Jochen Dittmann:
On the Riemannian geometry of finite dimensional mixed states
Journal of Lie Theory, vol. 3 (1), p. 73-87
We consider the Riemannian geometry of
the space of nonsingular
density matrices $\cD^1$ equipped with the Bures metric $g^B$. This
space is of certain physical relevance on the background of
a generalization of the Berry phase to mixed states.
We determine the covariant
derivative and the curvature tensor field related to the Levi-Cevita
connection of $(\cD^1,g^B)$, which allow us to calculate
other curvature quantities . It turns out
that $\cD^1$ is not a space of constant curvature and not even a
locally symmetric space, in contrast to what
the case of two-dimensional density matrices might suggest.
Moreover, we give a local
description of $\cD^1$ and explicit
formulae for the Bures metric in
terms of natural matrix operations containing
$\r$ and $\D\r$ only.