This Quantum World

Bell's theorem

The bare facts

Here is what we do:

Pairs of particles are launched in opposite directions.
Each particle is subjected to one of three possible measurements (1, 2, or 3).
Each time the two measurements are chosen at random.
Each measurement has two possible outcomes, indicated by a red or green light.

And here is what we find:

If both particles are subjected to the same measurement, identical outcomes are never obtained.
The two sequences of recorded outcomes are completely random. In particular, half of the time both lights are the same color.

If this does not bother you, then please explain how it is that the colors differ whenever identical measurements are performed.

The simplest version of Bell's theorem

The obvious explanation is that each particle arrives with an "instruction set" — a property (or set of properties) that pre-determines the outcome of each possible measurement. Let's see what this entails.

Each particle arrives with one of the following instruction sets:

RRR, RRG, RGR, GRR, RGG, GRG, GGR, GGG.

If a particle arrives with, say, RGG, then the apparatus flashes red if it is set to 1 and green if it is set to 2 or 3.

In order to explain why the outcomes differ whenever both particles are subjected to the same measurement, we have to assume that particles launched together arrive with opposite instruction sets. If one comes with, say, RRG, then the other comes with GGR.

Suppose that the instruction sets are RRG and GGR. In this case we observe different colors with the following five of the nine possible combinations of apparatus settings:

1-1, 2-2, 3-3, 1-2, and 2-1

...and we observe equal colors with the following four combinations:

1-3, 2-3, 3-1, and 3-2.

Because the apparatus settings are randomly chosen, this particular pair of instruction sets therefore results in different colors 5/9 of the time. The same is true of the other pairs of instruction sets except the pair RRR, GGG. If the two particles carry these instruction sets, we see different colors every time.

If follows that we see different colors at least 5/9 of the time.

But different colors are observed half of the time! In reality the probability of observing different colors is 1/2 (50 percent)!

It follows that the statistical predictions of quantum mechanics cannot (always) be explained with the help of instruction sets. In general, measurements cannot be interpreted as revealing pre-existent properties — properties that would be possessed even if they were not observed.

The conclusion that we see different colors at least 5/9 of the time is the simplest version of Bell's theorem, due to N. David Mermin ("Is the Moon there when nobody looks? Reality and the quantum theory," Physics Today, April 1985).

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