Hilbert-space-valued states on quantum logics
type: Item
instance of: scholarly article
title: Hilbert-space-valued states on quantum logics
zbMATH Open document ID: 0767.03034
published in: Applications of Mathematics
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/15700
review text: The authors study finitely additive orthogonal vector states (=normed measures) with values in a real Hilbert space (so called \(h\)-states). It is shown that there is no \(h\)-state on a Hilbert logic \(L(M)\) of projections in \(H\), \(\dim H \geq 3\), to a Hilbert space of a lesser dimension. Further, a Hilbert space \(H\), \(\dim H \geq 3\), has a finite dimension iff every \(h\)-state on \(L(H)\) to \(H\) is, up to a unitary mapping, concentrated on a one-dimensional projection. Finally, a characterization of logics with extension property with respect to \(h\)-states to a finite dimensional Hilbert space is given.
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46N50
zbMATH DE Number: 58780
zbMATH Keywords: Hilbert-space-valued state
zbMATH Keywords: finitely additive orthogonal vector states
zbMATH Keywords: normed measures
zbMATH Keywords: Hilbert logic
zbMATH Keywords: extension property
author: Jan Hamhalter
author: Pavel Pták
reviewed by: Josef Tkadlec
MaRDI profile type: Publication
cites work: Q3909265
cites work: Q3736359
cites work: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?
cites work: Q3245484
cites work: Orthomodular lattices admitting no states
cites work: Q3881378
cites work: Dispersion-Free States and the Exclusion of Hidden Variables
cites work: Measures in Boolean Algebras
cites work: Vector measures on the closed subspaces of a Hilbert space
cites work: Q3803173
cites work: Q4711790
cites work: Exotic logics
cites work: Weak dispersion-free states and the hidden variables hypothesis
cites work: Q3712337
cites work: A characterization of state spaces of orthomodular lattices
description: scientific article; zbMATH DE number 58780