3.8 Bell’s Theorem

Bell [8, 9] showed, using basically the same postulates as those of EPR, that no deterministic local hidden-variables theory can reproduce all statistical predictions of quantum mechanics. Initially, Bell’s argument has been applied to an EPR-type Gedanken experiment of Bohm; later, Bell’s analysis was extended to actual systems, and experimental tests were suggested and performed (see, for example, Clauser and Shimony [44]). Essentially, the particles on either side appear to be “more correlated” than can be expected by a classical analysis assuming locality (i.e. the impossibility of any kind of information or correlation transfer faster than light).
               
                                                                                        Figure 8: John Bell

In what follows we will use Odifreddi [107] in presenting an elementary analysis of Bell’s result. The setting is the following. We consider two physical systems; on one two types of measurements are made (), and on the other one two other types (). The results are binary, so they will be denoted by “" and “". We will repeat these measurements to ensure statistically relevant results. Correlations appear when measurements give the same outcome, that is, “” and “”. The basic result is that in almost all cases, more “” and “” (and less “” and “”) coincidences are recorded than one can explain by any local classical analysis.

Let be the probability that, by taking the measure on the first system, the outcome will be ; is the probability that by taking the measure on the first system and the measure on the second, the outcome of the first system alone will be ; is the probability that by taking the measure on the first system and measure on the second system, the outcomes will be respectively, and ; finally, is the probability that when taking the measures on the first system and on the second one, and having outcome on the second, the outcome of the first will be .

The main result can be stated as follows:
If the outcomes of the experiments on both systems are independent, that is

then the lack of correlation in one of the two types of measures cannot exceed the lack of correlation in the remaining types, that is, the following quadrangular inequality holds true:














(11)


It is remarkable that this inequality – and, of course, all inequalities obtained by systematic permutations – can be obtained with just an elementary manipulation of binary variables. The probabilistic hypothesis of independence can actually be decomposed in the conjunction of two hypotheses with more physical significance (see Jarett [76]):

Separability: The statistical outcomes performed on one system are independent of the outcomes performed on the other system:

Locality: The statistical outcomes performed an experiment on one system are independent of the types of experiments performed on the other system:

Separability says that the spatio-temporal separation between the two systems makes them reducible to individual parts, the “whole" is no more than the “sum of parts"; locality forbids any instantaneous interaction.


Separability and locality implies independence as

Consequently, if the outcomes of the experiments on both systems are separable and local, then the lack of correlation in one of the two types of measures cannot exceed the lack of correlation in the remaining types.

Probabilities can be interpreted as truth-values of elementary propositions, so the above analysis can be reformulated in the language of “classical logic". Indeed, let us write for and for , and similarly for . Further on, let us notice that the elementary operations with probabilities can be reformulated as logical operations, namely, conjunction will correspond to product, disjunction to sum, and implication to .

A “logical" version of the quadrangular inequality can be deduced:
If the conjunction is distributive with respect to disjunction for all propositions , that is,

then the following quadrangular implication holds true:




































Both quadrangular inequality and implication have been experimentally falsified, hence no theory satisfying their hypotheses can be physically correct. So, locality and separability cannot be simultaneously adopted. The failure of independence affects Reichenbach’s [116] causality principle: two correlated (non independent) events have a common cause, that there exists an event in their “past" with respect to which they are independent. So, we arrive at the idea of synchronicity that has important implication for quantum computing:
there exist events which are correlated in a way which is neither casual nor causal.
Finally, the failure of distributivity – the “mark" of quantum logic, has been proved to be more pervasive than the universe of quantum mechanics statements: it is excluded from any logic aiming to describe the physical world. Is any hope to rescue classical logic, which seems to be so brutally excluded …

Quantum mechanics has chosen to drop separability.