An Intensional Probability Theory: Investigating the Link between Classical and Quantum 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).

In this chapter we present a brief introductions to Kolmogorov’s measure-theoretic and von Mises’ frequency probability theories. These are classical probability theories. A rather common opinion is that classical probability can not describe the probabilistic structure of quantum mechanics. Such an opinion is based on belief that classical probability can not reproduce the main distinguishing probabilistic features of quantum formalism, e.g., interference of probabilities and Born’s rule. It is also believed that the violation of Bell’s inequality can not be explained in the framework of classical probability theory. The aim of this book is to show that such a viewpoint to the relation between classical and quantum probabilities is not justified. We shall see that the mathematical formalisms of Kolmogorov’s measure-theoretic [Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933; Shiryaev in Ann. Probab. 17:866–944, 1989; Shiryaev, Probability, 1984] and Mises’ frequency [Von Mises in Math. Z. 5:52–99, 1919; Von Mises, Probability, Statistics and Truth, 1957; Von Mises, The Mathematical Theory of Probability and Statistics, 1964] models can be represented in the QL way. However, these models should be interpreted in the contextual framework.

Quantum theory provides a comprehensive framework for quantifying uncertainty, often applied in quantum finance to explore the stochastic nature of asset returns. This perspective likens asset returns to microscopic particle motion, governed by quantum probabilities akin to physical laws. However, such approaches presuppose specific microscopic quantum effects in return changes, a premise criticized for lack of guarantee. This paper diverges by regarding that quantum probability as a mathematical extension of classical probability to complex numbers. It isn’t exclusively tied to microscopic quantum phenomena, bypassing the requirement for quantum effects in returns. By directly linking quantum probability’s mathematical structure to traders’ decisions and market behaviors, it avoids assuming quantum effects for returns and invoking the wave function. The complex phase of quantum probability, capturing transitions between long and short decisions while considering information interaction among traders, offers an inherent advantage over classical probability in characterizing the multimodal distribution of asset returns. Utilizing Fourier decomposition, we derive a Schödinger-like trading equation, where each term explicitly corresponds to implications of market trading. The equation indicates discrete energy levels in financial trading, with returns following a normal distribution at the lowest level. As the market transitions to higher trading levels, a phase shift occurs in the return distribution, leading to multimodality and fat tails. Empirical research on the Chinese stock market supports the existence of energy levels and multimodal distributions derived from this quantum probability asset returns model.

Quantum probability

There has been a growing trend to develop cognitive models based on the mathematics of quantum theory. A common theme in the motivation of such models has been findings which apparently challenge the applicability of classical formalisms, specifically ones based on classical probability theory. Classical probability theory has had a singularly important place in cognitive theory, because of its (in general) descriptive success but, more importantly, because in decision situations with low, equivalent stakes it offers a multiply justified normative standard. Quantum cognitive models have had a degree of descriptive success and proponents of such models have argued that they reveal new intuitions or insights regarding decisions in uncertain situations. However, can quantum cognitive models further benefit from normative justifications analogous to those for classical probability models? If the answer is yes, how can we determine the rational status of a decision, which may be consistent with quantum theory, but inconsistent with classical probability theory? In this paper, we review the proposal from Pothos, Busemeyer, Shiffrin, and Yearsley (2017), that quantum decision models benefit from normative justification based on the Dutch Book Theorem, in exactly the same way as models based on classical probability theory.

Contextual decisions and beliefs and their impact upon market outcomes are at the core of research in behavioural finance. We describe some of the notable probabilistic fallacies that underpin investor behaviour and the consequent deviation of asset prices from the rational expectations equilibrium. In real financial markets, the complexity of financial products and the surrounding ambiguity calls for a more general formalization of agents belief formation than offered by the standard probability theory and dynamic models based on classical stochastic processes. The main advantage of quantum probability (QP) is that it can capture contextuality of beliefs through the notion of non-commuting prospect observables. QP has the potential to model myopia in asset return evaluation, as well as inter-asset valuation. Moreover, the interference term of the agents’ comparison state can provide a quantitative description of their vacillating ambiguity perception characterized by non-additive beliefs of agents. Some of the implications of non-classicality in beliefs for the composite market outcomes can also be modelled with the aid of QP. As a final step we also discuss the contributions of the growing body of psychological studies that reveal a true (quantum type) contextuality in human preference statistics showing that the classical probability theory is too restrictive to capture the very strong non-classical correlations between preference outcomes and beliefs.

