For a quantum system with ad-dimensional Hilbert space, suppose a pure state|ψ〉is subjected to a complete orthogonal measurement. The measurement effectively maps|ψ〉to a point(p1,…,pd)in the appropriate probability simplex. It is a known fact—which depends crucially on the complex nature of the system’s Hilbert space—that if|ψ〉is distributed uniformly over the unit sphere, then the resulting ordered set(p1,…,pd)is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional todp1⋯dpd−1. In this paper we ask whether there is some foundational significance to this uniform measure. In particular, we ask whether it is the optimal measure for the transmission of information from a preparation to a measurement in some suitably defined scenario. We identify a scenario in which this is indeed the case, but our results suggest that an underlying real-Hilbert-space structure would be needed to realize the optimization in a natural way.