Originally, quantum probability theory was developed to analyze statistical phenomena in quantum systems, where classical probability theory does not apply, because the lattice of measurable sets is not necessarily distributive. On the other hand, it is well known that the lattices of concepts, that arise in data analysis, are in general also non-distributive, albeit for completely different reasons. In his recent book, van Rijsbergen argues that many of the logical tools developed for quantum systems are also suitable for applications in information retrieval. I explore the mathematical support for this idea on an abstract vector space model, covering several forms of data analysis (information retrieval, data mining, collaborative filtering, formal concept analysis...), and roughly based on an idea from categorical quantum mechanics. It turns out that quantum (i.e., noncommutative) probability distributions arise already in this rudimentary mathematical framework. We show that a Bell-type inequality must be satisfied by the standard similarity measures, if they are used for preference predictions. The fact that already a very general, abstract version of the vector space model yields simple counterexamples for such inequalities seems to be an indicator of a genuine need for quantum statistics in data analysis.
Originally, quantum probability theory was developed to analyze statistical phenomena in quantum systems, where classical probability theory does not apply, because the lattice of measurable sets is not necessarily distributive. On the other hand, it is well known that the lattices of concepts, that arise in data analysis, are in general also non-distributive, albeit for completely different reasons. In his recent book, van Rijsbergen argues that many of the logical tools developed for quantum systems are also suitable for applications in information retrieval. I explore the mathematical support for this idea on an abstract vector space model, covering several forms of data analysis (information retrieval, data mining, collaborative filtering, formal concept analysis...), and roughly based on an idea from categorical quantum mechanics. It turns out that quantum (i.e., noncommutative) probability distributions arise already in this rudimentary mathematical framework. We show that a Bell-type inequality must be satisfied by the standard similarity measures, if they are used for preference predictions. The fact that already a very general, abstract version of the vector space model yields simple counterexamples for such inequalities seems to be an indicator of a genuine need for quantum statistics in data analysis.
The three main points of this article are: 1. Quantum mechanical data differ from conventional data: for example, joint distributions usually cannot be defined conventionally; 2. rigorous methods have been developed for analyzing such data; the methods often use quantum-consistent analogs of classical statistical procedures; 3. with these procedures, statisticians, both data-analytic and more theoretically oriented, can become active participants in many new and emerging areas of science and biotechnology. In the physical realm described by quantum mechanics, many conventional statistical and probabilistic assumptions no longer hold. Probabilistic ideas are central to quantum theory but the standard Kolmogorov axioms are not uniformly applicable. Studying such phenomena requires an altered model for sample spaces, for random variables and for inference and decision making. The appropriate decision theory has been in development since the mid-1960s. It is both mathematically and statistically rigorous and conforms to the requirements of the known physical results. This article provides a tour of the structure and current applications of quantum-consistent statistical inference and decision theory. It presents examples, outlines the theory and considers applications and open problems. Certain central concepts of quantum theory are more clearly apprehended in terms of the quantum-consistent statistical decision theory. For example, the Heisenberg uncertainty principle can be obtained as a consequence of the quantum version of the Cramer-Rao inequality. This places concepts of statistical estimation and decision theory, and thus the statistician, at the center of the quantum measurement process. Quantum statistical inference offers considerable scope for participation by the statistical community, in both applications and foundational questions.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.
The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.
Statistical physical theories are frequently formulated in terms of probabilistic structures founded on a ’’logic of experimentally verifiable propositions.’’ It is argued that to each experimentally verifiable proposition there corresponds an experimental procedure which, in general, alters the state of the system, and is completely characterized by a ’’measurement transformation’’ or ’’operation.’’ An analysis of the relations among these experimental procedures leads us to a ’’logic of operations’’ which is quite different from the ’’lattice theoretic logics’’ that are often considered (albeit inadequate empirical justification), as models for the calculus of experimentally verifiable propositions of quantum theory. It is seen that the quantum probability theory based on the logic of operations provides the proper mathematical framework for discussing the statistics of successive observations in quantum theory. We also indicate how a theory of quantum stochastic processes can be formulated in a way similar to the Kolmogorov formulation of the classical theory.
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.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.