Quantum Probability Model Research Articles

Decision-Making & Quantum Mechanical Models of Cognitive Processing

There have been numerous neurobiological theories positing that quantum mechanics is importantly responsible for certain psychological phenomena, such as consciousness and conscious decision-making. In this respect, such theories understand the relevant aspects of the brain to be “quantum mechanical.” Although it is highly questionable whether any of these theories have the requisite or sufficient empirical evidence strictly within the domain of neuroscience to appropriately justify their views, a different approach to arguing that quantum mechanics is importantly relevant to human decisionmaking will be discussed. By putting various puzzle pieces together of empirical evidence in psychology that quantum probability models describe certain human cognitive decision-making processes, I contend that human decision-making overall is best described by quantum mechanical processes. Finally, I at times provide a novel defense of this view from a number of objections.

Elements of Quantum Finance Models

Both academic research and practical application of mathematical finance have been extremely fruitful since the seminal work of Black-Scholes-Merton in the early 1970s. In this framework, the prices of financial assets are modeled as stochastic processes in probability spaces inside which the machinery of stochastic calculus is a powerful tool. The fundamental asset pricing theorem states that the absence of arbitrage opportunities in a market is equivalent to the existence of a probability measure, equivalent to the objective probability, under which the discounted prices of the assets become local martingales. This linkage between finance on the one hand and probability theory on the other is the key to the success of mathematical finance. In this note, we show that it is possible to extend a classical probability model to a quantum probability model. Classical stochastic calculus is replaced by its quantum counterpart on a Boson Fock space. In particular, we show that the fundamental asset pricing theorem remains valid in this non-commutative setting. By its very nature of coping with non-commutative random variables however, many essential elements of classic mathematical finance but in quantum form have not been recovered in this note. A couple that have been established are not in their full generality. For instance, existence of non-trivial self-financing portfolios and preserving of self-financing property under change of numeraire can only be reclaimed subject to technical conditions. This writing puts together a few helpful results for the purpose of future studies of this potentially new research subject.

An investigation of a quantum probability model for the constructive effect of affective evaluation.

The idea that choices can have a constructive effect has received a great deal of empirical support. The act of choosing appears to influence subsequent preferences for the options available. Recent research has proposed a cognitive model based on quantum probability (QP), which suggests that whether or not a participant provides an affective evaluation for a positively or negatively valenced stimulus can also be constructive and so, for example, influence the affective evaluation of a second oppositely valenced stimulus. However, there are some outstanding methodological questions in relation to this previous research. This paper reports the results of three experiments designed to resolve these questions. Experiment 1, using a binary response format, provides partial support for the interaction predicted by the QP model; and Experiment 2, which controls for the length of time participants have to respond, fully supports the QP model. Finally, Experiment 3 sought to determine whether the key effect can generalize beyond affective judgements about visual stimuli. Using judgements about the trustworthiness of well-known people, the predictions of the QP model were confirmed. Together, these three experiments provide further support for the QP model of the constructive effect of simple evaluations.

Quantization and Quantum-Like Phenomena: A Number Amplitude Approach

Historically, quantization has meant turning the dynamical variables of classical mechanics that are represented by numbers into their corresponding operators. Thus the relationships between classical variables determine the relationships between the corresponding quantum mechanical operators. Here, we take a radically different approach to this conventional quantization procedure. Our approach does not rely on any relations based on classical Hamiltonian or Lagrangian mechanics nor on any canonical quantization relations, nor even on any preconceptions of particle trajectories in space and time. Instead we examine the symmetry properties of certain Hermitian operators with respect to phase changes. This introduces harmonic operators that can be identified with a variety of cyclic systems, from clocks to quantum fields. These operators are shown to have the characteristics of creation and annihilation operators that constitute the primitive fields of quantum field theory. Such an approach not only allows us to recover the Hamiltonian equations of classical mechanics and the Schrodinger wave equation from the fundamental quantization relations, but also, by freeing the quantum formalism from any physical connotation, makes it more directly applicable to non-physical, so-called quantum-like systems. Over the past decade or so, there has been a rapid growth of interest in such applications. These include, the use of the Schrodinger equation in finance, second quantization and the number operator in social interactions, population dynamics and financial trading, and quantum probability models in cognitive processes and decision-making. In this paper we try to look beyond physical analogies to provide a foundational underpinning of such applications.

