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arXiv:quant-ph/0309017 · Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

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

Jonathan Barrett, Adrian Kent

Published Mar 19, 2004• 60 citations
arXivHTMLPDF
DOI
10.1016/j.shpsb.2003.10.003

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We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision measurements on defining aspects of quantum mechanics such as contextuality and the Born rule. While foundational results such as the Kochen-Specker and Gleason theorems are valid in the context of infinite precision, they fail to hold in general in a world with limited resources. Here we employ an interval-valued framework to establish bounds on the validity of those theorems in realistic experimental environments. In this way, not only can we quantify the idea of finite-precision measurement within our theory, but we can also suggest a possible resolution of the Meyer-Mermin debate on the impact of finite-precision measurement on the Kochen-Specker theorem.
The claim of Meyer, Kent and Clifton (MKC) that finite precision measurement nullifies the Kochen-Specker theorem is criticised. It is argued that, although MKC have nullified the Kochen-Specker theorem strictly so-called, there are other, related propositions which are not nullified. The argument given is an elaboration of some of Mermin's critical remarks. Although MKC allow for the fact that the observables to be measured cannot be precisely specified, they continue to assume that the observables which are actually measured are strictly commuting. As Mermin points out, this assumption is unjustified. Consequently, the analysis of MKC is incomplete. To make it complete one needs to investigate the predictions their models make regarding approximate joint measurements of non-commuting observables. Such an investigation is carried out, using methods previously developed in connection with approximate joint measurements of position and momentum. It is shown that a form of contextuality then re-emerges.
The possibility of experimentally testing the Bell–Kochen–Specker theorem is investigated critically, following the demonstrations by Meyer, Kent, and Clifton–Kent that the predictions of quantum mechanics are indistinguishable (up to arbitrary precision) from those of a non-contextual model, and the subsequent debate about the extent to which these models are actually classical or non-contextual. The present analysis starts from a careful consideration of these 'finite-precision' approximations. A stronger condition for non-contextual models, dubbed ontological faithfulness , is exhibited. It is shown that this allows us to approximately formulate the constraints in Bell–Kochen–Specker theorems, such as to render the usual proofs robust. Consequently, one can experimentally test to finite precision ontologically faithful non-contextuality, and thus experimentally refute explanations from this smaller class. We include a discussion of the relation of ontological faithfulness to other proposals to overcome the finite precision objection. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.