Convergence of logarithmic trace inequalities via generalized Lie–Trotter formulae

• Published in: Linear algebra and its applications Vol. 396; pp. 353 - 372
• Main Author: Furuta, Takayuki
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.02.2005; Elsevier Science
• Subjects: Algebra; Exact sciences and technology; Generalized Lie–Trotter formula; Linear and multilinear algebra, matrix theory; Log majorization; Logarithmic trace inequality; Mathematics; Sciences and techniques of general use; Log majorization; Generalized Lie–Trotter formula; 47A63; Logarithmic trace inequality; Trace; Upper bound; Logarithmic function; Linear algebra; 47A63 Generalized Lie-Trotter formula; Entropy; Lie group; Generalized
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2004.09.007

We shall extend logarithmic trace inequalities shown by Bebiano et al. [N. Bebiano, R. Lemos, J. da Providencia, Inequalities for quantum relative entropy, preprint] and also by Hiai and Petz [The Golden–Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993) 153–185], by applying log majorization equivalent to an order preserving operator inequality. We shall generalize the Lie–Trotter formulae, which extend the original Lie–Trotter formula, and the α-mean variant of the original Lie–Trotter formula in Hiai–Petz [Linear Algebra Appl. 181 (1993) 153–185]. By using this generalized Lie–Trotter formulae, we shall consider the convergence of certain logarithmic trace inequalities, as some extensions of Bebiano et al. [N. Bebiano, R. Lemos, J. da Providencia, Inequalities for quantum relative entropy, preprint] and Hiai–Petz [The Golden–Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993) 153–185].