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Workshop

The $f \longleftrightarrow \tilde f$ correspondence and its applications in quantum information theory

  • Paolo Gibilisco (Università degli Studi di Roma "Tor Vergata", Roma, Italy)
University n.n. Universität Leipzig (Leipzig)

Abstract

Let ${\cal F}_{op}$=\{symmetric, normalized, operator monotone functions\}. If we set \[ {\cal F}^{\, r}_{op}:=\{f \in {\cal F}_{op} | f(0)>0 \}, \qquad \qquad {\cal F}^{\, n}_{op}:=\{f \in {\cal F}_{op} | f(0)=0 \}, \] trivially it holds $$ {\cal F}_{op}={\cal F}^{\, r}_{op} \, \dot\cup \, {\cal F}^{\, n}_{op}. $$ Define \[ \tilde{f}(x):=\frac{1}{2}\left[ (x+1)-(x-1)^2 \frac{f(0)}{f(x)} \right]\qquad x>0. \] It is possible to prove that the map $f \to \tilde f,$ is a bijection from ${\cal F}^{\, r}_{op}$ to ${\cal F}^{\, n}_{op}$, namely a bijection between regular and non-regular functions.

In the last years a number of consequences has been derived from this fact: 1) the dynamical uncertainty principle; 2) its generalization to von Neumann algebras; 3) a new proof of the fact that the Wigner-Yanase-Dyson is an example of a quantum Fisher information; 4) a new proof the monotonicity property for the WYD information; 5) a link between quantum relative entropy and metric adjusted skew information.

The purpose of my talk is to describe the above applications.

  • Andai, A., Uncertainty principle with quantum Fisher information, J. Math. Phys. 49 (2008), 012106.
  • Audenaert, K., Cai, L. and Hansen, F., Inequalities for quantum skew information, Lett. Math. Phys., 85: 135--146, 2008.
  • Gibilisco, P., Hansen, F. and Isola T., On a correspondence between regular and non-regular operator monotone functions. Linear Algebra Appl., 430: 2225-2232, 2009.
  • Gibilisco, P., Hiai F. and Petz, D., Quantum covariance, quantum Fisher information and the uncertainty relations. IEEE Trans. Inform. Theory, 55: 439-443, 2009.
  • Gibilisco, P. and Isola, T., A dynamical uncertainty principle in von Neumann algebras by operator monotone functions. J. Stat. Phys., 132: 937--944, 2008.
  • Luo, S., Quantum Fisher information and uncertainty relations. Lett. Math. Phys. 53: 243--251, 2000.
  • Petz, D. and Szabó, V. E. S., From quasi-entropy to skew information. International J. Math., 20:1421--1430, 2009.

Links

conference
02.08.10 06.08.10

Information Geometry and its Applications III

Universität Leipzig University 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