Quantum Probability Model Research Articles

Probability Updating in the Classical and the Quantum Model

Imagine a Bayesian decision agent who is keen to invest only in technologies or businesses that are conductive to achieving a sustainable economic future. Being initially keen to invest in a certain technology, subsequently two pieces of new information are received that both individually reduce the agent’s inclination to make that investment. In such a situation, it would be natural to assume that the simultaneous consideration of both pieces of information should further reduce the agent’s initial enthusiasm; after all the Bayesian method has been called “ nothing but common sense reduced to calculation.” Somewhat surprisingly, making general statements like this about the double conditional probability requires substantial additional assumptions in classical Bayesian probability theory. We investigate four schemes of assumptions that allow ‘reasonable’ conclusions about the double conditional probability. We compare this with two schemes for the quantum probability model where the double conditional results from sequential updating through projections.

Quantum Probabilistic Model Checking for Time-Bounded Properties

Probabilistic model checking (PMC) is a verification technique for analyzing the properties of probabilistic systems. However, existing techniques face challenges in verifying large systems with high accuracy. PMC struggles with state explosion, where the number of states grows exponentially with the size of the system, making large system verification infeasible. While statistical model checking (SMC) avoids PMC’s state explosion problem by using a simulation approach, it suffers from runtime explosion, requiring numerous samples for high accuracy. To address these limitations in verifying large systems with high accuracy, we present quantum probabilistic model checking (QPMC), the first method leveraging quantum computing for PMC with respect to time-bounded properties. QPMC addresses state explosion by encoding PMC problems into quantum circuits that superpose states within qubits. Additionally, QPMC resolves runtime explosion through Quantum Amplitude Estimation, efficiently estimating the probabilities of specified properties. We prove that QPMC correctly solves PMC problems and achieves a quadratic speedup in time complexity compared to SMC.

What is Quantum in probabilistic explanations of the sure-thing principle violation?

The Prisoner’s Dilemma game (PDG) is one of the simple test-beds for the probabilistic nature of the human decision-making process. Behavioral experiments have been conducted on this game for decades and show a violation of the so-called sure-thing principle, a key principle in the rational theory of decision. Quantum probabilistic models can explain this violation as a second-order interference effect, which cannot be accounted for by classical probability theory. Here, we adopt the framework of generalized probabilistic theories and approach this explanation from the viewpoint of quantum information theory to identify the source of the interference. In particular, we reformulate one of the existing quantum probabilistic models using density matrix formalism and consider different amounts of classical and quantum uncertainties for one player’s prediction about another player’s action in PDG. This enables us to demonstrate that what makes possible the explanation of the violation is the presence of quantum coherence in the player’s initial prediction and its conversion to probabilities during the dynamics. Moreover, we discuss the role of other quantum information-theoretical quantities, such as quantum entanglement, in the decision-making process. Finally, we propose a three-choice extension of the PDG to compare the predictive powers of quantum probability theory and a more general probabilistic theory that includes it as a particular case and exhibits third-order interference.

A Planck Radiation and Quantization Scheme for Human Cognition and Language.

As a result of the identification of “identity” and “indistinguishability” and strong experimental evidence for the presence of the associated Bose-Einstein statistics in human cognition and language, we argued in previous work for an extension of the research domain of quantum cognition. In addition to quantum complex vector spaces and quantum probability models, we showed that quantization itself, with words as quanta, is relevant and potentially important to human cognition. In the present work, we build on this result, and introduce a powerful radiation quantization scheme for human cognition. We show that the lack of independence of the Bose-Einstein statistics compared to the Maxwell-Boltzmann statistics can be explained by the presence of a ‘meaning dynamics,” which causes words to be attracted to the same words. And so words clump together in the same states, a phenomenon well known for photons in the early years of quantum mechanics, leading to fierce disagreements between Planck and Einstein. Using a simple example, we introduce all the elements to get a better and detailed view of this “meaning dynamics,” such as micro and macro states, and Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac numbers and weights, and compare this example and its graphs, with the radiation quantization scheme of a Winnie the Pooh story, also with its graphs. By connecting a concept directly to human experience, we show that entanglement is a necessity for preserving the “meaning dynamics” we identified, and it becomes clear in what way Fermi-Dirac addresses human memory. Within the human mind, as a crucial aspect of memory, in spaces with internal parameters, identical words can nevertheless be assigned different states and hence realize locally and contextually the necessary distinctiveness, structured by a Pauli exclusion principle, for human thought to thrive.

