Non-contextuality, finite precision measurement and the Kochen–Specker theorem

Abstract

Meyer originally raised the question of whether non-contextual hidden variable models can, despite the Kochen-Specker theorem, simulate the predictions of quantum mechanics to within any fixed finite experimental precision (Meyer, D. 1999. Phys. Rev. Lett., 83, 3751-3754). Meyer's result was extended by Kent (Kent, A. 1999. Phys. Rev. Lett., 83, 3755-3757). Clifton and Kent later presented constructions of non-contextual hidden variable theories which, they argued, indeed simulate quantum mechanics in this way (Clifton, R and Kent, A. 2000. Proc. Roy. Soc. Lond. A, 456, 2101-2114.These arguments have evoked some controversy. Among other things, it has been suggested that the Clifton-Kent models do not in fact reproduce correctly the predictions of quantum mechanics, even when finite precision is taken into account. It has also been suggested that careful analysis of the notion of contextuality in the context of finite precision measurement motivates definitions which imply that the Clifton-Kent 1 models are in fact contextual. Several critics have also argued that the issue can be definitively resolved by experimental tests of the KochenSpecker theorem or experimental demonstrations of the contextuality of Nature.One aim of this paper is to respond to and rebut criticisms of the Meyer-Clifton-Kent papers. We thus elaborate in a little more detail how the Clifton-Kent models can reproduce the predictions of quantum mechanics to arbitrary precision. We analyse in more detail the relationship between classicality, finite precision measurement and contextuality, and defend the claims that the Clifton-Kent models are both essentially classical and non-contextual. We also examine in more detail the senses in which a theory can be said to be contextual or non-contextual, and in which an experiment can be said to provide evidence on the point. In particular, we criticise the suggestion that a decisive experimental verification of contextuality is possible, arguing that the idea rests on a conceptual confusion.

Violation of the Leggett–Garg inequality with weak measurements of photons

“…Therefore some of the criticisms levelled at contextuality tests (see, e.g., ref. 24 and references therein) do not apply to the LGI. It may be that there is a deeper connection between contextuality and noninvasive measurement, but we leave that for future work.…”

Classical Probability and Quantum Outcomes

“…A link is also drawn with research dealing with a very specific countable collection of measurements over which quantum noncontextuality of results is assumed to be valid; see [8,9]. This work shows how a kind of rationals-only restriction of quantum noncontextuality can avoid contradiction with the standard quantum formalism.…”

“…It is useful to note that the proof of Theorem 3 invokes rotations about angles such that the cosine is transcendental. This suggests that the collection of rationals-only projectors considered in [8,9] would be outside the scope of the argument. More on this is presented in the Discussion.…”

“…This notion of QNC(Value) and its consequences has been extensively studied in the foundations literature: see [1][2][3][4][5][6][7][8][9]. It is a form of classical determinism and has been called the noncontextuality of results; see [3].…”

Compatibility and noncontextuality for sequential measurements

“…There have been several proposals to test the KS theorem [20][21][22][23][24], but there also have been debates about whether the KS theorem can be experimentally tested at all [25][26][27][28][29][30][31][32][33][34]. Nevertheless, first experiments have been performed, but these experiments required some additional assumptions [35][36][37][38].…”