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Bell-Fine theorems on complementarity

Elisa Masa, Laura Ares, Alfredo Luis

Published Mar 4, 2021• 0 citations
OpenAlex

Abstract

We provide a Bell-type analysis of complementarity via a suitably designed hidden-variables model that leads to a set of Bell-like inequalities that can be tested by easily measured observables. We show that this violation is equivalent to Fine-like theorems regarding the lack of a joint distribution for incompatible observables. This is illustrated by path-interference duality in a Young interferometer.

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We provide a Bell-type analysis of complementarity via a suitably designed hidden-variables model that leads to a set of Bell-like inequalities that can be tested by easily measured observables. We show that this violation is equivalent to Fine-like theorems regarding the lack of a joint distribution for incompatible observables. This is illustrated by path-interference duality in a Young interferometer.
Abstract By filtering out the philosophic component we can be said that the EPR-paper was directed against the straightforward interpretation of the Heisenberg’s uncertainty principle or more generally the Bohr’s complementarity principle. The latter expresses contextuality of quantum measurements: dependence of measurement’s output on the complete experimental arrangement. However, Bell restructured the EPR-argument against complementarity to justify nonlocal theories with hidden variables of the Bohmian mechanics’ type. Then this Bell’s kind of nonlocality - subquantum nonlocality - was lifted to the level of quantum theory - up to the terminology “quantum nonlocality”. The aim of this short note is to explain that Bell’s test is simply a special test of local incompatibility of quantum observables, similar to interference experiments, e.g., the two-slit experiment.
This paper is aimed to dissociate nonlocality from quantum theory. We demonstrate that the tests on violation of the Bell type inequalities are simply statistical tests of local incompatibility of observables. In fact, these are tests on violation of the Bohr complementarity principle. Thus, the attempts to couple experimental violations of the Bell type inequalities with “quantum nonlocality” is really misleading. These violations are explained in the quantum theory as exhibitions of incompatibility of observables for a single quantum system, e.g., the spin projections for a single electron or the polarization projections for a single photon. Of course, one can go beyond quantum theory with the hidden variables models (as was suggested by Bell) and then discuss their possible nonlocal features. However, conventional quantum theory is local.
This is a review devoted to the complementarity–contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. Bohr contextuality is the dependence of an observable’s outcome on the experimental context; on the system–apparatus interaction. Probabilistically, complementarity means that the joint probability distribution (JPD) does not exist. Instead of the JPD, one has to operate with contextual probabilities. The Bell inequalities are interpreted as the statistical tests of contextuality, and hence, incompatibility. For context-dependent probabilities, these inequalities may be violated. I stress that contextuality tested by the Bell inequalities is the so-called joint measurement contextuality (JMC), the special case of Bohr’s contextuality. Then, I examine the role of signaling (marginal inconsistency). In QM, signaling can be considered as an experimental artifact. However, often, experimental data have signaling patterns. I discuss possible sources of signaling—for example, dependence of the state preparation on measurement settings. In principle, one can extract the measure of “pure contextuality” from data shadowed by signaling. This theory is known as contextuality by default (CbD). It leads to inequalities with an additional term quantifying signaling: Bell–Dzhafarov–Kujala inequalities.
A simple classical, deterministic, local situation violating the Bell inequality is described. The detectors used in the experiment are ideal and the observers who decide which pair of measuring devices to choose for a given pair of particles have free will. The construction uses random variables which are not jointly measurable in a single run of an experiment and the hidden variables have a nonsymmetric probability density. Such random variables are complementary but still fully classical. An assumption that classical random variables cannot satisfy any form of complementarity principle is false, and this is the loophole used in this example. A relationship to the detector inefficiency loophole is discussed.