### Abstract

Umegaki's relative entropy S(ω,φ{symbol})=Tr D_{ω}(log D_{ω}-log D_{φ{symbol}}) (of states ω and φ{symbol} with density operators D_{ω} and D_{φ{symbol}}, 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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Pages (from-to) | 99-114 |

Number of pages | 16 |

Journal | Communications in Mathematical Physics |

Volume | 143 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1991 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**The proper formula for relative entropy and its asymptotics in quantum probability.** / Hiai, Fumio; Petz, D.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 143, no. 1, pp. 99-114. https://doi.org/10.1007/BF02100287

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TY - JOUR

T1 - The proper formula for relative entropy and its asymptotics in quantum probability

AU - Hiai, Fumio

AU - Petz, D.

PY - 1991/12

Y1 - 1991/12

N2 - Umegaki's relative entropy S(ω,φ{symbol})=Tr Dω(log Dω-log Dφ{symbol}) (of states ω and φ{symbol} with density operators Dω and Dφ{symbol}, 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.

AB - Umegaki's relative entropy S(ω,φ{symbol})=Tr Dω(log Dω-log Dφ{symbol}) (of states ω and φ{symbol} with density operators Dω and Dφ{symbol}, 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.

UR - http://www.scopus.com/inward/record.url?scp=0002499298&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002499298&partnerID=8YFLogxK

U2 - 10.1007/BF02100287

DO - 10.1007/BF02100287

M3 - Article

VL - 143

SP - 99

EP - 114

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -