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Quantum Information Processing

, Volume 15, Issue 8, pp 3393–3420 | Cite as

A family of generalized quantum entropies: definition and properties

  • G. M. Bosyk
  • S. Zozor
  • F. Holik
  • M. Portesi
  • P. W. Lamberti
Article

Abstract

We present a quantum version of the generalized \((h,\phi )\)-entropies, introduced by Salicrú et al. for the study of classical probability distributions. We establish their basic properties and show that already known quantum entropies such as von Neumann, and quantum versions of Rényi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicrú form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum \((h,\phi )\)-entropies under the action of quantum operations and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement and introduce a discussion on possible generalized conditional entropies as well.

Keywords

Quantum entropies Majorization relation Entanglement detection 

Notes

Acknowledgments

GMB, FH, MP and PWL acknowledge CONICET and UNLP (Argentina), and MP and PWL also acknowledge SECyT-UNC (Argentina) for financial support. SZ is grateful to the University of Grenoble-Alpes (France) for the AGIR financial support.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • G. M. Bosyk
    • 1
  • S. Zozor
    • 2
  • F. Holik
    • 1
  • M. Portesi
    • 1
  • P. W. Lamberti
    • 3
  1. 1.Instituto de Física La Plata, UNLP, CONICETFacultad de Ciencias ExactasLa PlataArgentina
  2. 2.Laboratoire Grenoblois d’Image, Parole, Signal et Automatique (GIPSA-Lab, CNRS)Saint Martin d’HèresFrance
  3. 3.Facultad de Matemática, Astronomía y FísicaUNC, CONICETCórdobaArgentina

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