Information Geometry and its Applications III

Abstract Paolo Gibilisco

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Paolo Gibilisco  (Università degli Studi di Roma "Tor Vergata", Italy)
Friday, August 06, 2010, room Hörsaal 2
The formula3 correspondence and its applications in quantum information theory

Let formula52={symmetric, normalized, operator monotone functions}. If we set
displaymath48
trivially it holds
displaymath54
Define
displaymath49
It is possible to prove that the map formula56 is a bijection from formula58 to formula60, namely a bijection between regular and non-regular functions.

In the last years a number of consequences has been derived from this fact: 1) the dynamical uncertainty principle; 2) its generalization to von Neumann algebras; 3) a new proof of the fact that the Wigner-Yanase-Dyson is an example of a quantum Fisher information; 4) a new proof the monotonicity property for the WYD information; 5) a link between quantum relative entropy and metric adjusted skew information.

The purpose of my talk is to describe the above applications.

  • Andai, A., Uncertainty principle with quantum Fisher information, J. Math. Phys. 49 (2008), 012106.
  • Audenaert, K., Cai, L. and Hansen, F., Inequalities for quantum skew information, Lett. Math. Phys., 85: 135-146, 2008.
  • Gibilisco, P., Hansen, F. and Isola T., On a correspondence between regular and non-regular operator monotone functions. Linear Algebra Appl., 430: 2225-2232, 2009.
  • Gibilisco, P., Hiai F. and Petz, D., Quantum covariance, quantum Fisher information and the uncertainty relations. IEEE Trans. Inform. Theory, 55: 439-443, 2009.
  • Gibilisco, P. and Isola, T., A dynamical uncertainty principle in von Neumann algebras by operator monotone functions. J. Stat. Phys., 132: 937-944, 2008.
  • Luo, S., Quantum Fisher information and uncertainty relations. Lett. Math. Phys. 53: 243-251, 2000.
  • Petz, D. and Szabó, V. E. S., From quasi-entropy to skew information. International J. Math., 20:1421-1430, 2009.

Date and Location

August 02 - 06, 2010
University of Leipzig
Augustusplatz
04103 Leipzig
Germany

Scientific Organizers

Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Information Theory of Cognitive Systems Group
Germany
Contact by Email

Paolo Gibilisco
Università degli Studi di Roma "Tor Vergata"
Facoltà di Economia
Italy
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František Matúš
Academy of Sciences of the Czech Republic
Institute of Information Theory and Automation
Czech Republic
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Scientific Committee

Shun-ichi Amari
RIKEN
Brain Science Institute, Mathematical Neuroscience Laboratory
Japan
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Imre Csiszár
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
Hungary
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Dénes Petz
Budapest University of Technology and Economics
Department for Mathematical Analysis
Hungary
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Giovanni Pistone
Collegio Carlo Alberto, Moncalieri
Italy
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Administrative Contact

Antje Vandenberg
Max Planck Institute for Mathematics in the Sciences
Contact by Email
Phone: (++49)-(0)341-9959-552
Fax: (++49)-(0)341-9959-555

05.04.2017, 12:42