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

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Abstract

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.

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Notes

  1. The reviewer of the paper kindly called our attention to the fact that an inequality stronger than (3) could be deduced using the techniques of Lemma 5 in [16].

  2. The reviewer of the paper pointed out that the convexity part of this statement follows also from a celebrated result of Lieb, see Corollary 2.1 in Convex trace functions and the Wigner–Yanase–Dyson conjecture, Advances in Math. 11 (1973), 267–288.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Hong-Yi Chen, Chih-Neng Liu & Ngai-Ching Wong

  2. MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged, 6720, Hungary

    György Pál Gehér

  3. MTA-DE “Lendület” Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, P.O. Box 400, Debrecen, 4002, Hungary

    György Pál Gehér, Lajos Molnár & Dániel Virosztek

  4. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged, 6720, Hungary

    Lajos Molnár

  5. Institute of Mathematics, Budapest University of Technology and Economics, Budapest, 1521, Hungary

    Lajos Molnár & Dániel Virosztek

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  1. Hong-Yi Chen
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  2. György Pál Gehér
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  3. Chih-Neng Liu
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  4. Lajos Molnár
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  5. Dániel Virosztek
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  6. Ngai-Ching Wong
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Correspondence to Lajos Molnár.

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Chen, HY., Gehér, G.P., Liu, CN. et al. Maps on positive definite operators preserving the quantum \(\chi _\alpha ^2\)-divergence. Lett Math Phys 107, 2267–2290 (2017). https://doi.org/10.1007/s11005-017-0989-0

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  • DOI: https://doi.org/10.1007/s11005-017-0989-0

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