From f-divergence to quantum quasi-entropies and their use

TL;DR

This paper reviews the extension of classical f-divergence to quantum quasi-entropies, exploring their mathematical properties and connections to quantum information geometry, including Fisher information and uncertainty principles.

Contribution

It provides a comprehensive overview of quantum quasi-entropies, their mathematical framework, and their applications in quantum information theory, including a conjecture on scalar curvature.

Findings

Quantum quasi-entropies generalize classical divergence measures.

Connections established between quantum quasi-entropies and Fisher information.

Discussion of scalar curvature conjecture in quantum information geometry.

Abstract

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.