Bell inequalities

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2020 Mathematics Subject Classification: Primary: 81P15 Secondary: 81P4081R1546L07 [MSN][ZBL]

Motivated by the desire to bring into the realm of testable hypotheses at least some of the important matters concerning the interpretation of quantum mechanics which were evoked in the controversy surrounding the Einstein–Podolsky–Rosen paradox [EPR], [Bo], J.S. Bell discovered the first version [B], [B2] of a series of related inequalities which are now generally called Bell's inequalities (for general reviews from different points of view, see [CS], [D], [P]). These inequalities commonly provide bounds on the strength of correlations between systems which are no longer interacting but have interacted in their past.

A typical class of correlation experiments involved in these inequalities can be briefly described. A source provides an ensemble of identically prepared systems, one after another, and, as part of the preparation, splits each system into two subsystems, directing these to separate arms of the experiment. At one arm the arriving subsystem is subjected to a measuring device chosen from a class $ {\mathcal A} $ of suitable devices, and at the other arm the incident subsystem interacts with a measuring device from a second class $ {\mathcal B} $. For each choice of devices $ A \in {\mathcal A} $ and $ B \in {\mathcal B} $ with outcome sets $ {\widehat{A} } $ and $ {\widehat{B} } $, the relative frequencies $ p ( \alpha, \beta ) $ of the measurement of $ \alpha \in {\widehat{A} } $ on one arm and $ \beta \in {\widehat{B} } $ on the other are determined. An operational condition of independence of the two arms of the experiment is required:

$$ \sum _ {\beta \in {\widehat{B} } } p ( \alpha, \beta ) \equiv p ( \alpha ) $$

$$ \left ( \textrm{ respectively, } \sum _ {\alpha \in {\widehat{A} } } p ( \alpha, \beta ) \equiv p ( \beta ) \right ) $$

must be independent of the choice of $ B \in {\mathcal B} $( respectively, independent of the choice of $ A \in {\mathcal A} $).

Bell's inequality, in the form of J.F. Clauser and M.H. Horne [CH], is:

$$ \tag{a1 } 0 \leq p ( \alpha _ {1} ) + p ( \beta _ {1} ) + p ( \alpha _ {2} , \beta _ {2} ) - p ( \alpha _ {1} , \beta _ {1} ) + $$

$$ - p ( \alpha _ {1} , \beta _ {2} ) - p ( \alpha _ {2} , \beta _ {1} ) \leq 1, $$

for all $ \alpha _ {i} \in {\widehat{A} } $, $ \beta _ {j} \in {\widehat{B} } $, $ A \in {\mathcal A} $, $ B \in {\mathcal B} $. Bell's theorem (and its many subsequent extensions) is a meta-theoretical theorem which states that all theories of a certain class $ {\mathcal C} $ describing such a correlation experiment must provide predictions satisfying (a1). Hence, if in a real experiment one measures correlation probabilities violating (a1), then one must conclude that there exist real physical processes which cannot be described by any theory in the class $ {\mathcal C} $. If a given theory predicts a violation of (a1), then Bell's theorem entails that no theory in the class $ {\mathcal C} $ can reproduce all predictions of this theory.

Which class of theories must yield predictions satisfying (a1)? As there are many forms of Bell's theorem in the literature, there is no unique answer to this question. One illustrative answer is the following. If there exists a measure space $ ( \Omega, \Sigma, \mu ) $ such that for all $ \alpha _ {i} \in {\widehat{A} } $, $ \beta _ {j} \in {\widehat{B} } $, $ A \in {\mathcal A} $, $ B \in {\mathcal B} $ there exist corresponding measurable sets $ {\widetilde \alpha } _ {i} , {\widetilde \beta } _ {j} \in \Sigma $ such that

$$ p ( \alpha _ {i} , \beta _ {j} ) = \int\limits _ \Omega {\chi _ { {\widetilde \alpha } _ {i} } \chi _ { {\widetilde \beta } _ {j} } } {d \mu } , $$

where $ \chi _ { {\widetilde \alpha } _ {i} } $ is the characteristic function for the set $ {\widetilde \alpha } _ {i} $, then (a1) must hold [W]. Classical physics and the so-called "local hidden-variable theorylocal hidden-variable theories" fall into the class $ {\mathcal C} $.

The interest of Bell's inequalities for quantum physics is that quantum mechanics predicts the existence of preparations and measuring devices such that the resulting correlation experiments violate (a1) (and these predictions have been verified in the laboratory). Indeed, quantum field theory (which is relativistic quantum mechanics) predicts that for every preparation there exist measuring devices such that (a1) is violated (see the review [S] and references cited there). Hence, the results of these theories cannot be reproduced by any theory in the class $ {\mathcal C} $. For the significance of this fact for the interpretation of quantum theory, see the above-mentioned reviews.

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