On contextuality in behavioural data

Open Systems, Quantum Probability, and Logic for Quantum-like Modeling in Biology, Cognition, and Decision-Making

The aim of this review is to highlight the possibility of applying the mathematical formalism and methodology of quantum theory to model behavior of complex biosystems, from genomes and proteins to animals, humans, and ecological and social systems. Such models are known as quantum-like, and they should be distinguished from genuine quantum physical modeling of biological phenomena. One of the distinguishing features of quantum-like models is their applicability to macroscopic biosystems or, to be more precise, to information processing in them. Quantum-like modeling has its basis in quantum information theory, and it can be considered one of the fruits of the quantum information revolution. Since any isolated biosystem is dead, modeling of biological as well as mental processes should be based on the theory of open systems in its most general form-the theory of open quantum systems. In this review, we explain its applications to biology and cognition, especially theory of quantum instruments and the quantum master equation. We mention the possible interpretations of the basic entities of quantum-like models with special interest given to QBism, as it may be the most useful interpretation.

On Applicability of Quantum Formalism to Model Decision Making: Can Cognitive Signaling Be Compatible with Quantum Theory?

This note is devoted to the problem of signaling (marginal inconsistency) in the Bell-type experiments with physical and cognitive systems. It seems that in quantum physics, this problem is still not taken seriously. Only recently have experimenters started to check the signaling hypothesis for their data. For cognitive systems, signaling was statistically significant in all experiments (typically for decision making) performed up to today. Here, one cannot simply ignore this problem. Since signaling contradicts the quantum theory of measurement for compatible observables, its statistical significance in experiments with humans can be considered as an objection for quantum-like modeling-applications of quantum theory to cognition, decision making, psychology, economics and finance, social and political science. In this paper, we point to two possible sources of signaling generation that are consistent with quantum measurement theory. Thus, the signaling objection for quantum-like modeling is not catastrophic. One of these sources is the direct physical signaling about selection of experimental settings, questions or tasks in quantum-like studies. Another possible source is a state modification dependent on experimental settings. The latter was a rather common source of signaling in quantum physics. Since the physical size of the brain is very small comparing with the light velocity, it seems to be impossible to prevent the direct physical signaling (with electromagnetic waves) between the brain's areas processing two questions a and b. However, if, for these questions, not the electromagnetic waves, but electrochemical communication plays the crucial role, the experimenter may hope to make signaling weaker by answering the questions faster. The problem of question-dependent mental state modification seems to be solvable via smarter experimental design. This paper can be useful both for physicists interested in quantum foundations and for researchers working in quantum-like studies, e.g., applying the quantum theory to model decision making or psychological effects. This paper is solely about quantum theory. Thus, we do not consider general contextual probabilistic models.

Contextuality, Complementarity, Signaling, and Bell Tests

This is a review devoted to the complementarity–contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. Bohr contextuality is the dependence of an observable’s outcome on the experimental context; on the system–apparatus interaction. Probabilistically, complementarity means that the joint probability distribution (JPD) does not exist. Instead of the JPD, one has to operate with contextual probabilities. The Bell inequalities are interpreted as the statistical tests of contextuality, and hence, incompatibility. For context-dependent probabilities, these inequalities may be violated. I stress that contextuality tested by the Bell inequalities is the so-called joint measurement contextuality (JMC), the special case of Bohr’s contextuality. Then, I examine the role of signaling (marginal inconsistency). In QM, signaling can be considered as an experimental artifact. However, often, experimental data have signaling patterns. I discuss possible sources of signaling—for example, dependence of the state preparation on measurement settings. In principle, one can extract the measure of “pure contextuality” from data shadowed by signaling. This theory is known as contextuality by default (CbD). It leads to inequalities with an additional term quantifying signaling: Bell–Dzhafarov–Kujala inequalities.

