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Petz D.,Alfred Renyi Institute of Mathematics
Entropy | Year: 2010

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. 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. © 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland.

Merai L.,Alfred Renyi Institute of Mathematics
Fundamenta Informaticae | Year: 2012

In the paper the pseudorandomness of binary sequences defined over elliptic curves is studied and both the well-distribution and correlation measures are estimated. The paper is based on the Kohel-Shparlinski bound and the Erdös-Turán-Koksma inequality.

Toth G.,University of the Basque Country | Toth G.,Ikerbasque | Toth G.,Hungarian Academy of Sciences | Petz D.,Hungarian Academy of Sciences | And 2 more authors.
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2013

We show that the variance is its own concave roof. For rank-2 density matrices and operators with zero diagonal elements in the eigenbasis of the density matrix, we prove analytically that the quantum Fisher information is four times the convex roof of the variance. Strong numerical evidence suggests that this statement is true even for operators with nonzero diagonal elements or density matrices with a rank larger than 2. We also find that within the different types of generalized quantum Fisher information considered in Petz and Gibilisco, Hiai, and Petz, after appropriate normalization, the quantum Fisher information is the largest. Hence, we conjecture that the quantum Fisher information is four times the convex roof of the variance even for the general case. © 2013 American Physical Society.

Furedi Z.,Alfred Renyi Institute of Mathematics
Journal of Combinatorial Theory. Series B | Year: 2015

Let Tn,p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Turán's theorem is proved. Every Kp+1-free graph with n vertices and e(Tn,p)-t edges contains a p-partite subgraph with at least e(Tn,p)-2t edges. As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemerédi's regularity lemma) for the classical stability result of Simonovits [25]. © 2015 Elsevier Inc.

Keszegh B.,Alfred Renyi Institute of Mathematics | Palvolgyi D.,Eotvos Lorand University
Discrete and Computational Geometry | Year: 2012

We prove that octants are cover-decomposable; i. e., any 12-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into two coverings. As a corollary, we obtain that any 12-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into two coverings. We also show that any 12-fold covering of the whole plane with the translates of a given open triangle can be decomposed into two coverings. However, we exhibit an indecomposable 3-fold covering with translates of a given triangle. © 2011 Springer Science+Business Media, LLC.

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