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.
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.
Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical probability and statistics. On the other hand, the unique character of quantum physics sets many of the questions addressed apart from those met classically in stochastics. Furthermore, concurrent advances in experimental techniques and in the theory of quantum computation have led to a strong interest in questions of quantum information, in particular in the sense of the amount of information about unknown parameters in given observational data or accessible through various possible types of measurements. This scenery is outlined (with an audience of statisticians and probabilists in mind).
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
Statistical learning theory provides the mathematical foundations for understanding generalization from finite data. As quantum computing and quantum machine learning continue to develop, there is increasing interest in whether analogous principles can be formulated for learning settings involving quantum data, quantum models, or quantum computation. This paper offers an introductory perspective on quantum statistical learning theory. We begin by recalling the basic ideas of classical statistical learning theory, including empirical risk minimization, hypothesis classes, generalization, and sample complexity, and then discuss how these concepts must be adapted in the quantum setting. In particular, we distinguish between learning with classical data using quantum algorithms, learning with quantum models, and learning directly from quantum data. We then outline the central learning-theoretic objects in the quantum setting, including quantum hypotheses, measurement-based predictions, population and empirical risk, shot complexity, and information-extraction limitations. A simple illustrative example is used to highlight how finite measurements and quantum uncertainty affect learning performance. We conclude with a discussion of open problems and directions toward a more unified theory of generalization and learnability in quantum learning.
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.
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 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.