How (Maximally) Contextual is Quantum Mechanics

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.