Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen-Specker theorem: the Peres-Mermin square

Abstract

When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen-Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp measurements, but also preparations as well, and for which a quantum no-go result still holds. According to this notion, the model need only specify a probability distribution over the outcomes of a measurement in a context-independent way, rather than specifying a particular outcome. It also implies novel constraints of context-independence for the representation of preparations. In this article, we describe a general technique for translating proofs of the Kochen-Specker theorem into inequality constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a noncontextual model. We focus on algebraic state-independent proofs, using the Peres-Mermin square as our illustrative example. Our technique yields the necessary and sufficient conditions for a particular set of correlations (between the preparations and the measurements) to admit a noncontextual model. The inequalities thus derived are demonstrably robust to noise. We specify how experimental data must be processed in order to achieve a test of these inequalities. We also provide a criticism of prior proposals for experimental tests of noncontextuality based on the Peres-Mermin square.

31 pages, 4 figures, comments welcome

References in corpus (8)

• Hidden Variables and the Two Theorems of John Bell
• Experimentally testable state-independent quantum contextuality
• State-independent experimental test of quantum contextuality
• State-independent quantum contextuality with single photons
• Preparation contextuality powers parity-oblivious multiplexing
• Experimental implementation of a Kochen-Specker set of quantum tests
• Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories
• Parity Oblivious d-Level Random Access Codes and Class of Noncontextuality Inequalities

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