Contextuality in composite systems: entanglement vs. the Kochen-Specker theorem

The Kochen-Specker (KS) theorem is often taken as a notion of nonclassicality that is independent of entanglement since it's provable on a three-dimensional Hilbert space. However, the smallest system on which both the KS theorem and entanglement are meaningful notions of nonclassicality is a two-qubit system.

I will present some recent work on the necessity of entanglement in proofs of the KS theorem on multiqubit systems. We show two key results: firstly, that any proof of the KS theorem that uses KS sets necessarily requires entangled measurements, and secondly, that a statistical proof of the KS theorem with unentangled measurements on a multiqubit state exists if and only if this state can witness a Bell inequality violation. We also obtain an overall understanding of the relationship between unentangled Gleason and KS theorems on  multiqudit systems in general. Time permitting, I will also discuss some implications of these results for the role of contextuality as a resource for multiqubit quantum computation with state injection. Based on arXiv:2109.13594.