Information Geometry and its Applications III

Abstract Shigeru Furuichi

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Shigeru Furuichi  (Nihon University, Japan)
Thursday, August 05, 2010, room Hörsaal 1
On trace inequalities related to skew informations and generalized relative entropies

My talk is composed by the following topics.

  • Sec.1: Schrödinger type uncertainty relation for mixed states (based on [1].) We shall give the Schrödinger type uncertainty relation for a quantity representing a quantum uncertainty, introduced by S.Luo in [2]. Our result improves the Heisenberg uncertainty type relation shown in [2] for a mixed state. We also discuss the relation between our result and the original Schrödinger uncertainty relation.
  • Sec.2: A matrix trace inequality and its application to entropy theory (based on [3].) We here give a complete and affirmative answer to a conjecture [4] on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices. Finally, we give a lower bound of the generalized relative entropy (Tsallis relative entropy [5, 6]) applying a slightly different variational expression [7, 8] and the generalized Golden-Thompson inequality.
  • Sec.3: Trace inequalities related to skew informations (based on [9].) (The talk of this section will be given, if we have an enough time.) We study some trace inequalities of the products of the matrices and the power of matrices, which are natural generalized forms related to the quantities constituting skew informations. See [10] for the similar problems and their answers.


  1. S.Furuichi, Schrödinger uncertainty relation for mixed states, arXiv:1005.2655v1.
  2. S.Luo, Heisenberg uncertainty relation for mixed states, Phys.Rev.A,Vol.72(2005), 042110.
  3. S.Furuichi and M.Lin, A matrix trace inequality and its application, to appear in Linear Alg. Appl.
  4. S.Furuichi, A mathematical review of the generalized entropies and their matrix trace inequalities, in: Proceedings of WEC2007, 2007, pp.840-845.
  5. C. Tsallis et al. In: S. Abe and Y. Okamoto, Editors, Nonextensive Statistical Mechanics and its Applications, Springer-Verlag, Heidelberg (2001). See also the comprehensive list of references at
  6. C.Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, Springer, 2009.
  7. F.Hiai and D.Petz, The Golden-Thompson trace inequality is complemented, Linear Alg. Appl. Vol. 181 (1993), pp.153-185.
  8. S.Furuichi, Trace inequalities in nonextensive statistical mechanics, Linear Alg. Appl., Vol.418(2006), pp.821-827
  9. S.Furuichi, K.Kuriyama and K.Yanagi, Trace inequalities for products of matrices, Linear Alg. Appl., Vol.430(2009),pp.2271-2276
  10. T. Ando, F. Hiai and K. Okubo, Trace inequalities for multiple products of two matrices, Math. Inequal. Appl. Vol.3(2000), pp.307-318

Date and Location

August 02 - 06, 2010
University of Leipzig
04103 Leipzig

Scientific Organizers

Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Information Theory of Cognitive Systems Group
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Paolo Gibilisco
Università degli Studi di Roma "Tor Vergata"
Facoltà di Economia
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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
Brain Science Institute, Mathematical Neuroscience Laboratory
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Imre Csiszár
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
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Dénes Petz
Budapest University of Technology and Economics
Department for Mathematical Analysis
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Giovanni Pistone
Collegio Carlo Alberto, Moncalieri
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Administrative Contact

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

05.04.2017, 12:42