Can There be Given Any Meaning to Contextuality Without Incompatibility?
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Can There be Given Any Meaning to Contextuality Without Incompatibility? Andrei Khrennikov1 Received: 19 May 2020 / Accepted: 16 November 2020 / © The Author(s) 2020
Abstract Our aim is to compare the fundamental notions of quantum physics - contextuality vs. incompatibility. One has to distinguish two different notions of contextuality, Bohrcontextuality and Bell-contextuality. The latter is defined operationally via violation of noncontextuality (Bell type) inequalities. This sort of contextuality will be compared with incompatibility. It is easy to show that, for quantum observables, there is no contextuality without incompatibility. The natural question arises: What is contextuality without incompatibility? (What is “dry-residue”?) Generally this is the very complex question. We concentrated on contextuality for four quantum observables. We shown that, for “natural quantum observables” , contextuality is reduced to incompatibility. But, generally contextuality without incompatibility may have some physical content. We found a mathematical constraint extracting the contextuality component from incompatibility. However, the physical meaning of this constraint is not clear. In Appendix 1, we briefly discuss another sort of contextuality based on Bohr’s contextuality-incompatibility principle. Bohr-contextuality plays the crucial role in quantum foundations. Incompatibility is, in fact, a consequence of Bohr-contextuality. Finally, we remark that outside of physics, e.g., in cognitive psychology and decision making Bell-contextuality distilled of incompatibility can play the important role. Keywords Contextuality · Incompatibility · Complementarity principle · Joint probability distribution · Noncontextual inequalities · Product of commutators
1 Introduction Contextuality formalized in the form of violation of noncontextuality inequalities, Bellcontextuality [1, 2], is a hot topic in quantum physics (see, e.g., [3, 4] and references herein). Unfortunately, it is typically presented in the mathematical framework and its physical meaning is unclear.
Andrei Khrennikov
[email protected] 1
International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, V¨axj¨o, SE-351 95, Sweden
International Journal of Theoretical Physics
We stress that, in fact, one has to distinguish two different notions of contextuality, Bohrcontextuality and Bell-contextuality. In this paper, we consider the latter, see Appendix 2 for for the former. Therefore we shall speak simply about contextuality (having in mind Bell-contextuality). We point out that discussions on this sort contextuality are typically started with heuristically more attractive definition going back to Bell [1, 2]. It can be called explicit contextuality: if A, B, C are three quantum observables, such that Acompatible with B and C, a measurement of A might give different result depending upon whether A is measured with B or with C. However, for incompatible observables B and C, this statement is not testable experimental
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