Connection between joint distribution and compatibility
type: Item
instance of: scholarly article
title: Connection between joint distribution and compatibility
zbMATH Open document ID: 0552.03041
DOI: 10.1016/0034-4877(84)90007-7
published in: Reports on Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: The paper dicusses the connections between the concept of joint distribution of observables on a quantum logic and that of compatibility of observables. Within the framework of axiomatic quantum theory a necessary and sufficient condition for the existence of the joint distribution of the observables \(x_ 1,...,x_ n\) is given. Namely, \(x_ 1,...,x_ n\) possess a joint distribution in a state m if and only if \[ \sum^{1}_{j_ 1,...,j_ n=0}m(\bigwedge^{n}_{i=1}x_ i(E_ i^{j_ i}))=1 \] for any Borel subsets \(E_ 1,...,E_ n\) of the real line \({\mathbb{R}}\) (by definition, \(E^ j=E\) if \(j=1\), and \(E^ j={\mathbb{R}}\setminus E\) if \(j=0)\). Lateron, a theorem is proved giving a necessary and sufficient condition for the existence of the joint distribution of the observables \(x_ 1,...,x_ n\) with pure point spectra. It is shown that the existence of the joint distribution in a state m implies that the corresponding measurements can be made simultaneously, due to the compatibility of the ''relativized observables''. In the final part of the paper three categories of compatibility of observables are defined: compatibility, partial compatibility, and total incompatibility. The paper ends with some interesting theorems on total incompatibility.
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 81P05
Mathematics Subject Classification ID: 06A99
Mathematics Subject Classification ID: 06C15
zbMATH DE Number: 3878925
zbMATH Keywords: joint distribution of observables
zbMATH Keywords: compatibility of observables
zbMATH Keywords: axiomatic quantum theory
author: Anatolij Dvurečenskij
author: Sylvia Pulmannová
MaRDI profile type: Publication
full work available at URL: 0034-4877(84)90007-7
OpenAlex ID: W2014772830
cites work: Q3853851
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cites work: Joint distributions of observables on quantum logics
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cites work: Q5537293
cites work: Axioms for non-relativistic quantum mechanics
description: scientific article; zbMATH DE number 3878925