EPR Paradox and Bell's Theorem

In 1935, Albert Einsten, Boris Podolsky, and Nathan Rosen (EPR) proposed an argument in which they attempted to show that quantum mechanics is an “incomplete” theory. The argument—now known as the EPR paradox—hinges upon the condition that a “complete” theory would require that “nothing that’s an ‘element of reality’ of the world, gets left out of that description” (P.61). The three colleagues further define the condition (in their own words): “if, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of reality corresponding to this physical quantity.” In the language and form used by Albert, this would mean that if you measure the color of electron 1 to be [color = +1>, you will also be able to (instantly) measure [color = -1> on electron 2. This is not due to the non-locality of quantum mechanics (to EPR), but because the color was an element of reality prior to its measurement on either electron 1 or electron 2.

In making this claim, and in the further extrapolation of this position, EPR make 2 assumptions:

1) That the predictions of quantum mechanics about results of experiments are correct (Albert’s words), and

2) Locality: things can be set up so that electrons 1 and 2 cannot share information (EPR will use this in attempt to show that experimental results must be actual elements of reality prior to any measurement).

Based on these two assumptions and the reality condition, EPR argued that if you separate electrons 1 and 2 so that they cannot easily exchange information and measure the color of electron 1 to be [color = +1>, you can know with certainty (probability 1) that the color of electron 2, if measured, would instantly be [color = -1>. Since they hold locality to be true, EPR concluded that the color must have existed prior to the act of measuring it. They further took advantage of the principle of locality by saying that you could measure the hardness of electron 2 instead of its color, without interfering with electron 1. Thus, since you know the color of electron 1 and you know the hardness of electron 2 you can simultaneously know both the color and hardness of each electron. According to the standard way of thinking about quantum mechanics, these observables are incompatible and to know both simultaneously is impossible. EPR concluded then, that the standard way of thinking about quantum mechanics must be false. For EPR, both observables (or any observables for that matter) are simultaneous elements of reality, unexplained by quantum mechanics.

In drawing this conclusion (and thus, illustrating the incompleteness of quantum mechanics), EPR relied upon 2 constraints:

1) The deterministic constraint: the value of every spin-space observable for electron 1 must be opposite the value of electron 2 (Albert’s words). For example, if electron 1 has [color = +1> then electron 2 must have [color = -1>, and

2) The statistical constraint: If we measure the color of electron 1 to be [color = +1> then we know with certainty (and without measurement) that the color of electron 2 is [color = -1>. Since EPR assumes locality and a color measurement is not made on electron 2, and if the 2 particles are sufficiently separated so as they cannot interact, then the measurements made on each particle would have no effect on the other particle. Therefore, the fact that electron 2 would have [color = -1> after measuring the color of electron 1, must be have been a preexisting element of reality as it is impossible for the particles to exchange information instantly (which would be in violation of the principle of locality). It is locality then, which allows us to use quantum formalism to show the probability of a simultaneous spin observable (say scrad) and to actually make that measurement, thereby knowing both observables (color and scrad) simultaneously.

The EPR paradox appeared to show that the contemporary version of quantum mechanics was incomplete, thus opening the door for a future local hidden variable theory to complete the picture. That door however, was slammed shut nearly thirty years later by theoretical physicist John Bell. In a paper, entitled “On the Einsten Podolsky Rosen Paradox”, Bell proposed what would (naturally) become known as Bell’s Theorem in which he showed that the deterministic constraint and the statistical constrain described above were mathematically inconsistent and furthermore, that not only was EPR’s conclusion false but one of their assumptions must too be false!

Before addressing the implications of Bell’s Theorem, it is worth briefly investigating the particulars of the theorem itself. To illustrate the (insurmountable) inconsistency between the two constraints, Bell first derived mathematical relationships to represent each constraint. The deterministic constraint is identified as:

P(a,b) = -a•b

where a and b are both vectors along which the spin of an electron will be measured (one vector per electron) and P(a,b) is the average value of the scalar product of their spins. The statistical constraint is represented by the Bell inequality, which is arrived at in a more involved (though similarly straightforward) method than the deterministic constraint. For the Bell inequality we have

│P(a,b) – P(a,c)│≤1 + P(b,c)

which can be shown—quite simply—to be “patently inconsistent with Bell’s inequality,” (Griffiths p.426). Experimental results are in excellent agreement with Bell’s Theorem and it has been nearly unanimously accepted in the physics community. What then does this mean?

In making their argument, EPR made two assumptions; Bell has now shown that one of those assumptions must be false! Either the predictions of quantum mechanics about experimental results are false and quantum mechanics is not only incomplete but it is wrong, or the physics governing the universe must be nonlocal in nature. With a perfect experimental track record (if electron 1 is spin up, electron 2 is always spin down) it was locality—not quantum mechanics—which was proved false. Ironically, the argument Einstein (and Podolsky and Rosen) proposed not only failed to show quantum mechanics as incomplete, but it overthrew the notion of absolute locality which he had championed, a result Einstein would have found more appalling than what he had started with.