A Quantum Finance Model

Both academic research and practical application of mathematical finance have been extremely fruitful since the seminal work of Black-Scholes-Merton in the early 1970s. In this framework, the prices of financial assets are modeled as stochastic processes in probability spaces inside which the machinery of stochastic calculus is a powerful tool. The fundamental asset pricing theorem states that the absence of arbitrage opportunities in a market is equivalent to the existence of a probability measure, equivalent to the objective probability, under which the discounted prices of the assets become local martingales. This linkage between finance on the one hand and the probability theory on the other is the key to the success of mathematical finance. In this note, we show that it is possible to extend the classical probability model to a quantum probability model. The classical stochastic calculus is replaced by its quantum counterpart on the Boson Fock space. In particular, we show that the fundamental asset pricing theorem remains valid in this non-commutative setting. As an application, prices of quantum European options are obtained.

Two-slit experiment: quantum and classical probabilities

Inter-relation between quantum and classical probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by classical probability theory with the viewpoint that quantum–classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of classical probability theory, the law of total probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the classical (Kolmogorov, 1933) probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-classicality of quantum probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincaré disk model for the Lobachvesky plane).

Macroscopic noncontextuality as a principle for almost-quantum correlations

Quantum mechanics allows only certain sets of experimental results (or ``probabilistic models'') for Bell-type quantum nonlocality experiments. A derivation of this set from simple physical or information theoretic principles would represent an important step forward in our understanding of quantum mechanics, and this problem has been intensely investigated in recent years. ``Macroscopic locality,'' which requires the recovery of locality in the limit of large numbers of trials, is one of several principles discussed in the literature that place a bound on the set of quantum probabilistic models. A similar question can also be asked about probabilistic models for the more general class of quantum contextuality experiments. Here, we extend the macroscopic locality principle to this more general setting, using the hypergraph approach of Ac\'{\i}n, Fritz, Leverrier, and Sainz [Comm. Math. Phys. 334(2), 533--628 (2015)], which provides a framework to study both phenomena of nonlocality and contextuality in a unified manner. We find that the set of probabilistic models allowed by our macroscopic noncontextuality principle is equivalent to an important and previously studied set in this formalism, which is slightly larger than the quantum set. In the particular case of Bell scenarios, this set is equivalent to the set of ``almost-quantum'' models, which is of particular interest since the latter was recently shown to satisfy all but one of the principles that have been proposed to bound quantum probabilistic models, without being implied by any of them (or even their conjunction). Our condition is the first characterization of the almost-quantum set from a simple physical principle.

Conjunction and negation of natural concepts: A quantum-theoretic modeling

We perform two experiments with the aim to investigate the effects of negation on the combination of natural concepts. In the first experiment, we test the membership weights of a list of exemplars with respect to two concepts, e.g., Fruits and Vegetables, and their conjunction Fruits And Vegetables. In the second experiment, we test the membership weights of the same list of exemplars with respect to the same two concepts, but negating the second, e.g., Fruits and Not Vegetables, and again their conjunction Fruits And Not Vegetables. The collected data confirm existing results on conceptual combination, namely, they show dramatic deviations from the predictions of classical (fuzzy set) logic and probability theory. More precisely, they exhibit conceptual vagueness, gradeness of membership, overextension and double overextension of membership weights with respect to the given conjunctions. Then, we show that the quantum probability model in Fock space recently elaborated to model Hampton’s data on concept conjunction (Hampton, 1988a) and disjunction (Hampton, 1988b) faithfully accords with the collected data. Our quantum-theoretic modeling enables to describe these non-classical effects in terms of genuine quantum aspects, namely ‘contextuality’, ‘superposition’, ‘interference’ and ‘emergence’. The obtained results confirm and strengthen the analysis in Aerts (2009a) and Sozzo (2014) on the identification of quantum structures in experiments on conceptual vagueness. And, more, they can be inserted within the general research on the identification of quantum structures in cognitive processes.

