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Ehtibar N. Dzhafarov

Publications and source records attributed to Ehtibar N. Dzhafarov.

At least 19 recordsLinked to original sources
arXiv2026

Degrees, Levels, and Profiles of Contextuality

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.

quant-ph↗
arXiv2023

Quantum mechanics is compatible with counterfactual definiteness

Counterfactual definiteness (CFD) means that if some property is measured in some context, then the outcome of the measurement would have been the same had this property been measured in a different context. A context includes all other measurements made together with the one in question, and the spatiotemporal relations among them. The proviso for CFD is non-disturbance: any physical influence of the contexts on the property being measured is excluded by the laws of nature, so that no one measuring this property has a way of ascertaining its context. It is usually claimed that in quantum mechanics CFD does not hold, because if one assigns the same value to a property in all contexts it is measured in, one runs into a logical contradiction, or at least contravenes quantum theory and experimental evidence. We show that this claim is not substantiated if one takes into account that only one of the possible contexts can be a factual context, all other contexts being counterfactual. With this in mind, any system of random variables can be viewed as satisfying CFD. The concept of CFD is closely related to but distinct from that of noncontextuality, and it is the latter property that may or may not hold for a system, in particular being contravened by some quantum systems

quant-ph↗
arXiv2023

Hidden variables, free choice, context-independence, and all that

This paper provides a systematic account of the hidden variable models (HVMs) formulated to describe systems of random variables with mutually exclusive contexts. Any such system can be described either by a model with free choice but generally context-dependent mapping of the hidden variables into observable ones, or by a model with context-independent mapping but generally compromised free choice. These two types of HVMs are equivalent, one can always be translated into another. They are also unfalsifiable, applicable to all possible systems. These facts, the equivalence and unfalsifiability, imply that freedom of choice and context-independent mapping are no assumptions at all, and they tell us nothing about freedom of choice or physical influences exerted by contexts as these notions would be understood in science and philosophy. The conjunction of these two notions, however, defines a falsifiable HVM that describes noncontextuality when applied to systems with no disturbance or to consistifications of arbitrary systems. This HVM is most adequately captured by the term "context-irrelevance," meaning that no distribution in the model changes with context.

quant-ph↗
arXiv2023

Hypercyclic systems of measurements and patterns of contextuality

We consider four measures of contextuality, chosen for being based on the fundamental properties of the notion of contextuality, and for being applicable to arbitrary systems of measurements, both without and with disturbance. We have previously shown that no two of them are functions of each other: as systems of measurements change, either of them can change, while the other remains constant. This means that they measure different aspects of contextuality, and we proposed that rather than picking just one measure of contextuality in one specific sense, one could use all of them to characterize a contextual system by its pattern of contextuality. To study patterns of contextuality, however, one needs a systematic way of varying systems of measurements, which requires their convenient parametrization. We have convenient parametrization within the class of cyclic systems that have played a dominant role in the foundations of quantum mechanics. However, they cannot be used to study patterns of contextuality, because within this class the four measures of contextuality have been shown to be proportional to each other. In this concept paper, we introduce hypercyclic systems of measurements. They generalize cyclic systems while preserving convenient parametrization. We show that within this class of systems, the same as for systems at large, no two of the measures of contextuality are functions of each other. This means that hypercyclic systems can be used to study patterns of contextuality.

quant-ph↗
arXiv2023

The Contextuality-by-Default View of the Sheaf-Theoretic Approach to Contextuality

The Sheaf-Theoretic Contextuality (STC) theory developed by Abramsky and colleagues is a very general account of whether multiply overlapping subsets of a set, each of which is endowed with certain "local'" structure, can be viewed as inheriting this structure from a global structure imposed on the entire set. A fundamental requirement of STC is that any intersection of subsets inherit one and the same structure from all intersecting subsets. I show that when STC is applied to systems of random variables, it can be recast in the language of the Contextuality-by-Default (CbD) theory, and this allows one to extend STC to arbitrary systems, in which the requirement in question (called "consistent connectedness'" in CbD) is not necessarily satisfied. When applied to deterministic systems, such as systems of logical statements with fixed truth values, a problem arises of distinguishing lack of consistent connectedness from contextuality. I show that it can be resolved by considering systems with multiple possible deterministic realizations as quasi-probabilistic systems with Bayesian priors assigned to the realizations. Although STC and CbD have distinct native languages and distinct aims and means, the conceptual modifications presented in this paper seem to make them essentially coextensive.

