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Physical Review A

Quantum violation of stochastic noncontextual hidden-variable theories

Shasanka Mohan Roy, Virendra Singh

Published Oct 1, 1993• 21 citations
Publisher

Abstract

We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state \ensuremath{\lambda} specifies not definite values, but expectation values A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}), . . . of observables A, . . . and noncontextuality means that A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (${2}^{\mathit{n}}$-1)/2, quantum violation of classical noncontextuality grows exponentially with n.

DOI
10.1103/physreva.48.3379

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We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state \ensuremath{\lambda} specifies not definite values, but expectation values A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}), . . . of observables A, . . . and noncontextuality means that A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (2n{2}^{\mathit{n}}2n-1)/2, quantum violation of classical noncontextuality grows exponentially with n.
We present two experiments testing the hypothesis of noncontextual hidden variables. The first one is based on observation of two-photon pseudo-Greenberger-Horne-Zeilinger correlations, with two of the originally three particles mimicked by the polarization degree of freedom and the spatial degree of freedom of a single photon. The second one, a single-photon experiment, utilizes the same trick to emulate two particle correlations, and is an "event ready" test of a Bell-like inequality, derived from the noncontextuality assumption. Modulo fair sampling, the data falsify noncontextual hidden variables.
The discussion on noncontextual hidden variable models as an underlying description for the quantum-mechanical predictions started in ernest with 1967 paper by Kochen and Specker. There, it was shown that no noncontextual hidden-variable model can give these predictions. The proof used in that paper is complicated, but recently, a paper by Yu and Oh [PRL, 2012] proposes a simpler statistical proof that can also be the basis of an experimental test. Here we report on a sharper version of that statistical proof, and also explain why the algebraic upper bound to the expressions used are not reachable, even with a reasonable contextual hidden variable model. Specifically, we show that the quantum mechanical predictions reach the maximal possible value for a contextual model that keeps the expectation value of the measurement outcomes constant. © 2012 American Institute of Physics.
The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors-the context-are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.
The quest for fundamental tests of quantum mechanics is an ongoing effort. We here address the question of what are the lowest possible moments needed to prove quantum nonlocality and noncontextuality without any further assumptions---in particular, without the often assumed dichotomy. We first show that second-order correlations can always be explained by a classical noncontextual local-hidden-variable theory. Similar third-order correlations also cannot violate classical inequalities in general, except for a special state-dependent noncontextuality. However, we show that fourth-order correlations can violate locality and state-independent noncontextuality. Finally we obtain a fourth-order continuous-variable Bell inequality for position and momentum, which can be violated and might be useful in Bell tests, closing all loopholes simultaneously.
Abstract Bell’s theorem (Bell 1964) establishes that local hidden variable theories are committed to Bell inequalities violated by quantum correlations. Since violations of Bell inequalities can be verified without requiring that the observables whose correlations figure in the inequalities be measured with arbitrarily high precision, Bell’s theorem yields a method of falsifying local hidden variable the­ ories. Moreover, the evidence for violations of Bell inequalities We thank J. Finklestein, N. D. Mermin, and an anonymous referee for helpful comments. A. K. was supported by a Royal Society University Research Fellow­ ship and thanks the Oxford Centre for Quantum Computation for much appreciated hospitality.