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arXiv:2306.01909

Separation theorems and Bell inequalities in algebraic quantum mechanics

Guido Bacciagaluppi

Published Jun 2, 2023• 7 citations
arXivHTMLPDF

Abstract

The paper discusses the concept of separation of quantum mechanical systems in the algebraic approach. We review known theorems, then establish a link between the C*-algebraic and the corresponding W*-algebraic concepts. A characterization of separation in terms of Bell inequalities, due to Raggio (1988), is given a C*-algebraic formulation. Finally, we comment on the implications for the understanding of the Bell inequalities.

DOI
10.48550/arxiv.2306.01909

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The paper discusses the concept of separation of quantum mechanical systems in the algebraic approach. We review known theorems, then establish a link between the C*-algebraic and the corresponding W*-algebraic concepts. A characterization of separation in terms of Bell inequalities, due to Raggio (1988), is given a C*-algebraic formulation. Finally, we comment on the implications for the understanding of the Bell inequalities.
There has been some recent interest in applying the techniques of Algebraic Quantum Field Theory (AQFT) to entanglement problems in perturbative QFT. In particular, the Hilbert space independence of this formulation makes it particularly interesting in the context of curved spacetimes and the emphasis on the algebra of observables makes the treatment of Bell inequalities in QFT resemble such treatment in non-relativistic Quantum Mechanics. In this work, we present the mathematical structures needed for formulating AQFT in terms of the Haag-Araki-Kastler (HAK) axioms and discuss their implications. Moreover, we discuss the algebraic approach to quantum entanglement in the form of Bell inequalities. We provide an extension of this formulation to general globally hyperbolic spacetimes using the so-called Locally Covariant approach to QFT, which extends the HAK axioms to general spacetimes by means of the Category Theory language.
Bell’s inequalities are briefly presented in the context of order-unit spaces and then studied in some detail in the framework of C*-algebras. The discussion is then specialized to quantum field theory. Maximal Bell correlations β(φ,𝒜(𝒪1), 𝒜(𝒪2)) for two subsystems localized in regions 𝒪1 and 𝒪2 and constituting a system in the state φ are defined, along with the concept of maximal Bell violations. After a study of these ideas in general, properties of these correlations in vacuum states of arbitrary quantum field models are studied. For example, it is shown that in the vacuum state the maximal Bell correlations decay exponentially with the product of the lowest mass and the spacelike separation of 𝒪1 and 𝒪2. This paper is also preparation for the proof in Paper II [S. J. Summers and R. Werner, J. Math. Phys. 28, 2448 (1987)] that Bell’s inequalities are maximally violated in the vacuum state.
This paper reviews some of our understanding of general quantum mechanics. It starts with the exposition of an abstract algebraic formalism useful to formulate classical and quantum-mechanical models of physical systems. It then highlights the essential differences between classical models (commutative algebra) and quantum-mechanical models (non-commutative algebra) of physical systems. It is explained in which sense classical models are “realistic” and deterministic, while quantum-mechanical models are intrinsically probabilistic – in spite of the fact that the Heisenberg time-evolution of operators representing physical quantities is “deterministic”. The quantum theory of time-ordered sequences of measurements is developed in some detail, and the crucial role of “decoherence” in the emergence of facts – or “(almost) consistent histories” – is explained. Some technical matters (Bell inequalities, quantum marginal problem) are discussed in appendices.
Abstract It is shown that the Bell inequalities are closely related to the triangle inequalities involving distance functions amongst pairs of random variables with values { 0 , 1 } . A hidden variables model may be defined as a mapping between a set of quantum projection operators and a set of random variables. The model is noncontextual if there is a joint probability distribution. The Bell inequalities are necessary conditions for its existence. The inequalities are most relevant when measurements are performed at space-like separation, thus showing a conflict between quantum mechanics and local realism (Bell's theorem). The relations of the Bell inequalities with contextuality, the Kochen–Specker theorem, and quantum entanglement are briefly discussed.
John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can be explained by Boole's ideas. Violations of Bell's inequality also call for a revision of the mathematical abstractions and corresponding algebra. It will be shown that this particular view of Bell's inequalities points toward an incompleteness of quantum mechanics, rather than to any superluminal propagation or influences at a distance.
John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can be explained by Boole's ideas. Violations of Bell's inequality also call for a revision of the mathematical abstractions and corresponding algebra. It will be shown that this particular view of Bell's inequalities points toward an incompleteness of quantum mechanics, rather than to any superluminal propagation or influences at a distance.