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Inconsistencies of Fine’s Prism Models

Roel Beernaert

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

The concept of a prism model was introduced by Arthur Fine to be used for local and realistic descriptions of quantum statistics. Recently Feintzeig and Fletcher have shown that a wide class of generalizations of the classical probability model are incapable of replicating the quantum statistics for certain experiments. We employ a modification of their strategy to see under what conditions prism models cannot replicate quantum statistics. In doing so we differentiate two interpretations: one which we can show to be inconsistent in replicating the statistics of certain experiments, and one which commits to unavoidable rejection rates for certain experiments.

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The concept of a prism model was introduced by Arthur Fine to be used for local and realistic descriptions of quantum statistics. Recently Feintzeig and Fletcher have shown that a wide class of generalizations of the classical probability model are incapable of replicating the quantum statistics for certain experiments. We employ a modification of their strategy to see under what conditions prism models cannot replicate quantum statistics. In doing so we differentiate two interpretations: one which we can show to be inconsistent in replicating the statistics of certain experiments, and one which commits to unavoidable rejection rates for certain experiments.
Arthur Fine's use of prism models to provide local and deterministic accounts of quantum correlation experiments is presented and analyzed in some detail. Fine's claim that “there is … no question of the consistency of prism models … with the quantum theory“ (forthcoming, p. 16) is disputed. Our criticisms are threefold: (1) consideration of the possibility of additional analyzer positions shows that prism models entail unacceptably high rejection rates in the relevant experiments; (2) similar considerations show that the models are at best only superficially local and deterministic; and (3) in any case, Fine extracts the quantum correlation statistics from prism models only by resurrecting conceptual problems similar to those that his models were to designed to solve.
In the first part of this two-part article (Aerts & Sassoli de Bianchi, 2014), we have introduced and analyzed a multidimensional model, called the general tension-reduction (GTR) model, able to describe general quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a universal measurement. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by uniform fluctuations on the measurement situation. Since quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied–mutatis mutandis–to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally robust descriptions of indeterministic reproducible experiments, and since quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the quantum statistics in the description of a large class of phenomena.
Recently, there has been some discussion of how Dutch book arguments might be used to demonstrate the rational incoherence of certain hidden variable models of quantum theory (Feintzeig [2015]; Feintzeig and Fletcher [2017]; Wroński and Godziszewski [2017]). In this paper, we argue that the ‘form of inconsistency’ underlying this alleged irrationality is deeply and comprehensively related to the more familiar ‘inconsistency’ phenomenon of contextuality. Our main result is that the hierarchy of contextuality due to Abramsky and Brandenburger ([2011]) corresponds to a hierarchy of additivity/convexity-violations that yields formal Dutch books of different strengths. We then use this result to provide a partial assessment of whether these formal Dutch books can be given a convincing normative interpretation.
We investigate how Dutch Book considerations can be conducted in the context of two classes of nonclassical probability spaces used in philosophy of physics. In particular we show that a recent proposal by B. Feintzeig to find so called “generalized probability spaces” which would not be susceptible to a Dutch Book and would not possess a classical extension is doomed to fail. Noting that the particular notion of a nonclassical probability space used by Feintzeig is not the most common employed in philosophy of physics, and that his usage of the “classical” Dutch Book concept is not appropriate in “nonclassical” contexts, we then argue that if we switch to the more frequently used formalism and use the correct notion of a Dutch Book, then all probability spaces are not susceptible to a Dutch Book. We also settle a hypothesis regarding the existence of classical extensions of a class of generalized probability spaces.
It was recently argued by Catani et al that it is possible to reproduce the phenomenology of quantum interference classically, by the double-slit experiment with a deterministic, local, and classical model (Quantum 7, 1119 (2023)). The stated aim of their argument is to falsify the claim made by Feynman (in his third book of Lectures on Physics) that quantum interference is ``impossible, absolutely impossible, to explain in any classical way'' and that it ``contains the only mystery'' of quantum mechanics. We here want to point out some problems with their argument.