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American Mathematical Monthly

Finite Quantum Measure Spaces

Stanley Gudder

Published Jan 1, 2010• 16 citations
Publisher

Abstract

Quantum measure spaces possess a certain “quantum weirdness” and lack some of the simplicity and intuitive nature of their classical counterparts. Much of this unusual behavior is due to a phenomenon called quantum interference, which is a recurrent theme in the present article. Because of this interference, quantum measures need not be additive but satisfy a more general condition called grade-2 additivity. Examples of quantum measure spaces such as “quantum coins” and particle-an tip article pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don't need quantum mechanics or measure theory to understand this article.

DOI
10.4169/000298910x492808

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Quantum measure spaces possess a certain “quantum weirdness” and lack some of the simplicity and intuitive nature of their classical counterparts. Much of this unusual behavior is due to a phenomenon called quantum interference, which is a recurrent theme in the present article. Because of this interference, quantum measures need not be additive but satisfy a more general condition called grade-2 additivity. Examples of quantum measure spaces such as “quantum coins” and particle-an tip article pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don't need quantum mechanics or measure theory to understand this article.
This pedagogical article solves an interesting problem in quantum measure theory. Although a quantum measure μμμ is a generalization of an ordinary probability measure, μμμ need not satisfy the usual additivity condition. Instead, μμμ satisfies a grade-2 additivity condition. Besides the quantum measure problem, we present three additional combinatorial problems. These are (1)\enspace A sum of binomial coefficients problem;\enspace (2)\enspace A recurrence relation problem; and\enspace (3)\enspace An interated vector problem. We show that these three problems are equivalent in that they have a common solution. We then show that this solves the original quantum measure problem.
Grade-d measures on a σ-algebra A⊆2X over a set X are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions initially introduced as interference operators in quantum mechanics. Every signed polymeasure λ on (X,A)d produces a grade-d measure as its diagonal λ˜(A):=λ(A,…,A), and we prove that as soon as d≥2, measures (as opposed to polymeasures) do not suffice: the separate σ-additivity of a λ producing μ=λ˜ cannot, generally, be amplified to global σ-additivity. This amends a result in the literature, asserting the contrary in case d=2.
Abstract Quantum physics is often perceived as a weird and abstract theory, which physicists must use in order to make correct predictions. But many recent experiments have shown that the weirdness of the theory simply mirrors the weirdness of phenomena: it is Nature itself, and not only our description of it, that behaves in an astonishing way. This book selects those, among these typical quantum phenomena, whose rigorous description requires neither the formalism, nor an important background in physics. The first part of the book deals with the phenomenon of single-particle interference, covering the historical questions of wave-particle duality, objective randomness and the boundary between the quantum and the classical world, but also the recent idea of quantum cryptography. The second part introduces the modern theme of entanglement, by presenting two-particle interference phenomena and discussing Bell’s inequalities. A concise review of the main interpretations of quantum physics is provided.
Measure theory is used in physics, not just to capture classical probability, but also to quantify the number of states. In previous works, we found that state quantification plays a foundational role in classical mechanics, and, therefore, we set ourselves to construct the quantum equivalent of the Liouville measure. Unlike the classical counterpart, this quantized measure is non-additive and has a unitary lower bound (i.e., no set of states can have less than one state). Conversely, requiring that state quantification is finite for finite continuous regions and that each state counts as one already implies non-additivity, which in turn implies the failure of classical theory. In this article we show these preliminary results and outline a new line of inquiry that may provide a different insight into the foundations of quantum theory. Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom.
In recent notes by Einstein, Podolsky and Rosen and by Bohr, attention has been called to the fact that certain results of quantum mechanics are not to be reconciled with the assumption that a system has independently real properties as soon as it is free from mechanical interference. We here investigate in general, and in abstract terms, the extent of this disagreement. When suitably formulated, such an assumption gives to certain types of questions the same answers as does quantum mechanics; this is true of the formulas usually given in discussions of the theory of measurement. There exists, however, a general class of cases in which contradictions occur. That such contradictions are not restricted to the abstract mathematical theory, but can be realized in the commonest physical terms, is shown by the working out of an example..