THE WORK OF GLEASON AND OF KOCHEN AND SPECKER

Ehud Hrushovski, Itamar Pitowsky

Karl-Peter Marzlin, Taylor Landry

We present an elementary proof of a reduced version of Gleason’s theorem and the Kochen–Specker theorem to provide a novel perspective on the relation between both theorems. The proof is based on a set of linear equations for the values of a function, m, on the unit sphere. In the case of Gleason’s theorem, the entire unit sphere needs to be considered, while a finite set of points suffices to prove the Kochen–Specker theorem.

Cristian S. Calude, Peter H. Hertling, Karl Svozil

We present two geometric proofs of the Kochen-Specker theorem. A quite similar argument has been used by Cooke, Keane, and Moran, as well as by Kalmbach in her book to derive the Gleason theorem.

Andrew Lenard

Victoria J Wright, Ravi Kunjwal

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.

Guillermo Morales Luna