Welcome to the page of Marie Curie project H2020-MSCA-IF-2015 grant 696331, awarded to Dr. Irina Basieva and Dr. Emmanuel Pothos, at City, University of London.


This project explores the potential of quantum theory to provide formal models for belief updating and rumination/ ambivalence in cognition. Quantum theory is used as a basis for a cognitive model, and no assumptions are made about the brain (we assume a classical brain). The slides from a recent talk summarizing the quantum cognition programme can be found here.








This project concerns two themes:

1)      What are the principles that guide belief updating, especially under circumstances when the decision maker receives surprising information.

2)     The structure of ambivalence or rumination, when decision makers are faced with complex problems.



Navigate down to the following sections:


*publications/ summaries

*lay summaries


We summarize each theme separately:


Belief updating.

We experimentally observed violations of classical, Bayesian updating of belief. As shown, updating on strong evidence can lead to a dramatic increase of confidence (from zero, practically denying the possibility) to almost complete confidence. We explained how and why quantum probability theory can be applied to describe the experimental results and resolved the zero-prior trap, in a way which is more efficient than following Cromwell’s rule (applying only non-zero and non-one probabilities to all the options). This work provides new insights potentially applicable in the experimental and theoretical study of the phenomenon of creativity, which can be interesting not only for cognitive psychology but also for more applied subjects, such as artificial intelligence (this work was primarily reported in Basieva et al., 2017).


Rumination/ ambivalence.

For the first time we explored ambivalence in decision making, using an innovative experimental paradigm and data collection via eye tracking and mouse tracking. (The data we collected from eye tracking corresponded to momentary shifts in attention during decision making, which under some assumptions, can be associated with propensities for different decisions.) The empirical data was modelled with a sophisticated open system dynamics quantum model, which is characterized by initial oscillatory behaviour (corresponding to an early period of ambivalence) followed by stabilization (corresponding to reaching a decision, which may still involve uncertainty). We also argued that the more common drift diffusion models are poorly suited to describe our empirical results, because of the multiple reversals in decision propensities. The open systems quantum model provided excellent fits and the model parameters revealed decision structure in the eye tracking dynamics (less so in mouse tracking dynamics, though for reasons which may relate to the fidelity of data).


The belief updating experimental investigation was based on a crime mystery with a surprising suspect!


The experimental work in this strand was based on eye-tracking methods.







Our collaborators

Dr. M. Asano, Tokuyama College of Technology (in relation to quantum dynamics).

Prof. F. Bagarello, University of Palermo (in relation to state dependence in Heisenberg inequality and number operators in quantum field theory).

Dr. A. Barque-Duran, City University (in relation to design of MouseTracking experiments)

Dr. J. Broekaert, Indiana University (in relation to open system dynamics; this association started while Dr. Broekaert was at City).

Dr. P. Blaziak, Institute of Nuclear Physics Polish Academy of Sciences,31342 Krakow, Poland (in relation to dynamics of quantum systems)

Dr. A. Gloeckner, University of Hagen (in relation to drift diffusion models)

Dr. A. Khrennikov, University of Linnaeus

Dr. A. Sholz, University of Zurich (in relation to ambivalence in decision making and eye tracking)

Dr. J. Trueblood, Vanderbilt University (in relation to quantum updating).

Dr. B. von Helversen, University of Zurich (in relation to ambivalence in decision making and eye tracking)



Publications/ summaries

Note: all publications are deposited at the City Research Online open access depository.



Irina Basieva, Emmanuel Pothos, Jennifer Trueblood, Andrei Khrennikov, Jerome Busemeyer, Quantum probability updating from zero priors (by-passing Cromwell’s rule), Journal of Mathematical Psychology 77 (2017), 58-69



Cromwell’s rule (also known as the zero priors paradox) refers to the constraint of classical probability theory that if one assigns a prior probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as well (this is a straightforward implication of how Bayes’ rule works). Relatedly, hypotheses with a very low prior cannot be updated to have a very high posterior without a tremendous amount of new evidence to support them (or to make other possibilities highly improbable). Cromwell’s rule appears at odds with our intuition of how humans update probabilities. In this work, we report two simple decision making experiments, which seem to be inconsistent with Cromwell’s rule. Quantum probability theory, the rules for how to assign probabilities from the mathematical formalism of quantum mechanics, provides an alternative framework for probabilistic inference. An advantage of quantum probability theory is that it is not subject to Cromwell’s rule and it can accommodate changes from zero or very small priors to significant posteriors. We outline a model of decision making, based on quantum theory, which can accommodate the changes from priors to posteriors, observed in our experiments.