What is the proper attitude toward what is expressed by a vague sentence in the face of borderline evidence? Some call this attitude “ambivalence” and distinguish it from uncertainty. It has been argued that Classical Epistemicism conjoined with classical probability theory fails to characterize this attitude, and that we must therefore abandon classical logic or classical probabilities in the presence of vagueness. In this paper, I give a characterization of ambivalence assuming a supervaluationist semantics for vague terms that does not revise either. The theory, which I call the theory of Superprobabilities, identifies the proper attitude toward a vague sentence, in the presence of exact borderline evidence, as the set of classical probabilities of the evidence on each member of the set of all precisifications of a vague sentence. I defend the use of sets of probabilities against objections by generalizing the theory of Superprobabilities to a decision theory called Superrationality. I then compare the merits of the theory of Superprobabilities to Classical Epistemicism and nonclassical probabilities theories with respect to the problem of ambivalence.

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications. Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one. Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications. Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.

Cognitive frameworks best represent cognitive information and knowledge. Several classical probability theory (CPT)-based cognitive frameworks were proposed in the literature. Recently, CPT has failed in explaining certain cognitive processes while quantum theories were successful in explaining the same. In this work, we integrate two cognitive frameworks namely conceptual spaces and 3-way formal concept analysis (3WFCA) to propose a high dimensional conceptual space (HDCS). Our new insights into the analysis of this proposal reveal its quantum characteristics. Among different cognitive processes that can be modelled, our interest is on phenomenal consciousness. Accordingly, we have proposed a formal method to achieve consciousness. We have also proposed an algorithm for the conceptual scaling of a cognitive scenario. HDCS represents a cognitive state using quality dimensions, attributes and their relations. Subsequently, the proposed model makes novel use of agent-environment interaction paradigm for guiding the interaction of HDCS with conceptually scaled cognitive scenario. Cognitive-state representation in HDCS is analogous to quantum state representation in a N-qubit system. Cognitive states are learnt through the parallel accumulation of evidences against all the attributes of multiple dimensions. The interaction between the HDCS and the scenario facilitates the identification and removal of uncertainties. The identified uncertainty coupled with its resolution time resembles Heisenberg-like uncertainty. The analogous of quantum two-slit self-interference would be comparison of evidence accumulation in HDCS with the preceding version of itself. Consequently, this research reveals that modelling consciousness requires a high dimensional space rather than set theoretic structures and support from quantum theories.

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

Recent experimental findings clearly suggest that classical probability theory is still not successful to explain modalities in human cognition, especially in connection to decision making. The major problem seems to be the presence of epistemic uncertainty and its effects on cognition at any time point. Moreover, the stochasticity in the model arises due to the unknown path or trajectory (the definite state of mind at each point in time) a person follows. In fact, there exists more ambiguity than clarity in some aspects of cognition, for example, regarding cognitive affective decision-making behavior and subsequent choices (George et al. in Academy of Management Review 31: 347–365, 2006). The concept of ambiguity, its typical role and its function are not only long-standing debatable concerns in the humanities but also assume a similar level of importance, and are the subject of much debate, in contemporary neurological and cognitive approaches. For example, biologist Fredrick Grinnell made an argument clarifying that ambiguity “is inherent in carrying out and reporting research” (Grinnell in Science and Engineering Ethics 5:205, 1999). Not only that, but he also pointed out some parameters of ambiguity that support his view, for instance, the difficulty of distinguishing between data and noise in research “at the edge of discovery” (ibid. 207). Interestingly, as a positive side of ambiguity, he observed “while ambiguity in method and procedure—as an aspect of data and its interpretation—is clearly a potential problem, posing the risk of distortion, it is also associated here with creative insight.” The consideration of “black box model of human mental functions” produces much ambiguity. So, as we are aware of the intertwined mathematical, neurological and cognitive mysteries of the brain, ultimately, these very characteristics are challenges faced by the human cognition and give rise to a very complicated and complex integrative system of cognitive computation and affective perception. A generalized version of probability theory, borrowing the idea from the quantum paradigm, may be a plausible approach. Quantum theory enables a person to be in an indefinite state (superposition state) in the context of neurobiology, especially in relation to central nervous system (CNS), and this allows all these states to be potentially (of course, with proper probability amplitudes) expressible at each moment. Thus, a superposition state seems to provide a better representation of the conflict, ambiguity or uncertainty that a person experiences at each moment. This situation can be described in somewhat poetic language, i.e., the dynamical laws, as demonstrated by nature, allow equations but are only sometimes posed by the Mind of the experimentalist. This way, consciousness (as expressed in his introduction to consciousness or cognition can be expressed as only a critical description of human activity and not as a theory of mind. Some critical remarks about handling the ambiguity and contradictions due to emotions within quantum probability framework are made in this chapter. Mental states and related decision making have been widely discussed in various ancient and medieval Indian traditions. The existence of an indefinite state of mind was discussed by many Indian philosophical schools, including Buddhism. The concept of the neutral mind, as well as equanimity, is discussed in this chapter, especially because it has a potential to be related to modern neuroscience, as well as to quantum theory.

