Maximal violation of Bell's inequalities is generic in quantum field theory
type: Item
instance of: scholarly article
title: Maximal violation of Bell's inequalities is generic in quantum field theory
zbMATH Open document ID: 0626.46056
DOI: 10.1007/BF01207366
author: Reinhard F. Werner
author: Stephen Jeffrey Summers
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: Under weak technical assumptions on a net of local von Neumann algebras \(\{\) \({\mathcal A}({\mathcal O})\}\) in a Hilbert space \({\mathcal H}\), which are fulfilled by any net associated to a quantum field satisfying the standard axioms, it is shown that for every vector space \(\phi\) in \({\mathcal H}\) there exist observables localized in complementary wedge-shaped regions in Minkowski space-time that maximally violate Bell's inequalities in the state \(\phi\). If, in addition, the algebras corresponding to wedge-shaped regions are injective (which is known to be true in many examples), then he maximal violation occurs in any state \(\phi\) on \({\mathcal B}({\mathcal H})\) given by a density matrix. Mathematically, the primary result is Theorem 2.3. Let \({\mathcal M}\) be a von Neumann algebra on a separable Hilbert space \({\mathcal H}\), \({\mathcal M}^ 1\) be its commutant, and \(\phi\) be a state on \({\mathcal B}({\mathcal H})\). Define \(\beta\) (\(\phi\),\({\mathcal M},{\mathcal M}^ 1)\equiv \sup \phi (A_ 1(B_ 1+B_ 2)+A_ 2(B_ 1-B_ 2))\), where the supremum is taken over all selfadjoint contractions \(A_ i\in {\mathcal M}\), \(B_ j\in {\mathcal M}^ 1\). The theorem 2.3 asserts (in part) that if \({\mathcal M}\) has a cyclic and separating vector in \({\mathcal H}\), then \({\mathcal M}\) is strongly stable (i.e. isomorphic to \({\mathcal M}\otimes {\mathcal R}_ 1\), where \({\mathcal R}_ 1\) is the hyperfinite type \(II_ 1\) factor) if and only if \(\beta\) (\(\phi\),\({\mathcal M},{\mathcal M}^ 1)=\sqrt{2}\) for all states \(\phi\in {\mathcal B}({\mathcal H})\).
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 81T05
zbMATH DE Number: 4017656
zbMATH Keywords: Bell's inequalities
zbMATH Keywords: net of local von Neumann algebras
zbMATH Keywords: quantum field
zbMATH Keywords: vector space
zbMATH Keywords: Minkowski space-time
zbMATH Keywords: maximal violation
zbMATH Keywords: density matrix
zbMATH Keywords: cyclic and separating vector
zbMATH Keywords: strongly stable
zbMATH Keywords: factor
MaRDI profile type: Publication
Wikidata QID: Q62124875
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description: scientific article; zbMATH DE number 4017656