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

TL;DR

This paper investigates the role of entanglement in multiqubit systems for proofs of the Kochen-Specker theorem, revealing that entanglement and non-locality are essential for state-independent proofs and linking contextuality to quantum computational resources.

Contribution

It demonstrates that unentangled measurements cannot prove KS-contextuality in multiqubit systems and establishes the necessity of entangled projections for such proofs, connecting entanglement with foundational quantum theorems.

Findings

Unentangled measurements cannot yield state-independent KS proofs in multiqubit systems.

Entangled projections are necessary for proofs of the KS theorem in multiqubit systems.

Multiqubit contextuality can serve as a resource in quantum computation with state injection.

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.