Varieties of Selective Influence: Toward a More Complete Taxonomy and Implications for Systems Identification

All science, including psychological science, is subject to what Townsend and Ashby have called the principle of correspondent change which ensures that experimental manipulations act as informed agents with respect to predictions and testing critical theoretical features. Mostly, this type of program goes unspoken. Within the general field known as the information processing approach, S. Sternberg invented the additive factors method in which the aforesaid feature plays a major and explicit role. We call this approach a theory driven methodology because the scientist formulates a set of theories or models and then formulates experimental variables that will permit strong tests among the hypothetical alternatives. Our term for the general approach is systems factorial technology. Often, these tests can be accomplished with qualitative, non-parametric, distribution free methods, but our so-called sieve method advocates, once the initial qualitative steps are accomplished, a move to assessing more detail parametric versions of the model classes. Over the decades, the meta-theory underpinning SFT and like approaches has evidenced dramatic growth in both expanse and depth. Particularly, the critical assumption of selective influence, testable to some extent, has received extensive and sophisticated treatment. The various central allied concepts are interlinked but do not form a simple linearly-ordered chain. This study carries on exploration of the central concepts and relationships and their implications for psychological research.

Application of Theory of Quantum Instruments to Psychology: Combination of Question Order Effect with Response Replicability Effect

Recently, quantum formalism started to be actively used outside of quantum physics: in psychology, decision-making, economics, finances, and social science. Human psychological behavior is characterized by a few basic effects; one of them is the question order effect (QOE). This effect was successfully modeled (Busemeyer–Wang) by representing questions A and B by Hermitian observables and mental-state transformations (back action of answering) by orthogonal projectors. However, then it was demonstrated that such representation cannot be combined with another psychological effect, known as the response replicability effect (RRE). Later, this no-go result was generalized to representation of questions and state transformations by quantum instruments of the atomic type. In light of these results, the possibility of using quantum formalism in psychology was questioned. In this paper, we show that, nevertheless, the combination of the QOE and RRE can be modeled within quantum formalism, in the framework of theory of non-atomic quantum instruments.

Perspectives on Correctness in Probabilistic Inference from Psychology

Research into decision making has enabled us to appreciate that the notion of correctness is multifaceted. Different normative framework for correctness can lead to different insights about correct behavior. We illustrate the shifts for correctness insights with two tasks, the Wason selection task and the conjunction fallacy task; these tasks have had key roles in the development of logical reasoning and decision making research respectively. The Wason selection task arguably has played an important part in the transition from understanding correctness using classical logic to classical probability theory (and information theory). The conjunction fallacy has enabled a similar shift from baseline classical probability theory to quantum probability. The focus of this overview is the latter, as it represents a novel way for understanding probabilistic inference in psychology. We conclude with some of the current challenges concerning the application of quantum probability theory in psychology in general and specifically for the problem of understanding correctness in decision making.

Resource theory of contextuality

In addition to the important role of contextuality in foundations of quantum theory, this intrinsically quantum property has been identified as a potential resource for quantum advantage in different tasks. It is thus of fundamental importance to study contextuality from the point of view of resource theories, which provide a powerful framework for the formal treatment of a property as an operational resource. In this contribution, we review recent developments towards a resource theory of contextuality and connections with operational applications of this property. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

Relating causal and probabilistic approaches to contextuality

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'.

Generalized Fock space and contextuality

This paper is devoted to linear space representations of contextual probabilities-in generalized Fock space. This gives the possibility to use the calculus of creation and annihilation operators to express probabilistic dynamics in the Fock space (in particular, the wide class of classical kinetic equations). In this way, we reproduce the Doi-Peliti formalism. The context-dependence of probabilities can be quantified with the aid of the generalized formula of total probability-by the magnitude of the interference term. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

Classical (Local and Contextual) Probability Model for Bohm-Bell Type Experiments: No-Signaling as Independence of Random Variables

We start with a review on classical probability representations of quantum states and observables. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The main part of the paper is devoted to the conditional probability model with conditioning on the selection of the pairs of experimental settings. From the viewpoint of quantum foundations, this is a local contextual hidden-variables model. Following the recent works of Dzhafarov and collaborators, we apply our conditional probability approach to characterize (no-)signaling. Consideration of the Bohm–Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science. The main message of this paper (rooted to Ballentine) is that quantum probabilities and more generally probabilities related to the Bohm–Bell type experiments (not only in physics, but also in psychology, sociology, game theory, economics, and finances) can be classically represented as conditional probabilities.