Are Words the Quanta of Human Language? Extending the Domain of Quantum Cognition.

In previous research, we showed that ‘texts that tell a story’ exhibit a statistical structure that is not Maxwell–Boltzmann but Bose–Einstein. Our explanation is that this is due to the presence of ‘indistinguishability’ in human language as a result of the same words in different parts of the story being indistinguishable from one another, in much the same way that ’indistinguishability’ occurs in quantum mechanics, also there leading to the presence of Bose–Einstein rather than Maxwell–Boltzmann as a statistical structure. In the current article, we set out to provide an explanation for this Bose–Einstein statistics in human language. We show that it is the presence of ‘meaning’ in ‘texts that tell a story’ that gives rise to the lack of independence characteristic of Bose–Einstein, and provides conclusive evidence that ‘words can be considered the quanta of human language’, structurally similar to how ‘photons are the quanta of electromagnetic radiation’. Using several studies on entanglement from our Brussels research group, we also show, by introducing the von Neumann entropy for human language, that it is also the presence of ‘meaning’ in texts that makes the entropy of a total text smaller relative to the entropy of the words composing it. We explain how the new insights in this article fit in with the research domain called ‘quantum cognition’, where quantum probability models and quantum vector spaces are used in human cognition, and are also relevant to the use of quantum structures in information retrieval and natural language processing, and how they introduce ‘quantization’ and ‘Bose–Einstein statistics’ as relevant quantum effects there. Inspired by the conceptuality interpretation of quantum mechanics, and relying on the new insights, we put forward hypotheses about the nature of physical reality. In doing so, we note how this new type of decrease in entropy, and its explanation, may be important for the development of quantum thermodynamics. We likewise note how it can also give rise to an original explanatory picture of the nature of physical reality on the surface of planet Earth, in which human culture emerges as a reinforcing continuation of life.

The cost of asking: How evaluations bias subsequent judgments.

A novel decision bias, called the evaluation bias (EB), was reported by White et al. (2014). In a sequence of two stimuli of opposite affective valence, evaluating the first stimulus leads to a more contrasting evaluation for the second one, compared to when the first stimulus is just observed. The EB is consistent with a long tradition of constructive influences or decision biases in questionnaire judgments. The prediction of the EB was based on the application of a quantum probability model, taking advantage of the unique role of evaluations in quantum probability. In the present work, we develop the quantum model so as to examine whether similar predictions are possible in the context of real questionnaires, where precise control over the relative valence of stimulus pairs is impossible. It is shown that an EB prediction can be extracted and we test this prediction in an organizational opinion survey, administered to a range of organizations across four experiments (total N = 868 and 84 organizations) and with two different languages. In all experiments, there was clear evidence for an EB. We examine the result with the quantum model and Hogarth and Einhorn’s (1992) belief-adjustment model. Both models can broadly capture the empirical findings and so offer promise for providing a formal understanding of constructive influences.

Quantum probability: A new method for modelling travel behaviour

There has been an increasing effort to improve the behavioural realism of mathematical models of choice, resulting in efforts to move away from random utility maximisation (RUM) models. Some new insights have been generated with, for example, models based on random regret minimisation (RRM, μ-RRM). Notwithstanding work using for example Decision Field Theory (DFT), many of the alternatives to RUM tested on real-world data have however only looked at only modest departures from RUM, and differences in results have consequently been small. In the present study, we address this research gap again by investigating the applicability of models based on quantum theory. These models, which are substantially different from the state-of-the-art choice modelling techniques, emphasise the importance of contextual effects, state dependence, interferences and the impact of choice or question order. As a result, quantum probability models have had some success in better explaining several phenomena in cognitive psychology. In this paper, we consider how best to operationalise quantum probability into a choice model. Additionally, we test the quantum model frameworks on a best/worst route choice dataset and demonstrate that they find useful transformations to capture differences between the attributes important in a most favoured alternative compared to that of the least favoured alternative. Similar transformations can also be used to efficiently capture contextual effects in a dataset where the order of the attributes and alternatives are manipulated. Overall, it appears that models incorporating quantum concepts hold significant promise in improving the state-of-the-art travel choice modelling paradigm through their adaptability and efficient modelling of contextual changes.

