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Quantum Correlations

Nuel Belnap, Thomas Müller, Tomasz Placek

Published Dec 16, 2021• 0 citations
Publisher

Abstract

Abstract This chapter analyzes the phenomenon of quantum mechanical correlations using the BST notions of transitions, propensities, and funny business. It considers two ways of understanding such correlations: First, as modal correlations (exhibited, e.g., in the GHZ setup) and, second, as probabilistic correlations (exhibited, e.g., in the Bell-Aspect setup). Having introduced the notion of structure extensions, it asks if it is possible to extend an initial BST structure harboring correlations (modal or probabilistic) in such a way that the resulting extended structure harbors no such correlations. An important distinction, rigorously stated in BST, is that between agents-induced indeterminism and Nature-induced indeterminism. A main result concerning structure extensions is that in the mentioned cases, the procedure is not possible unless the extended structure violates the initially given division between cases of agents-induced indeterminism and cases of Nature-induced indeterminism.

DOI
10.1093/oso/9780190884314.003.0008

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Abstract This chapter analyzes the phenomenon of quantum mechanical correlations using the BST notions of transitions, propensities, and funny business. It considers two ways of understanding such correlations: First, as modal correlations (exhibited, e.g., in the GHZ setup) and, second, as probabilistic correlations (exhibited, e.g., in the Bell-Aspect setup). Having introduced the notion of structure extensions, it asks if it is possible to extend an initial BST structure harboring correlations (modal or probabilistic) in such a way that the resulting extended structure harbors no such correlations. An important distinction, rigorously stated in BST, is that between agents-induced indeterminism and Nature-induced indeterminism. A main result concerning structure extensions is that in the mentioned cases, the procedure is not possible unless the extended structure violates the initially given division between cases of agents-induced indeterminism and cases of Nature-induced indeterminism.
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Abstract The chapter shows how local indeterminism underlying BST combines with relativistic space-times. First it defines particular BST structures in which histories are isomorphic to Minkowski space-times. It further argues that many general relativistic space-times are one-history structures of BST. It introduces the notion of non-Hausdorff differential manifolds and investigates if they can be interpreted modally, as structures of BST with multiple histories. It investigates bifurcating curves in non-Hausdorff manifolds, which are natural representations of alternative evolutions of point-like objects. As the required bifurcating curves are unlikely in General Relativity, whereas there are cases of indeterministic general relativistic space-times, the chapter concludes that General Relativity is globally indeterministic, but locally deterministic.
The paper develops models of statistical experiments that combine propensities with frequencies, the underlying theory being the branching space-times (BST) of Belnap (1992). The models are then applied to analyze Bell's theorem. We prove the so-called Bell-CH inequality via the assumptions of a BST version of Outcome Independence and of (non-probabilistic) No Conspiracy. Notably, neither the condition of probabilistic No Conspiracy nor the condition of Parameter Independence is needed in the proof. As the Bell-CH inequality is most likely experimentally falsified, the choice is this: contrary to the appearances, experimenters cannot choose some measurement settings, or some transitions, with spacelike related initial events, are correlated; or both.
This survey gives a comprehensive account of quantum correlations understood as a phenomenon stemming from the rules of quantization. Centered on quantum probability it describes the physical concepts related to correlations (both classical and quantum), mathematical structures, and their consequences. These include the canonical form of classical correlation functionals, general definitions of separable (entangled) states, definition and analysis of quantumness of correlations, description of entanglement of formation, and PPT states. This work is intended both for physicists interested not only in collection of results but also in the mathematical methods justifying them, and mathematicians looking for an application of quantum probability to concrete new problems of quantum theory.
Introduction In this chapter, we will focus on the phenomenon of quantum entanglement . Entanglement is a special property that some states of two or more quantum particles may possess. When particles are in an entangled state they are correlated. But entanglement is far more than mere correlation . Entanglement is a type of correlation that macroscopic objects cannot have, even in principle; it is what we call a ‘non-local’ correlation. Entanglement, and the non-locality that derives from it, is by now considered to be, arguably, the most important aspect of quantum mechanics. Philosophically it leads to many puzzles and to a complete change in our view of nature. At the same time, the practical implications of this type of correlation are very significant. In particular, they can lead to the dramatically enhanced information processing capabilities that quantum systems possess. The whole field of quantum information processing (Steane, 1998; Bennett and DiVincenzo, 2000; Zoller et al. , 2005) has entanglement at its core. Given the above, it is natural to ask whether biological systems make use and take advantage of the enhanced possibilities that entanglement offers. It seems natural to expect this, especially at biological molecular level, where the laws of quantum mechanics apply. Entanglement between two isolated particles is easy to obtain: almost every time two particles interact they get entangled. But entanglement is very fragile. When two entangled quantum systems interact with a third, they easily become entangled in a tri-partite way; however, any entanglement between any two of them is reduced or disappears altogether.