Quantum Rényi and f-Divergences from Integral Representations

authored by: 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.
External Organisation(s): Technical University of Munich (TUM)
National University of Singapore
Type: Article
Journal: Communications in Mathematical Physics
Volume: 405
No. of pages: 52
ISSN: 0010-3616
Publication date: 20.08.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Statistical and Nonlinear Physics, Mathematical Physics
Electronic version(s): https://doi.org/10.1007/s00220-024-05087-3 (Access: Closed)

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",
}