A Generalization of Powers–Størmer Inequality

  • Yoshiko Ogata


In this note, we prove the following inequality: \({2\Vert\Delta_{\eta\varphi}^{\frac s2}\xi_{\varphi}\Vert ^2 \ge \varphi(1)+\eta(1)- \vert\varphi-\eta\vert(1)}\) , where \({\varphi}\) and η are positive normal linear functionals over a von Neumann algebra. This is a generalization of the famous Powers–Størmer inequality (Powers and Størmer proved the inequality for \({L({\mathcal H})}\) in Commun Math Phys 16:1–33, 1970; Takesaki in Theory of Operator Algebras II, 2001). For matrices, this inequality was proven by Audenaert et al. (Phys Rev Lett 98:160501, 2007). We extend their result to general von Neumann algebras.

Mathematics Subject Classification (2010)



Powers–Størmer inequality Chernoff bound 


  1. 1.
    Araki H.: Relative entropy of states of von Neumann algebras. Pub. R.I.M.S., Kyoto Univ. 11, 809–833 (1976)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Araki H., Masuda T.: Positive cones and L p-spaces for von Neumann algebras. Pub. R.I.M.S., Kyoto Univ. 18, 339–411 (1982)MathSciNetGoogle Scholar
  3. 3.
    Audenaert K.M.R., Calsamiglia J., Masanes Ll., Munoz-Tapia R., Acin A., Bagan E., Verstraete F.: The quantum Chernoff bound. Phys. Rev. Lett. 98, 160501 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Jakšić, V., Ogata, Y., Pillet, C.-A., Seiringer, R.: (in preparation)Google Scholar
  5. 5.
    Powers R.T., Størmer E.: Free states of canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970)ADSMATHCrossRefGoogle Scholar
  6. 6.
    Takesaki M.: Theory of Operator Algebras II, Springer Encyclopedia of Mathematical Sciences, vol. 125. Springer, Berlin (2001)Google Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Graduate School of MathematicsUniversity of TokyoTokyoJapan

Personalised recommendations