### 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 language | English |
---|---|

Pages (from-to) | 691-747 |

Number of pages | 57 |

Journal | Reviews in Mathematical Physics |

Volume | 23 |

Issue number | 7 |

DOIs | |

Publication status | Published - Aug 2011 |

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### 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

*Reviews in Mathematical Physics*,

*23*(7), 691-747. https://doi.org/10.1142/S0129055X11004412

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

Research output: Contribution to journal › Article

*Reviews in Mathematical Physics*, vol. 23, no. 7, pp. 691-747. https://doi.org/10.1142/S0129055X11004412

}

TY - JOUR

T1 - Quantum f-divergences and error correction

AU - Hiai, Fumio

AU - Mosonyi, M.

AU - Petz, D.

AU - Bény, Cédric

PY - 2011/8

Y1 - 2011/8

N2 - 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.

AB - 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.

KW - Chernoff distance

KW - f-divergences

KW - Hoeffding distances

KW - operator convex functions

KW - quasi-entropy

KW - Rényi relative entropies

KW - Relative entropy

KW - Schwarz maps

KW - stochastic maps

KW - substochastic maps

UR - http://www.scopus.com/inward/record.url?scp=80052040382&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052040382&partnerID=8YFLogxK

U2 - 10.1142/S0129055X11004412

DO - 10.1142/S0129055X11004412

M3 - Article

VL - 23

SP - 691

EP - 747

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 7

ER -