  Csiszar's f-divergence of two probability distributions was extended to the
quantum case by the author in 1985. In the quantum setting positive
semidefinite matrices are in the place of probability distributions and the
quantum generalization is called quasi-entropy which is related to some other
important concepts as covariance, quadratic costs, Fisher information,
Cramer-Rao inequality and uncertainty relation. A conjecture about the scalar
curvature of a Fisher information geometry is explained. The described subjects
are overviewed in details in the matrix setting, but at the very end the von
Neumann algebra approach is sketched shortly.
