Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem

Abstract

The Kochen--Specker (KS) theorem reveals the nonclassicality of single quantum systems. In contrast, Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. Accordingly, unlike incompatibility, entanglement and Bell non-locality are not necessary to demonstrate KS-contextuality. However, here we find that for multiqubit systems, entanglement and non-locality are both essential to proofs of the Kochen--Specker theorem. Firstly, we show that unentangled measurements (a strict superset of local measurements) can never yield a logical (state-independent) proof of the KS theorem for multiqubit systems. In particular, unentangled but nonlocal measurements -- whose eigenstates exhibit "nonlocality without entanglement" -- are insufficient for such proofs.This also implies that proving Gleason's theorem on a multiqubit system necessarily requires entangled projections, as shown by Wallach [Contemp Math, 305: 291-298 (2002)]. Secondly, we show that a multiqubit state admits a statistical (state-dependent) proof of the KS theorem if and only if it can violate a Bell inequality with projective measurements. We also establish the relationship between entanglement and the theorems of Kochen--Specker and Gleason more generally in multiqudit systems by constructing new examples of KS sets. Finally, we discuss how our results shed new light on the role of multiqubit contextuality as a resource within the paradigm of quantum computation with state injection.

References in corpus (8)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• A simple test for hidden variables in spin-1 system
• Is the quantum state real? An extended review of $\psi$-ontology theorems
• On the role of entanglement and correlations in mixed-state quantum computation
• Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
• Uniqueness of noncontextual models for stabilizer subtheories
• Bell nonlocality is not sufficient for the security of standard device-independent quantum key distribution protocols
• Fine's theorem, noncontextuality, and correlations in Specker's scenario

Cited by in corpus (3)

• Trading causal order for locality
• Harvesting Contextuality from the Vacuum
• An Entanglement Monotone from the Contextual Fraction