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Superposition, transition probabilities and primitive observables in infinite quantum systems

Detlev Buchholz A, Erling Størmer

Published Aug 17, 2016• 13 citations
OpenAlex

Abstract

The concepts of superposition and of transition probability, familiar from pure states in quantum physics, are extended to locally normal states on funnels of type I∞ factors. Such funnels are used in the description of infinite systems, appearing for example in quantum field theory or in quantum statistical mechanics; their respective constituents are interpreted as algebras of observables localized in an increasing family of nested spacetime regions. Given a generic reference state (expectation functional) on a funnel, e.g. a ground state or a thermal equilibrium state, it is shown that irrespective of the global type of this state all of its excitations, generated by the adjoint action of elements of the funnel, can coherently be superimposed in a meaningful manner. Moreover, these states are the extreme points of their convex hull and as such are analogues of pure states. As further support of this analogy, transition probabilities are defined, complete families of orthogonal states are exhibited and a one-to-one correspondence between the states and families of minimal projections on a Hilbert space is established. The physical interpretation of these quantities relies on a concept of primitive observables. It extends the familiar framework of observable algebras and avoids some counter intuitive features of that setting. Primitive observables admit a consistent statistical interpretation of corresponding measurements and their impact on states is described by a variant of the von Neumann–Lüders projection postulate.

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The concepts of superposition and of transition probability, familiar from pure states in quantum physics, are extended to locally normal states on funnels of type I∞ factors. Such funnels are used in the description of infinite systems, appearing for example in quantum field theory or in quantum statistical mechanics; their respective constituents are interpreted as algebras of observables localized in an increasing family of nested spacetime regions. Given a generic reference state (expectation functional) on a funnel, e.g. a ground state or a thermal equilibrium state, it is shown that irrespective of the global type of this state all of its excitations, generated by the adjoint action of elements of the funnel, can coherently be superimposed in a meaningful manner. Moreover, these states are the extreme points of their convex hull and as such are analogues of pure states. As further support of this analogy, transition probabilities are defined, complete families of orthogonal states are exhibited and a one-to-one correspondence between the states and families of minimal projections on a Hilbert space is established. The physical interpretation of these quantities relies on a concept of primitive observables. It extends the familiar framework of observable algebras and avoids some counter intuitive features of that setting. Primitive observables admit a consistent statistical interpretation of corresponding measurements and their impact on states is described by a variant of the von Neumann–Lüders projection postulate.
Abstract Philosophical accounts of quantum theory commonly suppose that the observables of a quantum system form a Type-I factor von Neumann algebra. Such algebras always have atoms, which are minimal projection operators in the case of quantum mechanics. However, relativistic quantum field theory and the thermodynamic limit of quantum statistical mechanics make extensive use of von Neumann algebras of more general types. This chapter addresses the question whether interpretations of quantum probability devised in the usual manner continue to apply in the more general setting. Features of non-type I factor von Neumann algebras are cataloged. It is shown that these novel features do not cause the familiar formalism of quantum probability to falter, since Gleason's Theorem and the Lüders Rule can be generalized. However, the features render the problem of the interpretation of quantum probability more intricate.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.
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.
Transition probabilities are an important and useful tool in quantum mechanics. However, in their present form, they are limited in scope and only apply to pure quantum states. In this article we extend their applicability to mixed states and to transitions between quantum effects. We also present their dependence on a measured operation or instrument. We begin by defining our concepts on a general quantum effect algebra. These concepts are illustrated using Holevo operations and instruments. We then present transition probabilities in the special case of the Hilbert space formulation of quantum mechanics. We show that for pure states and particular types of operations the transition probabilities reduce to their usual form. We give examples in terms of Lüders operations and instruments.
Max Born's statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg's and others' uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.