# 1 Bell’s theorem

Consider the following setup: The device at the center launches a pair of particles in opposite directions. Each particle enters an apparatus capable of performing one of three measurements. Each measurement has two possible outcomes, which are indicated by either a red or a green light. In each run of the experiment the measurement performed is randomly selected for each apparatus. After a large number of runs, we have in our hands a long record of apparatus settings and responses. Like this one:

21RG 31GG 12RG 13GR 31GR 12RG 22GR 21GG 11RG 31RR 33GR 31RG 21RR 33GR 11GR 31GG 13GG 22RG 12RG 32GR 31GG 22RG 11GR 12GG 23RR 22RG 23GG 32RG 23RG …

Any such record displays the following characteristics (how these statistical properties can actually be obtained is explained on this page):

• Whenever both apparatuses perform the same measurement (settings 11, 22, or 33), equal colors (RR or GG) are never observed.

This means that, if the settings agree, the outcomes are correlated: the joint probability p(R & R), which equals 0, is different from the product of marginal probabilities p(R) × p(R), which equals 1/4, and p(R & G), which equals 1/2, is different from p(R) × p(G), which also equals 1/4.

• The pattern of R’s and G’s is completely random. In particular, the apparatuses flash different colors exactly half of the time.

Does this bother you? If not, then try to explain how it is that the colors differ whenever identical measurements are performed.

You might try to explain this by assuming that each particle arrives with an “instruction set” — a set of properties that determines how the apparatus it enters will respond. There would be eight such sets: RRR, RRG, RGR, GRR, RGG, GRG, GGR, and GGG. If, for instance, a particle arrived with RGG, the apparatus would flash red if it is set to 1, and green if it is set to 2 or 3.

On the basis of this assumption, the reason why the outcomes differ whenever both particles are subjected to the same measurement is that the particles are launched with opposite instruction sets: if one particle carries the instruction set RRG, then the other particle carries the instruction set GGR.

Let’s see how this pans out. Suppose that the instruction sets are RRG and GGR. In this case we expect to see different colors with five of the 9 possible combinations of apparatus settings (namely, 11, 22, 33, 12, 21), and we expect to see equal colors with four (namely, 13, 23, 31, and 32). Because the apparatus settings are randomly chosen, this pair of instruction sets produces different colors 5/9 of the time. The same will be true of the remaining pairs of instruction sets (since each contains exactly two equal colors) except the pair RRR, GGG, which contains exactly three equal colors. If the two particles carry these instruction sets, we see different colors every time, regardless of the apparatus settings.

The bottom line: we see different colors at least 5/9 of the time. On the basis of our assumption we predict that the probability of observing different colors is equal to or greater than 5/9. This is Bell’s inequality for the present setup.

If the particles did arrive with instruction sets, Bell’s inequality would be satisfied. But it isn’t, for, as said, the apparatuses flash different colors half of the time, and 1/2 is less that 5/9!

In this as in many similar experimental situations, the predictions of quantum mechanics cannot be explained with the help of instruction sets. These measurements do not merely reveal pre-existent properties or values. They create their outcomes.

But then how is it that the colors differ whenever identical measurements are made? What mechanism or process is responsible for these correlations? How does one apparatus or particle “know” which measurement is performed and which outcome is obtained by the other apparatus?

You understand this as well as anybody else!

As a distinguished Princeton physicist commented, “anybody who’s not bothered by Bell’s theorem has to have rocks in his head” (quoted in ).

Einstein was bothered, albeit not by Bell’s theorem, whose original version appeared in 1964, nine years after Einstein’s death.

The title of Bell’s paper, “On the Einstein–Podolsky–Rosen paradox,” refers to a seminal paper of 1935, in which Einstein, Podolsky, and Rosen made use of similar correlations to argue that quantum mechanics was incomplete. In 1947, Einstein wrote in a letter to Max Born that he could not seriously believe in the quantum theory “because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance”.

In his 1964 paper, Bell was led to conclude that, on the contrary, “there must be a mechanism whereby the setting of one measurement device can influence the reading of another instrument, however remote.”

Spooky actions at a distance are here to stay!

As Bell wrote in a subsequent paper, “the Einstein–Podolsky–Rosen paradox is resolved in a way which Einstein would have liked least.”

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1. [↑] [↑] Mermin, N.D. (1985). Is the Moon there when nobody looks? Reality and the quantum theory. Physics Today 38 (4), 38–47.

2. [↑] Bell, J.S. (1964). On the Einstein Podolsky Rosen paradox. Physics 1, 195–200.

3. [↑] Einstein, A., Podolsky, B., and Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review 47, 777–780.

4. [↑] Einstein, A. (1971). The Born–Einstein Letters with Comments by Max Born, Walker.

5. [↑] Bell, J.S. (1966). On the problem of hidden variables in quantum mechanics. Reviews of Modern Physics 38, 447–452.