Applications of quantum and classical Fisher information to two-level complex and quaternionic and three-level complex systems
Paul B. Slater
In the Bayesian theory of statistical inference, as first suggested by Harold Jeffreys in highly influential work, one can employ the square root of the determinant of an n×n Fisher information matrix as a reparametrization-invariant prior (generally unnormalized) measure over an n-dimensional family (Riemannian manifold) of probability distributions. Jeffreys’ ansatz is adopted here to the quantum context, that is, with regard to density matrices rather than probability distributions, by computing the quantum Fisher information matrices (associated with Helstrom and Holevo) for the three-, five-, and eight-dimensional convex sets of two-level complex, two-level quaternionic, and three-level complex systems, respectively. In both the two-level cases, the priors have been normalized to probability distributions over the 2×2 density matrices, while, in the much more computationally demanding three-level situation, no such normalization has been accomplished. An argument is made for the general form, in terms of eigenvalues, that the (unnormalized) prior should assume over the (n2−1)-dimensional convex set of n×n density matrices.
Journal of Mathematical Physics•Jun 1, 1996•17 citations