Optimal Information Transfer and the Uniform Measure over Probability Space - William K. Wootters (Entropy, 2023)
For a quantum system with a d-dimensional Hilbert space, suppose a pure state <semantics>|ψ⟩</semantics> is subjected to a complete orthogonal measurement. The measurement effectively maps <semantics>|ψ⟩</semantics> to a point <semantics>(p1,…,pd)</semantics> 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 <semantics>|ψ⟩</semantics> is distributed uniformly over the unit sphere, then the resulting ordered set <semantics>(p1,…,pd)</semantics> is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional to <semantics>dp1⋯dpd−1</semantics>. 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.