Free fisher information and amalgamated freeness

2004 ◽  
Vol 25 (10) ◽  
pp. 1100-1106
Author(s):  
Meng Bin ◽  
Guo Mao-zheng ◽  
Cao Xiao-hong
Keyword(s):  
Fisher Information ◽  
Free Fisher Information

A Free Logarithmic Sobolev Inequality on the Circle

2006 ◽  
Vol 49 (3) ◽  
pp. 389-406 ◽  
Author(s):  
Fumio Hiai ◽  
Dénes Petz ◽  
Yoshimichi Ueda
Keyword(s):  
Fisher Information ◽  
Sobolev Inequality ◽  
Large Deviation ◽  
Logarithmic Sobolev Inequality ◽  
Approximation Procedure ◽  
Matrix Approximation ◽  
Unitary Matrices ◽  
Free Entropy ◽  
Large Deviation Result ◽  
Free Fisher Information

AbstractFree 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.

scholarly journals Operator-valued free Fisher information and modular frames

2005 ◽  
Vol 133 (10) ◽  
pp. 3087-3096 ◽  
Author(s):  
Bin Meng ◽  
Maozheng Guo ◽  
Xiaohong Cao
Keyword(s):  
Fisher Information ◽  
Free Fisher Information

REMARKS ON THE FREE RELATIVE ENTROPY AND THE FREE FISHER INFORMATION DISTANCE

2010 ◽  
Vol 13 (02) ◽  
pp. 243-255
Author(s):  
HIROAKI YOSHIDA
Keyword(s):  
Relative Entropy ◽  
Fisher Information ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Semicircle Law ◽  
Information Distance ◽  
Poisson Law ◽  
Free Relative ◽  
Free Fisher Information

In this paper, we shall introduce the free Fisher information distance which is inspired by the estimation-theoretic representation of the free relative entropy investigated by Verdú. We shall see the free analogue of the logarithmic Sobolev inequality with respect to a centered semicircle law and also the semicircular approximation of the free Poisson law.

REMARKS ON A SEMICIRCULAR PERTURBATION OF THE FREE FISHER INFORMATION

2008 ◽  
Vol 11 (01) ◽  
pp. 97-108 ◽  
Author(s):  
HIROAKI YOSHIDA
Keyword(s):  
Fisher Information ◽  
Conditional Variance ◽  
Information Inequality ◽  
Free Entropy ◽  
Variance Formula ◽  
The Mean ◽  
Power Inequality ◽  
Free Fisher Information ◽  
Special Case

In this paper, we shall give the representation of a semicircular perturbation of the free Fisher information [Formula: see text] by the mean of the conditional variance [Formula: see text], where S is a standard semicircular element freely independent of X, and ε > 0. Using this representation, we will give alternative proofs of the free Fisher information inequality, in which the free analogue of Stam's inequality can be obtained as a special case, and of the free entropy power inequality in an infinitesimal approach to the free entropy.

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