Probabilistic Foundations of Contextuality

Abstract

Contextuality is usually defined as absence of a joint distribution for a set of measurements (random variables) with known joint distributions of some of its subsets. However, if these subsets of measurements are not disjoint, contextuality is mathematically impossible even if one generally allows (as one must) for random variables not to be jointly distributed. To avoid contradictions one has to adopt the Contextuality-by-Default approach: measurements made in different contexts are always distinct and stochastically unrelated to each other. Contextuality is reformulated then in terms of the (im)possibility of imposing on all the measurements in a system a joint distribution of a particular kind: such that any measurements of one and the same property made in different contexts satisfy a specified property, $\mathcal{C}$. In the traditional analysis of contextuality $\mathcal{C}$ means "are equal to each other with probability 1". However, if the system of measurements violates the "no-disturbance principle", due to signaling or experimental biases, then the meaning of $\mathcal{C}$ has to be generalized, and the proposed generalization is "are equal to each other with maximal possible probability" (applied to any set of measurements of one and the same property). This approach is illustrated on arbitrary systems of binary measurements, including most of quantum systems of traditional interest in contextuality studies (irrespective of whether "no-disturbance" principle holds in them).

To be published in Fortschritte der Physik - Progress of Physics. The last version corrects a typo

References in corpus (5)

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• Contextual inferences, nonlocality, and the incompleteness of quantum mechanics
• Incompatibility of observables, channels and instruments in information theories
• Classical causal models cannot faithfully explain Bell nonlocality or Kochen-Specker contextuality in arbitrary scenarios
• Replacing Nothing with Something Special: Contextuality-by-Default and Dummy Measurements
• Quantum formalism for cognitive psychology
• Advanced Analysis of Quantum Contextuality in a Psychophysical Double-Detection Experiment
• Assumption-free derivation of the Bell-type criteria of contextuality/nonlocality
• Epistemic Odds of Contextuality in Cyclic Systems
• On Universality of Classical Probability with Contextually Labeled Random Variables
• Contextuality and dichotomizations of random variables