According to classical probability, belief updating follows Bayes’s law: , so that the degree of belief revision depends on the ratio of the priors . This means that very unlikely hypotheses (low ) will be revised only to a limited extent, regardless of how strong the evidence is.

               According to quantum theory, belief updating follows Luder’s law: , where the ‘P’ objects are projection operators in quantum theory. Luder’s law allows jumps from priors to posteriors of any size, even when the priors are very low.

               We tested participants with a crime mystery: participants were presented with a hypothetical theft scenario. The initial information made some suspects more likely, others very unlikely. Following new information, participants were asked to re-evaluate the likelihood of different suspects being guilty. We modelled belief updating with both probability rules and concluded that the quantum one provided better description than the classical one.


Irina Basieva, Polina Khrennikova, Emmanuel M. Pothos, Masanari Asano, & Andrei Khrennikov (in press). Quantum-like model of subjective expected utility. Journal of Mathematical Economics.



We present a very general quantum-like model of lottery selection based on representation of beliefs of an agent by pure quantum states. Subjective probabilities are mathematically realized in the framework of quantum probability (QP). Utility functions are borrowed from the classical decision theory. But in the model they are represented not only by their values. Heuristically one can say that each value ui = u(xi) is surrounded by a cloud of information related to the event (A, xi). An agent processes this information by

using the rules of quantum information and QP. This process is very complex; it combines counterfactual reasoning for comparison between preferences for different outcomes of lotteries which are in general complementary. These comparisons induce interference type effects (constructive or destructive). The decision process is mathematically represented by the comparison operator and the outcome of this process is determined by the sign of the value of corresponding quadratic form on the belief state. This operational process can be decomposed into a few subprocesses. Each of them can be formally treated as a comparison of subjective expected utilities and interference factors (the latter express, in particular, risks related to lottery selection). The main aim of this paper is to analyze the mathematical structure of these processes in the most general situation: representation of lotteries by noncommuting operators.


Jan Broekaert, Irina Basieva, Pawel Blasiak, Emmanuel M. Pothos, Quantum-like dynamics applied to cognition: a consideration of available options, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375 (2017), 20160387



Quantum Probability Theory (QPT) has provided a novel, rich mathematical framework for cognitive modelling, especially for situations which appear paradoxical from classical perspectives. This work concerns the dynamical aspects of QPT, as relevant to cognitive modelling. We aspire to shed light on how the mind’s driving potentials (encoded in Hamiltonian and Lindbladian operators) impact the evolution of a mental state. Some existing QPT cognitive models do employ dynamical aspects when considering how a mental state changes with time, but it is often the case that several simplifying assumptions are introduced. What kind of modelling flexibility do QPT dynamics offer without any simplifying assumptions and is it likely that such flexibility will be relevant in cognitive modelling? We consider a series of nested QPT dynamical models, constructed with a view to accommodate results from a simple, hypothetical experimental paradigm on decision making. We consider Hamiltonians more complex than the ones which have traditionally been employed with a view to explore the putative explanatory value of this additional complexity. We then proceed to compare simple models with extensions regarding both the initial state (e.g., mixed state with a specific orthogonal decomposition; a general mixed state) and the dynamics (by introducing Hamiltonians which destroy the separability of the initial structure and by considering an open-systems extension). We illustrate the relations between these models mathematically and numerically.



This figure provides a helpful illustration of the dynamical options provided by various combinations of the Pauli spin matrices.


A link to a technical presentation outlining this work is here.


Sholz, A., Basieva, I., Barque-Duran, A., Gloeckner, A., von Helversen, B, & Pothos, E. M. (in preparation). Characterizing structure in eye tracking data.



