Quantum Hellinger distances revisited
Pitrik J, Virosztek D. 2020. Quantum Hellinger distances revisited. Letters in Mathematical Physics. 110(8), 2039–2052.
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Pitrik, Jozsef;
Virosztek, DanielISTA 
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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 ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of σ. 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.
Publishing Year
Date Published
2020-08-01
Journal Title
Letters in Mathematical Physics
Acknowledgement
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.
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 Erdös 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.
Volume
110
Issue
8
Page
2039-2052
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eISSN
IST-REx-ID
Cite this
Pitrik J, Virosztek D. Quantum Hellinger distances revisited. Letters in Mathematical Physics. 2020;110(8):2039-2052. doi:10.1007/s11005-020-01282-0
Pitrik, J., & Virosztek, D. (2020). Quantum Hellinger distances revisited. Letters in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s11005-020-01282-0
Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.” Letters in Mathematical Physics. Springer Nature, 2020. https://doi.org/10.1007/s11005-020-01282-0.
J. Pitrik and D. Virosztek, “Quantum Hellinger distances revisited,” Letters in Mathematical Physics, vol. 110, no. 8. Springer Nature, pp. 2039–2052, 2020.
Pitrik J, Virosztek D. 2020. Quantum Hellinger distances revisited. Letters in Mathematical Physics. 110(8), 2039–2052.
Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.” Letters in Mathematical Physics, vol. 110, no. 8, Springer Nature, 2020, pp. 2039–52, doi:10.1007/s11005-020-01282-0.
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