The conjunction fallacy, confirmation, and quantum theory: comment on Tentori, Crupi, and Russo (2013).

The conjunction fallacy refers to situations when a person judges a conjunction to be more likely than one of the individual conjuncts, which is a violation of a key property of classical probability theory. Recently, quantum probability (QP) theory has been proposed as a coherent account of these and many other findings on probability judgment "errors" that violate classical probability rules, including the conjunction fallacy. Tentori, Crupi, and Russo (2013) presented an alternative account of the conjunction fallacy based on the concept of inductive confirmation. They presented new empirical findings consistent with their account, and they also claimed that these results were inconsistent with the QP theory account. This comment proved that our QP model for the conjunction fallacy is completely consistent with the main empirical results from Tentori et al. (2013). Furthermore, we discuss experimental tests that can distinguish the 2 alternative accounts.

Structured representations in a quantum probability model of similarity

Recently, Busemeyer et al. (2011) presented a model for how the conjunction fallacy (Tversky & Kahneman, 1983) emerges, based on the principles of quantum probability (QP) theory. Pothos et al. (2013) extended this model to account for the main similarity findings of Tversky (1977), which have served as a golden standard for testing novel theories of similarity. However, Tversky’s (1977) empirical findings did not address the now established insight that, in comparing two objects, overlap in matching parts of the objects tends to have a greater impact on their similarity, than overlap in non-matching parts. We show how the QP similarity model can be directly extended to accommodate structure in similarity comparisons. Smolensky’s et al.’s (2014) proposal for modeling structure in linguistic representations, with tensor products, can be adapted ‘as is’ with the QP similarity model. The formal properties of the extended QP similarity model are analyzed, some indicative fits are presented, and, finally, a novel prediction is developed.

A quantum probability explanation in Fock space for borderline contradictions

The construction of a consistent theory for structuring and representing how concepts combine and interact is one of the main challenges for the scholars involved in cognitive studies. All traditional approaches are still facing serious hindrances when dealing with combinations of concepts and concept vagueness. One of the main consequences of these difficulties is the existence of borderline cases which is hardly explainable from the point of view of classical (fuzzy set) logic and probability theory. Resting on a quantum-theoretic approach which successfully models conjunctions and disjuncions of two concepts, we propound a quantum probability model in Fock space which faithfully reproduces the experimental data collected by Alxatib and Pelletier (2011) on borderline contradictions. Our model allows one to explain the occurrence of the latter contradictions in terms of genuine quantum effects, such as contextuality, superposition, interference and emergence. In particular, we claim that it is the specific mechanism of ‘emergence of a new concept’ that is responsible of these deviations from classical logical thinking in the cognitive studies on human thought. This result seems to be compatible with a recent interesting application of quantum probabilistic modeling in the study of borderline vagueness (Blutner, Pothos & Bruza, 2013), and analogies and differences with it are sketched here.

Conceptual Combination: Extension and Intension. Commentary on Aerts, Gabora, and Sozzo

Aerts et al. provide a valuable model to capture the interactive nature of conceptual combination in conjunctions and disjunctions. The commentary provides a brief review of the interpretation of these interactions that has been offered in the literature, and argues for a closer link between the more traditional account in terms of concept intensions, and the parameters that emerge from the fitting of the Quantum Probability model.