Eye tracking has been extensively employed to study momentary shifts in attention during decision making and several researchers have attempted to establish links between characteristics of eye tracking dynamics and more directly relevant behavioral variables (e.g., the actual decisions). However, to our knowledge there have not been any attempts to directly model eye tracking dynamics in a decision task, including based on the dominant framework for decision dynamics generally, drift diffusion models. Can drift diffusion models be employed to describe eye tracking dynamics in a decision task? We argue that such models are not ideally suited to the task at hand and so motivate an alternative framework, based on open systems quantum theory. We model eye tracking curves from a decision task and explore the interpretability of parameters.



We considered everyday ‘complex’ decision problems, for example the decision of whether a hypothetical couple would decide to keep a stray dog or not, based on information about the criteria relevant to that couple. For example, the arguments would be (arranged in the same way as would be presented to participants):



Our guiding consideration of interest was that the resolution of these (there were three) decision dilemmas would involve a process of back and forth – an intuition that one decision is better followed by an intuition that the other decision is better – for a number of cycles, before converging to the eventual decision.

               We employed eye tracking to monitor momentary attentional shifts, which are generally thought to be related to attentional propensities. The following figure provides a typical scanpath illustrating the assumption that for ‘complex’ problems of this sort there are multiple reversals in attentional focus (and, we believe, attentional propensities, though it is important to note that this was not directly tested in this work).




For the modelling we employed a quantum model based on open system dynamics. A prior, this is a suitable choice, because open system dynamics in quantum theory follow a pattern of initial oscillation (which cognitively can be associated with initial ambivalence) giving rise to a stable pattern (which cognitively can be associated with stabilization or resolution of opinion, which can still include some uncertainty). We note the form of the model, without commentary:


, where


We also present these diagrams as illustrations of model behaviour.



a12_a23_rainbow (002)

a12_a32_rainbow (002)

a12_d_rainbow (002)

Illustrating the six parameter; in all cases the vertical shows  at large times. The left figure illustrates the way dominance for yes response requires both drift for a yes response () and drift away from a no response (), even if the former influence is more important, with other parameters set as d=0.1,  and adjT=1 (in all cases). The middle figure shows how drift for a yes response () balances out drift for a no response (), with other parameters as . The right illustrates that high values of d prevent  strong dominance of (e.g.) yes response; other parameters .


Empirical results favoured the quantum model. A presentation outlining the key model elements can be found here.


Fabio Bagarello, Irina Basieva, Emmanuel M. Pothos, Andrei Khrennikov, Quantum like modeling of decision making: Quantifying uncertainty with the aid of Heisenberg–Robertson inequality, Journal of Mathematical Psychology 84 (2018) 49-56



This paper contributes to quantum-like modeling of decision making (DM) under uncertainty through application of Heisenberg's uncertainty principle (in the form of the Robertson inequality). In this paper we apply this instrument to quantify uncertainty in DM performed by quantum-like agents. As an example, we apply the Heisenberg uncertainty principle to the determination of mutual interrelation of uncertainties for “incompatible questions” used to be asked in political opinion pools. We also consider the problem of representation of decision problems, e.g., in the form of questions, by Hermitian operators, commuting and noncommuting, corresponding to compatible and incompatible questions respectively. Our construction unifies the two different situations (compatible versus incompatible mental observables), by means of a single Hilbert space and of a deformation parameter which can be tuned to describe these opposite cases. One of the main foundational consequences of this paper for cognitive psychology is formalization of the mutual uncertainty about incompatible questions with the aid of Heisenberg's uncertainty principle implying the mental state dependence of (in)compatibility of questions.


Fabio Bagarello, Irina Basieva, Andrei Khrennikov, Quantum field inspired model of decision making: Asymptotic stabilization of belief state via interaction with surrounding mental environment, Journal of Mathematical Psychology 82 (2018) 159 – 168


Irina Basieva, Andrei Khrennikov, Decision-Making and Cognition Modeling from the Theory of Mental Instruments, Chapter in a book The Palgrave Handbook of Quantum Models in Social Science (2017), pp 75-93.


