Strong Kochen-Specker theorem and incomputability of quantum randomness

TL;DR

This paper advances the Kochen-Specker theorem to identify specific observables that are provably value indefinite, and applies this to certify a quantum random number generator in a three-dimensional system.

Contribution

It improves the Kochen-Specker theorem to locate value indefinite observables and demonstrates their use in certifying quantum randomness.

Findings

Identified observables that are provably value indefinite.

Developed a certification method for quantum random number generation.

Validated the approach in a three-dimensional quantum system.

Abstract

The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is non-contextual and consistent with quantum mechanics. If we require non-contextuality, the consequence is that many observables must not have pre-existing definite values. However, the Kochen-Specker theorem does not allow one to determine which observables must be value indefinite. In this paper we present an improvement on the Kochen-Specker theorem which allows one to actually locate observables which are provably value indefinite. Various technical and subtle aspects relating to this formal proof and its connection to quantum mechanics are discussed. This result is then utilized for the proposal and certification of a dichotomic quantum random number generator operating in a three-dimensional Hilbert space.