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Generic Bell correlation between arbitrary local algebras in quantum field theory

Rob Clifton

Published Sep 23, 2009• 1 citation
Publisher

Abstract

Abstract There are many senses in which the phenomenon of Bell correlation, originally discovered and investigated in the context of elementary nonrelativistic quantum mechanics (Bell 1987, Clauser et al. 1969), is ‘generic’ in quantum field theory models. For example, it has been shown that every pair of commuting nonabelian von Neumann algebras possesses some normal state with maximal Bell correlation (Summers 1990; see also Landau 1987b). Moreover, in most standard quantum field models, all normal states are maximally Bell correlated across spacelike separated tangent wedges

DOI
10.1093/oso/9780199270156.003.0006

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Abstract There are many senses in which the phenomenon of Bell correlation, originally discovered and investigated in the context of elementary nonrelativistic quantum mechanics (Bell 1987, Clauser et al. 1969), is ‘generic’ in quantum field theory models. For example, it has been shown that every pair of commuting nonabelian von Neumann algebras possesses some normal state with maximal Bell correlation (Summers 1990; see also Landau 1987b). Moreover, in most standard quantum field models, all normal states are maximally Bell correlated across spacelike separated tangent wedges
The authors survey what is known about correlations in physical systems. Bell (1964-6) found that locally realistic theories are limited in the types of correlations which they allow. The more recent results of Greenberger, Horne and Zeilinger (1992) and of Hardy (1989) have shown that local realism imposes even stronger restrictions than those found by Bell. It is less well known that quantum mechanics is also limited in the kinds of correlations it can produce. This was first discovered by Tsirelson (1980). The authors discuss Tsirelson's results and related work by Landau (1987). They conclude by presenting further conditions which quantum mechanical correlations must satisfy.
We prove that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algebras is norm dense. We then apply this result to algebraic quantum field theory—where all local algebras are of infinite type—in order to show that for any two spacelike separated regions, there is an open dense set of field states that dictate Bell correlations between the regions. We also show that any vector state cyclic for one of a pair of commuting non-Abelian von Neumann algebras is entangled (i.e., nonseparable) across the algebras—from which it follows that every field state with bounded energy is entangled across any two spacelike separated regions.
We prove that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algebras is norm dense. We then apply this result to algebraic quantum field theory—where all local algebras are of infinite type—in order to show that for any two spacelike separated regions, there is an open dense set of field states that dictate Bell correlations between the regions. We also show that any vector state cyclic for one of a pair of commuting non-Abelian von Neumann algebras is entangled (i.e., nonseparable) across the algebras—from which it follows that every field state with bounded energy is entangled across any two spacelike separated regions.
We continue our study of Bell's inequalities and quantum field theory. It is shown in considerably broader generality than in our previous work that algebras of local observables corresponding to complementary wedge regions maximally violate Bell's inequality in all normal states. Pairs of commuting von Neumann algebras that maximally violate Bell's inequalities in all normal states are characterized. Algebras of local observables corresponding to tangent double cones are shown to maximally violate Bell's inequalities in all normal states in dilatation-invariant theories, in free quantum field models, and in a class of interacting models. Further, it is proven that such algebras are not split in any theory with an ultra-violet scaling limit On montre que les algebres d'observables locales correspondant a des domaines en coins violent au maximum les inegalites de Bell dans tous les etats normaux. On caracterise les paires d'algebres de Von Neumann qui commutent et qui violent maximalement les inegalites de Bell. On montre que les algebres d'observables correspondant a des cones doubles tangents violent aussi maximalement les inegalites de Bell dans tous les etats normaux des theories invariantes par dilatation, dans les modeles de champs libres et dans des modeles en interaction. On montre que ces algebres ne sont «split» dans aucune theorie avec une limite d'echelle UV
Employing mutually-commuting von Neumann algebras to represent the algebra of observables on quantum systems provides a framework for studying quantum information theory in systems with infinite degrees of freedom and quantum field theory, yielding many profound results that differ from non-relativistic quantum systems. In this paper, we establish a mutually-commuting von Neumann algebra model of quantum networks with arbitrary structures. We derive Bell-type inequalities on this model, and determine various bounds for Bell-type inequalities based on the structure of underline von Neumann algebras, and identify the algebraic structural conditions required for their violation. The conditions on the algebraic structure of observables for maximal violation of Bell-type inequalities, which we discovered in the context of von Neumann algebra models, can in turn guide the search for measurements in the non-relativistic setting.