Inequalities for quantum relative entropy
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.
scientific article; zbMATH DE number 2173147
scientific article