quant-ph↗
arXiv2022

Contextuality and Informational Redundancy

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.

quant-ph↗
arXiv2022

Context-independent mapping and free choice are equivalent

Free choice (or statistical independence) assumption in a hidden variable model (HVM) means that the settings chosen by experimenters do not depend on the values of the hidden variable. The assumption of context-independent (CI) mapping in an HVM means that the results of a measurement do not depend on settings for other measurements. If the measurements are spacelike separated, this assumption is known as local causality. Both free choice and CI mapping assumptions are considered necessary for derivation of the Bell-type criteria of contextuality/nonlocality. It is known, however, for a variety of special cases, that the two assumptions are not logically independent. We show here, in complete generality, for any system of random variables with or without disturbance/signaling, that an HVM that postulates CI mapping is equivalent to an HVM that postulates free choice. If one denies the possibility that a given empirical scenario can be described by an HVM in which measurements depend on other measurements' settings, free choice violations should be denied too, and vice versa. KEYWORDS: Contextuality; context-independent mapping; free choice; local causality; nonlocality.

quant-ph↗
arXiv2022

Contents, Contexts, and Basics of Contextuality

This is a non-technical introduction into theory of contextuality. More precisely, it presents the basics of a theory of contextuality called Contextuality-by-Default (CbD). One of the main tenets of CbD is that the identity of a random variable is determined not only by its content (that which is measured or responded to) but also by contexts, systematically recorded conditions under which the variable is observed; and the variables in different contexts possess no joint distributions. I explain why this principle has no paradoxical consequences, and why it does not support the holistic "everything depends on everything else" view. Contextuality is defined as the difference between two differences: (1) the difference between content-sharing random variables when taken in isolation, and (2) the difference between the same random variables when taken within their contexts. Contextuality thus defined is a special form of context-dependence rather than a synonym for the latter. The theory applies to any empirical situation describable in terms of random variables. Deterministic situations are trivially noncontextual in CbD, but some of them can be described by systems of epistemic random variables, in which random variability is replaced with epistemic uncertainty. Mathematically, such systems are treated as if they were ordinary systems of random variables.

math.PR↗
arXiv2021

Contextuality and dichotomizations of random variables

The Contextuality-by-Default approach to determining and measuring the (non)contextuality of a system of random variables requires that every random variable in the system be represented by an equivalent set of dichotomous random variables. In this paper we present general principles that justify the use of dichotomizations and determine their choice. The main idea in choosing dichotomizations is that if the set of possible values of a random variable is endowed with a pre-topology (V-space), then the allowable dichotomizations split the space of possible values into two linked subsets ("linkednes" being a weak form of pre-topological connectedness). We primarily focus on two types of random variables most often encountered in practice: categorical and real-valued ones (including continuous random variables, greatly underrepresented in the contextuality literature). A categorical variable (one with a finite number of unordered values) is represented by all of its possible dichotomizations. If the values of a random variable are real numbers, then they are dichotomized by intervals above and below a variable cut point.

quant-ph↗
arXiv2021

Assumption-free derivation of the Bell-type criteria of contextuality/nonlocality

Bell-type criteria of contextuality/nonlocality can be derived without any falsifiable assumptions, such as context-independent mapping (or local causality), free choice, or no-fine-tuning. This is achieved by deriving Bell-type criteria for inconsistently connected systems (i.e., those with disturbance/signaling), based on the generalized definition of contextuality in the Contextuality-by-Default approach, and then specializing these criteria to consistently connected systems.

quant-ph↗
arXiv2021

Fechnerian Scaling: Dissimilarity Cumulation Theory

This is a chapter for the third volume of the New Handbook of Mathematical Psychology. It presented mathematical foundations of Fechnerian Scaling, a method of metrizing stimulus spaces based on subjective measures of dissimilarity.

q-bio.QM↗
arXiv2021

Contextuality and Noncontextuality Measures and Generalized Bell Inequalities for Cyclic Systems

Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicoglu-Shumovky, 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. 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.[...]

quant-ph↗
arXiv2021

Epistemic Odds of Contextuality in Cyclic Systems

Beginning with the Bell theorem, cyclic systems of dichotomous random variables have been the object of many foundational findings in quantum mechanics. Here, we ask the question: if one chooses a cyclic system "at random" (uniformly within the hyperbox of all possible systems with given marginals), what are the odds that it will be contextual? We show that the odds of contextuality rapidly tend to zero as the size of the system increases. The result is based on the Contextuality-by-Default theory, in which we do not have to assume that the systems are subject to the no-disturbance/no-signaling constraints.

