pi-\theta$ from $$, and .

Products $+1$ for equal, $-1$ for opposite; $A=-1$ is the mirror case.

$\sin^{2}\tfrac{\theta}{2}-\cos^{2}\tfrac{\theta}{2}=-\cos\theta$.

For the singlet, the mean of the product of the two results is minus the cosine of the angle between the two axes.

Spin $A$ of a singlet is completely mixed, so along any axis it gives $+1$ or $-1$ at random, half the time each, whatever $B$ does.

At 90° the cosine is zero: equal and opposite results are equally likely, and the product averages to nothing.

The probability that the two results agree is the square of the sine of half the angle between the axes.

Great work! You now know how to:

• derive the singlet correlation $-\cos\theta$
• compute the probability of agreement at any angle
• compare it with a simple local model