Possibilities of using probabilistic models in theoretical physics

The article presents a historical overview of the path to understanding the probabilistic nature of the laws governing the behavior of microobjects and the creation of quantum mechanics. Its postulates are given and directions in the mathematization of quantum theory are described. The fundamental differences of the quantum probabilistic model from the classical one are presented in an overview.

Quantum Probabilistic Models Using Feynman Diagram Rules for Better Understanding the Information Diffusion Dynamics in Online Social Networks

This doctoral consortium presents an overview of my anticipated PhD dissertation which focuses on employing quantum Bayesian networks for social learning. The project, mainly, aims to expand the use of current quantum probabilistic models in human decision-making from two agents to multi-agent systems. First, I cultivate the classical Bayesian networks which are used to understand information diffusion through human interaction on online social networks (OSNs) by taking into account the relevance of multitude of social, psychological, behavioral and cognitive factors influencing the process of information transmission. Since quantum like models require quantum probability amplitudes, the complexity will be exponentially increased with increasing uncertainty in the complex system. Therefore, the research will be followed by a study on optimization of heuristics. Here, I suggest to use an belief entropy based heuristic approach. This research is an interdisciplinary research which is related with the branches of complex systems, quantum physics, network science, information theory, cognitive science and mathematics. Therefore, findings can contribute significantly to the areas related mainly with social learning behavior of people, and also to the aforementioned branches of complex systems. In addition, understanding the interactions in complex systems might be more viable via the findings of this research since probabilistic approaches are not only used for predictive purposes but also for explanatory aims.

Implementing perceptron models with qubits

We propose a method for learning a quantum probabilistic model of a perceptron. By considering a cross entropy between two density matrices we can learn a model that takes noisy output labels into account while learning. A multitude of proposals already exist that aim to utilize the curious properties of quantum systems to build a quantum perceptron, but these proposals rely on a classical cost function for the optimization procedure. We demonstrate the usage of a quantum equivalent of the classical log-likelihood, which allows for a quantum model and training procedure. We show that this allows us to better capture noisyness in data compared to a classical perceptron. By considering entangled qubits we can learn nonlinear separation boundaries, such as XOR.

On limit laws of multi-dimensional stochastic synchronization models

Lévy stochastic processes and related fine analytic properties of probability distributions such as infinite divisibility play an important role in construction of stochastic models of various distributed networks (e.g., local clock synchronization), of some physical systems (e.g., anomalous diffusions, quantum probability models), of finance etc. Nevertheless, little is known about limit probability laws resulted from the long time behavior of such stochastic systems. In this paper we will focus on the impact of interaction graph topologies on limit laws of multicomponent synchronization models.

Representation Lost: The Case for a Relational Interpretation of Quantum Mechanics.

Contemporary non-representationalist interpretations of the quantum state (especially QBism, neo-Copenhagen views, and the relational interpretation) maintain that quantum states codify observer-relative information. This paper provides an extensive defense of such views, while emphasizing the advantages of, specifically, the relational interpretation. The argument proceeds in three steps: (1) I present a classical example (which exemplifies the spirit of the relational interpretation) to illustrate why some of the most persistent charges against non-representationalism have been misguided. (2) The special focus is placed on dynamical evolution. Non-representationalists often motivate their views by interpreting the collapse postulate as the quantum mechanical analogue of Bayesian probability updating. However, it is not clear whether one can also interpret the Schrödinger equation as a form of rational opinion updating. Using results due to Hughes & van Fraassen as well as Lisi, I argue that unitary evolution has a counterpart in classical probability theory: in both cases (quantum and classical) probabilities relative to a non-participating observer evolve according to an entropy maximizing principle (and can be interpreted as rational opinion updating). (3) Relying on a thought-experiment by Frauchiger and Renner, I discuss the differences between quantum and classical probability models.

