Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 2 2 2021 12 01 The $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and the $n$-th operator valued divergence 71 79 144305 10.30504/jims.2022.305123.1039 EN H. Tohyama Department of Life Science and Informatics, Faculty of Engineering, Maebashi Institute of Technology, Maebashi, Japan E. Kamei 1-1-3, Sakuragaoka, Kanmakicho, Kitakaturagi-gun, Nara, 639-0202, Japan. M. Watanabe Maebashi Institute of Technology, 460-1, Kamisadori, Maebashi, Gunma, 371-0816, Japan. Journal Article 2021 09 16 We introduced the $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and showed its monotone property, for example, $\mathfrak{R}^{[n]}_{x,x}(A|B) \le \mathfrak{R}^{[n]}_{x,y}(A|B) \le \mathfrak{R}^{[n]}_{y,y}(A|B)$ and $\mathfrak{R}^{[n]}_{x,x}(A|B) \le \mathfrak{R}^{[n]}_{y,x}(A|B) \le \mathfrak{R}^{[n]}_{y,y}(A|B)$ for $x\le y$ if $A\le B$ or $n$ is odd.<br />The $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ is not symmetric on $x$ and $y$, that is, $\mathfrak{R}^{[n]}_{x,y}(A|B)\neq \mathfrak{R}^{[n]}_{y,x}(A|B)$ for $n\geq 2$ while $\mathfrak{R}^{[1]}_{x,y}(A|B)= \mathfrak{R}^{[1]}_{y,x}(A|B)$.<br />In this paper we compare $\mathfrak{R}^{[n]}_{x,y}(A|B)$ with $\mathfrak{R}^{[n]}_{y,x}(A|B)$ and give the relations between $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and the $n$-th operator divergence $\Delta_{i,x}^{[n]}(A|B)$.<br />In this process, we find another operator divergence ${\overline \Delta}_{i,x}^{[n]}(A|B)$ which is similar to $\Delta_{i,x}^{[n]}(A|B)$ but not the same. https://jims.ims.ir/article_144305_7344269d247681c6517cb6fda3661587.pdf