Maps on a quantum logic
type: Item
instance of: scholarly article
title: Maps on a quantum logic
zbMATH Open document ID: 1211.03094
author: Ľubica Valášková
author: Ol'ga Nánásiová
published in: Soft Computing
publication date: 2010-08-23T00:00:00Z
review text: Let \(L\) be an orthomodular lattice. The author introduces a class \(\Gamma\) of functions \(G: L \times L \to L\) and studies their properties. \(\Gamma\) splits into eight subclasses \(\Gamma_i\) according to the values of each \(G\) in the points \((O,O)\), \((O,I)\) and \((I,I)\). Three of them, \(\Gamma_2\), \(\Gamma_3\) and \(\Gamma_4\), are of special interest, for they consist, respectively, of the so-called maps for simultaneous measurement, join maps and difference maps, which have already been considered by the first author in earlier papers. More specifically, every function from \(\Gamma_2 \cup \Gamma_3 \cup \Gamma_4\) gives rise to a state \(m\) on \(L\): if \(G \in \Gamma_2\), then \(m(x) := G(x,I)\) and \(G(a,b) = m(a \wedge b)\); if \(G \in \Gamma_3\), then \(m(x) := G(x,x)\) and \(G(a,b) = m(a \vee b)\); if \(G \in \Gamma_4\), then \(m(x) := G(x,O)\) and \(G(a,b) = m(a \triangle b)\) (\(\triangle\) stands for the symmetric difference; \(a\) and \(b\) in these identities are assumed to be compatible).
reviewed by: Jānis Cīrulis
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 5774064
zbMATH Keywords: compatibility
zbMATH Keywords: orthogonality
zbMATH Keywords: orthomodular lattice
zbMATH Keywords: quantum logic
zbMATH Keywords: random event
MaRDI profile type: Publication
full work available at URL: s00500-009-0483-4
OpenAlex ID: W2076354429
cites work: Separating classical and quantum correlations
cites work: Q5427870
cites work: Ideals and filters in D-posets
cites work: Markov property in quantum logic: A reflection
cites work: Q2762622
cites work: Orthomodular lattices admitting no states
cites work: Q5389865
cites work: Map for simultaneous measurements for a quantum logic
cites work: An Orthomodular Lattice Admitting No Group-Valued Measure
cites work: Orthomodular structures as quantum logics. Transl. from the Slovak
cites work: Generalization of blocks for \(D\)-lattices and lattice-ordered effect algebras
cites work: Q5537293
DOI: 10.1007/S00500-009-0483-4
description: scientific article; zbMATH DE number 5774064