Inequalities for quantum relative entropy
type: Item
instance of: scholarly article
title: Inequalities for quantum relative entropy
zbMATH Open document ID: 1076.15019
published in: Linear Algebra and its Applications
publication date: 2005-06-01T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/4628
review text: Given positive semi-definite matrices \(A,B\in M_n(\mathbb{C})\), the authors prove the following log-majorization: \[ A^{(1+q)/2}\,B^q\,A^{(1+q)/2}\prec_{(\log)} A^{1/2}(A^{p/2}B^pA^{p/2})^{q/p}A^{1/2}, \] where \(0<q\leq p\). This inequality is then used to reprove a result from \textit{T. Ando} and \textit{F. Hiai} [Linear Algebra Appl. 197--198, 113--131 (1994; Zbl 0793.15011)], and this result is then used to prove several inequalities regarding \(\alpha\)-power means. The rest of the paper is devoted to proving a generalized thermodynamic inequality and its equivalence with the Peierls-Bogoliubov inequality and others.
reviewed by: Martín Argerami
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A90
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 82B10
zbMATH DE Number: 2173147
zbMATH Keywords: relative entropy
zbMATH Keywords: logarithmic trace inequalities
zbMATH Keywords: Peierls-Bogoliubov inequality
zbMATH Keywords: thermodynamic inequality
author: Natalia Bebiano
author: R. Lemos
author: João Da Providência
MaRDI profile type: Publication
OpenAlex ID: W1965267958
cites work: Log majorization and complementary Golden-Thompson type inequalities
cites work: Hölder type inequalities for matrices
cites work: On an inequality of Lieb and Thirring
cites work: Numerical ranges in physics
cites work: Matrix inequalities in statistical mechanics.
cites work: Q3041790
cites work: Q3477514
cites work: $A \geq B \geq 0$ Assures $(B^r A^p B^r)^{1/q} \geq B^{(p+2r)/q$ for $r \geq 0$, $p \geq 0$, $q \geq 1$ with $(1 + 2r)q \geq p + 2r$
cites work: Q5330291
cites work: The Golden-Thompson trace inequality is complemented
cites work: Equality cases in matrix norm inequalities of golden-thompson type
cites work: Means of positive linear operators
cites work: On Information and Sufficiency
cites work: Q3247767
cites work: Inequalities: theory of majorization and its applications
cites work: A note on the entropy for operator algebras
cites work: A Mathematical Theory of Communication
cites work: Conditional expectation in an operator algebra. IV. Entropy and information
DOI: 10.1016/J.LAA.2004.03.023
description: scientific article; zbMATH DE number 2173147