Quantum mechanics allows us to perform the following experiment (see the article by David Mermin in Physics Today Vol. 38 No. 4):

The device at the center is used to launch 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 (for example, 21) and responses (for example, RG). 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 ...

The particles are prepared in such a way that every such record displays the following characteristics:
  • Whenever both apparatuses perform the same measurement (settings 11, 22, or 33), equal colors (RR or GG) are never observed.
  • 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 will react to its arrival. Such an instruction set would specify one of two possible responses (R or G) for each of the three possible apparatus settings (1,2, or 3), and so there would be eight possible instruction sets: RRR, RRG, RGR, GRR, RGG, GRG, GGR, and GGG. If, for instance, a particle were to arrive with RGG, the apparatus would flash red if it is set to 1, and green if it is set to 2 or 3.

To explain why the outcomes differ whenever both particles are subjected to the same measurement, you would have to assume that the particles of each pair are launched with opposite instruction sets: if one particle carries RRG, then the other particle carries 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 will be 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!

So, at least in this one case, the predictions of quantum mechanics cannot be explained with the help of instruction sets. They cannot be explained by assuming that measurements merely reveal pre-existent properties or values. In a sense, measurements 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."

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 in which Einstein, Podolsky, and Rosen made use of a similar setup 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."

For those familiar with the quantum formalism, here is a possible way of realizing the experiment just described. The devise at the center launches pairs of spin-1/2 particles in the singlet state. The three quantities the apparatuses are designed to measure are spin components with respect to three coplanar axes - they lie in the same plane, with equal angles between them. Because the particles are in the singlet state, equal outcomes (RR or GG, corresponding to two "up"s or two "down"s) are never observed if the spins are measured with respect to the same axis (identical apparatus settings). What remains to be shown is that the outcomes are completely random if the apparatus settings are not taken into account.

If the spin of a spin-1/2 particle is found up with respect to a given axis, the probability of subsequently finding it up with respect to an axis that is rotated by an angle
α is cos2(α/2). This is also the probability of finding the spin of its singlet partner down with respect to the rotated axis. Because the apparatus settings are randomly selected, the probability with which both apparatuses measure the same spin component is 1/3, and in these cases the probability of obtaining opposite outcomes is 1. The probability with which the particle spins are measured with respect to different axes is 2/3, and in these cases the probability of obtaining opposite outcomes is cos2(60°) = 1/4. The probability of obtaining opposite outcomes is therefore (1/3) x 1 + (2/3) x (1/4) = 1/2. In other words, the apparatuses flash different colors exactly half of the time.

As a preparation to my next post, you are invited to ponder the following "game":

Three "players" (Andy, Bob, and Charles) compete agains three "inter­roga­tors", and the rules of the game are as follows:
  • Either all players are asked for the value of X, or one player is asked for the value of X while the two other players are asked for the value of Y.
  • The pos­sible values of X and Y are +1 and -1.
  • If all players are asked for the value of X, they win if (and only if) the product of their answers equals -1. Oth­er­wise they win if (and only if) the product of their answers equals +1.
Once the ques­tions are asked, the players are not allowed to com­mu­ni­cate with each other. Prior to that, they may work out a strategy. Is there a fail-safe strategy? Can they make sure that they will win? What says you?