quant-ph↗
arXiv2020

Systems of random variables and the Free Will Theorem

The title refers to the Free Will Theorem by Conway and Kochen whose flashy formulation is: if experimenters possess free will, then so do particles. In more modest terms, the theorem says that individual pairs of spacelike separated particles cannot be described by deterministic systems provided their mixture is the same for all choices of measurement settings. We reformulate and generalize the Free Will Theorem theorem in terms of systems of random variables, and show that the proof is based on two observations: (1) some compound systems are contextual (non-local), and (2) any deterministic system with spacelike separated components is non-signaling. The contradiction between the two is obtained by showing that a mixture of non-signaling deterministic systems, if they exist, is always noncontextual. The "experimenters' free will" (independence) assumption is not needed for the proof: it is made redundant by the assumption (1) above, critical for the proof. We next argue that the reason why an individual pair of particles is not described by a deterministic system is more elementary than in the Free Will Theorem. A system, contextual or not and deterministic or not, includes several choices of settings, each of which can be factually used without changing the system. An individual pair of particles can only afford a single realization of random variables for a single choice of settings. With this conceptualization, the "free will of experimenters" cannot be even meaningfully formulated, and the choice between the determinism and "free will of particles" becomes arbitrary and inconsequential.

quant-ph↗
arXiv2020

Contextuality Analysis of Impossible Figures

This paper has two purposes. One is to demonstrate contextuality analysis of systems of epistemic random variables. The other is to evaluate the performance of a new, hierarchical version of the measure of (non)contextuality introduced in earlier publications. As objects of analysis we use impossible figures of the kind created by the Penroses and Escher. We make no assumptions as to how an impossible figure is perceived, taking it instead as a fixed physical object allowing one of several deterministic descriptions. Systems of epistemic random variables are obtained by probabilistically mixing these deterministic systems. This probabilistic mixture reflects our uncertainty or lack of knowledge rather than random variability in the frequentist sense.

q-bio.NC↗
arXiv2020

True Contextuality Beats Direct Influences in Human Decision Making

In quantum physics there are well-known situations when measurements of the same property in different contexts (under different conditions) have the same probability distribution, but cannot be represented by one and the same random variable. Such systems of random variables are called contextual. More generally, true contextuality is observed when different contexts force measurements of the same property (in psychology, responses to the same question) to be more dissimilar random variables than warranted by the difference of their distributions. The difference in distributions is itself a form of context-dependence, but of another nature: it is attributable to direct causal influences exerted by contexts upon the random variables. The Contextuality-by-Default (CbD) theory allows one to separate true contextuality from direct influences in the overall context-dependence. The CbD analysis of numerous previous attempts to demonstrate contextuality in human judgments shows that all context-dependence in them can be accounted for by direct influences, with no true contextuality present. However, contextual systems in human behavior can be found. In this paper we present a series of crowdsourcing experiments that exhibit true contextuality in simple decision making.}{The design of these experiments is an elaboration of one introduced in the "Snow Queen" experiment (Decision 5, 193-204, 2018), where contextuality was for the first time demonstrated unequivocally.

q-bio.NC↗
arXiv2019

On contextuality and logic in scientific discourse: Reply to Aliakbarzadeh, Kitto, and Bruza

Aliakbarzadeh, Kitto, and Bruza (arXiv:1909.13048v1) criticize the Contextuality-by-Default (CbD) theory in three ways: (1) they claim an internal contradiction within the theory, that consists in both denying and accepting the equality of stochastically unrelated random variables; (2) they find CbD deficient because when one confines one's attention to consistently connected systems (those with no disturbance/signaling) one can dispense with the double-indexation of random variables; and (3) they see no relationship between double-indexation and ontic states. This commentary shows that (re$\:$1) the contradiction is inserted in the presentation of CbD by the authors of the paper; (re$\:$2) in the case of consistently connected systems CbD properly specializes to traditional understanding of contextuality; and (re$\:$3) if ontic states are understood as quantum states, they are simply not within the language of CbD, and if they are understood as domain probability spaces for context-sharing random variables, they can be trivially reconstructed.

quant-ph↗