The origin of black hole entropy and the black hole information problem pro- vide important clues for trying to piece together a quantum theory of gravity. Thus far, discussions on this topic have mostly assumed that in a consistent theory of gravity and quantum mechanics, quantum theory will be unmodied. Here, we examine the black hole information problem in the context of generalisations of quantum theory. In particular, we examine black holes in the setting of generalised probabilistic theories, in which quantum theory and classical probability theory are special cases. We compute the time it takes information to escape a black hole, assuming that information is preserved. We nd that under some very general assumptions, the arguments of Page (that information should escape the black hole after half the Hawking photons have been emitted), and the black- hole mirror result of Hayden and Preskill (that information can escape quickly) need to be modied. The modication is determined entirely by what we call the Wootters-Hardy pa- rameter associated with a theory. We nd that although the information leaves the black hole after enough photons have been emitted, it is fairly generic that it fails to appear outside the black hole at this point | something impossible in quantum theory due to the no-hiding theorem. The information is neither inside the black hole, nor outside it, but is delocalised. Our central technical result is an information decoupling theorem which holds in the generalised probabilistic framework.

All papers have been peer reviewed. This was the 4th conference arranged by ICMM on probabilistic foundations of classical and quantum physics. The first three conferences took place in 2000, 2002, and 2004. Some closely related conferences are Bohmian Mechanics 2000 and Quantum Theory: Reconsideration of Foundations 2001, 2003, and 2005. The main aim of these conferences is to understand the role that probability plays in the foundations of physics, theoretical as well as experimental, classical as well as quantum. In this conference, as well as during our previous conferences, we are glad to welcome a fruitful assembly of theoretical physicists, experimenters, mathematicians, and even philosophers interested in the foundations of probability and physics. Among important topics discussed during the conference were the probabilistic foundations of quantum mechanics, as well as the foundations of probability itself, the formation theory, quantum computing, quantum cryptography, quantum teleportation, quantum fluctuations in relation with stochastic electrodynamics, Bohmian mechanics, measurement theory, completeness and incompleteness of quantum mechanics, macroscopic quantum systems, experiments on quantum nonlocality and locality, Bell's inequality, entanglement; philosophical problems raised by quantum mechanics, and mathematical formalism. A special session devoted to the Bayesain approach to classical and quantum probability was organized.

I reflect on several aspects of the general claim that a quantum-like approach to Cognitive Science is advantageous over classical approaches. The classical approaches refer to the symbolic approaches including models using a classical (Kolmogorov) probability calculus. The general claim seems to be right from a descriptive viewpoint but not necessarily from an explanatory viewpoint. The explanatory perspective needs a more careful analysis since adding some additional arbitrary parameters (such as phase shift parameters in quantum probabilities) does not automatically increase the explanatory value of the approach; rather, it seems to decrease it. I argue further that there is another class of traditional models – the class of geometric models of cognition. These models have a much longer tradition than the symbolic models. Interestingly, quantum mechanics does not contradict the geometric models. Hence, real progress at the meeting between quantum mechanics and cognitive science could be made by unifying these geometric models with ideas from quantum theory.