Must hidden variables theories be contextual? Kochen & Specker meet von Neumann and Gleason

Abstract

It is a widespread belief that the Kochen-Specker theorem imposes a contextuality constraint on the ontology of beables in quantum hidden variables theories. On the other hand, after Bell's influential critique, the importance of von Neumann's wrongly called 'impossibility proof' has been severely questioned. However, Max Jammer, Jeffrey Bub and Dennis Dieks have proposed insightful reassessments of von Neumann's theorem: what it really shows is that hidden variable theories cannot represent their beables by means of Hermitian operators in Hilbert space. Hereby I show that i) the very same constraint can be derived from Gleason's theorem, and that ii) if we consider the import of von Neumann's and Gleason's theorems, the relevance of the Kochen-Specker theorem for hidden variables theories gets substantially weakened: it does not force them to be contextual in any interesting sense of the term.

Contextuality, Fine-Tuning, and Teleological Explanation

“…For realists, there is an obvious response to be made here: the demand for deterministic non-contextuality is a confusion arising from the naive classical view that measurements reveal pre-existing properties, and its lesson is simply that we should avoid taking the mathematical structure of the theory so literally. Indeed, an earlier result by von Neumann had already demonstrated the impossibility of deterministic hidden variable models whose variables are elements of a Hilbert space [15], and Ancuna has recently argued that Gleason's theorem leads to a similar conclusion [6]. As discussed at length in the subsequent literature [6,[16][17][18] these various results provide abundant evidence against straightforward deterministic hidden variable models which read ontic structure directly off mathematical structure, but they don't rule out the possibility of more general hidden variable models -indeed, since the Kochen-Specker theorem was proved a number of models, such as Spekkens' toy model [19], have been proposed to demonstrate that many of the predictions of quantum mechanics can be derived from 'psi-epistemic' models where the underlying ontic structure is different from the mathematical structure.…”

“…Indeed, an earlier result by von Neumann had already demonstrated the impossibility of deterministic hidden variable models whose variables are elements of a Hilbert space [15], and Ancuna has recently argued that Gleason's theorem leads to a similar conclusion [6]. As discussed at length in the subsequent literature [6,[16][17][18] these various results provide abundant evidence against straightforward deterministic hidden variable models which read ontic structure directly off mathematical structure, but they don't rule out the possibility of more general hidden variable models -indeed, since the Kochen-Specker theorem was proved a number of models, such as Spekkens' toy model [19], have been proposed to demonstrate that many of the predictions of quantum mechanics can be derived from 'psi-epistemic' models where the underlying ontic structure is different from the mathematical structure. Therefore these results are no great threat to the broader realist project or indeed to the possibility of determinism.…”

“…It can be shown [1] that the expectation value of Σ for any valid NCHV model is bounded above by the independence number α(G) of the exclusivity graph 5 . On the other hand, in models where the vertices are assigned probabilities derived from a quantum-mechanical representation in terms of density operators and projective measurements, the witness operator is instead upper bounded by the Lovasz number θ(G) of the exclusivity graph 6 . Finally, in models which obey no constraint other than Exclusivity -the requirement that the sum of the probabilities assigned to a set of measurement elements such that any two elements in the set are adjacent, and hence simultaneously measurable, is no greater than 1 -the witness operator is upper bounded by the fractional packing number v F (G) of the graph 7 .…”

“…a set such that no two two vertices in the set are adjacent [66]. 6 The Lovasz number of a graph G with weights {pv} is defined as v pvxv where {xv} is a set of real numbers such that v xv…”

“…On the mathematical side, we now have a rich understanding of the limits of quantum contextuality thanks to several powerful formalisms [1][2][3][4]. On the conceptual side, there are ongoing discussions both about the reasoning that leads us to prefer non-contextual theories [5,6] and the sorts of models which could naturally be expected to lead to contextuality [7,8]. My aim in this article is to clarify how these various approaches to contextuality relate to one another and to explore what this combined body of work means for a scientific realist.…”