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Dependency ordering of atomic observables. (English) Zbl 1329.81040

Summary: The notion of atomic observable was introduced by S.Gudder for effect test spaces in 1997. In this paper an observable is a \(\sigma\)-homomorphism from the Borel algebra on a line to some logic. Roughly, an observable on a logic is atomic, if it is completely determined by its restriction to one-element subsets of its point spectrum. In particular, every discrete observable is atomic. We study some elementary properties of such observables, and discuss a possible notion of functional dependency between them. Algebraically, a dependency is a certain preorder relation on the set of all atomic observables, which induces an order relation on the set of all maximal orthogonal subsets of the logic. Several properties, as well as characteristics in terms of the underlying logic, of these relations are stated.

MSC:

81P05 General and philosophical questions in quantum theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
Full Text: DOI

References:

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