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Entropy 2010, 12(3), 304-325; doi:10.3390/e12030304

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

Alfréd Rényi Institute of Mathematics, H-1364 Budapest, POB 127, Hungary
Received: 26 July 2009 / Revised: 20 February 2010 / Accepted: 25 February 2010 / Published: 1 March 2010
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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.
Keywords: ƒ-divergence; quasi-entropy; von Neumann entropy; relative entropy; monotonicity property; Fisher information; uncertainty ƒ-divergence; quasi-entropy; von Neumann entropy; relative entropy; monotonicity property; Fisher information; uncertainty
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Petz, D. From ƒ-Divergence to Quantum Quasi-Entropies and Their Use. Entropy 2010, 12, 304-325.

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