Contextuality, Complementarity, and Bell Tests

Andrei Yu. Khrennikov

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.

David Favrholdt

Marian Kupczyński

Genovese and Piacentini [] claim that, in a recent experiment [S. Virzì , ], the Bell parameter was measured on a single pair of photons, thus challenging several conclusions and discussions on the meaning of Bell inequalities as well as certain quantum mechanical interpretations. We explain that the parameter measured in the experiment of Virzì is not the Bell parameter S , which has been discussed and estimated in many loophole-free tests. Therefore, this experiment neither challenges our understanding of Bell tests nor allows having doubts about Bohr complementarity and contextuality.

Andrzej Grudka, Karol Horodecki, Michał Horodecki +5 more

Ehtibar N. Dzhafarov

Andrei Yu. Khrennikov

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.