A quantum probability account of individual differences in causal reasoning

We use quantum probability (QP) theory to investigate individual differences in causal reasoning. By analyzing datasets from Rehder (2014) on comparative judgments, and from Rehder and Waldmann (2016) on absolute judgments, we show that a QP model can both account for individual differences in causal judgments, and why these judgments sometimes violate the properties of causal Bayes nets. We implement this and previously proposed models of causal reasoning (including classical probability models) within the same hierarchical Bayesian inferential framework to provide a detailed comparison between these models, including computing Bayes factors. Analysis of the inferred parameters of the QP model illustrates how these can be interpreted in terms of putative cognitive mechanisms of causal reasoning. Additionally, we implement a latent classification mechanism that identifies subcategories of reasoners based on properties of the inferred cognitive process, rather than post hoc clustering. The QP model also provides a parsimonious explanation for aggregate behavior, which alternatively can only be explained by a mixture of multiple existing models. Investigating individual differences through the lens of a QP model reveals simple but strong alternatives to existing explanations for the dichotomies often observed in how people make causal inferences. These alternative explanations arise from the cognitive interpretation of the parameters and structure of the quantum probability model.

Multipartite Composition of Contextuality Scenarios

Contextuality is a particular quantum phenomenon that has no analogue in classical probability theory. Given two independent systems, a natural question is how to represent such a situation as a single test space. In other words, how separate contextuality scenarios combine into a joint scenario. Under the premise that the the allowed probabilistic models satisfy the No Signalling principle, Foulis and Randall defined the unique possible way to compose two contextuality scenarios. When composing strictly-more than two test spaces, however, a variety of possible composition methods have been conceived. Nevertheless, all these formally-distinct composition methods appear to give rise to observationally equivalent scenarios, in the sense that the different compositions all allow precisely the same sets of classical and quantum probabilistic models. This raises the question of whether this property of invariance-under-composition-method is special to classical and quantum probabilistic models, or if it generalizes to other probabilistic models as well, our particular focus being $\mathcal{Q}_1$ models. $\mathcal{Q}_1$ models are physically important since, when applied to scenarios constructed by a particular composition rule, coincide with the well-defined "Almost Quantum Correlations". In this work we see that some composition rules, however, give rise to scenarios with inequivalent allowed sets of $\mathcal{Q}_1$ models. We find that the non-trivial dependence of $\mathcal{Q}_1$ models on the choice of composition method is apparently an artifact of failure of those composition rules to capture the orthogonality relations given by the Local Orthogonality principle. We prove that $\mathcal{Q}_1$ models satisfy invariance-under-composition-method for all the constructive compositions protocols which do capture this notion of Local Orthogonality.

A quantum-probabilistic paradigm: Non-consequential reasoning and state dependence in investment choice

Seminal findings involving payoffs (Shafir and Tversky, 1992; Tversky and Shafir, 1992; Shafir, 1994) showed that individuals exhibit state-dependent behaviour in different informational contexts. In particular, in the condition of ambiguity as well as risk, individuals tend to exhibit ambiguity aversion. The core principle of rational (consequential) behaviour conceived by Savage (1954), that is the ‘Savage Sure Thing’ principle, has been shown to be violated. In mathematical language, this violation is equivalent to the violation of the “Law of total probability”, (Kolmogorov, 1933). Given the importance of original findings in the call for a generalization of classical expected utility, we perform in this paper a set of experiments related to expressing investment preferences: (i) under objective risk, (ii) after a preceding gain, or loss. In accordance with previous findings we detected state dependence in human judgement (previous gain or loss changed the preference state of the participants) as well as violation of consequential reasoning under risk. We propose a quantum probabilistic model of agents’ preferences, where non-consequentialism and state dependence can be well explained via interference of complex probability amplitudes. A geometric depiction of the experimental findings with a state reconstruction procedure from statistical data via the inverse Born’s rule (1926), allows for an accurate representation of agents’ preference formation in risky investment choice.