Open Parallel Cooperative and Competitive Decision Processes: A Potential Provenance for Quantum Probability Decision Models

In recent years quantum probability models have been used to explain many aspects of human decision making, and as such quantum models have been considered a viable alternative to Bayesian models based on classical probability. One criticism that is often leveled at both kinds of models is that they lack a clear interpretation in terms of psychological mechanisms. In this paper we discuss the mechanistic underpinnings of a quantum walk model of human decision making and response time. The quantum walk model is compared to standard sequential sampling models, and the architectural assumptions of both are considered. In particular, we show that the quantum model has a natural interpretation in terms of a cognitive architecture that is both massively parallel and involves both co-operative (excitatory) and competitive (inhibitory) interactions between units. Additionally, we introduce a family of models that includes aspects of the classical and quantum walk models.

Quantum probability, choice in large worlds, and the statistical structure of reality

Classical probability models of incentive response are inadequate in "large worlds," where the dimensions of relative risk and the dimensions of similarity in outcome comparisons typically differ. Quantum probability models for choice in large worlds may be motivated pragmatically - there is no third theory - or metaphysically: statistical processing in the brain adapts to the true scale-relative structure of the universe.

If quantum probability = classical probability + bounded cognition; is this good, bad, or unnecessary?

Quantum probability models may supersede existing probabilistic models because they account for behaviour inconsistent with classical probability theory that are attributable to normal limitations of cognition. This intriguing position, however, may overstate weaknesses in classical probability theory by underestimating the role of current knowledge states and may under-employ available knowledge about the limitations of cognitive processes. In addition, flexibility in model specification has risks for the use of quantum probability.

Does quantum uncertainty have a place in everyday applied statistics?

We are sympathetic to the general ideas presented in the article by Pothos & Busemeyer (P&B): Heisenberg's uncertainty principle seems naturally relevant in the social and behavioral sciences, in which measurements can affect the people being studied. We propose that the best approach for developing quantum probability models in the social and behavioral sciences is not by directly using the complex probability-amplitude formulation proposed in the article, but rather, more generally, to consider marginal probabilities that need not be averages over conditionals.

What are the mechanics of quantum cognition?

Pothos & Busemeyer (P&B) argue that quantum probability (QP) provides a descriptive model of behavior and can also provide a rational analysis of a task. We discuss QP models using Marr's levels of analysis, arguing that they make most sense as algorithmic level theories. We also highlight the importance of having clear interpretations for basic mechanisms such as interference.

What's the predicted outcome? Explanatory and predictive properties of the quantum probability framework

Quantum probability (QP) provides a new perspective for cognitive science. However, one must be clear about the outcome the QP model is predicting. We discuss this concern in reference to modeling the subjective probabilities given by people as opposed to modeling the choice proportions of people. These two models would appear to have different cognitive assumptions.

Quantum Decision Theory for Computational Psychiatry

Quantum probabilistic models (based on quantum decision theory) have recently been introduced into cognitive and behavioral sciences. However, to date, quantum decision theory has not been utilized in computational psychiatry, although mathematical models from neuroeconomics have already prevailed in neuropsychiatric research. We address, in the present paper, the importance of employing quantum decision theory in psychiatry. As an example, a possible application of quantum decision theory to autism research is focused. Future study directions in computational psychiatry employing quantum decision theory are discussed. NeuroQuantology | December 2012 | Volume 10 | Issue 4| Page 688-691

A Quantum Probability Model of Causal Reasoning

People can often outperform statistical methods and machine learning algorithms in situations that involve making inferences about the relationship between causes and effects. While people are remarkably good at causal reasoning in many situations, there are several instances where they deviate from expected responses. This paper examines three situations where judgments related to causal inference problems produce unexpected results and describes a quantum inference model based on the axiomatic principles of quantum probability theory that can explain these effects. Two of the three phenomena arise from the comparison of predictive judgments (i.e., the conditional probability of an effect given a cause) with diagnostic judgments (i.e., the conditional probability of a cause given an effect). The third phenomenon is a new finding examining order effects in predictive causal judgments. The quantum inference model uses the notion of incompatibility among different causes to account for all three phenomena. Psychologically, the model assumes that individuals adopt different points of view when thinking about different causes. The model provides good fits to the data and offers a coherent account for all three causal reasoning effects thus proving to be a viable new candidate for modeling human judgment.