Bures geometry of the three-level quantum systems. II

For the eight-dimensional Riemannian manifold comprised by the three-level quantum systems endowed with the Bures metric, we numerically approximate the integrals over the manifold of several functions of the curvature and of its (anti-)self-dual parts. The motivation for pursuing this research is to elaborate upon the findings of Dittmann in his paper, "Yang-Mills equation and Bures metric" (quant-ph/9806018).Comment: thirteen pages, LaTeX, four tables, two figures, this paper supersedes math-ph/0012031, "Numerical analyses of a quantum-theoretic eight-dimensional Yang-Mills fields," which will be withdrawn. For part I of this paper (to appear in J. Geom. Phys.), see quant-ph/000806

On the Curvature of Monotone Metrics and a Conjecture Concerning the Kubo-Mori Metric

It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its expected quantum statistical meaning. The scalar curvature is explained in more detail for three examples, the Bures metric, the largest monotone metric and the Kubo-Mori metric. In particular, we show an important conjecture of Petz concerning the Kubo-Mori metric up to a formal proof of the concavity of a certain function on R_+^3. This concavity seems to be numerically evident. The conjecture of Petz asserts that the scalar curvature of the Kubo-Mori metric increases if one goes to more mixed states.Comment: 20 pages, 4 figure

The Scalar Curvature of the Bures Metric on the Space of Density Matrices

The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we determine its scalar curvature and Ricci tensor and prove a lower bound for the curvature on the submanifold of trace one matrices. This bound is achieved for the maximally mixed state, a further hint for the quantum statistical meaning of the scalar curvature.Comment: Latex, 9 page