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A Free Logarithmic Sobolev Inequality on the Circle

Published online by Cambridge University Press:  20 November 2018

Fumio Hiai ,
Dénes Petz  and
Yoshimichi Ueda
Fumio Hiai
Affiliation:
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan e-mail: hiai@math.is.tohoku.ac.jp
Dénes Petz
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda u. 13-15, Hungary e-mail: petz@renyi.hu
Yoshimichi Ueda
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka 810-8560, Japan e-mail: ueda@math.kyushu-u.ac.jp
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Abstract

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Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

Keywords

46L5460E1594A17
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 389 - 406
Copyright
Copyright © Canadian Mathematical Society 2006

References

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