A reinforcement learning approach for UAV target searching and tracking

Owing to the advantages of Unmanned Aerial Vehicle (UAV), such as the extendibility, maneuverability and stability, multiple UAVs are having more and more applications in security surveillance. The object searching and trajectory planning become the important issues of uninterrupted patrol. We propose an online distributed algorithm for tracking and searching, while considering the energy refueling at the same time. The quantum probability model which describes the partially observable target positions is proposed. Moreover, the upper confidence tree algorithm is derived to resolve the best route, with the assistance of teammate learning model which handles the nonstationary problems in distributed reinforcement learning. Experiments and the analysis of the different situations show that the proposed scheme performs favorably.

Surprising rationality in probability judgment: Assessing two competing models

We describe 4 experiments testing contrasting predictions of two recent models of probability judgment: the quantum probability model (Busemeyer, Pothos, Franco, & Trueblood, 2011) and the probability theory plus noise model (Costello & Watts, 2014, 2016a). Both models assume that people estimate probability using formal processes that follow or subsume standard probability theory. One set of predictions concerned agreement between people’s probability estimates and standard probability theory identities. The quantum probability model predicts people’s estimates should agree with one set of identities, while the probability theory plus noise model predicts a specific pattern of violation of those identities. Experimental results show the specific pattern of violation predicted by the probability theory plus noise model. Another set of predictions concerned the conjunction fallacy, which occurs when people judge the probability of a conjunction P(A∧B) to be greater than one or other constituent probabilities P(A) or P(B), contrary to the requirements of probability theory. In cases where A causes B, the quantum probability model predicts that the conjunction fallacy should only occur for constituent B and not for constituent A: the noise model predicts that the fallacy should occur for both A and B. Experimental results show that the fallacy occurs equally for both, contrary to the quantum probability prediction. These results suggest that people’s probability estimates do not follow quantum probability theory. These results support the idea that people estimate probabilities using mechanisms that follow standard probability theory but are subject to random noise.

Parallel Interactive Processing as a Way to Understand Complex Information Processing: The Conjunction Fallacy and Other Examples.

Parallel interactive processing (PIP) represents an approach in which specific context generates interactive relationships between general attributes. This article summarizes previous research that demonstrates how such relationships influence inference making in categorization. This is followed by evidence that the approach can be extended to other areas of cognition, including probability judgments. PIP was successful in fitting data that revealed the prevalence of the conjunction fallacy as well as other probability estimation data. PIP provided better fits overall than the signed summation model and the configural weighted average model. The quantum probability model provided good fits for the conjunction fallacy data but not for other probability judgments.

The Detection of Series Arc Fault in Photovoltaic Systems Based on the Arc Current Entropy

This paper presents a series arc-fault detection algorithm for photovoltaic (PV) systems, which relies on the quantum probability model theory. The algorithm determines the presence of the arc by calculating the modified Tsallis entropy of the PV panel current. Based on the calculated entropy, the algorithm is able to differentiate an arc state (when the current variations are chaotic) from a no-arc state (when the current variations are ordered). The proposed algorithm enables arc-fault detection on a plug-and-play principle, requiring no prior information about the PV system in which it operates. The operation of the algorithm was first simulated on the prerecorded data from the system with the PV panel simulator and the commercial PV inverter. A laboratory prototype of the detector was then built and tested in a real 1.6-kW grid-connected PV system with different PV inverter, without any additional parameter adjustments. Both the sensitivity and the robustness of the detector were confirmed. The tests have shown that both the sustained series arcs and the small sparking series arcs were successfully detected, and there was no false detection due to the MPP tracker operation, PV current step change, or inverter turn on transient.

Bell Could Become the Copernicus of Probability

Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability, Bell could play the same role as Lobachevsky in geometry. In fact, violation of Bell’s inequality can be treated as implying the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus the quantum probabilistic model (based on Born’s rule) can be considered as the concrete example of the non-Kolmogorovian model of probability, similarly to the Lobachevskian model — the first example of the non-Euclidean model of geometry. This is the “probability model” interpretation of the violation of Bell’s inequality. We also criticize the standard interpretation—an attempt to add to rigorous mathematical probability models additional elements such as (non)locality and (un)realism. Finally, we compare embeddings of non-Euclidean geometries into the Euclidean space with embeddings of the non-Kolmogorovian probabilities (in particular, quantum probability) into the Kolmogorov probability space. As an example, we consider the CHSH-test.