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Workshop

Chernoff type distances on quantum state spaces

  • Arleta Szkoła (Max Planck Institute for Mathematics in the Sciences, Germany)
Raum n.n. Universität Leipzig (Leipzig)

Abstract

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
conference
8/2/10 8/6/10

Information Geometry and its Applications III

Universität Leipzig Raum n.n.

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Nihat Ay

Max Planck Institute for Mathematics in the Sciences, Germany

Paolo Gibilisco

Università degli Studi di Roma "Tor Vergata", Italy

František Matúš

Academy of Sciences of the Czech Republic, Czech Republic