### Abstract

Umegaki's relative entropy S(ω, φ) = Tr D_{ω}(log D_{ω} − log D_{φ}) (of states ω and φ with density operators D_{ω} and D_{φ}, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality Tr A log AB ≥ Tr A(log A + log B) is obtained for positive definite matrices A and B.

Original language | English |
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Title of host publication | Asymptotic Theory of Quantum Statistical Inference: Selected Papers |

Publisher | World Scientific Publishing Co. |

Pages | 43-63 |

Number of pages | 21 |

ISBN (Print) | 9789812563071, 9812560157, 9789812560155 |

DOIs | |

Publication status | Published - Jan 1 2005 |

### ASJC Scopus subject areas

- Physics and Astronomy(all)

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## Cite this

Hiai, F., & Petz, D. (2005). The proper formula for relative entropy and its asymptotics in quantum probability. In

*Asymptotic Theory of Quantum Statistical Inference: Selected Papers*(pp. 43-63). World Scientific Publishing Co.. https://doi.org/10.1142/9789812563071_0004