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How much contextuality

Karl Svozil

Published Nov 18, 2015• 21 citations
OpenAlex

Abstract

Abstract The amount of contextuality is quantified in terms of the probability of the necessary violations of noncontextual assignments to counterfactual elements of physical reality.

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In an ideal contextuality scenario it is assumed that, whenever two contexts overlap, the marginal distribution obtained for the intersection of these contexts must be the same. Although this nondisturbance idealization has been extremely useful in investigating the limitations of physical theories, in this paper, we will work out a more robust and possibly more experimentally friendly framework for contextuality. Simply put, we will restate the extended definition of noncontextuality provided by the noncontextuality by default and show that in this extended perspective it is possible to avoid weakening the nondisturbance hypothesis. By doing so, we also show how standard tools for characterizing and quantifying contextuality can also be applied to this extended scenario. In particular, we came up with a linear program to check whether or not a behavior is contextual in the extended sense. The connection with the usual definition of contextuality is also established because whenever nondisturbance is fully satisfied the extended conditions reduce to the original ones.
Contextuality is a defining feature that separates the quantum from the classical descriptions of physical systems. Within the marginal-scenario framework, noncontextual models are characterized by the existence of a single joint probability distribution consistent with all measurable contexts, while contextual models violate this condition. Building on this approach, we introduce a general method to analyze contextuality in terms of stochastic linear maps that effectively model invasive measurements on an otherwise classical statistics. These maps transform probabilities within the noncontextuality polytope, which includes all classical probabilities, into probabilities that may lie outside the polytope, while preserving the compatibility structure of the scenario at hand. We derive general consistency conditions that such maps must satisfy to represent admissible invasive measurements, and we fully identify them for a paradigmatic example of contextuality for a single three-level quantum system. Furthermore, we introduce a quantifier of contextuality based on the minimal invasiveness required to reproduce a given probability distribution, which offers a distinct approach on how to evaluate the degree of contextuality in a general scenario.
Previous experimental tests of quantum contextuality based on the Bell-Kochen-Specker (BKS) theorem have demonstrated that not all observables among a given set can be assigned noncontextual eigenvalue predictions, but have never identified which specific observables must fail such assignment. We now remedy this shortcoming by showing that BKS contextuality can be confined to particular observables by pre- and postselection, resulting in anomalous weak values that we measure using modern neutron interferometry. We construct a confined contextuality witness from weak values, which we measure experimentally to obtain a $5\ensuremath{\sigma}$ average violation of the noncontextual bound, with one contributing term violating an independent bound by more than $99\ensuremath{\sigma}$. This weakly measured confined BKS contextuality also confirms the quantum pigeonhole effect, wherein eigenvalue assignments to contextual observables apparently violate the classical pigeonhole principle.