While it is widely acknowledged that quantum theory is strange, and that the world view it implies is elusive, there is considerably less consensus as to why this is the case. This article will explore the possibility that the theory may be opaque because the mathematical language it employs is inarticulate. To address this possibility, three interconnected mathematical theories will be developed, each specifically designed to handle aspects of quantum theory's subject matter. The first will describe dynamic systems in terms that are general enough to be applicable to any kind dynamics, including both classical and quantum. That description will then be used to construct a theory of experiments, with the aim of understanding what types of experiments scientific methodology allows. Finally, the theory of experiments will found a theory of experimental probabilities applicable to any collection of experiments, so long as their probabilities are internally consistent. The resulting over-arching mathematical language will then be applied to quantum theory. In the initial application, it will be used to explain the core nature and structure of quantum measurement and probability. Following that, it will be used to explore quantum theory's models. Like any theory written in a mathematical language, quantum theory derives its meaning from its class of models. To understand the class quantum models, two models of central importance will be described. The first will be a conceptually simple construction that follows almost immediately from the above theory of dynamics. It fully satisfies both quantum theory and the above mentioned theory of experimentation; additionally, its macroscopic objects obey classical mechanics. The second will utilize a minimal amount of mathematical structure to satisfy the same conditions.
While it is widely acknowledged that quantum theory is strange, and that the world view it implies is elusive, there is considerably less consensus as to why this is the case. This article will explore the possibility that the theory may be opaque because the mathematical language it employs is inarticulate. To address this possibility, three interconnected mathematical theories will be developed, each specifically designed to handle aspects of quantum theory's subject matter. The first will describe dynamic systems in terms that are general enough to be applicable to any kind dynamics, including both classical and quantum. That description will then be used to construct a theory of experiments, with the aim of understanding what types of experiments scientific methodology allows. Finally, the theory of experiments will found a theory of experimental probabilities applicable to any collection of experiments, so long as their probabilities are internally consistent. The resulting over-arching mathematical language will then be applied to quantum theory. In the initial application, it will be used to explain the core nature and structure of quantum measurement and probability. Following that, it will be used to explore quantum theory's models. Like any theory written in a mathematical language, quantum theory derives its meaning from its class of models. To understand the class quantum models, two models of central importance will be described. The first will be a conceptually simple construction that follows almost immediately from the above theory of dynamics. It fully satisfies both quantum theory and the above mentioned theory of experimentation; additionally, its macroscopic objects obey classical mechanics. The second will utilize a minimal amount of mathematical structure to satisfy the same conditions.
Everybody agrees that quantum physics is strange, and that the world view it implies is elusive. However, it is rarely considered that the theory might be opaque because the mathematical language it employs is inarticulate. Perhaps, if a mathematical language were constructed specifically to handle the theory's subject matter, the theory itself would be clarified. This article explores that possibility. It presents a simple but rigorous language for the description of dynamics, experiments, and experimental probabilities. This language is then used to answer a compelling question: What is the set of allowed experiments? If an experiment is allowed, then the sum of the probabilities of its outcomes must equal 1. If probabilities are non-additive, there are necessarily sets of outcomes whose total probability is not equal to 1. Such experiments are therefore not allowed. That being the case, in quantum physics, which experiments are allowed, and why are the rest disallowed? What prevents scientists from performing the disallowed experiments? By phrasing these questions within our mathematical language, we will uncover answers that are complete, conceptually simple, and clearly correct. This entails no magic or sleight of hand. To write a rigorous mathematical language, all unnecessary assumptions must be shed. In this way, the thicket of ad hoc assumptions that surrounds quantum physics will be cleared. Further, in developing the theory, the logical consequences of the necessary assumptions will be laid bare. Therefore, when a question can be phrased in such a language, one can reasonably expect a clear, simple answer. In this way we will dispel much of the mystery surrounding quantum measurements, and begin to understand why quantum probabilities have their peculiar representation as products of Hilbert space projection operators.
The purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
We attempt to contribute some novel points of view to the "foundations of quantum mechanics", using mathematical tools from "quantum probability theory" (such as the theory of operator algebras). We first introduce an abstract algebraic framework for the formulation of mathematical models of physical systems that is general enough to encompass classical and quantum mechanical models. After a short discussion on the properties of classical systems (realism and determinism), we move on to the study of a general class of quantum-mechanical models of physical systems and explain what some of the key problems in a quantum theory of observations and measurements are. In the last part of our essay, we attempt to elucidate the roles played by entanglement between a system and its environment and of information loss in understanding "decoherence" and "dephasing", which are key mechanisms in a quantum theory of measurements and experiments. We also discuss the problem of "time in quantum mechanics" and sketch an answer to the question when an experiment can be considered to have been completed successfully.
Abstract Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.
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.
Abstract This review, of the understanding of quantum mechanics, is broad in scope, and aims to reflect enough of the literature to be representative of the current state of the subject. To enhance clarity, the main findings are presented in the form of a coherent synthesis of the reviewed sources. The review highlights core characteristics of quantum mechanics. One is statistical balance in the collective response of an ensemble of identically prepared systems, to differing measurement types. Another is that states are mathematical terms prescribing probability aspects of future events, relating to an ensemble of systems, in various situations. These characteristics then yield helpful insights on entanglement, measurement, and widely-discussed experiments and analyses. The review concludes by considering how these insights are supported, illustrated and developed by some specific approaches to understanding quantum mechanics. The review uses non-mathematical language precisely (terms defined) and rigorously (consistent meanings), and uses only such language. A theory more descriptive of independent reality than is quantum mechanics may yet be possible. One step in the pursuit of such a theory is to reach greater consensus on how to understand quantum mechanics. This review aims to contribute to achieving that greater consensus, and so to that pursuit.