The inequality that Bell derived applies to experiments that measure quantum mechanical correlations originating in pairs of entangled particle spins. Such correlations were calculated by Bell for the experimental schematic of Fig. 1. The quantities in the inequality are experimentally measured values consisting of ± 1’*s* corresponding to components of spin parallel and anti-parallel to the magnetic field directions on the *A* and *B* sides of the apparatus. The entanglement formalism furnishes joint probabilities allowing the calculation of count correlation *C*(*a*, *b*) = −cos (*θ*_{a} − *θ*_{b}) at field angles *θ*_{a} and *θ*_{b} used in the measurements on the two sides.

Figure 1 Schematic of Bell experiment in which a source sends two particles to two detectors having angular settings *θ*_{a} and *θ*_{b} and/or counterfactual settings *θ*_{a′} and *θ*_{b′}. While one measurement operation on the A-side, e.g. at setting *θ*_{a}, commutes with one on the B-side at *θ*_{b}, any additional measurements at either *θ*_{a′} or *θ*_{b′} are non-commutative with prior measurements at *θ*_{a} and *θ*_{b}, respectively.

Bell represented the count occurrences as a stochastic process [1, 6] by defining a function *A*(*a*, *λ*) that may take values ± 1 only, where *a* = *θ*_{a} is the magnetic field direction (or polarizer direction in the optical case) on the *A*-side of the apparatus. Parameter *λ* represents one or more random variables having probability density *ρ*(*λ*) that determine through *A*(*a*, *λ*) which values of are observed. The outcome of two measurements, one on each particle of a pair, is determined by the value of random variable *λ* occurring in that trial. Readouts at all possible angular settings are represented by *A*(*a*, *λ*).

The readout for the particle on the *B*-side of the apparatus is given by *B*(*b*, *λ*) = −*A*(*b*, *λ*), so that one stochastic process is used to describe all observations, and a correlation of -1 automatically occurs for equal settings on the two sides. The correlation of the detected values for the particle pairs may then be expressed as a probability average using a conventional probability integral (where triangular brackets on the left-hand-side indicate this average):
$$\u3008A(a,\lambda )B(b,\lambda )\u3009\equiv \int A(a,\lambda )B(b,\lambda )\rho (\lambda )d\lambda .$$(2.1)

Because three variables were necessary to obtain a condition on the correlations in the form of an inequality, Bell calculated the difference of two correlations arising from two different settings, *b* and *c*, on the B-side of the apparatus:
$$\begin{array}{}{\displaystyle \u3008A(a,\lambda )B(b,\lambda )-A(a,\lambda )B(c,\lambda )\u3009=\int \rho (\lambda )A(a,\lambda )\left(B(b,\lambda )\right.}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{\displaystyle \left.-B(c,\lambda )\right)d\lambda}\end{array}$$(2.2)

From Eq. (2.2)
$$\begin{array}{}{\displaystyle \u3008A(a,\lambda )B(b,\lambda )-A(a,\lambda )B(c,\lambda )\u3009=\int \rho (\lambda )A(a,\lambda )B(b,\lambda )}\\ \left(1-B(b,\lambda )B(c,\lambda )\right)d\lambda \end{array}$$(2.3)

since *B*(*b*,*λ*)^{2} = 1 . Taking absolute values of both sides produces
$$\begin{array}{}{\displaystyle \left|\u3008A(a,\lambda )B(b,\lambda )-A(a,\lambda )B(c,\lambda )\u3009\right|}\\ {\displaystyle =\left|\int \rho (\lambda )A(a,\lambda )B(b,\lambda )(1-(b,\lambda )B(c,\lambda )d\lambda )\right|}\\ {\displaystyle \u2a7d\rho (\lambda )(1-B(b,\lambda )B(c,\lambda ))d\lambda}\end{array}$$(2.4)

after bringing the absolute value inside the integral on the right-hand side. Defining *C*(*x*, *y*) = 〈*A*(*x*, *λ*)*B*(*y*, *λ*)〉, the Bell inequality is obtained:
$$|C(a,b)-C(a,c)|\u2a7d1+C(c,b),$$(2.5)

where
$$C(c,b)=\u3008-B(c,\lambda )B(b,\lambda )\u3009=\u3008A(c,\lambda )B(b,\lambda )\u3009$$(2.6)

All the correlations now occur between variables defined on opposite sides of the apparatus.

