From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities

TL;DR

This paper develops a method to derive noise-robust noncontextuality inequalities from Kochen-Specker theorem proofs, enabling experimental tests of quantum contextuality beyond idealized projective measurements.

Contribution

It extends previous techniques to derive noncontextuality inequalities from statistical proofs of the Kochen-Specker theorem, accommodating noisy, real-world measurements.

Findings

Derived inequalities applicable to noisy experimental data

Extended the framework from logical to statistical proofs

Provided a method to test noncontextuality without ideal projective measurements

Abstract

The Kochen-Specker theorem rules out models of quantum theory wherein projective measurements are assigned outcomes deterministically and independently of context. This notion of noncontextuality is not applicable to experimental measurements because these are never free of noise and thus never truly projective. For nonprojective measurements, therefore, one must drop the requirement that an outcome is assigned deterministically in the model and merely require that it is assigned a distribution over outcomes in a manner that is context-independent. By demanding context-independence in the representation of preparations as well, one obtains a generalized principle of noncontextuality that also supports a quantum no-go theorem. Several recent works have shown how to derive inequalities on experimental data which, if violated, demonstrate the impossibility of finding a generalized-noncontextual model of this data. That is, these inequalities do not presume quantum theory and, in particular, they make sense without requiring an operational analogue of the quantum notion of projectiveness. We here describe a technique for deriving such inequalities starting from arbitrary proofs of the Kochen-Specker theorem. It extends significantly previous techniques that worked only for logical proofs, which are based on sets of projective measurements that fail to admit of any deterministic noncontextual assignment, to the case of statistical proofs, which are based on sets of projective measurements that do admit of some deterministic noncontextual assignments, but not enough to explain the quantum statistics.