Details
| Original language | English |
|---|---|
| Article number | 208 |
| Number of pages | 52 |
| Journal | Communications in Mathematical Physics |
| Volume | 405 |
| Issue number | 9 |
| Publication status | Published - 20 Aug 2024 |
| Externally published | Yes |
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.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematics(all)
- Mathematical Physics
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In: Communications in Mathematical Physics, Vol. 405, No. 9, 208, 20.08.2024.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Quantum Rényi and f-Divergences from Integral Representations
AU - Hirche, Christoph
AU - Tomamichel, Marco
N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.
PY - 2024/8/20
Y1 - 2024/8/20
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85201685765&partnerID=8YFLogxK
U2 - 10.1007/s00220-024-05087-3
DO - 10.1007/s00220-024-05087-3
M3 - Article
AN - SCOPUS:85201685765
VL - 405
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 9
M1 - 208
ER -