Contextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to inconsistently connected systems (those with disturbance) by defining it in terms of the systems' couplings subject to certain constraints. Such extensions are sometimes met with skepticism. We pose the question of whether it is possible to develop a set of substantive requirements (i.e., those addressing a notion itself rather than its presentation form) such that (1) for any consistently connected system, these requirements are satisfied, but (2) they are violated for some inconsistently connected systems. We show that no such set of requirements is possible, not only for CbD but for all possible CbD-like extensions of contextuality. This follows from the fact that any extended contextuality theory T is contextually equivalent to a theory T' in which all systems are consistently connected. The contextual equivalence means the following: there is a bijective correspondence between the systems in T and T' such that the corresponding systems in T and T' are, in a well-defined sense, mere reformulations of each other, and they are contextual or noncontextual together.
Contextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to inconsistently connected systems (those with disturbance) by defining it in terms of the systems' couplings subject to certain constraints. Such extensions are sometimes met with skepticism. We pose the question of whether it is possible to develop a set of substantive requirements (i.e., those addressing a notion itself rather than its presentation form) such that (1) for any consistently connected system, these requirements are satisfied, but (2) they are violated for some inconsistently connected systems. We show that no such set of requirements is possible, not only for CbD but for all possible CbD-like extensions of contextuality. This follows from the fact that any extended contextuality theory T is contextually equivalent to a theory T' in which all systems are consistently connected. The contextual equivalence means the following: there is a bijective correspondence between the systems in T and T' such that the corresponding systems in T and T' are, in a well-defined sense, mere reformulations of each other, and they are contextual or noncontextual together.
We discuss three measures of the degree of contextuality in contextual systems of dichotomous random variables. These measures are developed within the framework of the Contextuality-by-Default (CbD) theory, and apply to inconsistently connected systems (those with 'disturbance' allowed). For one of these measures of contextuality, presented here for the first time, we construct a corresponding measure of the degree of non-contextuality in non-contextual systems. The other two CbD-based measures do not suggest ways in which degree of non-contextuality of a non-contextual system can be quantified. We find the same to be true for the contextual fraction measure developed by Abramsky, Barbosa and Mansfield. This measure of contextuality is confined to consistently connected systems, but CbD allows one to generalize it to arbitrary systems. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicioglu-Shumovsky, Einstein-Podolsky-RosenBell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected (i.e., it allows for all possible forms of "disturbance" or "signaling" in them). Consistently connected systems constitute a special case in which CbD essentially reduces to the conventional understanding of contextuality. We present a theoretical analysis of the degree of contextuality in cyclic systems (if they are contextual) and the degree of noncontextuality in them (if they are not). By contrast, all previously proposed measures of contextuality are confined to consistently connected systems, and most of them cannot be extended to measures of noncontextuality. Our measures of (non)contextuality are defined by the L-1-distance between a point representing a cyclic system and the surface of the polytope representing all possible noncontextual cyclic systems with the same single-variable marginals. We completely characterize this polytope, as well as the polytope of all possible probabilistic couplings for cyclic systems with given single-variable marginals. We establish that, in relation to the maximally tight Bell-type CbD inequality for (generally, inconsistently connected) cyclic systems, the measure of contextuality is proportional to the absolute value of the difference between its two sides. For noncontextual cyclic systems, the measure of noncontextuality is shown to be proportional to the smaller of the same difference and the L-1-distance to the surface of the box circumscribing the noncontextuality polytope. These simple relations, however, do not generally hold beyond the class of cyclic systems, and noncontextuality of a system does not follow from noncontextuality of its cyclic subsystems.
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky, Einstein-Podolsky-Rosen-Bell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected (i.e., it allows for all possible forms of ``disturbance'' or ``signaling'' in them). Consistently connected systems constitute a special case in which CbD essentially reduces to the conventional understanding of contextuality. We present a theoretical analysis of the degree of contextuality in cyclic systems (if they are contextual) and the degree of noncontextuality in them (if they are not). By contrast, all previously proposed measures of contextuality are confined to consistently connected systems, and most of them cannot be extended to measures of noncontextuality. Our measures of (non)contextuality are defined by the L1-distance between a point representing a cyclic system and the surface of the polytope representing all possible noncontextual cyclic systems with the same single-variable marginals. We completely characterize this polytope, as well as the polytope of all possible probabilistic couplings for cyclic systems with given single-variable marginals. We establish that, in relation to the maximally tight Bell-type CbD inequality for (generally, inconsistently connected) cyclic systems, the measure of contextuality is proportional to the absolute value of the difference between its two sides. For noncontextual cyclic systems, the measure of noncontextuality is shown to be proportional to the smaller of the same difference and the L1-distance to the surface of the box circumscribing the noncontextuality polytope. These simple relations, however, do not generally hold beyond the class of cyclic systems, and noncontextuality of a system does not follow from noncontextuality of its cyclic subsystems.
We show that the main idea behind contextuality-by-default (CbD), i.e., the assumption that a physical measurement has to be understood as a contextual collection of random variables, is implicit in the compatibility-hypergraph approach to contextuality (CA) and use this result to develop in the latter important concepts which were introduced in the former. We introduce in CA the non-degeneracy condition, which is the analogous of consistent connectedness, and prove that this condition is, in general, weaker than non-disturbance condition. The set of non-degenerate behaviours defines a polytope, implying that one can characterize consistent connectedness using linear inequalities. We introduce the idea of extended contextuality for behaviours and prove that a behaviour is non-contextual in the standard sense iff it is non-degenerate and non-contextual in the extended sense. Finally, we use extended scenarios and behaviours to shed new light on our results.
A noncontextual system of random variables may become contextual if one adds to it a set of new variables, even if each of them is obtained by the same context-wise function of the old variables. This fact follows from the definition of contextuality, and its demonstration is trivial for inconsistently connected systems (i.e., systems with disturbance). However, it also holds for consistently connected (and even strongly consistently connected) systems, provided one acknowledges that if a given property was not measured in a given context, this information can be used in defining functions among the random variables. Moreover, every inconsistently connected system can be presented as a (strongly) consistently connected system with essentially the same contextuality characteristics.
A primary goal in recent research on contextuality has been to extend this concept to cases of inconsistent connectedness, where observables have different distributions in different contexts. This article proposes a solution within the framework of probabi- listic causal models, which extend hidden-variables theories, and then demonstrates an equivalence to the contextuality-by-default (CbD) framework. CbD distinguishes contextuality from direct influences of context on observables, defining the latter purely in terms of probability distributions. Here, we take a causal view of direct influences, defining direct influence within any causal model as the probability of all latent states of the system in which a change of context changes the outcome of a measurement. Model-based contextuality (M-contextuality) is then defined as the necessity of stronger direct influences to model a full system than when considered individually. For consistently connected systems, M-contextuality agrees with standard contextuality. For general systems, it is proved that M-contextuality is equivalent to the property that any model of a system must contain 'hidden influences', meaning direct influences that go in opposite directions for different latent states, or equivalently signalling between observers that carries no information. This criterion can be taken as formalizing the 'no-conspiracy' principle that has been proposed in connection with CbD. M-contextuality is then proved to be equivalent to CbD-contextuality, thus providing a new interpretation of CbD-contextuality as the non-existence of a model for a system without hidden direct influences. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.