How (Maximally) Contextual is Quantum Mechanics?

TL;DR

This paper quantifies the extent of contextuality in quantum mechanics by establishing bounds on the fraction of measurements with context-dependent valuations, revealing that quantum mechanics is less contextual than some alternative theories.

Contribution

It introduces a quantitative bound on the fraction of projective measurements that must be context-dependent in quantum systems, extending to arbitrary rank projectors and comparing to other theories.

Findings

Quantum mechanics has a bounded fraction of context-dependent measurements.

Quantum contextuality is less extensive than in some alternative theories.

The results extend to projectors of arbitrary equal rank.

Abstract

Proofs of Bell-Kochen-Specker contextuality demonstrate that there exists sets of projectors that cannot each be assigned either 0 or 1 such that each basis formed from them contains exactly one 1-assigned projector. Instead, at least some of the projectors must have a valuation that depends on the \emph{context} in which they are measured. This motivates the question of \emph{how many} of the projectors must have contextual valuations. In this paper, we demonstrate a bound on what fraction of rank-1 projective measurements on a quantum system must be considered to have context-dependent valuations as a function of the quantum dimension, and show that quantum mechanics is not as contextual, by this metric, as other possible physical theories. Attempts to find quantum mechanical scenarios that yield a high value of this figure-of-merit can be thought of as generalisations or extensions of the search for small Kochen-Specker sets. We also extend this result to projector-valued-measures with projectors of arbitrary equal rank.