When *C*(*x*, *y*) = −cos(*θ*_{x} − *θ*_{y}) is inserted in Eq. (2.5) for each of the correlations, with *x* and *y* taking appropriate values among *a*, *b*, and *c*, the inequality is violated at certain angular differences. Reasons given for this have been the stated assumptions believed necessary for the derivation of Eq. (2.5). First, observed values on each side of the apparatus depend only on the setting on that side and are independent of the setting on the opposite side, e.g., *A*(*a*, *λ*) is independent of *b*, rather than dependent on both *a* and *b* as *A*(*a*, *b*, *λ*) would be. (Otherwise changing a value from *b* to *c* would cause a new value *A*(*a*, *c*, *λ*) that would disallow factoring *A*(*a*, *λ*) from the difference terms in Eq. (2.2) [7].) Second, the function *A*(*a*, *λ*), as used in the derivation, may be interpreted to imply that readouts exist at all instrument settings (values of *a*) before measurement. This would seem to conflict with the wave function collapse postulate of quantum mechanics. These two assumptions of the theorem are frequently referred to as locality and reality and are often cited as the reason for inequality violation by experimentally observed cosine correlations.

The meaning of the last step in Eq. (2.6) that defines *C*(*c*, *b*) must be considered in the context of application of the inequality to a Bell experiment. From quantum mechanics, the measurements on opposite sides of the apparatus on the two different particles commute. However, two spin measurements on the same particle, i.e., on the same side of the apparatus, do not commute. The principles of quantum mechanics prescribe that if variable *b* is measured followed by *b*′, the probability of different outcomes for *b*′ are conditional on the outcome at the previous setting *b*. Also, if *b* is measured again, its previous value is not necessarily obtained, since the probabilities of its outcomes are now conditional on that of *b*′. By contrast, if the observables in question commute, the same value of *b* as before must be observed if it is measured again after *b*′.

Thus, pairs of commutative measurements on two different particles on opposite sides of a Bell apparatus are described by qualitatively different probabilities than pairs of non-commutative measurements on the same particle on one side. It will be shown later in explicit examples that this leads to differing functional forms for correlations among the variables and to satisfaction of the Bell inequality.

In his book, Bell described the measurement *B*(*c*, *λ*) to be a predicted alternative value that allowed him to ignore the lack of commutation with measurement *B*(*b*,*λ*) [8]. *B*(*c*, *λ*) was the value that would have been observed had the angular setting been *c* rather *b* for the same particle in the same measurement trial. It was a counterfactual that commuted with *A*(*a*, *λ*) but not with *B*(*b*, *λ*), and could not be measured at the same time as *B*(*b*, *λ*). Its replacement by −*A*(*c*, *λ*)) was the corresponding counterfactual value that would have existed on the *A*-side at the same angular setting. Again, since this value now does not commute with *A*, it cannot be measured on the same *A*-side particle except by a second sequential measurement on the *A*-side.

After choosing to use a predicted value rather than a measurement for the third variable in the inequality, Bell assumed, without stating it, that the stochastic process represented by his function *A*(*a*, *λ*) was second order stationary [6], meaning that all pairs of correlations inserted in Eq. (2.5) have the same functional form dependent on coordinate differences. This will emerge as one of the most important flaws in the Bell theorem.

In Bell’s derivation, the inequality appears as a property of probabilistically computed correlations. As such, actual data fluctuations might be expected to cause some numerical disagreement with it. Further, the inequality results from the stochastic process represented by *A*(*a*, *λ*) and its assumed characteristics: locality and the definition of observables’ values independent of measurement. On this basis, it has been subject to experimental “test” by the physics community [9, 10].

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