QP-PQ, quantum probability and white noise analysis

ELEMENTS OF QUANTUM PROBABILITY

Burkhard Kümmerer, Hans Maassen

Published Oct 1, 1998• 17 citations

Abstract

This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is developed. It can describe, in particular, the polarization experiments. Some examples of quantum coin tosses are discussed, closely related to V.F.R. Jones approach to braid group representations, to spin relaxion, and to nuclear magnetic resonance. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability.

DOI: 10.1142/9789812816054_0003

Related papers

Introduction

James H. McGuire

This introductory chapter begins with a review of uncorrelated classical probabilities and then extends these concepts to correlated quantum systems. This is done both to establish notation and to provide a basis for those who are not experts to understand material in later chapters. Probability of a transition Seldom does one know with certainty what is going to happen on the atomic scale. What can be determined is the probability P that a particular outcome (i.e. atomic transition) will result when many atoms interact with photons, electrons or protons. A transition occurs in an atom when one or more electrons jump from their initial state to a different final state in the atom. The outcome of such an atomic transition is specified by the final state of the atom after the interaction occurs. Since there are usually many atoms in most systems of practical interest, we can usually determine with statistical reliability the rate at which various outcomes (or final states) occur. Thus, although one is unable to predict what will happen to any one atom, one may determine what happens to a large number of atoms. 1 Single particle probability A simple basic analogy is tossing a coin or dice. Tossing of a coin is analogous to interacting with an atom. In the case of a simple coin there are two outcomes: after the toss one side of the coin (‘heads’) will either occur or it will not occur.

Cambridge University Press eBooksMar 20, 19970 citations

The Noiseless Quantum Theory

Mark M. Wilde

Cambridge University Press eBooksApr 18, 20130 citations

Introduction to Quantum Theory

Andrei Yu. Khrennikov

This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.

Jul 30, 20200 citations

Contributions to quantum probability

T. A. Fritz

Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.

Jan 1, 20100 citations

The Noiseless Quantum Theory

Mark M. Wilde

The simplest quantum system is the physical quantum bit or qubit . The qubit is a two-level quantum system—example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. We do not worry too much about physical implementations in this chapter, but instead focus on the mathematical postulates of the quantum theory and operations that we can perform on qubits. From qubits we progress to a study of physical qudits . Qudits are quantum systems that have d levels and are an important generalization of qubits. Again, we do not discuss physical realizations of qudits. Noise can affect quantum systems, and we must understand methods of modeling noise in the quantum theory because our ultimate aim is to construct schemes for protecting quantum systems against the detrimental effects of noise. In Chapter 1, we remarked on the different types of noise that occur in nature. The first, and perhaps more easily comprehensible type of noise, is that which is due to our lack of information about a given scenario. We observe this type of noise in a casino, with every shuffle of cards or toss of dice. These events are random, and the random variables of probability theory model them because the outcomes are unpredictable. This noise is the same as that in all classical information-processing systems. On the other hand, the quantum theory features a fundamentally different type of noise. Quantum noise is inherent in nature and is not due to our lack of information, but is due rather to nature itself. An example of this type of noise is the “Heisenberg noise” that results from the uncertainty principle. If we know the momentum of a given particle from performing a precise measurement of it, then we know absolutely nothing about its position—a measurement of its position gives a random result. Similarly, if we know the rectilinear polarization of a photon by precisely measuring it, then a future measurement of its diagonal polarization will give a random result. It is important to keep the distinction clear between these two types of noise.

Cambridge University Press eBooksNov 2, 20160 citations

ON THE COMBINATION OF PROBABILITY AND RELATIVITY

Yrjö Ahmavaara

Classes of mathematic models in which both probability and relativity can be said to be involved are discussed in probabilistic terminology. Mathematically, this implies combination of a group notation with probabilistic structure in a way that may be useful in applied mathematics. The best known example of a model, where both relativity and probability are involved, is the formalism of quantum theory. (C.E.S.)

Jan 1, 19620 citations