1 Bell’s theorem

Con­sider the fol­lowing setup[1]:

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Bell's theorem (the simplest version)

Figure 1.1.1 The sim­plest ver­sion of Bell’s the­orem (setup)

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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 14.

  • 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 59 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 59 of the time. On the basis of our assump­tion we pre­dict that the prob­a­bility of observing dif­ferent colors is equal to or greater than 59. 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 12 is less that 59!

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 [1]).

Ein­stein was both­ered, albeit not by Bell’s the­orem, whose orig­inal ver­sion[2] 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,[3] 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”.[4]

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,[5] “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.