Matrix trace inequalities related to the Hiai-Petz inequality

Abstract

In this paper, we show matrix trace inequalities related to the Hiai-Petz inequality: For positive definite matrices \(A\) and \(B\), and each \( -1\leq \alpha <0\)

\[
\frac{1}{p} \operatorname{Tr}[B^{-\frac{p}{2}}AB^{-\frac{p}{2}}T_{-\frac{\alpha}{p}} (B^p|A^p)] \leq D_{\alpha}(A|B) \quad \mbox{for all $1-\alpha \geq p\geq -\alpha >0$},
\]

where the Tsallis relative entropy \(D_{\alpha}(A|B)\) is defined by \(D_{\alpha}(A|B)=\frac{1}{\alpha} \operatorname{Tr}[A-A^{1-\alpha}B^{\alpha}]\) and the Tsallis relative operator entropy \(T_{\alpha}(A|B)\) is defined by \(T_{\alpha}(A|B)=\frac{1}{\alpha} (A \natural_{\alpha}\ B-A)\), where \(A\natural_{\alpha} B = A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\alpha}A^{\frac{1}{2}}\) for \(\alpha \in {\Bbb R}\).