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

中文翻译：

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如果没有不兼容，是否可以赋予上下文任何意义？

我们的目标是比较量子物理学的基本概念——上下文与不兼容。人们必须区分两种不同的语境概念，即玻尔语境和贝尔语境。后者是通过违反非上下文（贝尔型）不等式在操作上定义的。这种上下文将与不兼容进行比较。很容易证明，对于量子可观测量，没有不兼容就没有上下文。自然的问题出现了：什么是没有不兼容的上下文？（什么是“干渣”？）通常这是一个非常复杂的问题。我们专注于四个量子可观测的上下文。我们表明，对于“自然量子可观测值”，上下文被简化为不兼容。但是，通常没有不兼容的上下文可能有一些物理内容。我们发现了一个数学约束，可以从不兼容性中提取上下文成分。但是，该约束的物理意义尚不清楚。在附录 1 中，我们简要讨论了基于玻尔的上下文不兼容原则的另一种上下文。玻尔语境在量子基础中起着至关重要的作用。事实上，不兼容是玻尔语境的结果。最后，我们指出，在物理学之外，例如，在认知心理学和决策制定中，从不相容性中提取的贝尔语境可以发挥重要作用。我们根据玻尔的语境-不兼容原则简要讨论另一种语境。玻尔语境在量子基础中起着至关重要的作用。事实上，不兼容是玻尔语境的结果。最后，我们指出，在物理学之外，例如，在认知心理学和决策制定中，从不相容性中提取的贝尔语境可以发挥重要作用。我们根据玻尔的语境-不兼容原则简要讨论另一种语境。玻尔语境在量子基础中起着至关重要的作用。事实上，不兼容是玻尔语境的结果。最后，我们指出，在物理学之外，例如，在认知心理学和决策制定中，从不相容性中提取的贝尔语境可以发挥重要作用。