Comparative noninformativities of quantum priors based on monotone metrics

Abstract

We consider a number of prior probability distributions of particular interest, all being defined on the three-dimensional convex set of two-level quantum systems. Each distribution is — following recent work of Petz and Sudar — taken to be proportional to the volume element of a monotone metric on that Riemannian manifold. We apply and entropy-based test (a variant of one recently developed by Clarke) to determine which of two priors is more noninformative in nature. This involves converting them to posterior probability distributions based on some set of hypothesized outcomes of measurements of the quantum system in question. It is, then, ascertained whether or not the relative entropy (Kullback-Leibler statistic) between a pair of priors increases or decreases when one of them is exchanged with its corresponding posterior. The findings lead us to assert that the maximal monotone metric yields the most noninformative prior distribution and the minimal monotone (that is, the Bures) metric, the least. Our conclusions both agree and disagree, in certain respects, with ones recently reached by Hall, who relied upon a less specific test criterion than our entropy-based one.