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Quantum Hellinger distances revisited

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Abstract

This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form \(\phi (A,B)=\mathrm {Tr} \left( (1-c)A + c B - A \sigma B \right) ,\) where \(\sigma \) is an arbitrary Kubo–Ando mean, and \(c \in (0,1)\) is the weight of \(\sigma .\) We note that these divergences belong to the family of maximal quantum f-divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case.

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Acknowledgements

We are grateful to Milán Mosonyi for drawing our attention to Ref.’s [6, 14, 15, 17, 20, 21], for comments on earlier versions of this paper, and for several discussions on the topic. We are also grateful to Miklós Pálfia for several discussions; to László Erdős for his essential suggestions on the structure and highlights of this paper, and for his comments on earlier versions; and to the anonymous referee for his/her valuable comments and suggestions.

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  1. MTA-BME Lendület (Momentum) Quantum Information Theory Research Group, Department of Analysis, Institute of Mathematics, Budapest University of Technology and Economics, Budapest, 1521, Hungary

    József Pitrik

  2. Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Dániel Virosztek

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  1. József Pitrik
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Correspondence to Dániel Virosztek.

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J. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant for Quantum Information Theory, No. 96 141, and by the Hungarian National Research, Development and Innovation Office (NKFIH) via Grants Nos. K119442, K124152 and KH129601. D. Virosztek was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (Project Code IC1027FELL01), by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via Grants Nos. K124152 and KH129601.

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Pitrik, J., Virosztek, D. Quantum Hellinger distances revisited. Lett Math Phys 110, 2039–2052 (2020). https://doi.org/10.1007/s11005-020-01282-0

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