Information Geometry and its Applications III

Abstract Arleta Szkoła

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Arleta Szkoła  (Max Planck Institute for Mathematics in the Sciences, Germany)
Friday, August 06, 2010, room Hörsaal 1
Chernoff type distances on quantum state spaces

The Chernoff distance represents a symmetrized version of the Kullback-Leibler distance/relative entropy between two probability distributions. Since it does not satisfy the triangle inequality it does not define a distance measure on the probability simplex in a strictly mathematical meaning. On the other hand it is an important measure of distinguishability among probability distributions. In particular, it is known to provide a sharp bound on exponential error rates in binary simple hypothesis testing. In the context of multiple hypothesis testing the corresponding optimal error exponent has been identified by Salikhov as the minimum of Chernoff distances over the different pairs of distributions from the finite set considered. This minimum is refered to as generalized Chernoff distance.
We want to present results of our earlier and recent work - see the list of references below - which settle Chernoff type bounds in the context of quantum hypothesis testing, where the hypotheses are represented by density operators associated to states of a finite quantum system. Further, we intend to address some naturally arising information-geometric questions concerning the quantum version of Hellinger arc, and more general, the structure of exponential families in state spaces of non-commutative algebras of observables.

  1. M. Nussbaum, A. Szkoła, "The Chernoff Lower Bound for Symmetric Quantum Hypothesis Testing", The Annals of Statistics Vol. 37, No. 2, 1040-1057 (2009)
  2. K. M. R. Audenaert, M. Nussbaum, A. Szkoła, and F. Verstraete, "Asymptotic Error Rates in Quantum Hypothesis Testing", Commun. Math. Phys. Vol. 279, No. 1, 251-283 (2008),
    springerlink.metapress.com/content/e1463367803n505g/fulltext.pdf
  3. M. Nussbaum, A. Szkoła, "Asymptotic optimal discrimination between pure quantum states", to appear in TQC Proceedings (2010), MPI MiS preprint 1/2010
  4. M. Nussbaum, A. Szkoła, "Exponential error rates in multiple state discrimination on a quantum spin chain", submitted to Commun. Math. Phys. (2010), MPI MiS preprint 3/2010,
    xxx.lanl.gov/abs/1001.2651

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
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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