User:EdC~enwiki/Bell's theorem

Description of Bell's theorem

Continuing on from the situation explored in the EPR paradox, consider that again a source produces paired particles, one sent to Alice and another to Bob. When Alice and Bob measure the spin of the particles in the same axis, they will get identical results; when Bob measures at right angles to Alice's measurements they will get the same results 50% of the time, the same as a coin toss. This is expressed mathematically by saying that in the first case, their results have a correlation of 1, or perfect correlation; in the second case they have a correlation of 0; no correlation. (A correlation of -1 would indicate getting opposite results the whole time.)

So far, this can be explained by positing local hidden variables; each pair of particles is sent out with instructions on how to behave when measured in the x axis and the z axis, generated randomly. Clearly, if the source only sends out particles whose instructions are correlated for each axis, then when Alice and Bob measure on the same axis, they are bound to get identical results; but (if all four possible pairs of instructions are generated equally) when they measure on perpendicular axes they will see zero correlation.

Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pair, one particle sent to Alice another to Bob. Each performs one of the two spin measurements.

Now consider that B rotates their apparatus (by 45 degrees, say) relative to that of Alice. Rather than calling the axes $x_{A}$, etc., henceforth we will call Alice's axes $a$ and $a'$, and Bob's axes $b$ and $b'$. The hidden variables (supposing they exist) would have to specify a result in advance for every possible direction of measurement. It would not be enough for the particles to decide what values to take just in the direction of the apparatus at the time of leaving the source, because either Alice or Bob could rotate their apparatus by a random amount any time after the particles left the source.

Next, we define a way to "keep score" in the experiment. Alice and Bob decide that they will record the directions they measured the particles in, and the results they got; at the end, they will tally up, scoring +1 for each time they got the same result and -1 for an opposite result - except that if Alice measured in $a$ and Bob measured in $b'$, they will score +1 for an opposite result and -1 for the same result. It turns out (see the mathematics below) that however the hidden variables are contrived, Alice and Bob cannot average more than 50% overall. (For example, suppose that for a particular value of the hidden variables, the $a$ and $b$ directions are perfectly correlated, as are the $a'$ and $b'$ directions. Then, since $a$ and $a'$ are at right angles and so have zero correlation, $a'$ and $b$ have zero correlation, as do $a$ and $b'$. The unusual "scoring system" is designed in part to ensure this holds for all possible values of the hidden variables.)

The question is now whether Alice and Bob can score higher if the particles behave as predicted by quantum mechanics. It turns out they can; if the apparatuses are rotated at 45° to each other, then the predicted score is 71%. In detail: when observations at an angle of $\theta$ are made on two entangled particles, the predicted correlation between the measurements is $\cos \theta$. In one explanation, the particles behave as if when Alice makes a measurement (in direction $x$, say), Bob's particle instantaneously switches to take that direction. When Bob makes a measurement, the correlation (the averaged-out value, taking +1 for the same measurement and -1 for the opposite) is equal to the length of the projection of the particle's vector onto his measurement vector; by trigonometry, $\cos \theta$. $\theta$ is 45°, and $\cos \theta$ is ${\frac {\sqrt {2}}{2}}$, for all pairs of axes except $(a,b')$ – where they are 135° and $-{\frac {\sqrt {2}}{2}}$ – but this last is taken in negative in the agreed scoring system, so the overall score is ${\frac {\sqrt {2}}{2}}$; 0.707, or 71%. If experiment shows - as it appears to - that the 71% score is attained, then hidden variable theories cannot be correct; not unless information is being transmitted between the particles faster than light, or the experimental design is flawed.