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Our aim is to compare the fundamental notions of quantum physics - contextuality vs. incompatibility. One has to distinguish two different notions of contextuality, {it Bohr-contextuality} and {it 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 {it 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 in the CHSH-scenarios (for natural quantum observables) {it contextuality is reduced to incompatibility.} However, 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 the Bohrs complementarity principle which is treated as the {it 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 cleaned of incompatibility can play the important role.
The existence of incompatible measurements is often believed to be a feature of quantum theory which signals its inconsistency with any classical worldview. To prove the failure of classicality in the sense of Kochen-Specker noncontextuality, one doe
A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible mea
We propose a new measure of relative incompatibility for a quantum system with respect to two non-commuting observables, and call it quantumness of relative incompatibility. In case of a classical state, order of observation is inconsequential, hence
One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher ord
We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when th