Masanari Asano, Irina Basieva, Emmanuel Pothos, Andrei Khrennikov, State Entropy and Differentiation Phenomenon, Entropy 20/6 (2018), 394.



In the formalism of quantum theory, a state of a system is represented by a density operator. Mathematically, a density operator can be decomposed into a weighted sum of (projection) operators representing an ensemble of pure states (a state distribution), but such decomposition is not unique. Various pure states distributions are mathematically described by the same density operator. These

distributions are categorized into classical ones obtained from the Schatten decomposition and other, non-classical, ones. In this paper, we define the quantity called the state entropy. It can be considered as a generalization of the von Neumann entropy evaluating the diversity of states constituting a distribution. Further, we apply the state entropy to the analysis of non-classical states created at the intermediate stages in the process of quantum measurement. To do this, we employ the model of differentiation, where a system experiences step by step state transitions under the influence of environmental factors. This approach can be used for modeling various natural and mental phenomena: cell’s differentiation, evolution of biological populations, and decision making.


Emmanuel M. Pothos, Irina Basieva, Albert Barque-Duran, Katy Tapper, Andrei Khrennikov, Information overflow and persistent disagreement (in preparation)



There have been concerns that modern political debate involves less truth and more truthiness. A key aspect of truthiness is persistent disagreement. We focus on factual questions (i.e., ones not depending on personal preference or values) and for well-meaning individuals (who try to mitigate the influence of careless processing, emotions, or other biases). We recognize information overflow as an important characteristic of modern political debate. Heuristics and biases research provides several insights for why individuals may disagree, but less if they are well-meaning. Classical probability theory (CPT) can explain why reasoners would be challenged under circumstances of information overflow, because of the requirement of constructing large Boolean algebras: it makes sense to break a complex question, e.g. Brexit, into smaller themes, however, CPT does not provide a prescription for doing so. Quantum probability theory (QPT), the rules for probabilistic assignment from quantum mechanics, formalizes the way a Boolean algebra can be simplified into a partial Boolean algebra, through the QPT notion of incompatibility (incompatible questions cannot have a joint probability distribution). The price for this simplification is a picture of the world which may be inaccurate. We discuss further implications and possible ameliorating procedures for reducing this kind of truthiness in modern political debate.


Andrei Khrennikov, Irina Basieva, Emmanuel M. Pothos, & I. Yamato (in press). Quantum probability in decision making from quantum information representation of neuronal states. Scientific Reports.



The recent wave of interest to modeling the process of decision making with the aid of the quantum formalism gives rise to the following question: ‘How can neurons generate quantum-like statistical data?’ (There is a plenty of such data in cognitive psychology and social science.) Our model is based on quantum-like representation of uncertainty in generation of action potentials. This uncertainty is a consequence of complexity of electrochemical processes in the brain; in particular, uncertainty of triggering an action potential by the membrane potential. Quantum information state spaces can be considered as extensions of classical information spaces corresponding to neural codes; e.g., 0/1, quiescent/firing neural code. The key point is that processing of information by the brain involves superpositions of such states. Another key point is that a neuronal group performing some psychological function F is an open quantum system. It interacts with the surrounding electrochemical environment. The process of decision making is described as decoherence in the basis of eigenstates of F: A decision state is a steady state. This is a linear representation of complex nonlinear dynamics of electrochemical states. Linearity guarantees exponentially fast convergence to the decision state.



Lay summaries


The research carried out concerns belief updating and rumination/ ambivalence. Belief updating is about how we update our beliefs, in light of presented information. A simple example is this: suppose you are interested in whether it is likely to rain or not (this is your belief). It is autumn and you have some expectations of whether it is likely to rain or not. But then you look out of the window and receive some information, say it is really sunny. How does this new information change your belief? Rumination/ ambivalence concerns how we deal with complex decisions. By complex decisions, we mean decisions such that the best option is just not (perhaps immediately) obvious. Consider the issue of trying to get a jumper and you narrow down your choice between a green and a blue one. There is no simple ‘utility’ to make you decide, since the considerations that favour either option may be complex and not matched. We believe that such decision problems involve a process of going back and forth, before eventually settling on a particular decision.

            We have created a brief article expanding on these ideas, which can be found here.