1.8 EPR with hidden variables
You may be somewhat underwhelmed by the outcome of the previous step. After all, if hidden variables really do underlie quantum mechanics, then we can suppose that the particle generator produces pairs of particles with definite spins (contrary to Heisenberg) which are directly opposed to each other, with each pair of spins aligned in a random direction (including vertical, horizontal and in-between). To check this supposed hidden-variables theory, the most natural spin-filter is one that registers “spin-up” if the incoming particle has a spin that points within ±90º of the spin-filter up-arrow, and “spin-down” otherwise.
In a local, hidden-variables theory, the two electrons have a “hidden” spin even before they are detected, and this shows that measurements in such a theory would always show matched spins if the detectors are parallel-opposed, in agreement with what we actually observe.
The figure shows how this would work in the hidden-variables theory. Alice’s spin-filter is shown schematically with its arrow pointing upwards. This means that, according to our hidden-variables supposition, it detects any spin that points within ±90º of the upward direction, represented by the grey semicircle. Bob’s spin-filter is parallel-opposed, inclined at 180º to Alice’s, with its arrow pointing downwards, along with its grey detection semicircle. We suppose that the particles are also emitted with hidden parallel-opposed spins in randomly orientated directions, so that, for example, Alice’s particle #1, labelled “1” in the circle, points upwards to the right and is in the grey region of her detector, while Bob’s corresponding particle #1 points downwards to the left and is also in the grey region of his detector. So, Alice and Bob will register matching measurements (because each of their spins is in their spin-filter’s grey semicircle). Indeed, since the spin-filters are mutually inclined at 180º, this guarantees that all measurements made by Alice and Bob will match. This theory doesn’t rely on any nonlocal influence – the mechanism is that, when a particle arrives at the detector, then the detector responds appropriately to that spin. It is a local action and so the theory is a local hidden-variables theory.
The remarkable thing that Bell did was to prove that any local hidden-variables theory – not just the theory that I’ve presented here – is incapable of reproducing all of the predictions of quantum mechanics. To do this, he investigated the case where the two spin-filters may not be just parallel-opposed, but can be orientated at any angle to each other. To get a feel for the essence of his paper, look at the figure, which shows the specific case of the two detectors being mutually inclined at 120º rather than the 180º of the previous figure.
I have drawn the same four pairs of spins as in the previous figure, but this time, because Bob’s spin-filter is not rotated all the way round to 180º – it is now at 120º – his spin #1 particle no longer falls in the grey region that matched Alice’s: it has landed in the white region, and so it no longer matches Alice’s result. Similarly, his spin #3 particle no longer falls in the white region that matched Alice’s: it has landed in the grey region, again no longer matching Alice’s result.
In general, as you can see, only if the spin of Alice’s particle falls within the two 120º arcs shown in the figure, will Bob’s result match Alice’s. Since the experiment is repeated many times, and since, each time, the orientation of the pair of spins is random, then the average number of times in which Alice’s result matches Bob’s will be (120º + 120º)/ 360º = 240º/360º = ⅔. In other words, the matching probability predicted by our local hidden-variables theory is ⅔.
When the detectors are mutually inclined as shown, not all detected spins of the electron pairs will match. In the arrangement above, the spins #2 match (both in the white semicircle) and the spins #4 match (both in the shaded semicircle), but spins #1 and #3 don’t match.
