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arXiv:quant-ph/9912081

A Kochen-Specker Theorem for Imprecisely Specified Measurement

N. David Mermin

Published Dec 16, 1999• 18 citations
arXivHTMLPDF

Abstract

A recent claim that finite precision in the design of real experiments ``nullifies'' the impact of the Kochen-Specker theorem, is shown to be unsupportable, because of the continuity of probabilities of measurement outcomes under slight changes in the experimental configuration.
DOI
10.48550/arxiv.quant-ph/9912081

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Kent's conclusion that ``non-contextual hidden variable theories cannot be excluded by theoretical arguments of the Kochen-Specker type once the imprecision in real world experiments is taken into account'' [Phys. Rev. Lett. 83, 3755 (1999)], is criticized. The Kochen-Specker theorem just points out that it is impossible even conceive a hidden variable model in which the outcomes of all measurements are pre-determined; it does not matter if these measurements are performed or not, or even if these measurements can be achieved only with finite precision.
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Meyer [Phys. Rev. Lett. 83, 3751 (1999)] recently showed that the Kochen-Specker theorem, which demonstrates the impossibility of a deterministic hidden variable description of ideal spin measurements on a spin 1 particle, can be effectively nullified if only finite precision measurements are considered. We generalize this result: one can ascribe consistent outcomes to a dense subset of the set of projection valued measurements, or to a dense subset of the set of positive operator valued measurements, on any finite dimensional system. Hence no Kochen-Specker-like contradiction can rule out hidden variable theories indistinguishable from quantum theory by finite precision measurements in either class.
It has recently been questioned whether the Kochen-Specker theorem is relevant to real experiments, which by necessity only have finite precision. We give an affirmative answer to this question by showing how to derive hidden-variable theorems that apply to real experiments, so that noncontextual hidden variables can indeed be experimentally disproved. The essential point is that for the derivation of hidden-variable theorems one does not have to know which observables are really measured by the apparatus. Predictions can be derived for observables that are defined in an entirely operational way.
We review non-classicality of quantum datum. We consider whether we can assign the predetermined ^^ ^^ hidden'' result to numbers 1 and −1-1−1 as in results of measurements in a thought experiment. We assume the number of measurements is two. If we detect ∣↑⟩|\uparrow\rangle∣↑⟩ as 1 and detect ∣↓⟩|\downarrow\rangle∣↓⟩ as −1-1−1, then we can derive the Kochen-Speker theorem. The same situation occurs when we use a finite-precision measurement theory that the results of measurements are either 1−ϵ1-\epsilon1−ϵ or −1+ϵ-1+\epsilon−1+ϵ.