Abstract
Csiszár’s ƒ-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, Cram´er-Rao inequality and uncertainty relation. It is remarkable that in the quantum case theoretically there are several Fisher information and variances. Fisher information are obtained as the Hessian of a quasi-entropy. A conjecture about the scalar curvature of a Fisher information geometry is explained. The described subjects are overviewed in details in the matrix setting. The von Neumann algebra approach is also discussed for uncertainty relation.
Highlights
Let X be a finite space with probability measures p and q
A possible generalization of the relative entropy is the f -divergence introduced by Csiszár: ( p(x) )
This paper first gives a rather short survey about f -divergence and we turn to the non-commutative generalization. Speaking this means that the positive n-tuples p and q are replaced by positive semidefinite n × n matrices and the main questions in the study remain rather similar to the probabilistic case
Summary
Let X be a finite space with probability measures p and q This paper first gives a rather short survey about f -divergence and we turn to the non-commutative (algebraic, or quantum) generalization. Speaking this means that the positive n-tuples p and q are replaced by positive semidefinite n × n matrices and the main questions in the study remain rather similar to the probabilistic case. The quantum generalization was originally called quasi-entropy, but quantum f -divergence might be a better terminology This notion is related to some other important concepts as covariance, quadratic costs, Fisher information, Cramér-Rao inequality and uncertainty relation.
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