Quantum contextuality for rational vectors

TL;DR

This paper demonstrates that quantum contextuality persists even when restricting to rational vectors, challenging the idea that noncontextual hidden variables could explain quantum predictions in such cases.

Contribution

It proves that the Kochen-Specker theorem's contradiction holds for rational vectors, extending the understanding of quantum contextuality.

Findings

Rational vectors can violate noncontextual inequalities

Quantum contextuality persists with rational projectors

The inconsistency with hidden variables remains for rational vectors

Abstract

The Kochen-Specker theorem states that noncontextual hidden variable models are inconsistent with the quantum predictions for every yes-no question on a qutrit, corresponding to every projector in three dimensions. It has been suggested [D. A. Meyer, Phys. Rev. Lett. 83, 3751 (1999)] that the inconsistency would disappear when we are restricted to projectors on unit vectors with rational components; that noncontextual hidden variables could reproduce the quantum predictions for rational vectors. Here we show that a qutrit state with rational components violates an inequality valid for noncontextual hidden-variable models [A. A. Klyachko et al., Phys. Rev. Lett. 101, 020403 (2008)] using rational projectors. This shows that the inconsistency remains even when using only rational vectors.