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arXiv:1411.4583 · Canadian Journal of Physics

On the connection between the theorems of Gleason and of Kochen and Specker

Karl-Peter Marzlin, Taylor Landry

Published Jul 14, 2015• 4 citations
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Abstract

We present an elementary proof of a reduced version of Gleason’s theorem and the Kochen–Specker theorem to provide a novel perspective on the relation between both theorems. The proof is based on a set of linear equations for the values of a function, m, on the unit sphere. In the case of Gleason’s theorem, the entire unit sphere needs to be considered, while a finite set of points suffices to prove the Kochen–Specker theorem.

DOI
10.1139/cjp-2014-0631

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We present an elementary proof of a reduced version of Gleason’s theorem and the Kochen–Specker theorem to provide a novel perspective on the relation between both theorems. The proof is based on a set of linear equations for the values of a function, m, on the unit sphere. In the case of Gleason’s theorem, the entire unit sphere needs to be considered, while a finite set of points suffices to prove the Kochen–Specker theorem.
The purpose of this note is to complete the interesting review on quantum contextuality [1] that appeared recently. In particular we will introduce and discuss the ideas of extracontextuality and extravalence, that allow one to relate Kochen-Specker's and Gleason's theorems, and also to shift the emphasis from the first to the second one. We will also argue that whereas Kochen-Specker's is essentially a negative result (a no-go theorem), Gleason's is a positive one since it provides a mathematical justification of Born's rule. The link between these issues is provided by a specific quantum feature that we call extravalence.