فهرست نشریات علمی دسترسی آزاد ایران

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.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 $\mathfrak{R}^{[1]}_{x,y}(A|B)= \mathfrak{R}^{[1]}_{y,x}(A|B)$.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 $\Delta_{i,x}^{[n]}(A|B)$.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.