Quantum f-divergences and error correction

Fumio Hiai, M. Mosonyi, D. Petz, Cédric Bény

Research output: Contribution to journalArticle

73 Citations (Scopus)

Abstract

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

Original languageEnglish
Pages (from-to)691-747
Number of pages57
JournalReviews in Mathematical Physics
Volume23
Issue number7
DOIs
Publication statusPublished - Aug 2011

Fingerprint

F-divergence
Error Correction
divergence
Monotonicity
Relative Entropy
entropy
operators
Equality
Operator
Quantum Error Correction
Quantum Entropy
convexity
Matrix Analysis
Reversibility
Quantum State
ingredients
Integral Representation
Monotonic
Convex function
Convexity

Keywords

  • Chernoff distance
  • f-divergences
  • Hoeffding distances
  • operator convex functions
  • quasi-entropy
  • Rényi relative entropies
  • Relative entropy
  • Schwarz maps
  • stochastic maps
  • substochastic maps

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Quantum f-divergences and error correction. / Hiai, Fumio; Mosonyi, M.; Petz, D.; Bény, Cédric.

In: Reviews in Mathematical Physics, Vol. 23, No. 7, 08.2011, p. 691-747.

Research output: Contribution to journalArticle

Hiai, Fumio ; Mosonyi, M. ; Petz, D. ; Bény, Cédric. / Quantum f-divergences and error correction. In: Reviews in Mathematical Physics. 2011 ; Vol. 23, No. 7. pp. 691-747.
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