From ƒ-Divergence to Quantum Quasi-Entropies and Their Use

Csiszár'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, Cramér-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. Let X be a finite space with probability measures p and q. Their relative entropy or divergence D(p||q) = x∈X p(x) log p(x) q(x) was introduced by Kullback and Leibler in 1951 [27]. More precisely, if p(x) = q(x) = 0, then log(p(x)/q(x)) = 0 and if p(x) = 0 but q(x) = 0 for some x ∈ X , then log(p(x)/q(x)) = +∞. A possible generalization of the relative entropy is the f-divergence introduced by Csiszár: D f (p||q) = x∈X q(x)f p(x) q(x) (1)