A structure for quantum measurements
type: Item
instance of: scholarly article
title: A structure for quantum measurements
zbMATH Open document ID: 1098.81014
DOI: 10.1016/S0034-4877(05)80031-X
published in: Reports on Mathematical Physics
publication date: 2006-03-28T00:00:00Z
review text: We are told at the outset that effects are 2-valued measures, ``yes/no'' experiments or ideally precise measurement outcomes, represented in Hilbert space by projection operators. An effect algebra is generated by a partial binary ``orthosum'' relation over effects, where \(a\oplus b\) represents ``a parallel measurement of \(a\) and \(b\)'' (p. 249). A sequential effect algebra has in addition a binary ``sequential sum'' relation such that \(a\circ b\) represents ``a sequential measurement first of \(a\) then of \(b\)''. The author seeks to extend these definitions to ``unsharp'' measurements ``with a finite or even infinite number of real values'' (p. 250), represented in Hilbert space by the positive operators bounded above by 1. In section 2 generalized definitions introduce effect algebras as structures \(({\mathcal E},0,1,\oplus)\) where \(0\), \(1\) are zero and unit elements, and \(\oplus\) is an orthosum. Each effect \(a\) in \({\mathcal E}\) has a unique complement \(a'\). Effects are now ``quantum events that may be imprecise or fuzzy'' (p. 251). A partial ordering is defined by \(a\leq b\) if there is a \(c\) in \({\mathcal E}\) such that \(a\oplus c= b\), and ``sharp'' effects are now those where \(a\vee a'= 0\). The \(\sigma\)-effect algebras are effect algebras ``where \(a_1\leq a_2\leq\cdots\) implies that the \(\bigvee a_i\) exists.'' (Ibid) Sequential effect algebras (SEA) have an binary operator o defined by conditions that for example allow interval \([0,1]\), a subset of reals, to be a \(\sigma\)-SEA with \(a\circ b= ab\) (p. 253). Measurements are introduced in section 3 as ``normalized, effect valued measures on the \(\sigma\)-effect algebra \({\mathcal B}(\mathbb{R})\)'' of Borel subsets of the reals (p. 253). So a measurement \(X\) is a morphism \(X: {\mathcal B}(\mathbb{R})\to{\mathcal E}\). The author presents a series of results about the structure \({\mathcal M}({\mathcal E})\), the set of measurements on \({\mathcal E}\). Theorem 3.2 sh…
Mathematics Subject Classification ID: 81P15
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 81P05
zbMATH DE Number: 5015613
zbMATH Keywords: quantum measurements
zbMATH Keywords: effect algebras
zbMATH Keywords: sequential products
author: Stan Gudder
MaRDI profile type: Publication
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full work available at URL: s0034-4877(05)80031-x
OpenAlex ID: W1995494917
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description: scientific article; zbMATH DE number 5015613