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, scriptC. In the traditional analysis of contextuality scriptC 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 scriptC 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 the “no‐disturbance” principle holds in them).

Advanced analysis of quantum contextuality in a psychophysical double-detection experiment

“…What is the right way of generalizing the maximal coupling in this case? There is a compelling reason [25,27] to consider the three conteNt-sharing random variables one pair at a time, and to compute maximal couplings for them separately. This means finding a jointly distributed triple T is the maximal coupling of R As shown in [25,27], such a coupling (called multimaximal in CbD ) always exists, and it is unique (as all the random variables here are binary).…”

“…There is a compelling reason [25,27] to consider the three conteNt-sharing random variables one pair at a time, and to compute maximal couplings for them separately. This means finding a jointly distributed triple T is the maximal coupling of R As shown in [25,27], such a coupling (called multimaximal in CbD ) always exists, and it is unique (as all the random variables here are binary). The above-mentioned compelling reason for maximizing the couplings pairwise is that then, if the system is noncontextual, it will remain noncontextual after one deletes from it one or more random variables.…”

“…So each random variable in a system is double-indexed, R c q . According to the CbD theory's main principle [18,20,23,24,27,28], two random variables R q with the same conteNt in different conteXts are stochastically unrelated (which implies, among other things, that they can never be considered to be one and the same random variable). Their individual distributions may be the same but they need not be.…”

“…This experiment was reported previously [14], but its analysis was confined to extracting from it a large number of cyclic subsystems and showing all of them to be noncontextual. It is mathematically possible, however, that a system is contextual with all its cyclic subsystems being noncontextual.A satisfactory way to expand the contextuality analysis beyond cyclic systems was proposed in a recent modification of CbD, dubbed "CbD 2.0" [25,27]: it is essentially 2 the original CbD in which the measurements of the same property (say, responses to the same stimulus) are analyzed in pairs only. This modification has compelling reasons behind it, The main one is that in the modified theory a subsystem of a noncontextual system is always noncontextual.…”

“…A satisfactory way to expand the contextuality analysis beyond cyclic systems was proposed in a recent modification of CbD, dubbed "CbD 2.0" [25,27]: it is essentially 2 the original CbD in which the measurements of the same property (say, responses to the same stimulus) are analyzed in pairs only. This modification has compelling reasons behind it, The main one is that in the modified theory a subsystem of a noncontextual system is always noncontextual.…”