Contextuality Analysis of the Double Slit Experiment (with a Glimpse into Three Slits)

The Contextuality-by-Default theory is illustrated on contextuality analysis of the idealized double-slit experiment. The experiment is described by a system of contextually labeled binary random variables each of which answers the question: Has the particle hit the detector, having passed through a given slit (left or right) in a given state (open or closed)? This system of random variables is a cyclic system of rank 4, formally the same as the system describing the Einsten-Podolsky-Rosen-Bell paradigm with signaling. Unlike the latter, however, the system describing the double-slit experiment is always noncontextual, i.e., the context-dependence in it is entirely explainable in terms of direct influences of contexts (closed-open arrangements of the slits) upon the marginal distributions of the random variables involved. The analysis presented is entirely within the framework of abstract classical probability theory (with contextually labeled random variables). The only physical constraint used in the analysis is that a particle cannot pass through a closed slit. The noncontextuality of the double-slit system does not generalize to systems describing experiments with more than two slits: in an abstract triple-slit system, almost any set of observable detection probabilities is compatible with both a contextual scenario and a noncontextual scenario of the particle passing though various combinations of open and closed slits (although the issue of physical realizability of these scenarios remains open).

Contextuality in canonical systems of random variables

Random variables representing measurements, broadly understood to include any responses to any inputs, form a system in which each of them is uniquely identified by its content (that which it measures) and its context (the conditions under which it is recorded). Two random variables are jointly distributed if and only if they share a context. In a canonical representation of a system, all random variables are binary, and every content-sharing pair of random variables has a unique maximal coupling (the joint distribution imposed on them so that they coincide with maximal possible probability). The system is contextual if these maximal couplings are incompatible with the joint distributions of the context-sharing random variables. We propose to represent any system of measurements in a canonical form and to consider the system contextual if and only if its canonical representation is contextual. As an illustration, we establish a criterion for contextuality of the canonical system consisting of all dichotomizations of a single pair of content-sharing categorical random variables.This article is part of the themed issue 'Second quantum revolution: foundational questions'.

Spin and Wind Directions II: A Bell State Quantum Model

In the first half of this two-part article (Aerts et al. in Found Sci. doi:10.1007/s10699-017-9528-9, 2017b), we analyzed a cognitive psychology experiment where participants were asked to select pairs of directions that they considered to be the best example of Two Different Wind Directions, and showed that the data violate the CHSH version of Bell's inequality, with same magnitude as in typical Bell-test experiments in physics. In this second part, we complete our analysis by presenting a symmetrized version of the experiment, still violating the CHSH inequality but now also obeying the marginal law, for which we provide a full quantum modeling in Hilbert space, using a singlet state and suitably chosen product measurements. We also address some of the criticisms that have been recently directed at experiments of this kind, according to which they would not highlight the presence of genuine forms of entanglement. We explain that these criticisms are based on a view of entanglement that is too restrictive, thus unable to capture all possible ways physical and conceptual entities can connect and form systems behaving as a whole. We also provide an example of a mechanical model showing that the violations of the marginal law and Bell inequalities are generally to be associated with different mechanisms.

Spin and Wind Directions I: Identifying Entanglement in Nature and Cognition

We present a cognitive psychology experiment where participants were asked to select pairs of spatial directions that they considered to be the best example of Two different wind directions. Data are shown to violate the CHSH version of Bell's inequality with the same magnitude as in typical Bell-test experiments with entangled spins. Wind directions thus appear to be conceptual entities connected through meaning, in human cognition, in a similar way as spins appear to be entangled in experiments conducted in physics laboratories. This is the first part of a two-part article. In the second part (Aerts et al. in Found Sci, 2017) we present a symmetrized version of the same experiment for which we provide a quantum modeling of the collected data in Hilbert space.

Preface of the special issue quantum foundations: information approach

This special issue is based on the contributions of a group of top experts in quantum foundations and quantum information and probability. It enlightens a number of interpretational, mathematical and experimental problems of quantum theory.