Contextuality

Andrei Yu. Khrennikov

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 Bohr's 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.

Ehtibar N. Dzhafarov, Victor H. Cervantes

We introduce a new notion, that of a contextuality profile of a system of random variables. Rather than characterizing a system's contextuality by a single number, its overall degree of contextuality, we show how it can be characterized by a curve relating degree of contextuality to level at which the system is considered. A system is represented at level n if one only considers the joint distributions with no more than n variables, ignoring higher-order joint distributions. We show that the level-wise contextuality analysis can be used in conjunction with any well-constructed measure of contextuality. We present a method of concatenated systems to explore contextuality profiles systematically, and we apply it to the contextuality profiles for three major measures of contextuality proposed in the literature.

Andrei Yu. Khrennikov

Abstract Our aim is to compare the fundamental notions of quantum physics - contextuality vs. incompatibility. One has to distinguish two different notions of contextuality,Bohr-contextualityandBell-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 isno 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.

Travis Norsen