[Paper Review] Noncontextuality, Finite Precision Measurement and the Kochen-Specker Theorem

Jonathan Barrett, Adrian Kent|ArXiv.org|Sep 1, 2003

Quantum Mechanics and Applications31 references4 citations

TL;DR

This paper defends the Clifton-Kent non-contextual hidden variable models against criticisms that they are contextual or fail to reproduce quantum mechanics under finite precision measurement. It argues that these models can simulate quantum predictions to arbitrary precision using classical mechanics, and that operational definitions of contextuality cannot unambiguously distinguish classical from quantum behavior, thus preserving the logical possibility of non-contextual simulations despite the Kochen-Specker theorem.

ABSTRACT

Meyer recently queried 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. Clifton and Kent have presented constructions of non-contextual hidden variable theories which, they argued, indeed simulate quantum mechanics in this way. These arguments have evoked some controversy. One aim of this paper is to respond to and rebut criticisms of the MCK papers. We thus elaborate in a little more detail how the CK 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 CK 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.

Motivation & Objective

• To respond to criticisms that the Clifton-Kent models are contextual or fail to reproduce quantum predictions under finite precision measurement.
• To clarify the conceptual relationship between classicality, finite precision, and contextuality in hidden variable theories.
• To argue that operational definitions of contextuality cannot definitively distinguish classical from quantum behavior, undermining claims of experimental verification of contextuality.
• To reaffirm that non-contextual hidden variable models can, in principle, reproduce quantum mechanics to any fixed finite precision, challenging the necessity of contextuality in fundamental physics.

Proposed method

• Analyzes the Clifton-Kent models in detail, showing how they assign values to observables using finite subsets of projectors that reproduce quantum probabilities within a fixed precision ε.
• Demonstrates that these models are constructed from classical mechanics principles and involve only finitely many degrees of freedom, making them physically realizable.
• Revisits the definition of contextuality in the context of finite precision, arguing that the standard KS criteria do not apply when measurement precision is limited.
• Critiques operational definitions of contextuality (e.g., SBZ, Larsson) as theory-relative and unable to distinguish classical from quantum behavior in practice.
• Uses logical and conceptual analysis to show that finite sub-models of the Clifton-Kent construction are not inherently contextual, even if they approximate quantum behavior.
• Argues that no experiment can definitively verify contextuality because operational definitions conflate measurement disturbance with contextuality, leading to conceptual confusion.

Experimental results

Research questions

• RQ1 Can non-contextual hidden variable models simulate quantum mechanics to within any fixed finite experimental precision, despite the Kochen-Specker theorem?
• RQ2 Are the Clifton-Kent models truly non-contextual when finite precision measurement is taken into account?
• RQ3 Can operational definitions of contextuality unambiguously distinguish classical from quantum theories in practice?
• RQ4 Is it possible to definitively test contextuality through experiment, as some critics claim?
• RQ5 What is the correct conceptual relationship between classical mechanics, finite precision, and contextuality in hidden variable theories?

Key findings

• The Clifton-Kent models successfully reproduce quantum predictions to arbitrary finite precision ε by using finite subsets of projectors that are not KS-uncolourable.
• These models are physically realizable as classical systems with finitely many degrees of freedom, demonstrating that classical mechanics can simulate quantum behavior under finite precision.
• The models are non-contextual under the standard KS definition, as they assign values to observables without dependence on the measurement context, even when the full set is KS-uncolourable.
• Operational definitions of contextuality, such as those based on knob-setting perturbations, fail to distinguish classical from quantum behavior because classical systems can also exhibit similar sensitivity.
• No experiment can definitively verify contextuality because operational definitions conflate measurement disturbance with contextuality, leading to conceptual confusion.
• The paper concludes that the Kochen-Specker theorem does not rule out non-contextual simulations of quantum mechanics under finite precision, preserving a logical loophole in the argument against non-contextual hidden variables.