While mathematical modeling beyond physics—in biology, neuroscience, psychology, and economics—has been and is still dominated by classical mathematical models (C-models), primarily of probabilistic and statistical nature, quantum mathematical models (Q-models), mathematical models based in the mathematical formalism of quantum theory, have recently acquired currency in mathematical modeling in these areas. This chapter examines some of the reasons for using such models. In order to do so, the author considers the fundamental principles behind these models in quantum physics itself, taking as its point of departure Einstein’s distinction between “constructive” and “principle” theories. Two types of principle thinking in quantum theory will be considered, those defining the initial development of quantum mechanics in 1920s and those of quantum information theory, a more recent and still ongoing development. The principle perspective, the author argues, may help us to understand better a possible and possibly necessary role for quantum mathematical models beyond physics.
While mathematical modeling beyond physics—in biology, neuroscience, psychology, and economics—has been and is still dominated by classical mathematical models (C-models), primarily of probabilistic and statistical nature, quantum mathematical models (Q-models), mathematical models based in the mathematical formalism of quantum theory, have recently acquired currency in mathematical modeling in these areas. This chapter examines some of the reasons for using such models. In order to do so, the author considers the fundamental principles behind these models in quantum physics itself, taking as its point of departure Einstein’s distinction between “constructive” and “principle” theories. Two types of principle thinking in quantum theory will be considered, those defining the initial development of quantum mechanics in 1920s and those of quantum information theory, a more recent and still ongoing development. The principle perspective, the author argues, may help us to understand better a possible and possibly necessary role for quantum mathematical models beyond physics.
The history of mathematical modeling outside physics has been dominated by the use of classical mathematical models, C-models, primarily those of a probabilistic or statistical nature. More recently, however, quantum mathematical models, Q-models, based in the mathematical formalism of quantum theory have become more prominent in psychology, economics, and decision science. The use of Q-models in these fields remains controversial, in part because it is not entirely clear whether Q-models are necessary for dealing with the phenomena in question or whether C-models would still suffice. My aim, however, is not to assess the necessity of Q-models in these fields, but instead to reflect on what the possible applicability of Q-models may tell us about the corresponding phenomena there, vis-à-vis quantum phenomena in physics. In order to do so, I shall first discuss the key reasons for the use of Q-models in physics. In particular, I shall examine the fundamental principles that led to the development of quantum mechanics. Then I shall consider a possible role of similar principles in using Q-models outside physics. Psychology, economics, and decision science borrow already available Q-models from quantum theory, rather than derive them from their own internal principles, while quantum mechanics was derived from such principles, because there was no readily available mathematical model to handle quantum phenomena, although the mathematics ultimately used in quantum did in fact exist then. I shall argue, however, that the principle perspective on mathematical modeling outside physics might help us to understand better the role of Q-models in these fields and possibly to envision new models, conceptually analogous to but mathematically different from those of quantum theory, helpful or even necessary there or in physics itself. I shall suggest one possible type of such models, singularized probabilistic, SP, models, some of which are time-dependent, TDSP-models. The necessity of using such models may change the nature of mathematical modeling in science and, thus, the nature of science, as it happened in the case of Q-models, which not only led to a revolutionary transformation of physics but also opened new possibilities for scientific thinking and mathematical modeling beyond physics.
This article examines some experimental conditions that invite and possibly require recourse to quantum-mechanical-like mathematical models (QMLMs), models based on the key mathematical features of quantum mechanics, in scientific fields outside physics, such as biology, cognitive psychology, or economics. In particular, I consider whether the following two correlative features of quantum phenomena that were decisive for establishing the mathematical formalism of quantum mechanics play similarly important roles in QMLMs elsewhere. The first is the individuality and discreteness of quantum phenomena, and the second is the irreducibly probabilistic nature of our predictions concerning them, coupled to the particular character of the probabilities involved, as different from the character of probabilities found in classical physics. I also argue that these features could be interpreted in terms of a particular form of epistemology that suspends and even precludes a causal and, in the first place, realist description of quantum objects and processes. This epistemology limits the descriptive capacity of quantum theory to the description, classical in nature, of the observed quantum phenomena manifested in measuring instruments. Quantum mechanics itself only provides descriptions, probabilistic in nature, concerning numerical data pertaining to such phenomena, without offering a physical description of quantum objects and processes. While QMLMs share their use of the quantum-mechanical or analogous mathematical formalism, they may differ by the roles, if any, the two features in question play in them and by different ways of interpreting the phenomena they considered and this formalism itself. This article will address those differences as well.
Quantum mechanics, one of the most successful theories in the history of science, was created to account for physical systems not describable by classical physics. Though it is consistent with all experiments conducted thus far, many of its core concepts (amplitudes, global phases, etc.) can not be directly accessed and its interpretation is still the subject of intense debate, more than 100 years since it was introduced. So, a fundamental question is why this particular mathematical model is the one that nature chooses, if indeed it is the correct model. In the past two decades there has been a renewed effort to determine what physical or informational principles define quantum mechanics. In this paper, recent attempts at establishing reasonable physical principles are reviewed and their degree of success is tabulated. An alternative approach using joint quasi-probability distributions is shown to provide a common basis of representing most of the proposed principles. It is argued that having a common representation of the principles can provide intuition and guidance to relate current principles or advance new principles. The current state of affairs, along with some alternative views are discussed.
The emergence of quantum computing has prompted researchers to examine potential parallels between quantum mechanical principles and human cognitive processes. This article explores the theoretical framework of quantum cognition, examining how concepts from quantum mechanics, including superposition, entanglement, and measurement, might inform our understanding of human decision-making, memory, and consciousness. While quantum computing operates on fundamentally different physical principles than biological neural networks, the mathematical formalisms of quantum mechanics offer novel perspectives on cognitive phenomena that classical probability theory struggles to explain. This review critically examines the current state of quantum cognition research, its applications in computational modeling, and the ongoing debates regarding the validity of quantum mechanical analogies in psychological theory. The analysis reveals both promising theoretical insights and significant limitations in applying quantum principles to human cognition, highlighting the need for empirical validation and careful distinction between mathematical analogy and physical reality.
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.
In this chapter, we will discuss the ontological status of the notion of quantuminformation. In the first part, we review the mathematical framework ofa generalized information theory. In this framework, probability theory will play a central role and, thus, we will devote Section 2 to it.We argue that the existence of probabilistic models that go beyond the classical and quantum realms, and the possibility of performing informational protocols in those models, allow us to claim that a generalized information theory can be conceived. Notice that this perspective is complementary to that of other research programs (see, e.g., Holik et al. 2015; Holik et al. 2016, and references therein). In the second part, we will consider the question about the ontology of the possible models satisfying the different instances of that generalized formalism. This will be the content of Section 3. Specifically, we want to study the role quantum mechanics plays in quantum information protocols; the question is whether there are other possible ontologies that can perform the same tasks. We will relate this problem to the existence of many examples of physical systems that are actually constructed using an essentially classical ontology but, on the other hand, are modeled by structures resembling the laws of quantum mechanics (e.g. Aerts 1998; Kwiat et al. 2000; Spreeuw 2001;Couder et al. 2005; Couder and Fort 2006; Qian et al. 2015). Their significancefor us relies in the fact that they can be used to perform quantum information protocols (though, in an inefficient way) (Man?ko et al. 2001; Bhattacharya et al. 2002; Francisco et al. 2006a, 2006b; Francisco and Ledesma 2008; Goldin et al. 2010). On this basis, we will explore the ontological implications of the existence of these simulations. Finally, in Section 4 we will draw our conclusions.