Bayesian inference for complex and quaternionic two-level quantum systems

Jeffreysʹ approach for generating reparameterization-invariant prior distributions is applied to the three-dimensional convex set of complex two-level quantum systems. For this purpose, such systems are identified with bivariate complex normal distributions over the vectors of two-dimensional complex Hilbert space. The trivariate prior obtained is improper or non-normalizable over the convex set. However, its three bivariate marginals are — through a limiting procedure — normalizable to probability distributions and are, consequently, suitable for the Bayesian inference of two-level systems. Analogous results hold for the five-dimensional convex set of quaternionic two-level systems. The complex univariate and quaternionic trivariate marginals of the improper priors are uniform distributions. The bivariate marginals in the two cases are opposite in character.