Letters in Mathematical Physics

, Volume 107, Issue 12, pp 2267–2290 | Cite as

Maps on positive definite operators preserving the quantum \(\chi _\alpha ^2\)-divergence

  • Hong-Yi Chen
  • György Pál Gehér
  • Chih-Neng Liu
  • Lajos MolnárEmail author
  • Dániel Virosztek
  • Ngai-Ching Wong


We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum \(\chi _\alpha ^2\)-divergence for some \(\alpha \in [0,1]\). We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.


Positive definite operators Quantum \(\chi _\alpha ^2\)-divergence Preservers 

Mathematics Subject Classification

Primary 46N50 47B49 



Gy. P. Gehér and L. Molnár were supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office - NKFIH, Grant No. K115383. D. Virosztek was supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences, by the National Research, Development and Innovation Office - NKFIH, Grant No. K104206, and by the “For the Young Talents of the Nation” scholarship program (NTP-EFÖ-P-15-0481) of the Hungarian State. The project was also supported by the joint venture of Taiwan and Hungary MOST-HAS, Grant No. 104-2911-1-110-508.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Hong-Yi Chen
    • 1
  • György Pál Gehér
    • 2
    • 3
  • Chih-Neng Liu
    • 1
  • Lajos Molnár
    • 3
    • 4
    • 5
    Email author
  • Dániel Virosztek
    • 3
    • 5
  • Ngai-Ching Wong
    • 1
  1. 1.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan
  2. 2.MTA-SZTE Analysis and Stochastics Research Group, Bolyai InstituteUniversity of SzegedSzegedHungary
  3. 3.MTA-DE “Lendület” Functional Analysis Research Group, Institute of MathematicsUniversity of DebrecenDebrecenHungary
  4. 4.Bolyai InstituteUniversity of SzegedSzegedHungary
  5. 5.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary

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