Quantum Rényi and f-Divergences from Integral Representations

verfasst von: Christoph Hirche, Marco Tomamichel
Abstract: Smooth Csiszár f-divergences can be expressed as integrals over so-called hockey stick divergences. This motivates a natural quantum generalization in terms of quantum Hockey stick divergences, which we explore here. Using this recipe, the Kullback–Leibler divergence generalises to the Umegaki relative entropy, in the integral form recently found by Frenkel. We find that the Rényi divergences defined via our new quantum f-divergences are not additive in general, but that their regularisations surprisingly yield the Petz Rényi divergence for α<1 and the sandwiched Rényi divergence for α>1, unifying these two important families of quantum Rényi divergences. Moreover, we find that the contraction coefficients for the new quantum f divergences collapse for all f that are operator convex, mimicking the classical behaviour and resolving some long-standing conjectures by Lesniewski and Ruskai. We derive various inequalities, including new reverse Pinsker inequalites with applications in differential privacy and explore various other applications of the new divergences.
Externe Organisation(en): Technische Universität München (TUM)
National University of Singapore
Typ: Artikel
Journal: Communications in Mathematical Physics
Band: 405
Anzahl der Seiten: 52
ISSN: 0010-3616
Publikationsdatum: 20.08.2024
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Statistische und nichtlineare Physik, Mathematische Physik
Elektronische Version(en): https://doi.org/10.1007/s00220-024-05087-3 (Zugang: Geschlossen)

BibTeX

@article{2cfa6589ca224710a1a0487aa26a9a90,
title = "Quantum R{\'e}nyi and f-Divergences from Integral Representations",
abstract = "Smooth Csisz{\'a}r f-divergences can be expressed as integrals over so-called hockey stick divergences. This motivates a natural quantum generalization in terms of quantum Hockey stick divergences, which we explore here. Using this recipe, the Kullback–Leibler divergence generalises to the Umegaki relative entropy, in the integral form recently found by Frenkel. We find that the R{\'e}nyi divergences defined via our new quantum f-divergences are not additive in general, but that their regularisations surprisingly yield the Petz R{\'e}nyi divergence for α<1 and the sandwiched R{\'e}nyi divergence for α>1, unifying these two important families of quantum R{\'e}nyi divergences. Moreover, we find that the contraction coefficients for the new quantum f divergences collapse for all f that are operator convex, mimicking the classical behaviour and resolving some long-standing conjectures by Lesniewski and Ruskai. We derive various inequalities, including new reverse Pinsker inequalites with applications in differential privacy and explore various other applications of the new divergences.",
author = "Christoph Hirche and Marco Tomamichel",
note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.",
year = "2024",
month = aug,
day = "20",
doi = "10.1007/s00220-024-05087-3",
language = "English",
volume = "405",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "9",
}