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 <link http: tsallis.cat.cbpf.br biblio.htm external>tsallis.cat.cbpf.br/biblio.htm.
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